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Hypothesis in Machine Learning

• Bayes Theorem in Machine learning
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The concept of a hypothesis is fundamental in Machine Learning and data science endeavours. In the realm of machine learning, a hypothesis serves as an initial assumption made by data scientists and ML professionals when attempting to address a problem. Machine learning involves conducting experiments based on past experiences, and these hypotheses are crucial in formulating potential solutions.

It’s important to note that in machine learning discussions, the terms “hypothesis” and “model” are sometimes used interchangeably. However, a hypothesis represents an assumption, while a model is a mathematical representation employed to test that hypothesis. This section on “Hypothesis in Machine Learning” explores key aspects related to hypotheses in machine learning and their significance.

Table of Content

How does a Hypothesis work?

Hypothesis space and representation in machine learning, hypothesis in statistics, faqs on hypothesis in machine learning.

A hypothesis in machine learning is the model’s presumption regarding the connection between the input features and the result. It is an illustration of the mapping function that the algorithm is attempting to discover using the training set. To minimize the discrepancy between the expected and actual outputs, the learning process involves modifying the weights that parameterize the hypothesis. The objective is to optimize the model’s parameters to achieve the best predictive performance on new, unseen data, and a cost function is used to assess the hypothesis’ accuracy.

In most supervised machine learning algorithms, our main goal is to find a possible hypothesis from the hypothesis space that could map out the inputs to the proper outputs. The following figure shows the common method to find out the possible hypothesis from the Hypothesis space:

Hypothesis Space (H)

Hypothesis space is the set of all the possible legal hypothesis. This is the set from which the machine learning algorithm would determine the best possible (only one) which would best describe the target function or the outputs.

Hypothesis (h)

A hypothesis is a function that best describes the target in supervised machine learning. The hypothesis that an algorithm would come up depends upon the data and also depends upon the restrictions and bias that we have imposed on the data.

The Hypothesis can be calculated as:

[Tex]y = mx + b [/Tex]

• m = slope of the lines
• b = intercept

To better understand the Hypothesis Space and Hypothesis consider the following coordinate that shows the distribution of some data:

Say suppose we have test data for which we have to determine the outputs or results. The test data is as shown below:

We can predict the outcomes by dividing the coordinate as shown below:

So the test data would yield the following result:

But note here that we could have divided the coordinate plane as:

The way in which the coordinate would be divided depends on the data, algorithm and constraints.

• All these legal possible ways in which we can divide the coordinate plane to predict the outcome of the test data composes of the Hypothesis Space.
• Each individual possible way is known as the hypothesis.

Hence, in this example the hypothesis space would be like:

The hypothesis space comprises all possible legal hypotheses that a machine learning algorithm can consider. Hypotheses are formulated based on various algorithms and techniques, including linear regression, decision trees, and neural networks. These hypotheses capture the mapping function transforming input data into predictions.

Hypothesis Formulation and Representation in Machine Learning

Hypotheses in machine learning are formulated based on various algorithms and techniques, each with its representation. For example:

• Linear Regression : [Tex] h(X) = \theta_0 + \theta_1 X_1 + \theta_2 X_2 + … + \theta_n X_n[/Tex]
• Decision Trees : [Tex]h(X) = \text{Tree}(X)[/Tex]
• Neural Networks : [Tex]h(X) = \text{NN}(X)[/Tex]

In the case of complex models like neural networks, the hypothesis may involve multiple layers of interconnected nodes, each performing a specific computation.

Hypothesis Evaluation:

The process of machine learning involves not only formulating hypotheses but also evaluating their performance. This evaluation is typically done using a loss function or an evaluation metric that quantifies the disparity between predicted outputs and ground truth labels. Common evaluation metrics include mean squared error (MSE), accuracy, precision, recall, F1-score, and others. By comparing the predictions of the hypothesis with the actual outcomes on a validation or test dataset, one can assess the effectiveness of the model.

Hypothesis Testing and Generalization:

Once a hypothesis is formulated and evaluated, the next step is to test its generalization capabilities. Generalization refers to the ability of a model to make accurate predictions on unseen data. A hypothesis that performs well on the training dataset but fails to generalize to new instances is said to suffer from overfitting. Conversely, a hypothesis that generalizes well to unseen data is deemed robust and reliable.

The process of hypothesis formulation, evaluation, testing, and generalization is often iterative in nature. It involves refining the hypothesis based on insights gained from model performance, feature importance, and domain knowledge. Techniques such as hyperparameter tuning, feature engineering, and model selection play a crucial role in this iterative refinement process.

In statistics , a hypothesis refers to a statement or assumption about a population parameter. It is a proposition or educated guess that helps guide statistical analyses. There are two types of hypotheses: the null hypothesis (H0) and the alternative hypothesis (H1 or Ha).

• Null Hypothesis(H 0 ): This hypothesis suggests that there is no significant difference or effect, and any observed results are due to chance. It often represents the status quo or a baseline assumption.
• Aternative Hypothesis(H 1 or H a ): This hypothesis contradicts the null hypothesis, proposing that there is a significant difference or effect in the population. It is what researchers aim to support with evidence.

Q. How does the training process use the hypothesis?

The learning algorithm uses the hypothesis as a guide to minimise the discrepancy between expected and actual outputs by adjusting its parameters during training.

Q. How is the hypothesis’s accuracy assessed?

Usually, a cost function that calculates the difference between expected and actual values is used to assess accuracy. Optimising the model to reduce this expense is the aim.

Q. What is Hypothesis testing?

Hypothesis testing is a statistical method for determining whether or not a hypothesis is correct. The hypothesis can be about two variables in a dataset, about an association between two groups, or about a situation.

Q. What distinguishes the null hypothesis from the alternative hypothesis in machine learning experiments?

The null hypothesis (H0) assumes no significant effect, while the alternative hypothesis (H1 or Ha) contradicts H0, suggesting a meaningful impact. Statistical testing is employed to decide between these hypotheses.

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Introduction to the hypothesis space and the bias-variance tradeoff in machine learning.

In this post, we introduce the hypothesis space and discuss how machine learning models function as hypotheses. Furthermore, we discuss the challenges encountered when choosing an appropriate machine learning hypothesis and building a model, such as overfitting, underfitting, and the bias-variance tradeoff.

The hypothesis space in machine learning is a set of all possible models that can be used to explain a data distribution given the limitations of that space. A linear hypothesis space is limited to the set of all linear models. If the data distribution follows a non-linear distribution, the linear hypothesis space might not contain a model that is appropriate for our needs.

To understand the concept of a hypothesis space, we need to learn to think of machine learning models as hypotheses.

The Machine Learning Model as Hypothesis

Generally speaking, a hypothesis is a potential explanation for an outcome or a phenomenon. In scientific inquiry, we test hypotheses to figure out how well and if at all they explain an outcome. In supervised machine learning, we are concerned with finding a function that maps from inputs to outputs.

But machine learning is inherently probabilistic. It is the art and science of deriving useful hypotheses from limited or incomplete data. Our functions are not axioms that explain the data perfectly, and for most real-life problems, we will never have all the data that exists. Accordingly, we will not find the one true function that perfectly describes the data. Instead, we find a function through training a model to map from known training input to known training output. This way, the model gradually approximates the assumed true function that describes the distribution of the data. So we treat our model as a hypothesis that needs to be tested as to how well it explains the output from a given input. We do this using a test or validation data set.

The Hypothesis Space

During the training process, we select a model from a hypothesis space that is subject to our constraints. For example, a linear hypothesis space only provides linear models. We can approximate data that follows a quadratic distribution using a model from the linear hypothesis space.

Of course, a linear model will never have the same predictive performance as a quadratic model, so we can adjust our hypothesis space to also include non-linear models or at least quadratic models.

The Data Generating Process

The data generating process describes a hypothetical process subject to some assumptions that make training a machine learning model possible. We need to assume that the data points are from the same distribution but are independent of each other. When these requirements are met, we say that the data is independent and identically distributed (i.i.d.).

Independent and Identically Distributed Data

How can we assume that a model trained on a training set will perform better than random guessing on new and previously unseen data? First of all, the training data needs to come from the same or at least a similar problem domain. If you want your model to predict stock prices, you need to train the model on stock price data or data that is similarly distributed. It wouldn’t make much sense to train it on whether data. Statistically, this means the data is identically distributed . But if data comes from the same problem, training data and test data might not be completely independent. To account for this, we need to make sure that the test data is not in any way influenced by the training data or vice versa. If you use a subset of the training data as your test set, the test data evidently is not independent of the training data. Statistically, we say the data must be independently distributed .

Overfitting and Underfitting

We want to select a model from the hypothesis space that explains the data sufficiently well. During training, we can make a model so complex that it perfectly fits every data point in the training dataset. But ultimately, the model should be able to predict outputs on previously unseen input data. The ability to do well when predicting outputs on previously unseen data is also known as generalization. There is an inherent conflict between those two requirements.

If we make the model so complex that it fits every point in the training data, it will pick up lots of noise and random variation specific to the training set, which might obscure the larger underlying patterns. As a result, it will be more sensitive to random fluctuations in new data and predict values that are far off. A model with this problem is said to overfit the training data and, as a result, to suffer from high variance .

To avoid the problem of overfitting, we can choose a simpler model or use regularization techniques to prevent the model from fitting the training data too closely. The model should then be less influenced by random fluctuations and instead, focus on the larger underlying patterns in the data. The patterns are expected to be found in any dataset that comes from the same distribution. As a consequence, the model should generalize better on previously unseen data.

But if we go too far, the model might become too simple or too constrained by regularization to accurately capture the patterns in the data. Then the model will neither generalize well nor fit the training data well. A model that exhibits this problem is said to underfit the data and to suffer from high bias . If the model is too simple to accurately capture the patterns in the data (for example, when using a linear model to fit non-linear data), its capacity is insufficient for the task at hand.

When training neural networks, for example, we go through multiple iterations of training in which the model learns to fit an increasingly complex function to the data. Typically, your training error will decrease during learning the more complex your model becomes and the better it learns to fit the data. In the beginning, the training error decreases rapidly. In later training iterations, it typically flattens out as it approaches the minimum possible error. Your test or generalization error should initially decrease as well, albeit likely at a slower pace than the training error. As long as the generalization error is decreasing, your model is underfitting because it doesn’t live up to its full capacity. After a number of training iterations, the generalization error will likely reach a trough and start to increase again. Once it starts to increase, your model is overfitting, and it is time to stop training.

Ideally, you should stop training once your model reaches the lowest point of the generalization error. The gap between the minimum generalization error and no error at all is an irreducible error term known as the Bayes error that we won’t be able to completely get rid of in a probabilistic setting. But if the error term seems too large, you might be able to reduce it further by collecting more data, manipulating your model’s hyperparameters, or altogether picking a different model.

Bias Variance Tradeoff

We’ve talked about bias and variance in the previous section. Now it is time to clarify what we actually mean by these terms.

Understanding Bias and Variance

In a nutshell, bias measures if there is any systematic deviation from the correct value in a specific direction. If we could repeat the same process of constructing a model several times over, and the results predicted by our model always deviate in a certain direction, we would call the result biased.

Variance measures how much the results vary between model predictions. If you repeat the modeling process several times over and the results are scattered all across the board, the model exhibits high variance.

In their book “Noise” Daniel Kahnemann and his co-authors provide an intuitive example that helps understand the concept of bias and variance. Imagine you have four teams at the shooting range.

Team B is biased because the shots of its team members all deviate in a certain direction from the center. Team B also exhibits low variance because the shots of all the team members are relatively concentrated in one location. Team C has the opposite problem. The shots are scattered across the target with no discernible bias in a certain direction. Team D is both biased and has high variance. Team A would be the equivalent of a good model. The shots are in the center with little bias in one direction and little variance between the team members.

Generally speaking, linear models such as linear regression exhibit high bias and low variance. Nonlinear algorithms such as decision trees are more prone to overfitting the training data and thus exhibit high variance and low bias.

