# Questions tagged [ergodicity]

A system is said to be ergodic if time averages are, for a sufficient long time, equivalent to phase space averages. This "ergodic hypothesis" is taken by many authors as the foundation of statistical mechanics.

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### About the ergodicity of a system formed by two isolated sub-systems

Imagine we have an isolated gas with volume $V$, energy $E$, formed by $N$ particles. Suppose $H(q,p)$ is the Hamiltonian, where $(q,p)$ are the $3N$ coordinates and $3N$ momenta of all the particles ...
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### Why is the time average equal to the ensemble average?

Ensembles can be defined in two ways (see here). In statistical mechanics, it is assumed that the time average of some property, e.g., the energy, is equal to the ensemble average of the same property....
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### “Non-analytic interaction”…what does it mean?

Reading an article about Hamiltonian chaos, I found this passage: Importantly, the few Hamiltonian systems for which the KAM theorem does not apply, and for which one can prove ergodicity and the ...
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### How do we use ergodic theory in physics?

I am Mathematics student taking a graduate Ergodic Theory class. We are going over a lot of mathematical theory, but I would like to understand (at least at a superficial level) the connection with ...
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### Accessible States in the Ergodic Hypothesis

According to Wikipedia, the ergodic hypothesis is the assumption that all accessible microstates are equiprobable over a long period of time. My question is about the precise meaning of "...
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### Are there non-ergodic statistical theories?

From what I understand, all statistical theory proceeds by assigning probabilities to microstates based off some ergodic-like assumption, and then looking at the implications of this on properties of ...
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### Liouville's theorem to show ergodicity?

a) Consider a harmonic oscillator with Hamiltonian $H=(1/2)(p^2+q^2)$ show that any phase space trajectory $x(t)$ with energy $E$, on the average, spend equal time in all regions of the constant ...
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### Resonant and non-resonant tori density in non-degenerate system

I'm following the discussion on the page 290 of Mathematical Methods of Classical Mechanics by V. I. Arnol'd (you can download it here), and I've encountered the fact that in a nondegenerate system, ...
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### What does equipartition of modes mean in ray optics?

Consider light as rays interacting with refractive boundaries - no polarization or diffraction, but with scattering - a fixed probability of a ray changing angle per unit time. First of all, what ...
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### Why doesn't the ergodic hypothesis hold for most systems?

Is there a physical (intuitive) explanation for why most systems are not ergodic? As my book states, it is a natural assumption that a system is at least quasi-ergodic; it then proceeds to state that ...
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### Chaos and Ergodicity in Hamiltonian Field Theory?

In classical mechanics, one intuitive formulation of chaos/ergodicity (in the loose sense) is that most trajectories should fill up phase space densely over infinite time. A classic example of such a ...
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### How can ergodicity explain thermalization

I am reading up on thermalization in classical systems. As most systems are ergodic, mostly through the mechanism of dynamical chaos, they will explore their whole allowed phasespace and we can ...
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### Thermalization in non-disordered systems

The eigenstate thermalization hypothesis explains the mechanism of the thermalization of generic many-body quantum systems. The presence of disorder, on the other hand, provides an elegant example of ...
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### How is the universe both non-ergodic and quantum?

From Ted Jacobson we know relativity is thermodynamical. This also (I think?) must mean relativity, as a classical analysis, assumes space-time to be ergodic, i.e. a system that can reach thermal ...
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### Is it possible for a system to be chaotic but not ergodic? If so, how?

In a recent lecture on ergodicity and many-body localization, the presenter, Dmitry Abanin, mentioned that it is possible for a classical dynamical system to be chaotic but still fail to obey the ...
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### Ergodic Hypothesis in cosmology

I'm studying primordial fluctuations of the Universe from a statistical point of view and I'am aware of the following problem: A fundamental limitation arises in cosmology – because there is only ...
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### Violation of Virial theorem as indication to ergodicity breaking

Under which conditions the break of virial theorem implies break of ergodicity? I've seen this question, but it is very limited and not sufficient. To constrain the discussion I'm interested in 1D ...
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### Derivation of 2nd law of Thermodynamics from ergodicity assumption

In Wikipedia it is claimed that: Assumption of the ergodic hypothesis allows proof that certain types of perpetual motion machines of the second kind are impossible. Since perpetual motion ...
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### Question about ergodicity and the evolution of the probability distribution under Liouville's theorem

According to Liouville's theorem, the probability distribution function $\rho$ evolve in phase space with $$\frac{d \rho}{d t} = \frac{\partial \rho}{\partial t}+\left\{\rho,H\right\}_{P.B} =0$$ ...
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### When is the ergodic hypothesis reasonable?

Consider an Hamiltonian system. In which circumstances is it possible to assume that all the states belonging to the hypersurface $H=E_0$ are equally visited? Is it necessary to have a very high ...
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### Are interactions with the environment unnecessary to attain thermodynamic equilibrium?

