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Entropy productions in dissipative systems


Authors: Hong Qian, Shirou Wang and Yingfei Yi
Journal: Proc. Amer. Math. Soc. 147 (2019), 5209-5225
MSC (2010): Primary 37H99, 34D45; Secondary 80A99
DOI: https://doi.org/10.1090/proc/14618
Published electronically: June 10, 2019
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Abstract: The present study aims to find connections between two types of entropy production rates for continuous dynamical systems, one defined in the general dissipation context and the other one defined in a thermodynamics context. On one hand, the entropy production rate $ e_p$ introduced by Andrey and Ruelle in the general dynamics context is based on Shannon's entropy in the phase space that characterizes volume contraction relating to dissipation. On the other hand, the Helmholtz-Boltzmann entropy production rate $ \dot {Q}/T$ introduced in the context of mechanical theory of thermodynamics, where $ \dot {Q}$ is the rate of heat generations and $ T$ is the temperature, captures the thermodynamic entropy production relating to the mechanical energy dissipation. For certain cases of energy dissipative systems, we show that $ e_p$ is indeed related to $ \dot {Q}/T$. This consistency between the general mathematical notion of dissipation and the thermodynamic entropy production in dissipative systems suggests a unifying representation of nonequilibrium phenomenon in deterministic systems based on the theory of nonlinear dynamical systems.


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Additional Information

Hong Qian
Affiliation: Department of Applied Mathematics, University of Washington, Seattle, Washington 98195
Email: hqian@u.washington.edu

Shirou Wang
Affiliation: Department of Mathematical & Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G2G1
Email: shirou@ualberta.ca

Yingfei Yi
Affiliation: Department of Mathematical & Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G2G1; and School of Mathematics, Jilin University, Changchun 130012, People’s Republic of China
Email: yingfei@ualberta.ca

DOI: https://doi.org/10.1090/proc/14618
Keywords: Dissipation, Shannon entropy, entropy production
Received by editor(s): November 21, 2018
Received by editor(s) in revised form: February 12, 2019, and March 6, 2019
Published electronically: June 10, 2019
Additional Notes: The second author was partially supported by NSFC 11771026, NSFC 11471344 and acknowledges PIMS PTCS and the PIMS at University of Washington supported by NSF DMS-1712701 and NDF DMS-1444084.
The third author was partially supported by NSERC discovery grant 1257749, a faculty development grant from the University of Alberta, and a scholarship from Jilin University.
All authors were supported in part by a PIMS CRG grant.
Communicated by: Wenxian Shen
Article copyright: © Copyright 2019 American Mathematical Society