A linear model used with non-linear data would exhibit a bias to predict data points along a straight line instead of accomodating the curves. But they are not as susceptible to random fluctuations in the data. A nonlinear algorithm that is trained on noisy data with lots of deviations would be more capable of avoiding bias but more prone to incorporate the noise into its predictions. As a result, a small deviation in the test data might lead to very different predictions.

To get our model to learn the patterns in data, we need to reduce the training error while at the same time reducing the gap between the training and the testing error. In other words, we want to reduce both bias and variance. To a certain extent, we can reduce both by picking an appropriate model, collecting enough training data, selecting appropriate training features and hyperparameter values. At some point, we have to trade-off between minimizing bias and minimizing variance. How you balance this trade-off is up to you.

The Bias Variance Decomposition

Mathematically, the total error can be decomposed into the bias and the variance according to the following formula.

Remember that Bayes’ error is an error that cannot be eliminated.

Our machine learning model represents an estimating function \hat f(X) for the true data generating function f(X) where X represents the predictors and y the output values.

Now the mean squared error of our model is the expected value of the squared difference of the output produced by the estimating function \hat f(X) and the true output Y.

The bias is a systematic deviation from the true value. We can measure it as the squared difference between the expected value produced by the estimating function (the model) and the values produced by the true data-generating function.

Of course, we don’t know the true data generating function, but we do know the observed outputs Y, which correspond to the values generated by f(x) plus an error term.

The variance of the model is the squared difference between the expected value and the actual values of the model.

Now that we have the bias and the variance, we can add them up along with the irreducible error to get the total error.

A machine learning model represents an approximation to the hypothesized function that generated the data. The chosen model is a hypothesis since we hypothesize that this model represents the true data generating function.

We choose the hypothesis from a hypothesis space that may be subject to certain constraints. For example, we can constrain the hypothesis space to the set of linear models.

When choosing a model, we aim to reduce the bias and the variance to prevent our model from either overfitting or underfitting the data. In the real world, we cannot completely eliminate bias and variance, and we have to trade-off between them. The total error produced by a model can be decomposed into the bias, the variance, and irreducible (Bayes) error.

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Version Spaces

References :

• T. Mitchell, 1997.
• P. Winston, "Learning by Managing Multiple Models", in P. Winston, Artificial Intelligence , Addison-Wesley Publishing Company, 1992, pp. 411-422.

A version space is a hierarchial representation of knowledge that enables you to keep track of all the useful information supplied by a sequence of learning examples without remembering any of the examples.

The version space method is a concept learning process accomplished by managing multiple models within a version space.

Version Space Characteristics

Tentative heuristics are represented using version spaces.

A version space represents all the alternative plausible descriptions of a heuristic. A plausible description is one that is applicable to all known positive examples and no known negative example.

A version space description consists of two complementary trees:

• One that contains nodes connected to overly general models, and
• One that contains nodes connected to overly specific models.

Node values/attributes are discrete .

Fundamental Assumptions

• The data is correct; there are no erroneous instances.
• A correct description is a conjunction of some of the attributes with values.

Diagrammatical Guidelines

There is a generalization tree and a specialization tree.

Each node is connected to a model .

Nodes in the generalization tree are connected to a model that matches everything in its subtree.

Nodes in the specialization tree are connected to a model that matches only one thing in its subtree.

Links between nodes and their models denote

• generalization relations in a generalization tree, and
• specialization relations in a specialization tree.

Diagram of a Version Space

In the diagram below, the specialization tree is colored red , and the generalization tree is colored green .

Generalization and Specialization Leads to Version Space Convergence

The key idea in version space learning is that specialization of the general models and generalization of the specific models may ultimately lead to just one correct model that matches all observed positive examples and does not match any negative examples.

That is, each time a negative example is used to specialilize the general models, those specific models that match the negative example are eliminated and each time a positive example is used to generalize the specific models, those general models that fail to match the positive example are eliminated. Eventually, the positive and negative examples may be such that only one general model and one identical specific model survive.

Version Space Method Learning Algorithm: Candidate-Elimination

The version space method handles positive and negative examples symmetrically.

• A representation language.
• A set of positive and negative examples expressed in that language.

Compute: a concept description that is consistent with all the positive examples and none of the negative examples.

• Initialize G, the set of maximally general hypotheses, to contain one element: the null description (all features are variables).
• Initialize S, the set of maximally specific hypotheses, to contain one element: the first positive example.
• The new specific models involve minimal changes.
• Each new specific model is a specialization of some general model.
• No new specific model is a generalization of some other specific model.
• Prune away all the general models that fail to match the positive example.
• The new general models involve minimal changes.
• Each new general model is a generalization of some specific model.
• No new general model is a specialization of some other general model.
• Prune away all the specific models that match the negative example.
• if they are identical, output their value and halt.
• if they are different, the training cases were inconsistent. Output this result and halt.
• else continue accepting new training examples.

The algorithm stops when:

• It runs out of data.
• 0 - no consistent description for the data in the language.
• 1 - answer (version space converges).
• 2 + - all descriptions in the language are implicitly included.

Comments on the Version Space Method

The version space method is still a trial and error method. The program does not base its choice of examples, or its learned heuristics, on an analysis of what works or why it works, but rather on the simple assumption that what works will probably work again.

Unlike the decision tree ID3 algorithm,

• Candidate-elimination searches an incomplete set of hypotheses (ie. only a subset of the potentially teachable concepts are included in the hypothesis space).
• Candidate-elimination finds every hypothesis that is consistent with the training data, meaning it searches the hypothesis space completely .
• Candidate-elimination's inductive bias is a consequence of how well it can represent the subset of possible hypotheses it will search. In other words, the bias is a product of its search space .
• No additional bias is introduced through Candidate-eliminations's search strategy .

Advantages of the version space method:

• Can describe all the possible hypotheses in the language consistent with the data.
• Fast (close to linear).

Disadvantages of the version space method:

• Inconsistent data (noise) may cause the target concept to be pruned.
• Learning disjunctive concepts is challenging.

Click on the links below for version space method exercises:

Life Is an Accident of Space and Time

Even if life existed on every planet that could support it, living matter in the universe would amount to only a few grains of sand in the Gobi Desert.

Like many people on planet Earth, I have been spellbound by the first pictures from the James Webb Space Telescope : the lacelike windings of galaxies, the apricot filaments of nebulae , the remnants of exploded stars . A less picturesque, but still revolutionary, part of Webb’s mission is the search for signs of life elsewhere in the universe. The telescope goes about this momentous quest by analyzing the starlight passing through the atmospheres of distant planets. Each kind of molecule leaves its own telltale imprints on traversing light, and some molecules, such as oxygen, carbon dioxide, and methane, may indicate life forms on the planet below. Indeed, Webb has already found evidence of carbon dioxide on at least one planet beyond our solar system.

Considering the billions of planets in our galaxy, and the billions of galaxies in the observable universe, few scientists believe that our planet is the only habitat with life. Nonetheless, finding definite evidence of living things elsewhere in the cosmos would have deep emotional and psychological import, as well as philosophical and theological meaning. Such a finding would force us humans to reconsider some of our fundamental beliefs: How do we define “life”? What are the possible varieties of life? Where did we living things come from? Is there some kind of cosmic community?

In fact, recent scientific research suggests that life in the universe is rare. A few years ago, using results from the Kepler satellite to estimate the fraction of stars with possibly habitable planets, I calculated that , even if all potentially habitable planets do in fact harbor life, the fraction of matter in the universe in living form is exceedingly small: about one-billionth of one-billionth. That’s like a few grains of sand on the Gobi Desert. Evidently, we living things are a very special arrangement of atoms and molecules.

Life may be even rarer than that. In the mid-1970s, the Australian physicist Brandon Carter pointed out that our universe seems particularly fine-tuned for the emergence of life. For example, if the nuclear force holding the centers of atoms together were a little weaker, then the complex atoms needed for life could never form. If it were a little stronger, all of the hydrogen in the infant universe would have fused to become helium. Without hydrogen, water (H 2 O) would not exist, and most biologists believe that water is necessary for life. As another example of fine-tuning: If the observed “dark energy” that fills the cosmos, discovered in 1998, were a little larger than it actually is, the universe would have expanded so rapidly that matter could never have pulled itself together to make stars, the essential nursery for all the complex atoms thought necessary for life. But with a slightly smaller value of dark energy, the universe would have expanded and recollapsed so quickly that stars wouldn’t have had time to form.

Carter’s observation that our universe is finely tuned for the emergence of life has been called the anthropic principle. A profound question raised by the principle is: Why? Why should the universe care whether it contains animate matter? The theological answer to this question is a cosmic form of intelligent design: Our universe was created by an all-powerful and purposeful being, who wanted it to have life. Another explanation, more scientific, is that our universe is but one of a huge number of universes, called the multiverse , which have a wide range of values for the strength of the nuclear force, the amount of dark energy, and many other fundamental parameters. In most of those universes, these values would not lie within the narrow range permitting life to emerge. We live in one of the life-friendly universes because otherwise we wouldn’t be here to ask the question . Our existence, and our universe itself, is simply an accident, one throw of the cosmic dice.

Read: Why Earth’s history appears so miraculous

A similar line of thinking could explain why planet Earth has such favorable conditions for life: liquid water, moderate temperatures (at the moment), plentiful oxygen for higher-level metabolism. The obvious explanation is that there are many planets, even in our own solar system, that do not have liquid water or pleasant temperatures or oxygen atmospheres. Those planets do not harbor life. We are here, to build houses and write novels and ask questions about our own existence, because we live on one of the small fraction of planets that have the right conditions for life. In sum, animate matter is not only rare in our particular universe, but seems to be nonexistent in most possible universes.

At the time that Carter published his paper, I had recently finished my graduate work and was doing research in astrophysics at Cornell University.

During my two years at Cornell, I lived in an apartment with a large picture window facing Cayuga Lake. Every day, the lake looked different, as if painted by a new artist. I spent hours when I should have been pondering equations staring out at the lake, its fluctuating colors and textures.

At Cornell, I met a number of scientific titans, such as Edwin Salpeter , Thomas Gold, and Hans Bethe . Gold, a theoretical astrophysicist and biophysicist born in Vienna in 1920, I got to know pretty well. Tommy was not particularly adept at mathematical calculations, but he was a brilliant and daring intuitionist. Barrel-chested and ruddy-faced when I knew him, with a broad smile, he had strong opinions about nearly everything and did not shirk from thumbing his nose at the scientific establishment. He rapidly threw out new ideas, like darts at a dartboard. Most of them missed the bull’s-eye, but not all.

In 1948, Gold partnered with other astrophysicists to challenge the Big Bang theory with a counter theory called the “steady state ” theory of cosmology. That theory proposed that the universe never had a beginning. It appears unchanging, even while expanding, because of a hypothesized constant creation of new matter. Steady state was eventually proved wrong. In 1968, Gold correctly hypothesized that the newly discovered pulsing radio waves from space were produced by rapidly rotating neutron stars . In the 1970s, Gold argued that the oil found on Earth did not originate from the decomposition of organic material, as most geologists believe, but was present deep underground when the planet first formed. He even persuaded the Swedish national power company to drill an exploratory well in a meteor crater. The interpretation of the sludge brought forth was highly controversial, and the company went bankrupt.

Read: Where science and miracles meet

I have vivid memories of standing in Tommy’s office, attempting to solve a problem with equations on the blackboard, when he would brush me away with exasperation and pronounce the answer simply by visualizing the problem in his head. Such physical intuition can be found in most scientists, but Tommy possessed it to an astounding degree.

Tommy was also good with his hands. He once showed me a beautiful, three-legged chair he’d designed and built, and he explained that all chairs should be made that way. Even if the three legs are not the same length, their ends will sit stably on the floor because three points (the ends of the legs of the chair) define a unique plane (the floor). Add a fourth leg—a fourth point—and, unless it is cut exactly to the right length, its end will not lie in the same plane as the first three. The chair can then wobble back and forth among its four legs, the ends of any three of them lying in the plane of the floor, but the fourth being out of place. In other words, three legs allow only one solution for the position of the chair, but four allow for several.

