Ergodic theory: from Dynamics to Gibbs measure

I'm trying to understand the ergodic theory approach to statistical mechanics, namely how ergodic measure preserving dynamics lead to the Gibbs measure.

I have a compact space $X$, a probability measure $\mu$ on $X$ and a one parameter family of transformations $T_t:X\rightarrow X$ ergodic on $\mu$. As I understand, ergodic theory states that $\mu$ is unique. Which result in ergodic theory shows that $\mu$ is the Gibbs measure?

More specifically, if my dynamics conserves average energy $\langle E\rangle$, how, in the framework of ergodic theory, do I show that $\mu$ looks something like this: $$\mu(x) \propto \exp[-\beta E(x)],$$ where $\beta$ is the inverse temperature?

From what I've seen so far in physics and maths textbooks, some sort of entropy maximisation hypothesis is invoked to construct $\mu$. If this is the only way to go about it, is there a justification for the principle of maximum entropy within ergodic theory?

• I suggest you to read the book Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms'' by Bowen and edited by Chazottes. Any answer I could give you will be strictly contained in this book. – user39115 Nov 18 '15 at 14:26
• From a quick glance: p. 16 - "We will describe Gibbs measures as those maximizing a certain quantity ...". Is this somehow justified by ergodic theory or is it an assumption? – Prof. Dombledurr Nov 18 '15 at 14:45
• This is justified by functional analysis. The main idea is this: Suppose that $(p_1,\ldots,p_n)$ is a probability vector. Then for any vector $(a_1,\ldots,a_n)$ of real numbers we have that $\sum_i p_i (a_i-\log p_i)\leq \log \left(\sum_i e^{a_i} \right)$ and it is equal iff $p_i=e^{a_i}/(\sum_i e^{a_i}).$ – user39115 Nov 18 '15 at 15:08
• A continuous $\mathbb{R}_{>0}$-action on a compact metric space by continuous transformations almost never has exactly one ergodic Borel probability measure. This is a very special condition called `unique ergodicity'. – Ian Morris Nov 18 '15 at 15:09
• I still don't get it. The fact that $\sum_ip_i(a_i - \ln p_i) \leq\ln\left(\sum_ie^{a_i}\right)$ just states that the Gibbs measure maximises the negative free energy. The fact that $\mu$ should do the same still seems like an assumption. I guess I'm operating under the assumption of unique ergodicity. Thanks for pointing out that it's not always satisfied. – Prof. Dombledurr Nov 18 '15 at 15:21

That's an excellent but highly unresolved question. The problem is that the physicists tend to be not so interested in mathematical foundations once a theory (statistical mechanics in this case) is successful and there is some plausible heuristic justification for it, whereas ergodic theorists quite often have little clue about the physical relevance and have an altogether different reason for being interested in Gibbs measures (starting with a "non-physical" analogy with statistical mechanics made by Sinai, Ruelle, ...). Meanwhile mathematical physicists have been busy with already highly challenging mathematical problems that arise if we take the physical meaning of Gibbs measures for granted.

I will tell you my take (à la Boltzmann) on how to approach the problem of the dynamical justification of Gibbs measures as "equilibrium states". Others will correct me if they don't agree.

Statistical mechanics is all about the relationship between the microscopic and macroscopic properties of systems. Let us say we have a set $X$ containing all the possible instantaneous microscopic states of a physical system, and a transformation $T:X\to X$ describing the time dynamics of the system, which we take to be discrete for simplicity. We also have a collection $\mathscr{O}$ of macroscopic observables, which are simply functions $f:X\to\mathbb{R}$. Knowing the macroscopic state of the system amounts to knowing the value of all the macroscopic observables, that is, a map $\pi:\mathscr{O}\to\mathbb{R}$. Each macroscopic state $\pi$ corresponds to a set $X_\pi$ of microscopic states that realize $\pi$, that is, $X_\pi=\lbrace x\in X: \text{$f(x)=\pi(f)$for each$f\in\mathscr{O}$}\rbrace$.

