Ergodic Theory in Statistical Mechanics

Farquhar, I.E.; Seeger, Raymond J.

doi:10.1119/1.1971105

Cited by 59 publications

(47 citation statements)

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“…In general, this path will be very complicated. If it succeeds in getting arbitrarily close to every point on this surface, then the system is called "ergodic" [19,20]. Ergodicity says, loosely speaking, that Γ(t) will get arbitrarily close to every point on the constant energy surface given a long enough time.…”

Section: Classical Thermalizationmentioning

confidence: 99%

Eigenstate thermalization hypothesis

2018

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The emergence of statistical mechanics for isolated classical systems comes about through chaotic dynamics and ergodicity. Here we review how similar questions can be answered in quantum systems. The crucial point is that individual energy eigenstates behave in many ways like a statistical ensemble. A more detailed statement of this is named the eigenstate thermalization hypothesis (ETH). The reasons for why it works in so many cases are rooted in the early work of Wigner on random matrix theory and our understanding of quantum chaos. The ETH has now been studied extensively by both analytic and numerical means, and applied to a number of physical situations ranging from black hole physics to condensed matter systems. It has recently become the focus of a number of experiments in highly isolated systems. Current theoretical work also focuses on where the ETH breaks down leading to new interesting phenomena. This review of the ETH takes a somewhat intuitive approach as to why it works and how this informs our understanding of many body quantum states.

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Section: Classical Thermalizationmentioning

confidence: 99%

Eigenstate thermalization hypothesis

2018

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“…Here, ρ single is the invariant measure that gives the temporal average of any magnitude and it is a fully objective distribution for a single system [9]. H is the ignorance or uncertainty that we have about the macrostate of the system.…”

Section: Entropy and Macroscopic Uncertaintymentioning

confidence: 99%

Entropy, macroscopic information, and phase transitions

Parrondo¹

2000

AIP Conference Proceedings

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Abstract. The relationship between entropy and information is reviewed, taking into account that information is stored in macroscopic degrees of freedom, such as the order parameter in a system exhibiting spontaneous symmetry breaking. It is shown that most problems of the relationship between entropy and information, embodied in a variety of Maxwell demons, are also present in any symmetry breaking transition.

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“…The relevance of dynamical instability and ergodicity 2 to Statistical Mechanics (SM) has been sometimes questioned, [Farquhar 1964], but it is by now generally accepted that the unpredictability implied by Chaos is a natural ingredient to justify the statistical description of mdf dynamical systems. As stressed in the introduction, however, there is no rigorous, or even convincing, prove of any direct relationship between the instability times, as can be derived from dynamics (Lyapunov exponents, Kolomogorov-Sinai entropy,...) and the time-scales related to a Statistical Mechanical treatment, i.e., those belonging under the name of relaxation times.…”

Section: Introductionmentioning

confidence: 99%