Thinking back on Tommy’s three-legged chair, I realize that it was a perfect metaphor for the single and unique universe that most scientists dreamed of. Physicists, and especially theoretical physicists, would like to think that there is only one possible universe consistent with the fundamental laws of nature, like one unique solution to a crossword puzzle or a chair with only three legs. If so, we would be able to calculate why our universe must be as it is.

The possibility that there may be many other universes with different properties, many different solutions to the same fundamental laws of nature, deeply disturbs many scientists. It’s a bit like going into a shoe store and finding that size 3 fits you, but sizes 6 and 11 fit equally well.

Modern physicists take great pride in being able to calculate everything from “first principles”—that is, from a few fundamental laws. For example, a physicist can calculate how fast a ball hits the floor when dropped from a height of three feet using a principle known as the “conservation of energy”: The total energy in a closed system is constant, even though that energy may change form. The conservation of energy, in turn, follows from an even deeper principle called “time invariance”: The laws of nature do not change from one moment to the next.

Using basic principles, physicists have been able to calculate the color of the sky, the detailed orbits of planets, the strength of magnetism in an electron, and many other phenomena. But if there are many different universes consistent with the same starting principles and laws, then the fundamental nature of our universe is incalculable . Some basic properties of our universe would have to be accidents. Physicists hate accidents. If there were too many accidents, nothing would be predictable. Wheelbarrows might suddenly float in the air. The sun might come up some days and not others. The world would be a frightening place.

There’s one more disturbing aspect of the multiverse idea. Even if this multitude of other universes are real, there may well be no way to prove or disprove their existence. By definition, a universe is a self-contained region of space and time that cannot send a signal to another such region even into the infinite future. Thus, a universe cannot communicate with another universe. The hypothesized boatload of universes must be accepted or rejected as a matter of faith. Just as scientists do not like accidents, they dislike being forced to accept things they cannot prove. But the multiverse, and other aspects of this strange cosmos we find ourselves in, may be not only unknown to us at this moment, but fundamentally unknowable . Although such a notion goes against the long tradition of science, it does offer a bit of humility, which is good medicine for any profession.

Read: Don’t be afraid of the multiverse

The multiverse hypothesis is not accepted by all scientists. But one thing is almost certain: Life in our universe is extremely rare. I have already explained that life is rare in space—only a small fraction of matter exists in living form. Life is also rare in time, in the long unfolding history of the universe. At some point in the future, in perhaps a few hundred billion years, after all of the stars have burned out and all sources of energy have been exhausted, life in our universe will end—not just life similar to that on Earth, but life of all kinds. The “era of life” will have passed.

What should we make of this realization? For me, it offers a feeling of kinship with all living things. We living things are the only mechanism by which the universe can observe itself. We living things, a few grains of sand on the desert, are that special arrangement of atoms and molecules that can attempt to fathom and record this dazzling spectacle of existence. In a limited but real sense, we living things help give the universe meaning. Without us, the cosmos would merely be.

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• Idea behind hypothesis testing

Examples of null and alternative hypotheses

• Writing null and alternative hypotheses
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How to Write a Great Hypothesis

Hypothesis Definition, Format, Examples, and Tips

Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

Amy Morin, LCSW, is a psychotherapist and international bestselling author. Her books, including "13 Things Mentally Strong People Don't Do," have been translated into more than 40 languages. Her TEDx talk,  "The Secret of Becoming Mentally Strong," is one of the most viewed talks of all time.

Verywell / Alex Dos Diaz

• The Scientific Method

Hypothesis Format

Falsifiability of a hypothesis.

• Operationalization

Hypothesis Types

Hypotheses examples.

• Collecting Data

A hypothesis is a tentative statement about the relationship between two or more variables. It is a specific, testable prediction about what you expect to happen in a study. It is a preliminary answer to your question that helps guide the research process.

Consider a study designed to examine the relationship between sleep deprivation and test performance. The hypothesis might be: "This study is designed to assess the hypothesis that sleep-deprived people will perform worse on a test than individuals who are not sleep-deprived."

At a Glance

A hypothesis is crucial to scientific research because it offers a clear direction for what the researchers are looking to find. This allows them to design experiments to test their predictions and add to our scientific knowledge about the world. This article explores how a hypothesis is used in psychology research, how to write a good hypothesis, and the different types of hypotheses you might use.

The Hypothesis in the Scientific Method

In the scientific method , whether it involves research in psychology, biology, or some other area, a hypothesis represents what the researchers think will happen in an experiment. The scientific method involves the following steps:

• Forming a question
• Performing background research
• Creating a hypothesis
• Designing an experiment
• Collecting data
• Analyzing the results
• Drawing conclusions
• Communicating the results

The hypothesis is a prediction, but it involves more than a guess. Most of the time, the hypothesis begins with a question which is then explored through background research. At this point, researchers then begin to develop a testable hypothesis.

Unless you are creating an exploratory study, your hypothesis should always explain what you  expect  to happen.

In a study exploring the effects of a particular drug, the hypothesis might be that researchers expect the drug to have some type of effect on the symptoms of a specific illness. In psychology, the hypothesis might focus on how a certain aspect of the environment might influence a particular behavior.

Remember, a hypothesis does not have to be correct. While the hypothesis predicts what the researchers expect to see, the goal of the research is to determine whether this guess is right or wrong. When conducting an experiment, researchers might explore numerous factors to determine which ones might contribute to the ultimate outcome.

In many cases, researchers may find that the results of an experiment  do not  support the original hypothesis. When writing up these results, the researchers might suggest other options that should be explored in future studies.

In many cases, researchers might draw a hypothesis from a specific theory or build on previous research. For example, prior research has shown that stress can impact the immune system. So a researcher might hypothesize: "People with high-stress levels will be more likely to contract a common cold after being exposed to the virus than people who have low-stress levels."

In other instances, researchers might look at commonly held beliefs or folk wisdom. "Birds of a feather flock together" is one example of folk adage that a psychologist might try to investigate. The researcher might pose a specific hypothesis that "People tend to select romantic partners who are similar to them in interests and educational level."

Elements of a Good Hypothesis

So how do you write a good hypothesis? When trying to come up with a hypothesis for your research or experiments, ask yourself the following questions:

• Is your hypothesis based on your research on a topic?
• Can your hypothesis be tested?
• Does your hypothesis include independent and dependent variables?

Before you come up with a specific hypothesis, spend some time doing background research. Once you have completed a literature review, start thinking about potential questions you still have. Pay attention to the discussion section in the  journal articles you read . Many authors will suggest questions that still need to be explored.

How to Formulate a Good Hypothesis

To form a hypothesis, you should take these steps:

• Collect as many observations about a topic or problem as you can.
• Evaluate these observations and look for possible causes of the problem.
• Create a list of possible explanations that you might want to explore.
• After you have developed some possible hypotheses, think of ways that you could confirm or disprove each hypothesis through experimentation. This is known as falsifiability.

In the scientific method ,  falsifiability is an important part of any valid hypothesis. In order to test a claim scientifically, it must be possible that the claim could be proven false.

Students sometimes confuse the idea of falsifiability with the idea that it means that something is false, which is not the case. What falsifiability means is that  if  something was false, then it is possible to demonstrate that it is false.

One of the hallmarks of pseudoscience is that it makes claims that cannot be refuted or proven false.

The Importance of Operational Definitions

A variable is a factor or element that can be changed and manipulated in ways that are observable and measurable. However, the researcher must also define how the variable will be manipulated and measured in the study.

Operational definitions are specific definitions for all relevant factors in a study. This process helps make vague or ambiguous concepts detailed and measurable.

For example, a researcher might operationally define the variable " test anxiety " as the results of a self-report measure of anxiety experienced during an exam. A "study habits" variable might be defined by the amount of studying that actually occurs as measured by time.

These precise descriptions are important because many things can be measured in various ways. Clearly defining these variables and how they are measured helps ensure that other researchers can replicate your results.

Replicability

One of the basic principles of any type of scientific research is that the results must be replicable.

Replication means repeating an experiment in the same way to produce the same results. By clearly detailing the specifics of how the variables were measured and manipulated, other researchers can better understand the results and repeat the study if needed.

Some variables are more difficult than others to define. For example, how would you operationally define a variable such as aggression ? For obvious ethical reasons, researchers cannot create a situation in which a person behaves aggressively toward others.

To measure this variable, the researcher must devise a measurement that assesses aggressive behavior without harming others. The researcher might utilize a simulated task to measure aggressiveness in this situation.

Hypothesis Checklist

• Does your hypothesis focus on something that you can actually test?
• Does your hypothesis include both an independent and dependent variable?
• Can you manipulate the variables?
• Can your hypothesis be tested without violating ethical standards?

The hypothesis you use will depend on what you are investigating and hoping to find. Some of the main types of hypotheses that you might use include:

• Simple hypothesis : This type of hypothesis suggests there is a relationship between one independent variable and one dependent variable.
• Complex hypothesis : This type suggests a relationship between three or more variables, such as two independent and dependent variables.
• Null hypothesis : This hypothesis suggests no relationship exists between two or more variables.
• Alternative hypothesis : This hypothesis states the opposite of the null hypothesis.
• Statistical hypothesis : This hypothesis uses statistical analysis to evaluate a representative population sample and then generalizes the findings to the larger group.
• Logical hypothesis : This hypothesis assumes a relationship between variables without collecting data or evidence.

A hypothesis often follows a basic format of "If {this happens} then {this will happen}." One way to structure your hypothesis is to describe what will happen to the  dependent variable  if you change the  independent variable .

The basic format might be: "If {these changes are made to a certain independent variable}, then we will observe {a change in a specific dependent variable}."

A few examples of simple hypotheses:

• "Students who eat breakfast will perform better on a math exam than students who do not eat breakfast."
• "Students who experience test anxiety before an English exam will get lower scores than students who do not experience test anxiety."​
• "Motorists who talk on the phone while driving will be more likely to make errors on a driving course than those who do not talk on the phone."
• "Children who receive a new reading intervention will have higher reading scores than students who do not receive the intervention."

Examples of a complex hypothesis include:

• "People with high-sugar diets and sedentary activity levels are more likely to develop depression."
• "Younger people who are regularly exposed to green, outdoor areas have better subjective well-being than older adults who have limited exposure to green spaces."

Examples of a null hypothesis include:

• "There is no difference in anxiety levels between people who take St. John's wort supplements and those who do not."
• "There is no difference in scores on a memory recall task between children and adults."
• "There is no difference in aggression levels between children who play first-person shooter games and those who do not."

Examples of an alternative hypothesis:

• "People who take St. John's wort supplements will have less anxiety than those who do not."
• "Adults will perform better on a memory task than children."
• "Children who play first-person shooter games will show higher levels of aggression than children who do not." 

Collecting Data on Your Hypothesis

Once a researcher has formed a testable hypothesis, the next step is to select a research design and start collecting data. The research method depends largely on exactly what they are studying. There are two basic types of research methods: descriptive research and experimental research.

Descriptive Research Methods

Descriptive research such as  case studies ,  naturalistic observations , and surveys are often used when  conducting an experiment is difficult or impossible. These methods are best used to describe different aspects of a behavior or psychological phenomenon.

Once a researcher has collected data using descriptive methods, a  correlational study  can examine how the variables are related. This research method might be used to investigate a hypothesis that is difficult to test experimentally.

Experimental Research Methods

Experimental methods  are used to demonstrate causal relationships between variables. In an experiment, the researcher systematically manipulates a variable of interest (known as the independent variable) and measures the effect on another variable (known as the dependent variable).

Unlike correlational studies, which can only be used to determine if there is a relationship between two variables, experimental methods can be used to determine the actual nature of the relationship—whether changes in one variable actually  cause  another to change.

The hypothesis is a critical part of any scientific exploration. It represents what researchers expect to find in a study or experiment. In situations where the hypothesis is unsupported by the research, the research still has value. Such research helps us better understand how different aspects of the natural world relate to one another. It also helps us develop new hypotheses that can then be tested in the future.

