Essentially Ergodic Behaviour

Bibliographic Data

ID8399117
AuthorsPaula Reichert (0000-0003-3291-4088, Ludwig-Maximilians-Universität München, corresponding author)
Year2023
Volume74
Issue1
Pages57-73
Publication date2023-03-01
Peer ReviewedYes
Open AccessYes
TypeARTICLE
VenueThe British Journal for the Philosophy of Science (JOURNAL)
Journal identifiersISSN: 0007-0882 • E-ISSN: 1464-3537
PublisherOxford University Press (PUBLISHER • GB)
DOI10.1093/bjps/axaa007
OpenAlexW3000956584
LanguageEN
Citations received1
References cited16

I prove a theorem on the precise connection of the time and phase-space average of the Boltzmann equilibrium showing that the behaviour of a dynamical system with a stationary measure and a dominant equilibrium state is qualitatively ergodic. Explicitly, I show that given a dynamical system with a stationary measure and a region of overwhelming phase-space measure, almost all trajectories spend almost all of their time in that region. Conversely, given that almost all trajectories spend almost all of their time in a certain region, that region is of overwhelming phase-space measure. In total, the time and phase-space average of the equilibrium state approximately coincide. Consequently, equilibrium can be defined equivalently in terms of the time or the phase-space average. Moreover, since the two averages are almost equal, the behaviour of the system is essentially ergodic. While this does not explain the approach to equilibrium, it provides a means to estimate the fluctuation rates.

Connection (principal bundle) · Dynamical system (definition) · Ergodic theory · Measure (data warehouse) · State (computer science) · Stationary ergodic process · Thermodynamic equilibrium · Advanced Thermodynamics and Statistical Mechanics · Gas Dynamics and Kinetic Theory · Statistical Mechanics and Entropy

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