A typical setting in statistical mechanics is a lattice model, in which the microscopic states of a physical system are represented by symbolic configurations on the lattice $\mathbb{Z}^d$, that is, by functions $x:\mathbb{Z}^d\to\Sigma$ for some finite set of symbols $\Sigma$. A natural candidate for the time dynamics $T$ is a cellular automaton, that is, a continuous map on $X=\Sigma^{\mathbb{Z}^d}$ that has all the translations as its symmetries (i.e., it commutes with the shifts). The choice of the macroscopic observables in this setting is somewhat debatable, but let's say the frequency of a finite word in the configuration is a macroscopic observable, and everything that can be written as a linear combination of such frequencies is also a macroscopic observable. Then, the macroscopic states can be represented by shift-invariant probability measures on $X$ (or better, shift-ergodic probability measures).

Suppose now that the system has a collection of conserved quantities like energy, that is, macroscopic observables $e\in\mathscr{O}$ such that $e(Tx)=e(x)$ "for all" $x\in X$. What can we say about the macroscopic state of the system? The second law of thermodynamics suggests that if the system is "in equilibrium", then the macroscopic state of the system has to maximize the "entropy" subject to the constraints imposed by the conservation laws, and if the system is not "in equilibrium", it gradually approaches the "equilibrium", provided the system is sufficiently "chaotic".

The physicists' explanation of the second law of thermodynamics in a finite-state system is that if the dynamics goes through all the configuration space in one cycle (it is "ergodic"), then in equilibrium (long long time after it has started its evolution) it is "most likely" to be found in a configuration whose macroscopic state encompasses the largest portion of the state space (i.e., $|X_\pi|$ is the largest), and if it is not in equilibrium, it is "most likely" to evolve in time towards macroscopic states with larger volume. In this case, the entropy of a macroscopic state $\pi$ is $\log|X_\pi|$.

Translating this heuristic to infinite systems (say in our lattice setting with cellular automaton dynamics) is not obvious. One is inclined to replace $|X_\pi|$ with the configuration space volume of $X_\pi$ according to some notion of volume, say the uniform Bernoulli measure in $\Sigma^{\mathbb{Z}^d}$ in case of a lattice model. Although this is how the mathematical notion of ergodicity came to being, it doesn't help with the problem of identifying macroscopic equilibrium states, because for all but a single macroscopic state $\pi$, the measure of $X_\pi$ with respect to the uniform Bernoulli measure is zero.

The correct way is to measure the the size of $X_\pi$ is by the amount of information per site required to describe a microscopic configuration in $X_\pi$, that is, by the entropy per site of $\pi$. Note that in the language of dynamical systems, this is the Kolmogorov-Sinai entropy of $\pi$ under the dynamics of the shift action (i.e., the space dynamics), and a priori has nothing to do with the time dynamics. Equilibrium statistical mechanics (e.g., the book of Robert Israel) tells us that the macroscopic states maximizing entropy under the constraints of the conservation laws are Gibbs measures.

A reasonable ergodic-theoretic justification of considering Gibbs measures as equilibrium states involves (1) showing that they are invariant under the time dynamics (this is the easy part) and (2) showing that starting from other macroscopic states the system evolves towards states with larger entropy at least under reasonable assumptions on the starting state, and provided the dynamics has a sufficiently strong chaotic behaviour.

Like I said at the beginning, this is largely open, but there is a beautiful example (and its extensions) for which some mathematical results have been found. The XOR cellular automaton is defined as $T:\lbrace 0,1\rbrace^{\mathbb{Z}}\to\lbrace 0,1\rbrace^{\mathbb{Z}}$ with $(TX)_i:=x_i+x_{i+1}\pmod{2}$. This system has no conserved quantity, so the second law suggests maximum entropy measure (the uniform Bernnoulli measure) as the only macroscopic equilibrium state. It is easy to see that the uniform Bernoulli measure is indeed invariant under $T$. Miyamoto and Lind (and later others) have shown that starting from any macroscopic state that has sufficient mixing property under the space dynamics (e.g., any Bernoulli or Markov measure), the system gradually approaches under $T$ towards the uniform Bernoulli measure (in a suitable sense).

