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Essentially Ergodic Behaviour

Reichert, Paula (2020) Essentially Ergodic Behaviour. The British Journal for the Philosophy of Science.

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Abstract

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 will show that, given a measure-preserving dynamical system and a region of overwhelming phase space measure, almost all trajectories spend almost all of their time in that region. The other way round, 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 equivalently be defined in terms of the time or the phase space average. Even more, 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.


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Item Type: Published Article or Volume
Creators:
CreatorsEmailORCID
Reichert, Paulapaula.reichert@gmail.com
Keywords: Boltzmann equilibrium, Time Average, Phase Space Average, Ergodicity, Stationary Measure
Depositing User: Dr. Paula Reichert
Date Deposited: 22 Jan 2020 05:23
Last Modified: 22 Jan 2020 05:23
Item ID: 16841
Journal or Publication Title: The British Journal for the Philosophy of Science
Date: 17 January 2020
URI: https://philsci-archive-dev.library.pitt.edu/id/eprint/16841

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