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AN INVESTIGATION INTO RECENT THEORIES OF RELAXATION TOWARDS EQUILIBRIUM

Part of: Time-dependent statistical mechanics (dynamic and nonequilibrium)

Published online by Cambridge University Press:  27 September 2024

MATTHEW ROGER KING Open the ORCID record for MATTHEW ROGER KING [Opens in a new window]
MATTHEW ROGER KING*
Affiliation:
School of Environment and Science, Griffith University, Nathan, Queensland 4111, Australia
*
e-mail: [email protected]
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Abstract

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Keywords

dissipation functionnonequilibrium statistical mechanicsrelaxation to equilibriumfluctuation relation

MSC classification

Primary: 82C05: Classical dynamic and nonequilibrium statistical mechanics (general)
Secondary: 82C22: Interacting particle systems 82C99: None of the above, but in this section
Type
PhD Abstract
Information
Bulletin of the Australian Mathematical Society , First View , pp. 1 - 2
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

There are many open questions surrounding the statistical mechanical characterisation of systems out of equilibrium, particularly regarding their relaxation towards equilibrium or nonequilibrium steady states, as well as the fluctuations which occur throughout their dynamics. The dissipation function of Evans and Searles [Reference Evans, Searles and Williams3] has found successful applications in both of these areas. In this work, we investigate the use of the dissipation function to characterise these behaviours in a number of dynamical systems.

In a system which allows us to tune the chaoticity of its dynamics, we use the dissipation function to study the necessary conditions for fluctuation relations to be observed. This section of work teases apart the conditions of chaotic dynamics and transient mixing (T-mixing) [Reference Evans, Searles and Williams4, Reference Searles, Johnston, Evans and Rondoni6], showing that fluctuation relations hold in nonchaotic systems, counter to the contemporary theories of the necessary conditions. Our results suggest that the Evans–Searles fluctuation theorem [Reference Evans and Searles1, Reference Evans and Searles2] may provide a more fruitful path forward for the study of fluctuations in dynamical systems.

We have found that the dissipation function captures apparent relaxation in the absence of particle–particle interactions. Without interactions between particles, this relaxation is akin to relaxation in a single-particle system. It is not clear whether there is any way to give a physical interpretation of this relaxation or whether this should be interpreted as dissipation in any sense. The absence of particle–particle interactions also makes it impossible for the system to relax in the momentum distribution. Despite this, the system appears to undergo the same degree of dissipation in the absence of particle–particle interactions, suggesting that the dissipation function may not capture all aspects of relaxation within a system.

Finally, we have applied the dissipation function to the study of relaxation in a microcanonical system. To the best of our knowledge, this represents the first application of the dissipation function to a microcanonical system. One of the few conditions which can be placed on the study of relaxation with the dissipation function is that the equilibrium state of the system will be dissipationless. However, the system we have studied appears to undergo a large number of apparently dissipationless states beyond which relaxation continues. These dissipationless states occur both instantaneously ( $\Omega _t^f = 0$ ) and when measured over long time averages ( $d\Omega _{0,t}^f/dt = 0$ ). We also find that the state of maximal dissipation does not necessarily indicate the maximally relaxed state of a system. These results raise deep questions about the use of the dissipation function in the study of equilibration.

Some of this research has been published in [Reference King and Jepps5].

Footnotes

Thesis submitted to Griffith University in February 2023; degree approved on 17 November 2023; supervisor Owen G. Jepps.

References

Evans, D. J. and Searles, D. J., ‘Equilibrium microstates which generate second law violating steady states’, Phys. Rev. E 50(2) (1994), Article no. 1645.CrossRefGoogle ScholarPubMed
Evans, D. J. and Searles, D. J., ‘The fluctuation theorem’, Adv. Phys. 51(7) (2002), 1529–1585.CrossRefGoogle Scholar
Evans, D. J., Searles, D. J. and Williams, S. R., ‘Dissipation and the relaxation to equilibrium’, J. Stat. Mech. Theory Exp. 2009(7) (2009), Article no. P07029.CrossRefGoogle Scholar
Evans, D. J., Searles, D. J. and Williams, S. R., ‘On the probability of violations of Fourier’s law for heat flow in small systems observed for short times’, J. Chem. Phys. 132(2) (2010), Article no. 024501.CrossRefGoogle ScholarPubMed
King, M. R. and Jepps, O. G., ‘Non-monotonic relaxation in a harmonic well’, Comput. Methods Sci. Technol. 23(3) (2017), 199–209.Google Scholar
Searles, D. J., Johnston, B. M., Evans, D. J. and Rondoni, L., ‘Time reversibility, correlation decay and the steady state fluctuation relation for dissipation’, Entropy 15(5) (2013), 1503–1515.CrossRefGoogle Scholar