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.
Questions tagged [ergodicity]
75 questions
120
votes
6 answers
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 be changed to CW then I'll let the mods change…
Logan M
- 4,614
17
votes
3 answers
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 entropy increases over the time. I was wondering to…
mathvc_
- 533
15
votes
2 answers
What would be non-ergodic physics processes?
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 any non-ergodic process in physics..
user50374
- 229
13
votes
1 answer
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 ergodic hypothesis, which is frankly a pretty…
Emilio Pisanty
- 137,480
13
votes
5 answers
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 hypothesis, both in its general formulation as the…
JackI
- 1,764
- 10
- 26
12
votes
1 answer
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 oscillators put to oscillate in a determinate…
Diracology
- 18,168
8
votes
2 answers
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 statistical distribution for small parts (subsystems) of…
Lourenco Entrudo
- 1,096
8
votes
2 answers
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 number of degrees of freedom?
On the other hand, if…
AndreaPaco
- 1,322
8
votes
3 answers
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 system evolves in time. Thus formally, one could…
user929304
- 4,910
7
votes
1 answer
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 averages of a quantity, in the case of…
Chill2Macht
- 554
7
votes
4 answers
Physical distinction between mixing and ergodicity
How can one in a very contrasting manner distinguish between the physical meaning of mixing dynamics and that of ergodic dynamics? More precisely, is one a stronger condition than the other? (which begs to ask questions such as: are ergodic systems…
user929304
- 4,910
6
votes
2 answers
Canonical ensemble, ergodicity and Liouville’s equation
I understand that in Statistical Mechanics Liouville’s equation applies to the probability density of ensembles where microstates’ trajectories are governed by Hamiltonian dynamics. However I’m confused about its application to microstates in the…
n1ps
- 71
6
votes
2 answers
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 foundational aspects of SM are discussed in textbooks,…
Ignacio
- 1,330
6
votes
4 answers
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 same energy is proportional to the volume of this…
matori82
- 943
- 3
- 11
- 28
5
votes
0 answers
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 questions here, and here). Thus, for a solid with…
KF Gauss
- 8,314