Thompson WH, Skau S. On the scope of scientific hypotheses .  R Soc Open Sci . 2023;10(8):230607. doi:10.1098/rsos.230607

Taran S, Adhikari NKJ, Fan E. Falsifiability in medicine: what clinicians can learn from Karl Popper [published correction appears in Intensive Care Med. 2021 Jun 17;:].  Intensive Care Med . 2021;47(9):1054-1056. doi:10.1007/s00134-021-06432-z

Eyler AA. Research Methods for Public Health . 1st ed. Springer Publishing Company; 2020. doi:10.1891/9780826182067.0004

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Nevid J. Psychology: Concepts and Applications. Wadworth, 2013.

By Kendra Cherry, MSEd Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

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• Published: 19 April 2024

Asteroid Kamo‘oalewa’s journey from the lunar Giordano Bruno crater to Earth 1:1 resonance

• Yifei Jiao   ORCID: orcid.org/0000-0003-1097-0521 1 ,
• Bin Cheng   ORCID: orcid.org/0000-0002-8025-9113 1 ,
• Yukun Huang   ORCID: orcid.org/0000-0003-1215-4130 1 , 2 ,
• Erik Asphaug   ORCID: orcid.org/0000-0003-1002-2038 3 ,
• Brett Gladman 2 ,
• Renu Malhotra   ORCID: orcid.org/0000-0002-1226-3305 3 ,
• Patrick Michel   ORCID: orcid.org/0000-0002-0884-1993 4 ,
• Yang Yu   ORCID: orcid.org/0000-0001-9329-7015 5 &
• Hexi Baoyin   ORCID: orcid.org/0000-0002-6389-8677 1 , 6  

Nature Astronomy ( 2024 ) Cite this article

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• Asteroids, comets and Kuiper belt
• Rings and moons

Among the nearly 30,000 known near-Earth asteroids (NEAs), only tens possess Earth co-orbital characteristics with semi-major axes ~1 au. In particular, 469219 Kamo‘oalewa (2016 HO3), an upcoming target of China’s Tianwen-2 asteroid sampling mission, exhibits a meta-stable 1:1 mean-motion resonance with Earth. Intriguingly, recent ground-based observations show that Kamo‘oalewa has spectroscopic characteristics similar to space-weathered lunar silicates, hinting at a lunar origin instead of an asteroidal one like the vast majority of NEAs. Here we use numerical simulations to demonstrate that Kamo‘oalewa’s physical and orbital properties are compatible with a fragment from a crater larger than 10–20 km formed on the Moon in the last few million years. The impact could have ejected sufficiently large fragments into heliocentric orbits, some of which could be transferred to Earth 1:1 resonance and persist today. This leads us to suggest the young lunar crater Giordano Bruno (22 km diameter, 1–10 Myr age) as the most likely source, linking a specific asteroid in space to its source crater on the Moon. The hypothesis will be tested by the Tianwen-2 mission when it returns a sample of Kamo‘oalewa. And the upcoming NEO Surveyor mission may help us to identify such a lunar-derived NEA population.

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The raw simulation data are available from the corresponding authors upon reasonable request. Source data are provided with this paper.

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Acknowledgements

B.C. is supported by the National Natural Science Foundation of China (no. 12202227) and the Postdoctoral Innovative Talent Support Program of China (no. BX20220164). This work is also supported by the National Natural Science Foundation of China under grant 62227901. We thank W. F. Bottke and others for valuable discussions on this work at the Asteroids, Comets, Meteors Conference 2023. We thank M. Connors, T. Santana-Ros and F. Ferrari for providing helpful comments to improve and clarify the manuscript. We acknowledge the use of imagery from Lunar QuickMap ( https://quickmap.lroc.asu.edu ), a collaboration between NASA, Arizona State University and Applied Coherent Technology Corp.

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Yifei Jiao, Bin Cheng, Yukun Huang & Hexi Baoyin

Department of Physics and Astronomy, University of British Columbia, Vancouver, British Columbia, Canada

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Lunar and Planetary Laboratory, University of Arizona, Tucson, AZ, USA

Erik Asphaug & Renu Malhotra

Laboratoire Lagrange, Observatoire de la Côte d’Azur, Université Côte d’Azur, CNRS, Nice, France

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Contributions

Y.J. performed the SPH and N -body numerical simulations and analysed the numerical results. B.C. and H.B. initiated the project, designed the simulations and led the research. Y.H., B.G. and R.M. contributed to the discussion of the dynamical evolution of lunar ejecta and the spectral comparison. E.A., P.M. and Y.Y. contributed to the discussion of the lunar impact ejection process. All authors contributed to interpretation of the results and preparation of the paper.

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Correspondence to Bin Cheng or Hexi Baoyin .

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Nature Astronomy thanks Martin Connors, Fabio Ferrari and Toni Santana-Ros for their contribution to the peer review of this work.

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Extended data

Extended data fig. 1 comparison of kamo‘oalewa’s spectral slope with bus-demeo asteroid taxonomies and lunar materials..

Due to errors in the infrared reflectance, Kamo‘oalewa’s slope possibly ranges from 76 to 101%/ μ m 3 . Typical asteroid slopes are plotted with the mean values and the standard deviations, based on 371 asteroid samples 10 . The slopes of lunar meteorite and samples, whose spectra have been shown in Fig. 1 , are measured using the online Bus-DeMeo taxonomy tool 10 .

Source data

Extended data fig. 2 spectral comparison using the mixture of common factor analysers (mcfa) model..

The latent scores are computed as a lower-dimensional representation of the reflectance spectrum, using the classy package 51 . The MCFA results suggest that Kamo‘oalewa is spectrally similar to lunar materials, but incompatible with any typical asteroid spectrum in public repositories.

Extended Data Fig. 3 Simulated rotation distribution of escaping SPH particles with L \({}_{\max }\) larger than 36 m, for a Kepler-sized crater forming event.

Here we use the SPH particle vorticity to approximate the rotation state of the sub-resolution fragments. The upper panel presents the cumulative fraction of these high-velocity and low-shocked particles versus the rotation period. Note that about 65% of these particles, when ejected, are spinning faster than Kamo‘oalewa which has a period of 28.3 min 3 . The lower panel is a boxplot of the period distribution, suggesting a median of about 6 minutes and an interquartile range (IQR, the box range from the first quartile Q1 to the third quartile Q3) from a few to several tens of minutes. The whiskers are bounded at Q1-1.5*IQR and Q3+1.5*IQR, with the flier points removed. There are 466 and 94 SPH particles used for k=10 30 m −3 and k=10 33 m −3 , respectively.

Extended Data Fig. 4 The location and topography of lunar crater Giordano Bruno.

Left is a map of the lunar farside using the Lunar QuickMap ( https://bit.ly/45Ftwjh ). Right is the topographic map of GB crater from the Lunar Reconnaissance Orbiter Camera (LROC) data 69 .

Extended Data Fig. 5 Initial condition of N-body simulations.

We start each set of simulation at a random lunar phase ϕ , which indicates the relative position of the Sun-Earth-Moon system, and launched 300 particles along a θ  = 45 ∘ cone at random azimuths ζ and with a given velocity magnitude distribution v 0 (following a power law distribution ranging from 2.38 to 6.0 km/s, and with the power of -4.0 according to Supplementary Fig. 7 ).

Extended Data Fig. 6 GB ejecta delivered to Earth, normalized with the background lunar meteorite flux.

It has been estimated that the GB ejecta is comparable to the total lunar ejecta produced by other craters over 10 Myr 42 , thus the background lunar ejecta per Myr is about one-tenth of the GB total ejecta. Assuming the same delivery efficiency (Earth collision fraction over time) as the GB ejecta, we can integrate the product of the ejecta volume and the delivery efficiency of all previous craters, to estimate the background flux delivered to Earth per Myr. The result shows a ten-fold spike of GB meteorites than the background flux in the first Myr after GB formation (several million years ago). However, this spike is currently unobservable due to the short terrestrial preservation period of lunar meteorites, which only lasts a few hundred thousand years 8 . Presently, GB ejecta contributes to roughly 10% of the background flux of lunar meteorites, implying that our current collection of lunar meteorites likely contains only a handful of GB ejecta, for example, the possible GB meteorites Yamato-82192/82193/86032 42 .

Supplementary information

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Jiao, Y., Cheng, B., Huang, Y. et al. Asteroid Kamo‘oalewa’s journey from the lunar Giordano Bruno crater to Earth 1:1 resonance. Nat Astron (2024). https://doi.org/10.1038/s41550-024-02258-z

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Jupiter killed Earth's sister Theia, but helped in giving birth to our Moon: Study

Produced by: Tarun Mishra Designed by: Manoj Kumar

Jupiter's Role in Solar System Chaos

Scientists suggest that Jupiter played a significant role in the chaotic early years of the solar system, potentially contributing to the formation of Earth's moon.

Timeframe of Instability

The so-called "great instability" event, which saw the gas giants shift positions, likely occurred between 60 and 100 million years after the solar system's formation.

Source : NASA

The Moon Formation Hypothesis

During this tumultuous period, Jupiter's migration may have destabilized the orbit of a protoplanet called Theia, leading to a collision with Earth and the formation of the moon.

Puzzle of Asteroid Composition

Scientists have puzzled over the compositions of asteroids and comets, which contain material that doesn't align with their expected locations in the solar system's formation disk.

Investigating Meteorites

Researchers focused on a specific type of meteorite, enstatite chondrites, which share similarities with Earth's composition and likely originated from the same region of the early solar system.

Connection to Asteroid Families

These meteorites have been linked to the Athor family of asteroids in the asteroid belt, suggesting they were once part of a larger body that was disrupted by an ancient collision.

Timing of Jupiter's Influence

Dynamical simulations indicate that Jupiter's migration could have scattered the progenitor of the Athor family into the asteroid belt as early as 60 million years after the solar system's formation.

Link to Moon Formation

The timeframe of Jupiter's instability coincides with the collision between Earth and Theia, providing a plausible link to the formation of Earth's moon.

Challenges of Proof

While evidence suggests Jupiter 's involvement in moon formation, proving it definitively remains challenging due to the distant timeframe and limited data available.

Publication and Presentation

The findings were published in the journal Science and presented at the European Geological Union General Assembly in Vienna on April 16, shedding light on the complex dynamics of the early solar system .

Gradient Flow of the Sinai–Ruelle–Bowen Entropy

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• Published: 25 April 2024
• Volume 405 , article number  118 , ( 2024 )

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• Miaohua Jiang   ORCID: orcid.org/0000-0002-0755-3818 1  

Motivated by an extension to Gallavotti–Cohen Chaotic Hypothesis, we study local and global existence of a gradient flow of the Sinai–Ruelle–Bowen entropy functional in the space of transitive Anosov maps. For the space of expanding maps on the unit circle, we equip it with a Hilbert manifold structure using a Sobolev norm in the tangent space of the manifold. Under the additional measure-preserving assumption and a slightly modified metric, we show that the gradient flow exists globally and every trajectory of the flow converges to a unique limiting map where the SRB entropy attains the maximal value. In a simple case, we obtain an explicit formula for the flow’s ordinary differential equation representation. This gradient flow has close connection to a nonlinear partial differential equation, a gradient-dependent diffusion equation.

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1 Introduction: Motivation and Problem Formulation

There are two sources of motivation for the study of the gradient flow of the entropy of Anosov maps with respect to the Sinai–Ruelle–Bowen (SRB) invariant measure. One is purely mathematical: Let U ( L ) denote a family of \(C^3\) transitive Anosov systems on a torus of dimension \(n\ge 2\) that are topologically conjugate to a linear Anosov map L . We know that every transitive Anosov map belongs to one of these families [ 20 ] (Page 587). Each map \(f \in U(L)\) possesses a unique SRB measure \(\rho _f\) supported on the phase space. It is now well-known that the Kolmogorov–Sinai entropy of f with respect to the SRB measure \(\rho _f\) , \(H(\rho _f)\) depends on the Anosov map f differentiably [ 27 ]. The value of this entropy \( H(\rho _f), f \in U(L)\) can vary between zero and the map’s topological entropy [ 14 ]. Because of the significance of the SRB measure [ 29 ], one naturally wishes to know what properties this entropy functional may have and whether it induces a gradient flow in U ( L ).