First of all I apologize for the lenght of this question. I have some basic statistical mechanics facts that I am confused about, and in this subject it is probably better to be precise. When ...
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### One consequence of the ergodic hypothesis?

this is my first question here, and I'm trying to self-learn physics from Kip Thorne's 2017 textbook "Classical Physics". IF I understand the ergodic hypothesis correctly, it is simply the statement ...
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### Does the ergodic hypothesis provide a uniquely determined definition of entropy?

One can distinguish between two schools of thought regarding thermodynamic entropy: (a) Thermodynamic entropy is a measure of the "amount of hidden information" in a system. Therefore, the entropy ...
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### What is the relationship between the integrability of a quantum many-body system and thermalization?

If a quantum many-body system is integrable, does it imply the system would always thermalized or many-body localized?
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### Why does the Coarse-grained Entropy increase?

It is a simple fact the entropy in the exact meaning in dynamical system does not change over time if the system is measure-preserving and ergodic. However, it is often said that the coarse-grained ...
2answers
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### Two Independent Harmonic Oscillators is NOT Ergodic!

I read on a book that the system of two or more independent harmonic oscillators in classical mechanics is not ergodic. I want to know why a harmonic oscillator is actually ergodic but two or more ...
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### Relation between ergodic hypothesis and fundamental postulate of statistical mechanics?

The fundamental postulate states, roughly, that each microstate of a system consistent with certain known constraints (e.g. on the total energy of the system), is equally likely to occur. The ...
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### Thermalization in Classical Mechanics: A Paradox

I was wondering to know whether there is any argument shows how a classical system (a system with a Hamiltonian and Poisson bracket) approaches its equilibrium and how the entropy increases. Newton's ...
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### Necessary vs sufficient condition for thermodynamic equilibrium in statistical mechanics

This question came to me after a discussion on another post on the site, which you can find here. I will restrict the following discussion to Hamiltonian systems. Also, I will use the abbreviated ...
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### Microcanonical ensemble, ergodicity and symmetry breaking

In a brief introduction to statistical mechanics, that is a part of a wider course on Solid State Physics I am taking, the teacher introduced the concept of microcanonical ensemble and the ergodic ...
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### Are there necessary and sufficient conditions for ergodicity?

What are the necessary and sufficient conditions (if any) for ergodicity (or non-ergodicity)? I see for instance that some integrable systems are not ergodic. For instance a linear chain of harmonic ...
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### Questions on ergodicity

The "definition" of an ergodic system is that the average of its states in phase space is the same as its average over time. So I was thinking to myself Is a pendulum an ergodic system? In this case,...
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### Is ergodic hypothesis in contradiction with the notion of equilibrium?

From wikipedia: In physics and thermodynamics, the ergodic hypothesis1 says that, over long periods of time, the time spent by a system in some region of the phase space of microstates with the ...
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### Why is the Virial Theorem not a Special Case of the Ergodic Theorem? What is their Relationship?

The virial theorem involves the time-averages of the potential and kinetic energies if the motion of the system is bounded to a finite region of space. An ergodic theorem relates the time and space ...
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### Extending the ergodic theorem to non-equilibrium systems

I try to make this as short and concise as possible. For equilibrium systems in statistical mechanics, we have the Liouville's theorem which says that the volume in phase space is conserved when the ...
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### What is the cause for the inclusion of 'thermal equilibrium' in the statement of Ergodic hypothesis?

This is the fundamental assumption of statistical mechanics: In an isolated system in thermal equilibrium $^1$, all accessible microstates are equally probable. But why does it mention the ...
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### Can binary sequences generated from ergodic maps be chaotic?

Briefly, the way symbols are generated is: Consider a one-dimensional chaotic map $T: [0,1]→[0,1]$ and a time series $\{x_n\}_{n=1}^N$ generated with this map. Define a threshold $A$ and a ...
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### Can an Ergodic dynamical system approach equilibrium?

An ergodic dynamical system $(\Omega,\phi^t,\mu)$ is such that the time average $\bar{f}$ of every function $f\in L_1(\Omega,\mu)$ equal the space average $\langle f \rangle_\mu$, i.e. the system ...
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### Ergodic Hypothesis; canonical ensemble

I'm currently studying for an exam in thermodynamics/classic statistical mechanics and 2 things came up which are confusing me. First the Ergodic Hypothesis states that it is equal to take the mean ...
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### What would be non-ergodic physics processes? [closed]

As the title says, what would be non-ergodic processes that occur in statistical physics? Many textbooks do not really cover ergodicity really well so I ask this question. I can't suddenly remember ...
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### Which quantum models violate the third law of thermodynamics and have an ergodic ground state?

Which any condensed matter quantum spin models over a finite lattice of size N which violates the third law of thermodynamics with ergodic mixings for the ground state?
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### What are the justifying foundations of statistical mechanics without appealing to the ergodic hypothesis?

This question was listed as one of the questions in the proposal (see here), and I didn't know the answer. I don't know the ethics on blatantly stealing such a question, so if it should be deleted or ...