New remarks on the Kolmogorov entropy of certain coarse-grained deterministic systems

Michel Moreau; Bernard Gaveau; Michel Moreau; Bernard Gaveau

doi:10.3934/math.20231343

AIMS Mathematics

2023, Volume 8, Issue 11: 26328-26342. doi: 10.3934/math.20231343

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New remarks on the Kolmogorov entropy of certain coarse-grained deterministic systems

Michel Moreau ^{1
,
,},
Bernard Gaveau ²

1.
Laboratory of Theoretical Physics of Condensed Matter, Faculty of Sciences, Sorbonne Université, Paris, France
2.
Former Professor at the Faculty of Mathematics, Sorbonne Université, Paris, France

Received: 30 March 2023 Revised: 21 August 2023 Accepted: 28 August 2023 Published: 14 September 2023
MSC : 60G99

Unless an appropriate dissipation mechanism is introduced in its evolution, a deterministic system generally does not tend to equilibrium. However, coarse-graining such a system implies a mesoscopic representation which is no longer deterministic. The mesoscopic system should be addressed by stochastic methods, but they lead to practically infeasible calculations. However, following the pioneering work of Kolmogorov, one finds that such mesoscopic systems can be approximated by Markov processes in relevant conditions, mainly, if the microscopic system is ergodic. So, the mesoscopic system tends to stationarity in specific situations, as expected from thermodynamics. Kolmogorov proved that in the stationary case, the instantaneous entropy of the mesoscopic process, conditioned by its past trajectory, tends to a finite limit at infinite times. Thus, one can define the Kolmogorov entropy. It can be shown that in certain situations, this property remains true even in the nonstationary case. We anticipated this important conclusion in a previous article, giving some elements of a justification, whereas it is precisely derived below in relevant conditions and in the case of a discrete system. It demonstrates that the Kolmogorov entropy is linked to basic aspects of time, such as its irreversibility. This extends the well-known conclusions of Boltzmann and of more recent researchers and gives a general insight to the fascinating relation between time and entropy.
- stochastic processes,
- coarse-graining,
- Kolmogorov entropy,
- Markov processes,
- martingales
Citation: Michel Moreau, Bernard Gaveau. New remarks on the Kolmogorov entropy of certain coarse-grained deterministic systems[J]. AIMS Mathematics, 2023, 8(11): 26328-26342. doi: 10.3934/math.20231343

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Abstract

Unless an appropriate dissipation mechanism is introduced in its evolution, a deterministic system generally does not tend to equilibrium. However, coarse-graining such a system implies a mesoscopic representation which is no longer deterministic. The mesoscopic system should be addressed by stochastic methods, but they lead to practically infeasible calculations. However, following the pioneering work of Kolmogorov, one finds that such mesoscopic systems can be approximated by Markov processes in relevant conditions, mainly, if the microscopic system is ergodic. So, the mesoscopic system tends to stationarity in specific situations, as expected from thermodynamics. Kolmogorov proved that in the stationary case, the instantaneous entropy of the mesoscopic process, conditioned by its past trajectory, tends to a finite limit at infinite times. Thus, one can define the Kolmogorov entropy. It can be shown that in certain situations, this property remains true even in the nonstationary case. We anticipated this important conclusion in a previous article, giving some elements of a justification, whereas it is precisely derived below in relevant conditions and in the case of a discrete system. It demonstrates that the Kolmogorov entropy is linked to basic aspects of time, such as its irreversibility. This extends the well-known conclusions of Boltzmann and of more recent researchers and gives a general insight to the fascinating relation between time and entropy.

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