But the main source of motivation for studying this gradient flow comes from a new perspective on mathematical modeling of the evolution process of an isolated nonequilibrium thermodynamic system of many-particles based on Gallavotti–Cohen Chaotic Hypothesis [ 5 , 6 , 7 , 8 ]. In their hypothesis, one transitive Anosov map is used to represent a thermodynamic system for the study of macroscopic properties: A reversible many-particle system in a stationary state can be regarded as a transitive Anosov system for the purpose of computing the macroscopic properties of the system. We extend the chaotic hypothesis to modeling the process in which an isolated thermodynamic system evolves from its nonequilibrium state to an equilibrium and propose macroscopic properties of the process that we can investigate with this mathematical model.

Extension of Gallavotti–Cohen Chaotic Hypothesis: The process in which an isolated nonequilibrium thermodynamic system evolves to its equilibrium can be regarded as a gradient flow of an entropy functional in a family of transitive Anosov systems for the purpose of investigating macroscopic properties of the process.

This extension of the chaotic hypothesis to the evolution process of an isolated thermodynamic system is quite natural based on the following three considerations:

It extends the chaotic hypothesis to irreversible many-particle systems. As an isolated thermodynamic system evolves to its equilibrium, the entropy increases. Thus, the process is irreversible.

It makes the maximal entropy production principle (MEPP) [ 25 ] an integral part of the hypothesis.

It will allow us to explore whether laws of thermodynamics can be proved rigorously in this mathematical model. Proving the basic postulate of thermodynamics that every system evolves to a unique equilibrium would mean to establish the global existence of a gradient flow in which every trajectory converges to a unique limiting Anosov system. Proving the second law of thermodynamics would mean to show that the entropy increases along any trajectory of the flow.

Before we can begin our investigation, we need to clarify a couple of technical issues. First, we need to decide which entropy to choose for a transitive Anosov system corresponding to that of a nonequilibrium thermodynamic system.

The choice of an entropy. We choose the entropy for a transitive Anosov system to be the Kolmogorov–Sinai (a.k.a measure-theoretic or metric) entropy of its SRB measure for several reasons. As Goldstein et al have pointed out [ 11 ] that there are essentially two notions of entropy for an isolated thermodynamic system: the Gibbs entropy which is defined for every equilibrium measure that is invariant under the dynamics and the other is the Boltzmann entropy that is defined for the system. They explained the differences between these two notions and why the Boltzmann entropy is the right choice for nonequilibrium systems. Choosing the entropy for a transitive Anosov system to be the Kolmogorov–Sinai entropy of the SRB measure combines the essential features of both notions: it is defined through an invariant measure and it is uniquely defined for each transitive Anosov system.

There is another significant reason why the SRB measure is chosen: it reflects the asymptotic distribution of orbits starting from a typical initial point with respect to the Lebesgue measure. Within the family of transitive Anosov systems, the value of the SRB entropy can take any value greater than zero and less than or equal to the topological entropy. Roughly speaking, when the value of the entropy is near zero, it often indicates that many orbits will spend a disproportional amount of time near a fixed point with a small expansion rate while a large entropy value indicates that orbits are more evenly distributed over the phase space.

There are many other possible choices of invariant measures for the entropy. The SRB entropy is the equilibrium state for the potential function \(\varphi = -\log J^uf \) , essentially, the rate of expansion along the unstable subspace. Theoretically, any f -dependent potential function could be chosen as long as it is Hölder continuous and its dependence on f is differentiable. For example, one may choose the sum of the rates along both the stable and unstable subspaces, \(\varphi = -\log J^uf + \log J^s f \) . The resulting gradient flow may have different behaviors.

Differentiable conjugacy: After this SRB entropy is chosen, we need to address the issue that different Anosov systems may have the same entropy. Indeed, two systems that are differentiably conjugate always have the same SRB entropy. Thus, the entropy functional is defined over equivalent classes. Systems that are differentiably conjugate can be considered as reversible systems since the entropy does not change.

There are unaddressed issues concerning this extension of the chaotic hypothesis: If Anosov systems are truly good coarse mathematical models for thermodynamic systems, we wish that such systems as a collection would satisfy the axioms proposed by Lieb and Yngvason in [ 22 ]. Properties such as the scaling property would hold for the entropy. However, at the moment, it is unclear how to address such issues. Another related question is about the role of the dimension of the phase space. Ideally, the phase space’s dimension should be very large, even infinite as in the coupled systems over an infinite lattice.

Under this extended version of Gallavotti–Cohen Chaotic Hypothesis with the aforementioned choice of the entropy functional, we now are ready to formulate mathematical problems that reflect laws of thermodynamics. We first describe briefly the space where the SRB entropy functional will be defined. Let \(f_0\) be a \(C^r, r \ge 3\) transitive Anosov map or an expanding map on a closed Riemannian manifold M . The \(C^3\) condition is used to guarantee the differentiability of the entropy functional for Anosov maps. For expanding maps, the \(C^3\) condition can be relaxed to \(C^{1+\alpha }\) for some \(\alpha > 0\) . Since we will later consider the differentiability of the entropy in a different norm (Sobolev norm), for convenience, we will require expanding maps to have at least the third-order derivative, but not necessarily \(C^3\) . Let \(U(f_0)\) be the connected open component of all \(C^r\) maps in either family topologically conjugate to \(f_0\) . It is well-known that there exists a unique Sinai–Ruelle–Bowen (SRB) measure \(\rho _f\) for every map \(f \in U(f_0)\) [ 29 ] and the entropy of the map with respect to the SRB measure \(\rho _f\) is given by the formula \(H(f)= \int _M \ln J^uf \ d \rho _f\) [ 26 ] (Pages 230 & 294), where \(J^uf\) is the Jacobian of f along the unstable subspace. This entropy is a Fréchet differentiable functional in f with respect to the \(C^r\) topology on \(U(f_0)\) [ 2 , 3 , 15 , 27 ]. The map \(f_0\) can also be a piecewise \(C^r\) expanding map on an interval or an Axiom A diffeomorphism on a closed Riemannian manifold. For simplicity, we limit our formulation of problems to the Anosov and expanding map cases.

Problem one : Does the entropy functional \(H(f): U(f_0) \rightarrow {\mathbb {R}}\) induce a gradient flow in \(U(f_0)\) ? If it does, what properties does the gradient flow have? In particular, does every trajectory of the flow converge to a limiting map?

We note that local or global existence of a gradient flow does not follow from the analyticity of the entropy functional defined on the space of Hölder potential functions. In the traditional setup, a fixed transitive Anosov system f is used to model a thermodynamic system. An equilibrium state is then defined via the variational principal for every Hölder potential function. This equilibrium state can also be defined via a transfer operator induced by the Anosov map [ 12 ]. The Kolmogorov–Sinai entropy of the equilibrium state is analytic in terms of potential functions. Thus, the entropy functional defines a gradient flow after an adjustment of the norm in the potential space [ 9 , 23 ]. We note that this gradient flow is not defined in the space of Anosov systems. It is true that the space \(U(f_0)\) can be embedded into the space of Hölder continuous potential functions via symbolic representation since all maps in \(U(f_0)\) share a common Markov partition. But \(U(f_0)\) is in general, not an invariant submanifold of the gradient flow defined on the potential space.

In our setup, if the global existence of the gradient flow of the SRB entropy functional holds and each trajectory is shown to converge to a unique Anosov system, it can be regarded as a rigorous proof of the basic postulate of the thermodynamics: an isolated thermodynamic nonequilibrium system evolves to an unique equilibrium.

Problem Two. Does the SRB entropy functional have any critical point, for example, a local maximum, where its value is strictly less than the global maximum, i.e., the topological entropy?

Such critical point is supposed to be absent in the evolution of an isolated thermodynamic system. If we can show that such critical point does not exist for the gradient flow and every trajectory converges to a limiting map, then the limiting map’s SRB entropy must be the maximum.

Problem Three. Would it be possible to have a differential equation description of the diffusion process induced by the gradient flow of the SRB entropy functional in the space of Anosov maps or the expanding maps?

The process of evolution of an isolated thermodynamic system from nonequilibrium system to an equilibrium is a diffusive process as the heat, density, or pressure diffuses. If the gradient flow exists and the gradient vector can be explicitly calculated, it may potentially be expressed as differential equations in an infinite-dimensional space. One would like to compare such equations with diffusion equations such as the heat equation derived at the macroscopic level by making assumptions on the heat flux. Considering the gradient flow in the space of Anosov maps or expanding maps will lead to alternative mathematical descriptions of this diffusion process. With possible physics experiments, we can gain insights on the relation between the choice of the entropy and the resulting diffusion process.

At the moment, rigorous results are only proven for low dimensional systems. In [ 16 ], it has been proven that for both families of expanding maps and Markov transformations on a closed interval semi-conjugating to a full shift, the SRB entropy functional does not have any nontrivial critical point. If f is a critical point, then f has to be a linear map up to a differentiable conjugacy. In [ 17 , 28 ], this property is proven for families of general transitive Markov transformations and Anosov maps on a torus of dimension two. Proofs of these results, with the exception in the the Anosov system case, involve only well-established techniques.

In this article we prove global existence of the gradient flow induced by the SRB entropy on the family of expanding maps on the circle preserving a common SRB measure. We also prove that every trajectory of the flow starting from a nonequilibrium map, i.e., its SRB entropy is strictly smaller than its topological entropy, will converge to an expanding map that is differentiably conjugate to the linear expanding map. Furthermore, we obtain an ordinary differential equation representation of the gradient flow that has a close connection to a gradient-dependent diffusion equation.

When the unstable subspace of an Anosov or expanding map is one dimensional, increasing the SRB entropy forces the map to become linear. The mechanism is similar to the situation of maximizing the Shannon entropy \(- \sum _{i=1}^n p_i \log p_i\) in a probability space. When the dimension of the unstable subspace is greater than one, the nonlinearity of the map will not disappear by simply increasing the SRB entropy since the entropy is equal to a sum of positive Lyapunov exponents. However, local and global existence of the gradient flow is expected to be true. But trajectories of the flow will unlikely converge to linear maps.

In next section, we first extend the family \(U(f_0)\) (still denoted by \(U(f_0)\) ) to include all expanding maps whose r th derivative is \(L^2\) . We show that \(U(f_0)\) is then equipped with a natural Hilbert manifold structure with a Sobolev norm in the tangent space. Under this Hilbert manifold structure, we consider the submanifold of \(U(f_0)\) consisting of maps preserving a common SRB measure and show that the SRB entropy functional H ( f ) remains Fréchet differentiable and thus the functional gives a gradient vector field over \(U(f_0)\) . In Sects.  2 and  3 , we show the gradient vector field is at least Lipschitz continuous, which guarantees the existence of a local gradient flow. We note that a differentiable function’s gradient is metric dependent. Under a slightly modified metric, we show in Sect.  4 the global existence of a gradient flow of the SRB entropy functional and the convergence of the flow to the linear expanding map as time approaches infinity. In the last section, via harmonic analysis, we obtain an ordinary differential equation representation of the gradient flow over the Hilbert manifold equipped with the Sobolev norm and give an example of a typical orbit using numerical approximation. The gradient flow also leads to a gradient-dependent diffusion equation on the circle.

