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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 ...
Pedro Huot's user avatar
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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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Equipartition theorem for continous medium

The equipartition theorem states that if $x_i$ is a canonical variable (either position or momentum), then $$\left\langle x_i \frac{\partial \mathcal{H}}{\partial x_j}\right\rangle = \delta_{ij}\ k T.$...
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Sequential updating breaks ergodicity of Metropolis–Hastings algorithm of the Ising model

I believe I have this question under control but I am puzzled at why I have not found people pointing this out. For simplicity consider a 1D ferromagnetic Ising model with periodic boundary conditions:...
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Quantum analog of mixing time correlation functions

Classical ergodic theory predicts that in maximally chaotic systems correlation functions relax to the long time limit \begin{equation} \langle A(0) B(t)\rangle_0 \to_{t\to+\infty} \langle A\rangle_0 \...
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Ergodicity and the 2D classic harmonic oscillation

In my script the case of the 2D classic harmonic oscillation is taken into consideration. We are given an example as to how it is related to ergodicity/ an ergotic system. This example might be a bit ...
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Understanding ergodicity and what an ergodic system is

I am trying to understand the concept of ergodicity/ergodic system in physics, but because my understanding of phase space, its elements is a bit unclear,I have trouble understanding the former. ...
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Ergodic theorem with more conserved quantities

Ergodic theory is constructed by fixing the dynamics on a surface of the phase space with constant energy. In case a non-integrable system conserved more additional quantities apart from the energy, ...
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Time taken for a system to return to it's original state

Consider the following system: There are N particles (point-like particles) of $1$ Kg each in a Sphere of radius $R$ centered at origin in three dimensions. Randomly assign these N particles their ...
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Ergodicity in quantum statistical mechanics

Is there an ergodicity assumption in quantum statistical mechanics ? The classical statistical mechanics derives its main results from the assumption that all the states with the same energy (and ...
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Is Mechanical Equilibrium Really Driven by Entropy Increase?

It is a standard result in statistical mechanics that when two interacting systems are free to exchange energy and volume, then in the macrostate of maximum entropy the systems will have equal ...
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Eigenstate thermalization in chaotic Floquet systems

Background In closed time-independent Hamiltonian systems, the eigenstate thermalization hypothesis (ETH) states, roughly speaking, that energy eigenstates "look thermal". More precisely, ...
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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 ...
Lo Scrondo's user avatar
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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 ...
Aakash Lakshmanan's user avatar
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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, ...
Lo Scrondo's user avatar
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1 answer
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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 ...
Emilio Pisanty's user avatar
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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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2 answers
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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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3 answers
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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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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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