# 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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### Does time average induce phase space propability distribution?

Lets say we have a trajectory (positions and momenta) $(x(t), p(t))$ that is the solution of the equation of motion for a system with Hamiltonian $H(x,p)$. For some function $A(x,p)$, the time average ...
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### How is this transformation measure preserving? Example from Birkhoff, George David (1942), "What is the ergodic theorem?"

I'm reading Birkhoff, George David (1942), "What is the ergodic theorem?", doi:10.2307/2303229, and I'm stuck on his 2nd example: the line segment is divided into the infinite set of ...
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### Is there an equipartition theorem for diatomic gases at transitional temperatures?

Context If you have a gas, you can insert a bit of energy $E$ and measure the resulting increase $K$ in the average kinetic energy in your favourite direction. For monatonic gases, $K=E/3$, as the ...
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### Meaning of an ergodic trajectory

I'm trying to understand the concept of an ergodic trajectory in the context of dynamical systems. I think that I have a reasonable idea of the word "ergodic". This question links to a ...
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### Spontaneous symmetry breaking and ergodicity

I am studying the spontaneous symmetry breaking in the mean-field Ising model and it's clear to me the necessity of taking first the thermodynamic limit and then the zero-field limit to see the phase ...
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### How do Landau and Lifshitz avoid the ergodicity problem?

In the preface to Landau and Lifshitz's Statistical Physics, they comment the following In the discussion of the foundations of classical statistical physics, we consider from the start the ...
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### From Newtonian mechanics to Boltzmann (or statistical) mechanics

Classical mechanical systems observable on a dynamical scale are subject to Newton's laws. In this case, knowledge of the Hamiltonian allows us to minimize energy taking into account inertia. This ...
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### Ergodicity in "unphysical" parts of Hilbert space

We know from Quantum complexity theory, that the vast majority of states in Hilbert space for physically relevant Hamiltonians cannot be accessed except in exponentially long time (see related ...
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### Ergodicity and spin connections at the event horizon

Hawking famously relates the entropy $S$ to the surface area of a black hole $A$ as $S=A/4$. Should I be thinking of the entropy as the number of possible configurations of a spin connection at the ...
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### Poincaré Recurrence Theorem in Quantum Mechanics

The recurrence theorem of Poincaré tells us that EVERY open set in the phase space will be crossed infinitely often. It doesnt matter if the open set is a neighbourhood of the initial data set or not. ...
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### Are solid materials ergodic systems?

It is stated that a system is considered ergodic if it can access all available states with the same energy in the phase space over long periods of time and that time average and ensemble average of ...
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### Ergodic and Non-ergodic (Equilibrium)

According to my textbook "Thermodynamics and Statistical Mechanics: An Integrated Approach (Cambridge by M. Scott Shell": systems that are at equilibrium and isolated, are systems that obey ...
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### Poincaré Recurrence Theorem and the 2º Law of Thermodynamics [duplicate]

I am currently working on a 15 pages project about ergodicity and I wanted to include some discussion about the Poincaré Recurrence Theorem (PRT) and, as far as I know, it contradicts the 2° Law of ...
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### Ensemble average and law of large numbers

In order to calculate the average of a macroscopic quantity such as energy, we need to average over all microstates of the system: $$\langle E \rangle = \sum_{i=1}^n p_i E_i$$ where $n$ is the number ...
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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 ...
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