• Sorry, I tend to over-explain things and end up writing an essay. ;-) – Algernon Nov 22 '15 at 13:29
• Thanks. I guess it all goes back to the controversial H-theorem: under given dynamics, the system evolves towards states with greatest entropy, but the definition of entropy and its connection to dynamics just isn't something that drops out of ergodic theory. – Prof. Dombledurr Nov 23 '15 at 16:42

In the statistical mechanics context, the starting point is somewhat different. For a finite discrete system having energy function $E$, physics tells us that the equilibrium probability of a state at inverse temperature $\beta$ should be proportional to $\exp(- \beta E)$. This turns out to satisfy a variational principle. Now finite systems don't have phase transitions, so we try to generalize this to infinite systems (typically describing "spins" on an infinite lattice such as $\mathbb Z^d$, with a translation-invariant interaction): since the total energy becomes infinite, this is not trivial. The infinite system has a compact configuration space $\Omega$ (the cartesian product over $\mathbb Z^d$ of the set of possible values for each individual spin), and $\mathbb Z^d$ acts on this by translations. Then we have various ways of characterizing "equilibrium" for translation-invariant probability measures on $\Omega$, which under suitable assumptions turn out to be equivalent. The set of equilibrium measures for a particular interaction turns out to be a Choquet simplex whose extreme points are ergodic for this action of translation. Note that this action is not dynamics in the physical sense. We can then investigate when the equilibrium measure is or is not unique.

• Thanks. Unfortunately I can't work from the statement that "physics tells us that the equilibrium probability of a state at inverse temperature $\beta$ should be proportional to $\exp(-\beta E)$" because there are a lot of ways to get there in the framework of statistical mechanics, all of which involve some assumptions. The variational principle has already been discussed above. – Prof. Dombledurr Nov 19 '15 at 9:35

I think this has been said already, but as Ian Morris said, for hyperbolic dynamical systems (the shift is a typical example), there are very very many invariant measures. Ruelle introduced a "thermodynamic formalism" into ergodic theory, giving loose dynamical analogues of quantities that appear in thermodynamics. In particular, he defined equilibrium states. These are states maximizing the pressure or free energy as you have called it. The principal justification for doing this is that equilibrium states show up naturally in a wide variety of contexts: information transmission, geometry, probability, differentiable dynamical systems etc.

In other words, the thermodynamic formalism is a method for producing interesting invariant measures.

• Thanks. I think this gives me enough to work with. Is there any reason why any invariant probability measure cannot be called an equilibrium state? Intuitively, if $\mu$ is any invariant probability measure, then it is the stationary distribution, which is an equilibrium state in a statistical sense. – Prof. Dombledurr Nov 19 '15 at 9:44
• I believe the terminology "equilibrium state" was specifically introduced by Ruelle and the correspondence between equilibrium states and Gibbs states was studied. Authors often concentrate on equilibrium states for Holder continuous potentials. Robert Israel wrote a nice book that includes a result related to your question: I believe he showed for any collection of invariant measures, one can find a potential for which they are the collection of equilibrium states (he will correct me if I'm wrong). – Anthony Quas Nov 19 '15 at 10:16
• Thank you. Could you tell me the title of the book? All I can find is "Convexity in the Theory of Lattice Gases". – Prof. Dombledurr Nov 19 '15 at 12:21
• Yes, that's the title. It's now available in the Princeton Legacy Library – Robert Israel Nov 19 '15 at 21:26
• For every finite set of ergodic states, there is a potential that has these as equilibrium states. The set of equilibrium states has to be a Choquet simplex whose extreme points are ergodic, and the mean entropy has to be continuous on it. – Robert Israel Nov 19 '15 at 21:30