2 Hilbert Manifold Structure on the Family of Circle Expanding Maps

The SRB entropy is a differentiable functional in the space of \(C^r, (r \ge 3)\) expanding maps on the unit circle. But when we consider the gradient of the entropy functional, the \(C^r\) norm may not be the most convenient one. So, a Sobolev norm becomes a more natural choice instead of the \(C^r\) -norm. We first give the definition of a gradient vector for any Gateaux differentiable functional on the Hilbert manifold \({\mathcal {M}}\) with a tangent space \(T_p{\mathcal {M}}\) at each point \(p \in {\mathcal {M}}\) and a Hilbert metric (inner product) \(<\cdot >_{\mathcal {M}}\) .

Definition 1

A vector \(V \in T_p{\mathcal {M}}\) is called a gradient vector of a Gateaux differentiable functional H if the Gateaux derivative of H at p defines a bounded linear functional on \(T_p{\mathcal {M}}\) : \( \nabla H(p): T_p{\mathcal {M}} \rightarrow {\mathbb {R}}\) and the Riesz representation of \(\nabla H(p)\) is V .

We now describe a Hilbert manifold structure on a family of circle expanding maps.

2.1 Hilbert manifold of expanding maps on the circle

First of all, by considering its lifts to the real line \({\mathbb {R}}\) , we identify every continuous map f on \(S^1=\{ e^{ i 2\pi x}, x \in [0, 1)\}\) with a unique function \({{\tilde{f}}}\) defined on \({\mathbb {R}}\) satisfying the conditions \( {{\tilde{f}}}(0) \in [0, 1)\) and \(\tilde{f}(x+1) = {{\tilde{f}}} (x) +n\) , where n is the degree of the map. For simplicity of statements, we limit our exposition to the orientation preserving expanding maps where \({{\tilde{f}}} '(x) > 0, x \in {\mathbb {R}}\) . In the case of orientation reversing expanding maps where \({{\tilde{f}}} '(x) < 0, x \in {\mathbb {R}}\) , the results of this paper remain valid but the exposition needs to be modified. Since expanding maps are defined on the circle, there must be a fixed point and we may assume 0 is a fixed point for both f and its lift \({{\tilde{f}}}\) : \(f(0)=0={{\tilde{f}}}(0)\) and \({{\tilde{f}}}(1) =n\) . Thus, each \(C^r, r \ge 1\) map f on the circle is identified with its lift \({{\tilde{f}}}\) on \({\mathbb {R}}\) , the universal covering of \(S^1\) , with the following properties: \( {\tilde{f}}(0)=0, {\tilde{f}}(1) = n, {\tilde{f}}'(x) > 1, x \in {\mathbb {R}}\) and \( {\tilde{f}}^{(k)}(0)= {\tilde{f}}^{(k)}(1), k=1,2,\ldots , r.\) We now define a family of \(C^{r-1}\) expanding maps \(F_r, r\ge 3\) via properties of their lift maps where we consider the gradient flow of the entropy functional:

The family \(F_r\) is slightly larger than the \(C^r \) family of expanding maps since we only require the r th derivative \(f^{(r)}\) to be \(L^2\) , instead of being continuous. \(F_r\) is a separable Hilbert manifold modeled on the Sobolev space \(H^r\) . For any given map \( f\in F_r\) , its open neighborhood is given with an open neighborhood of the origin of the following Hilbert space

equipped with the Sobolev norm

Notice that \(\Phi _r\) can be identified with a Sobolev sequence space

equipped with the corresponding Sobolev norm

where \(\{a_n\}_{n=0}^\infty \) and \(\{b_n\}_{n=1}^\infty \) are Fourier coefficients of \(\phi \) .

When \(r \ge 3\) , each expanding map \( f \in F_r\) possesses a unique invariant probability measure absolutely continuous with respect to the Lebesgue measure on \(S^1\) . Its probability density function \(\rho _f(x)\) is at least \(C^1\) [ 1 ] and depends on f differentiably in \(C^{r-1}\) topology. It yields the Fréchet differentiability of the entropy of f with respect to the measure \(\rho _f \) , i.e., \(H(f) = \int _{S^1} \ln f'(x) \rho _f(x) dx.\) We want to prove that H ( f ) is also a Fréchet differentiable functional with respect to the new Hilbert metric on \(F_r\) . In general, this can be done by using the imbedding theorem (Theorem 2.7 [ 13 ]). Since in this paper, we restrict our study to a simpler case when the SRB measure \(\rho _f\) is preserved by the perturbation of f , \(\rho _f(x)\) is independent of f , the differentiability of the entropy with respect to the Hilbert metric is much easier to prove. We can directly calculate the derivative operator and prove the Fréchet differentiability with respect to the Sobolev norm.

2.2 Hilbert manifold of expanding maps preserving the Lebesgue measure

We now define a Hilbert manifold \(F_r(\rho ), r\ge 3\) to be the subset of \(F_r\) consisting of maps that preserve the same invariant measure with a density function \(\rho (x)\) . We may assume \(\rho (x) =\rho _0(x)=1\) by changing the Riemannian metric on the circle [ 4 , 16 ]. The corresponding subset is denoted by \(F_r(\rho _0)\) .

Given any map \(f\in F_r(\rho _0)\) , the invariance of the Lebesgue measure under f is characterized by the equation

where \( 0\le y_1< y_2< \ldots< y_n < 1\) are n preimages of \(x \in [0,1):\) \({{\tilde{f}}}(y_i) = x, \text {mod\ 1}, i=1,2, \ldots , n.\)

We see that not only the equation ( 2 ) is nonlinear in f , the points \(\{y_i\}\) , preimages of x , also depend on f . Thus, it is not convenient when we calculate Gateaux derivatives of the entropy functional with respect to f . Instead, we now identify the subset \(F_r(\rho _0)\) with another Hilbert manifold with a Sobolev tangent space where the same SRB entropy functional’s properties are much easier to study.

For each map \(f \in F_r(\rho _0)\) , we consider its lift \({{\tilde{f}}}\) ’s inverse map \(g(y), y \in [0, n]\) . The r th derivative of \({{\tilde{f}}}\) is in \(L^2[0,1]\) if and only if g ( y )’s r th derivative is in \(L^2[0,n]\) since we have \({{\tilde{f}}}'(x) > 1, x \in [0,1]\) . Moreover, \({{\tilde{f}}}^{(k)}(0)= {{\tilde{f}}}^{(k)}(1), k=1,\dots r\) if and only if g ( y ) is differentiable in [0,  n ] up to order r and \(g^{(k)}(0^+) = g^{(k)}(n^-)\) , \(k=1,\dots r\) . That is, g ( y ) can be extended to a function whose k th derivative \(1 \le k \le r\) is a period n function. Given any \(x \in [0, 1)\) , \({{\tilde{f}}}\) maps n preimages of \(x\in [0,1)\) under f to \(x, x+1, \ldots , x + (n-1)\) in the universal covering space: \({{\tilde{f}}}(y_i) = x+ i-1, i=1,2,\ldots , n\) . Thus, we have

and the invariance of the Lebesgue measure becomes an equation linear in g :

We now define the Hilbert manifold where we consider the SRB entropy’s gradient for \(r \ge 3\) .

For any \(g_0, g \in G_r\) , \(g-g_0 \) belongs to the following Sobolev space, still denoted by \(\Phi _r\) :

The Sobolev norm on \(\Phi _r\) is defined in the same way:

On the other hand, given any \(g \in G_r\) , when \(\epsilon >0\) is sufficiently small, \(g +\epsilon \phi \in G_r\) for all \(\phi \in \Phi _r\) with \( \phi (x) \Vert ^2_{H^r} \le 1\) . Thus, for each \(g \in G_r\) , its open neighborhood can be identified with an open neighborhood of 0 in \(\Phi _r\) .

For convenience, we denote the even larger family of functions without the constraint of preservation of Lebesgue measure by \(\bar{G}_r\) and \({{\bar{\Phi }}}_r\) :

Indeed, \(G_r\) is a submanifold of \({{\bar{G}}}_r\) and \(\Phi _r\) a subspace of \({{\bar{\Phi }}}_r\) since the constraint is a linear affine equation.

On the Hilbert manifold \({{\bar{G}}}_r\) , there is a natural metric \(d(g_1, g_2)\) that is consistent with the Sobolev norm in the tangent space of \({{\bar{G}}}_r\) :

Indeed, \(g_1-g_2 \in {{\bar{\Phi }}}_r\) for any \(g_1, g_2 \in {{\bar{G}}}_r \) . Thus,

Let \(a_i\) denote the preimage of i under \({{\tilde{f}}}\) for \(i=0, 1, \ldots , n-1\) . The SRB entropy defined for every \(f \in F_r(\rho _0)\) becomes

We point out an interesting connection between the SRB entropy of measure-preserving expanding maps and the Gibbs entropy of a probability measure with a density. Any function \(g(y) \in {{\bar{G}}}_r\) can be considered as a probability measure on [0,  n ] with a density function \(0<g'(y)<1\) . H ( g ) is then precisely the Gibbs entropy of a probability measure.

We also point out similarities and differences between our approach and the approach of Jordan, Kinderleherer, and Otto (JKO) [ 18 , 19 ] in their study of the gradient flow of an entropy functional (see also [ 24 ]). The main similarity is that both approaches start from the Gibbs entropy. But two approaches have major differences: In JKO’s approach, the entropy (or the Gibbs–Boltzmann entropy as it is called in [ 18 , 19 ]. For more discussions on Gibbs and Boltzmann entropy, see [ 11 ]) is defined for probability density functions on \({\mathbb {R}}^n\) not associated with any dynamical system. They use a discretized process to obtain an approximate orbit from an initial density and then show that the orbit converges to an orbit from the heat equation as the step-size approaches zero. In our approach, the SRB entropy is defined for a chaotic dynamical system. The SRB entropy changes as the underlying chaotic dynamical system varies and we directly calculate the entropy functional’s gradient under the Sobolev norm. Our approach leads to a system of countably many ordinary differential equations where the vector field is defined via integrals. The system has a close connection to a nonlinear partial differential equation, a gradient-dependent diffusion equation on the unit circle. See Sect.  5 for detail.

During the reviewing process of the paper, referees bought to our attention another closely related work [ 9 ] on the gradient flow of another entropy functional. For a transitive Anosov map on a closed Riemannian manifold, for every Hölder continuous function \(\varphi \) , there exists a unique equilibrium measure \(\mu _\phi \) that satisfies the variational principal. The entropy functional \( \varphi \rightarrow h_{\mu _\phi } (f)\) is analytic on the space of Hölder continuous functions. This gradient flow from this functional is different since the map f is fixed. Since all maps in \(U(f_0)\) share a common Markov partition and the potential function for the SRB measure is Hölder continuous, \(U(f_0)\) can be embedded into the space of Hölder continuous functions. But \(U(f_0)\) , in general, is not an invariant set of the gradient flow induced by the functional \( \varphi \rightarrow h_{\mu _\phi } (\Sigma _A)\) , where \(\Sigma _A\) is the subshift of finite type defined by the Markov partition. The differentiability of these two functionals are also different: \( \varphi \rightarrow h_{\mu _\phi } (f)\) is analytic while the differentiability of \( f \rightarrow H (f)\) depends on that of \(f \in U(f_0)\) . See also [ 23 ] for related work.

We now state main results on the SRB entropy functional H ( g ) on the Hilbert manifold \(G_r\) .

The SRB entropy functional H ( g ) is Fréchet differentiable on \(G_r\) .

The gradient vector field of H ( g ) is well-defined: for each \(g \in G_r\) , there exists a unique vector \(X(g) \in \Phi _r\) such that the directional derivative \(<DH(g), X(g)/\Vert X\Vert _{H^r}>\) is the unique maximum among all directional derivatives.

The gradient vector field \(X(g) \in G_r\) is Lipschitz continuous in g .

There is a unique critical point for the gradient vector field X ( g ) at the point where g is a linear function on [0,  n ].

An immediate consequence of Theorem  1 is that the differential equation defined on \(G_r\) by \( \frac{d {\mathcal {F}}_t}{d t }\big |_{t=0} = X(g)\) has a unique local solution for \(t \in (-\epsilon , \epsilon )\) : \({\mathcal {F}}_t(g) = {\mathcal {F}}(t, g) \) is a local flow defined on \((-\epsilon _g, \epsilon _g) \times G_r \) .

3 Proof of Theorem  1

Proof of theorem 1 (1).

We need to show that H ( g ) is Fréchet differentiable. Since \(G_r\) is a submanifold of \({{\bar{G}}}_r\) . We can just prove differentiability in \({{\bar{G}}}_r\) . That means we do not need to consider the constraint of preservation of the Lebesgue measure.

We first calculate the first order term in \(\epsilon \) of the difference \(H(g + \epsilon \phi ) - H(g)\) .

Thus, the first order term of \(H(g + \epsilon \phi ) - H(g)\) in \(\epsilon \) is

Since \(\phi (0)=\phi (n) =0\) , via integration by parts, we have the derivative operator formula

We now show that H ( g ) is Fréchet differentiable on \({{\bar{G}}}_r\) , i.e, for any given \(g \in {{\bar{G}}}_r\) ,

According to our earlier calculation,

\(\square \)

We now need a simple lemma on the upper bound of \(|\phi '(y)|\) :

Given any function \(\phi \in \Phi _r\) , \(r\ge 2\) , \(|\phi '(y)| \le M \Vert \phi (y) \Vert _{H^2}\) . In general, \(| \phi ^{(k)} | \le M \Vert \phi (y) \Vert _{H^{k+1}}\) , where \(1 \le k < r \) and M is a constant independent of \(\phi \) .

Proof of Lemma 1

Since \(\phi \) is of period n and \(\phi ^{(r)} \in L^2[0,n]\) , we have \(\phi \) ’s Fourier expansion

where the Fourier coefficients \(a_k, b_k, k\ge 1\) satisfy the condition

For convenience, we may assume \(a_0=0.\) Thus, by the Cauchy-Schwartz inequality,

Let \(M= 2 \left[ \sum _{k=1}^\infty \frac{1}{k^2} \right] ^{\frac{1}{2}}\) and apply the inequality above to \(\phi '(y)\) . We have

Since \(g'(y)>0\) is bounded away from 0 and \(\phi ^{(2)} \in L^2[0,n]\) , the integrand \( O([\phi '(y)]^2/g'(y)) + \epsilon O( [\phi '(y)]^3/[g'(y) ]^2)\) is a bounded function over [0,  n ]. We have

H ( g ) is Fréchet differentiable at any \( g \in G_r\) .

Proof of Theorem 1 (2) & (4)

We pick an orthonormal basis of the separable Hilbert space \(\Phi _r\) : \(\{{{\textbf {e}}}_i\}_{i=1}^\infty \) . Via Riesz representation, every linear functional on \(\Phi _r\) can be identified with an element in \(\Phi _r\) . Define

where \(b_i= DH_g ({{\textbf {e}}}_i), i=1,2, \ldots .\)

Given any \(\phi \in \Phi _r\) with \(\Vert \phi \Vert _{H^r} =1\) and \(\phi = \sum _{i=1}^\infty <\phi , {{\textbf {e}}}_i>_{H^r} {{\textbf {e}}}_i \) , where \(<\cdot >_{H^r}\) denotes the inner product of \(\Phi _r\) , we have

By the Cauchy-Schwartz inequality, \(| DH_g(\phi ) |\) reaches its maximum if and only if \( <\phi , e_i>_{H^r} =C b_i, i = 1,2, \ldots \) for some constant C , i.e. \(\phi = C X(g)\) . Since \(\Vert \phi \Vert _{H^r} =1 \) , we have \(C = [ \Vert X(g)\Vert _{H^r} ]^{-1}.\)

Thus, the gradient vector field X ( g ) is well-defined on Hilbert manifold \(G_r\) . Since we know that \(X(g)=0\) if and only if g is linear [ 16 ], \(X(g) \not =0\) for all \(g \in G_r\) except for g linear. \(\square \)

Proof of Theorem 1 (3)

We now prove that the derivative operator \(DH_g\) is Lipschitz, which leads to the local existence of the gradient flow [ 21 ].

Given any two maps \(g_1\not = g_2 \in G_r\) , denote \(\psi = g_2 - g_1 \in \Phi _r\) . We now estimate the distance between two derivative operators \(DH_{g_1}\) and \(DH_{g_1+\psi }\) . When \(g_1\) and \(g_2\) are close, their open neighborhoods overlap. Thus, we can assume two derivative operators \(DH_{g_1}\) and \(DH_{g_2}\) are acting on the same Sobolev space \(\Phi _r\) . For simplicity of notation, we drop the subscript in \(g_1\) . We have

Since \(g' > 0\) is bounded from below and \(\psi \in \Phi _r, r\ge 2\) , we may assume that \(|\psi '|\) is sufficiently small and \(|\frac{\psi '}{g'}|< \delta < 1\) . So, there is a constant \(K>0 \) such that

Thus, we have

where \(C = K[{\displaystyle \min _{y \in [0,n]} g'(y) }]^{-1/2}.\) We conclude that \(DH_g\) is Lipschitz continuous over \(G_r\) . \(\square \)

4 Global Existence of the Gradient Flow

We now prove the global existence of the gradient flow of the SRB entropy for \(t \in [0, \infty )\) and show that every trajectory converges to the unique equilibrium where the expansion rate of the map is a constant and H ( f ) attains its global maximum. We note that under the Sobolev norm, the gradient vector at point \(g \in G_r\) is defined by the integral \( \int _0^n \frac{g''(y)}{g'(y)} \ \phi (y) dy\) . While the integral does define a linear functional in the tangent space \(\Phi _r\) for each \( g \in G_r\) , its Riesz representation in \(\Phi _r\) is, in general, not \( \frac{g''(y)}{g'(y)}\) since it may not be a vector in \(\Phi _r\) . This poses an obstacle for proving the global existence directly through the integral. We instead reconsider the global existence of the gradient flow in a different Hilbert metric on a slightly different Hilbert manifold.

We first expand the domain of the entropy functional to a larger Hilbert space where the gradient vector’s Riesz representation can be obtained explicitly.

For the Hilbert manifold

each map \(g \in G_r \) is uniquely defined by its derivative: \(g(y) = \int _0^y g'(\tau ) d \tau \) . So, we can embed \( G_r\) into another Hilbert manifold \(G'\) :

Given any \(g \in G_r\) , the embedding map is defined by

\(G'\) is clearly a Hilbert manifold with a tangent space

equipped with a common Hilbert norm \(\Vert \phi \Vert = \int _0^n \psi ^2(y) dy.\)

Note that the condition \( \int _0^n \psi (y) dy = 0 \) in ( 10 ) can be removed since it is implied by the condition \( \sum _{i=0}^{n-1} \psi (y+i) =0\) :

The tangent space \(\Psi \) is a subspace of \( L^2[0, n]\) .

Note also that the entropy functional,

on \( G_r\) becomes

which is well defined on entire \(G'\) and the Gateaux derivative of H ( h ) in the direction of \(\psi \in \Psi \) exists and has the same formula (see ( 7 )):

This Gateaux derivative defines a bounded linear functional on the tangent space \(\Psi \) .

A direct calculation will confirm that this linear functional’s Riesz representation is given by

where h ( y ) is extended into a period n function over \([0, \infty )\) . Indeed, we can easily verify that \(R_h(y) \in \Psi \) . We only need to verify that \( \int _0^n R_h(y) \psi dy = - \int _0^n \ln h(y) \psi dy\) for all \(\psi \in \Psi \) since the identity

for all \(y \in [0, 1]\) clearly holds due to the periodicity of h ( y ).

To see that \( \int _0^n R_h(y) \psi dy = - \int _0^n \ln h(y) \psi dy\) for all \(\psi \in \Psi \) , we first extend \(\psi \) to a period n function and calculate the following integral applying integration by substitution and periodicity of both functions h ( y ) and \(\psi (y)\) :

Let \(z= y+i\) in each integral, we have

We summarize the properties of the entropy functional \(H(h)=- \int _0^n \ln h(y)\ h(y) dy \) over the Hilbert manifold \(G'\) in the following proposition.

Proposition 1

H ( h ) is Gateaux differentiable at every \(h \in G'\) and the derivative formula in the direction of \(\psi \in \Psi \) is given by a continuous linear functional on \(\Psi \) :

The Riesz representation of the derivative operator \(DH_h \) over \(\Psi \) is

where h ( y ) is extended periodically to \([0, \infty )\) .

The maximum value of \(DH_h(\psi )\) over \(\psi \in \Psi \) with \( \int _0^n \psi ^2(y) d y = 1 \) is reached at the unit vector \( R_h(y) [ \int _0^n R_h^2(y) dy]^{-\frac{1}{2}}.\)

We denote this gradient vector field over \(G'\) by

It is Lipschitz continuous in terms of h under the \(L^2\) norm, thus, locally integrable.

We now prove the following theorem on the global existence of the gradient flow of the SRB entropy on \(G'\) and the convergence of every flow trajectory to a unique equilibrium as \( t\rightarrow \infty .\) Recall \(F_r(\rho _0)\) denotes the Hilbert manifold composed of Lebesgue-measure preserving \(C^r, r\ge 3\) expanding maps.

The SRB entropy functional \(f \rightarrow H(f) = \int _0^1 \ln f'(x) dx, f \in F_r(\rho _0)\) induces a gradient flow on \(G'\) , the space of derivatives of inverse map of \(f \in F_r(\rho _0)\) under the \(L^2\) -norm. This gradient flow exists globally for all \(t \in [0, \infty )\) and every trajectory converges to the unique equilibrium which corresponds to the linear expanding map in \(F_r(\rho _0)\) where H ( f ) attains its maximum value.

For any fixed initial map \(h(y) \in G'\) , let \({{\mathcal {G}}}_t(h)=g(t, h)\) denote the local flow defined by the gradient vector field Y ( h ) on \(G'\) for \(t \in (- \epsilon _h, \epsilon _h)\) . For any \(y \in [0,1]\) , We have

By periodicity of h ( y ), for all \(k=1,2, \ldots , n-1\) , we also have

Introducing new variables \(x_k = h(y+k-1)\) , \( k=1,2,\ldots , n\) , we have a system of n ordinary differential equations

subject to the condition \( \sum _{k=1}^n x_k =1\) . Consequently, we have

Notice that when \(0< x< y < 1\) , \(0< \ln y - \ln x \le \frac{1}{c} (y-x)\) for some \(c \in (x,y)\) . Thus, \( \ln y - \ln x > y-x \) . The differential equation \({\dot{y}} - {\dot{x}} = - ( \ln y - \ln x )\) implies \({\dot{y}} - {\dot{x}} < - (y-x)\) . So, \(y-x\) converges to zero exponentially fast. Therefore, the solution to the system exists globally for all initial values in the region \(0< x_1, x_2, \ldots , x_k < 1 \) on the invariant plane \( x_1+x_2 + \cdots + x_n =1\) and all solutions converge to the unique equilibrium \(x_1=x_2 =\cdots = x_n=\frac{1}{n}\) . \(\square \)

5 Differential Equation Representation of the Gradient Flow

We now explore the possibility of representing the gradient flow \({\mathcal {F}}_t(g)\) from Sect.  2 as explicit differential equations.

Let \({{\mathcal {F}}}_t\) denote the gradient flow defined by the vector field X ( g ) over \(G_r\) ,i.e, \( {{\mathcal {F}}}_t(g) \) is a map from \( (- \epsilon , \epsilon ) \times G_r\rightarrow G_r\) differentiable in t and \({{\mathcal {F}}}_0(g)=g\) and

where \(X(g) \in \Phi _r\) is defined by an integral operator

We see that maps in \( {{\bar{G}}}_r\) can be easily represented as a series. In the simple case when \(n=2\) , we can obtain a system of ordinary differential equations that generates the flow. Numerical methods such as Euler’s method [ 10 ] can then be used to obtain typical approximate trajectories of the flow.

For any given \( g \in {{\bar{G}}}_r\) , \(g(y) - \frac{y}{n} \in {{\bar{\Phi }}}_r \) is a continuous periodic function of period n and its derivative is bounded. Thus, its Fourier series converges to itself both pointwise and in the Sobolev norm. Thus, Hilbert manifold \({{\bar{G}}}_r\) can be represented as

where \(a_k, b_k\) are Fourier coefficients of \(g(y) - \frac{y}{n}\) satisfying the condition

Notice that we have replaced the condition \(g(0)=0\) by dropping the constant term in the Fourier series since the entropy is a function of \(g'(x)\) . This adjustment is also made to the tangent space \(\bar{\Phi }_r\) .

Maps in the submanifold \(G_r\) will have to satisfy an additional linear equation:

Suppose that we can find an orthonormal basis of \(\Phi _r\) : \(\{ {{\textbf {e}}}_k \}_{k=1}^\infty \) . Then, any trajectory of the flow \( u(t,y)= {{\mathcal {F}}}_t(g) \) can be written in the form

with \(u(y, 0)=\sum _{k=1}^\infty c_k(0) \textbf{e}_k = g -\frac{y}{n} \) . Thus the flow equation \(\frac{d {\mathcal {F}}_t(g)}{dt}\big |_{t=0} = X_g\) becomes

We have a system of countably many ordinary differential equations:

While it is easy to obtain a set of orthonormal basis for \(\bar{\Phi }_r\) since the set \(\{ \cos \frac{2\pi k }{n} y, \sin \frac{2\pi k }{n} y \}_{k=1}^\infty \) is clearly an orthogonal basis, the linear constraint ( 11 ) poses an obstacle to finding orthogonal basis for \(\Phi _r\) . Fortunately, in the simple case when \(n=2\) , an orthonormal basis for \(\Phi _r\) can be obtained directly from this set. That will allow us to obtain a system of countably many ordinary differential equations explicitly and thus, to approximate numerically typical trajectories of the flow.

5.1 Ordinary differential equation representation when \(n=2\)

We now look at the case when \(n=2\) . \(r \ge 2\) can be any number. In this case, the linear constraint ( 11 ) becomes

So, we can conclude that \(a_k=b_k =0\) when k is even. For simplicity, we also let \(r=2\) . Since

So, the set \(\{ c_{2\,m-1} \cos (2\,m-1) \pi y, c_{2\,m-1} \sin (2\,m-1) \pi y\}_{m=1}^\infty \) is an orthonormal basis of \(\Phi _2\) .

Let u ( t ,  y ) be a trajectory of the flow \({\mathcal {F}}_t(g)\) . For each t ,

For a fixed value of t , the gradient vector at u ( x ,  t ) is

Notice that

We obtain explicitly a system of ordinary differential equations defined on \(G_2\) that generates the gradient flow.i.e., the local flow \({\mathcal {F}}_t(g)\) is the solution to the system of differential equations:

5.2 The partial differential equation connection

Since the gradient vector X at \(u(t,y)\in G_2\) is defined by

there is a close connection between the gradient flow \({\mathcal {F}}_t(g)\) and the solution to the nonlinear partial differential equation \(w_t = \frac{w_{yy}}{w_y}\) , a gradient-dependent diffusion equation defined on the unit circle. Assume that

is a solution to \(w_t = \frac{w_{yy}}{w_y}\) in some open interval of t and \(w_t \) and \( \frac{w_{yy}}{w_y}\) are both in \(L^2[0, 2]\) for each t . We have

as functions of \(L^2[0,2]\) for each t . By orthogonality of the set

The systems in ( 15 ) and ( 13 ) differ only by a constant coefficient in front of each equation.

Graphs of the deviation of the derivative of the inverse map from its equilibrium for various values of t along a trajectory of the gradient flow

The dotted line is the graph of a cosine curve. The solid line is the graph of the deviation of the derivative of the inverse map from its equilibrium when t is large. The vertical axis is re-scaled with a factor of 1000

5.3 Numerical approximation of a flow trajectory

We limit the scope of our numerical exploration to the case when \(g'\) is an even function:

We have \(\dot{a}_{2\,m-1}= 0\) for all \(m \in {\mathbb {N}}\) . Thus, the system of ODEs in ( 14 ) is reduced to

Let \(\pi y=\tau . \) We have

where \(g' = \frac{1}{2} + \pi \sum _{k=1}^\infty b_{2k-1} (2k-1) \cos [(2k-1)\tau \) and \(g''=- \pi ^2 \sum _{k=1}^\infty (2k-1)^2 b_{2k-1}\sin [(2k-1)\tau ]\) . Denote \( B_k=\pi b_{2k-1}(2k-1)\) , \(h(\tau ) = g' = \frac{1}{2} + \sum _{k=1}^\infty B_k \cos [ (2k-1)\tau ]\) and \(h'(\tau ) = - \sum _{k=1}^\infty B_k (2k-1) \cos [ (2k-1)\tau ]\) . So, the system ( 17 ) becomes

Replacing \( \dot{b}_{2m-1}\) in ( 18 ) by \(\frac{\dot{B}_m}{\pi (2\,m-1)}\) , we have

Or, in a single complete formula,

Let \(F_m( \{B_k\})\) denote the right hand side of the equation ( 19 ).

We use Euler’s method to approximate a trajectory of this system of ODEs:

where \(\epsilon >0\) is the step size and the initial point is \(\{B_m(0)\}\) .

We choose \(B_1(0)= \frac{1}{4}\) and \(B_m(0)=0, m>1\) , i.e., the initial map’s derivative is \(h(y)= \frac{1}{2} + \frac{1}{4} \cos y.\) We have

These values can be easily computed using numeric integration. We see that \(B_m(\epsilon )\) is generally not zero for all \(m \ge 1\) , regardless how small the step size \(\epsilon >0\) is.

The numerical simulation of solutions to the system ( 19 ) is carried out on Maple by Maplesoft. Since \(B_m(k\epsilon )\) decays very fast in m , we have only kept three terms in the Galerkin method. The step size \(\epsilon =0.1\) in Euler’s method [ 10 ].

In Fig.  1 , graphs of \(\frac{d}{dy} {\mathcal {F}}_t(g) - \frac{1}{2} \) are shown for three values of t , \(t=0, \) \(t= 10\) , and \(t= 20\) :

In Fig.  2 , we show the differences between \(c_m \cos (y)\) (from the trajectory of the heat equation \(u_t =u_{xx}\) with the same initial value) and \(\frac{d}{dy} {\mathcal {F}}_t(g) - \frac{1}{2} \) for an even larger t . The vertical axis is re-scaled by a factor of 1000. \(t=50,\)

Ending Remarks

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Acknowledgements

The author thanks John Gemmer and Sarah Raynor for discussions on Sobolev norms and gradient flows in Hilbert spaces. Special thanks to Yunping Jiang and Yun Yang for many insightful discussions related to the behavior of the SRB entropy of expanding circle maps. Additionally, the author acknowledges the referees for their effort in helping clarify many statements in the original manuscript and for bringing related work in literature, especially, the work on the gradient flow in potential spaces to the attention of the author.

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Jiang, M. Gradient Flow of the Sinai–Ruelle–Bowen Entropy. Commun. Math. Phys. 405 , 118 (2024). https://doi.org/10.1007/s00220-024-05003-9

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DOI : https://doi.org/10.1007/s00220-024-05003-9

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Life Found On Mars – Spacecraft Spots ‘Spiders’ Crawling On The Surface Of The Red Planet

Posted: April 26, 2024 | Last updated: April 26, 2024

ESA’s Recent Post On X

In their latest post, the ESA, asked the question, “Spiders on Mars?”

The post went on to state that those weren’t real spiders. The post said, “No, these dark spots, seen by ESA’s Mars Express on the outskirts of an area nicknamed 'Inca City' in the southern polar region of Mars, have a more 'down to Earth' explanation...”

The Phenomenon That Led To The Formations

According to a press release from the ESA, the Mars Express orbiter recently photographed the "spiders," which are actually just small, dark features. These formations take shape when sunlight hits carbon dioxide deposits that accumulate over the cold winter months.

The sunlight triggers the transformation of the carbon dioxide ice at the base of the deposits and turns it into gas. Eventually, this gas erupts through the ice, which can be as thick as three feet. This creates geyser-like blasts of dust that settle on the planet's surface.

Not As Small As They Appear From Space

Although these spots may appear small when viewed from space, they are quite large. According to the ESA, these patches can range in size from as small as 145 feet wide to potentially over half a mile wide at their largest.

The ESA stated that under the larger spots, the arachnid-like pattern get etched into the surface beneath the carbon dioxide ice.

The Spider Patterns Were Detected by The ExoMars Trace Gas Orbiter

The spider patterns were detected by the ExoMars Trace Gas Orbiter that was launched in 2016 to investigate Mars for potential signs of previously existent life.

Most of the dark spots identified by the orbiter are situated on the periphery of a planet's region dubbed as "Inca City."

Resemblance To The Incan Ruins

The "Inca City" region of Mars is recognized for its linear, almost geometric ridges resembling the iconic Incan ruins.

Originally spotted by a NASA probe in 1972, the "Inca City" area, also known as Angustus Labyrinthus, is located near the southern polar cap of the planet.

Scientists Don’t Know Yet How Inca City Was Formed

Researchers haven't been able to understand yet how Inca City was formed.

According to different hypotheses proposed so far, the Inca City formation may stem from the gradual petrification of sand dunes or the leakage of magma through fracture's in Mars' surface.

The Curious Features Of Inca City

Inca City showcases a plethora of interesting features, including enigmatic swirls reminiscent of marble patterns. It also has mounds with flat tops and hills with steep sides and flat summits, reaching heights exceeding 1500 meters above the adjacent landscape.

Additionally, the images revealed a ground blanketed in light-colored smooth-textured dust.

Another Alternative Theory Offers A Different Explanation

According to another alternative theory, Inca City could reside within the remnants of an ancient impact crater, where the aftermath of the collision may have led to the formation of faults in the nearby region.

Subsequently, the leaking of lava through these faults could have sculpted the distinctive walls of the Inca City structure

The Spiders Aren’t The Only Peculiar Formation On Mars’ Surface

The spiders aren't the only peculiar formations on the surface of Mars. In 2015, Mars' Curiosity rover captured an odd spoon-shaped rock.

Back in 1976, the American Viking 1 orbiter and lander photographed a bizarre face-like feature on the planet's surface.

Scientists Tested The Hypothesis Of How The Spiders Were Formed By Replicating The Phenomenon In A Laboratory

In 2021, scientists tested out the hypothesis of how the spiders were formed by replicating the phenomenon in a laboratory.

Scientists achieved success in recreating miniature versions of Martian spiders by employing dry ice and a special machine that can effectively simulate the atmosphere of Mars.

Statement Explaining How The Scientists Replicated The Phenomenon In A Laboratory

In 2021, Open University planetary scientist and lead study author Lauren McKeown, explained the conclusions of the experiment in a statement.

McKeown wrote, “The experiments show directly that the spider patterns we observe on Mars from orbit can be carved by the direct conversion of dry ice from solid to gas.”

Spring-Like Weather On Mars

NASA reports that Mars is presently encountering weather that resembles spring-time.

NASA also stated in a news release that its Curiosity Rover has been relishing the milder conditions on the planet, engaging in environmental surveillance and exploring Mars' Gale Crater.

NASA’s Curiosity Rover Has Consistently Been Detecting Methane On Mars

From 2012 onwards, NASA's Curiosity rover has consistently been detecting methane on Mars.

Methane is more prominently present in the vicinity of Mars' landing site within the 96-mile-wide (154 kilometers) Gale Crater.

Erratic Behavior Of Methane On Mars

The behavior of methane on Mars has been quite erratic. The gas emerges solely at night and it fluctuates with the seasons. Methane sporadically surges to levels 40 times greater than what's normal.

All this adds to the mystery as the gas is not significantly present in the upper atmosphere or Mars. It has also not been detected close to the surface in other regions of the Red Planet.

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