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Journal of Chemical Physics / Volume 136 / Issue 2 / COMMUNICATIONS
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J. Chem. Phys. 136, 021101 (2012); http://dx.doi.org/10.1063/1.3675847 (4 pages)

Communication: Beyond Boltzmann's H-theorem: Demonstration of the relaxation theorem for a non-monotonic approach to equilibrium

James C. Reid1, Denis J. Evans2, and Debra J. Searles1

1Queensland Micro- and Nanotechnology Centre, School of Biomolecular and Physical Sciences, Griffith University, Brisbane Qld 4111, Australia
2Research School of Chemistry, Australian National University, Canberra ACT 0200, Australia

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(Received 13 November 2011; accepted 19 December 2011; published online 11 January 2012)

Relaxation of a system to equilibrium is as ubiquitous, essential, and as poorly quantified as any phenomena in physics. For over a century, the most precise description of relaxation has been Boltzmann's H-theorem, predicting that a uniform ideal gas will relax monotonically. Recently, the relaxation theorem has shown that the approach to equilibrium can be quantified in terms of the dissipation function first defined in the proof of the Evans-Searles fluctuation theorem. Here, we provide the first demonstration of the relaxation theorem through simulation of a simple fluid system that generates a non-monotonic relaxation to equilibrium.

© 2012 American Institute of Physics

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KEYWORDS and PACS

Keywords

Boltzmann equation, entropy, free energy, relaxation theory

PACS

  • 05.60.-k

    Transport processes

  • 05.70.Ce

    Thermodynamic functions and equations of state

  • 02.60.-x

    Numerical approximation and analysis

ARTICLE DATA

Digital Object Identifier
http://dx.doi.org/10.1063/1.3675847

PUBLICATION DATA

ISSN

0021-9606 (print)  
1089-7690 (online)

Publisher

$abstract.society.value is a Member of CrossRef American Institute of Physics

  1. K. Huang, Statistical Mechanics (Wiley, New York, 1963).
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  3. S. Goldstein and J. Lebowitz, Physica D 193, 53 (2004).
  4. V. Arnold and A. Avez, Ergodic Problems of Classical Mechanics (Benjamin, New York, 1968).
  5. J. Dorfman, An Introduction to Chaos in Nonequilibrium Statistical Mechanics (Cambridge University Press, Cambridge, England, 1999).
  6. E. M. Sevick, R. Prabhakar, S. R. Williams, and D. J. Searles, Annu. Rev. Phys. Chem.59, 603 (2008). [MEDLINE]
  7. D. J. Evans and D. J. Searles, Phys. Rev. E 50, 1645 (1994).
  8. D. J. Evans and D. J. Searles, Phys. Rev. E 52, 5839 (1995). [MEDLINE]
  9. D. J. Evans, D. J. Searles, and S. R. Williams, J. Chem. Phys. 128, 014504 (2008)JCPSA6000128000001014504000001. [MEDLINE]
  10. D. J. Searles and D. J. Evans, Aust. J. Chem. 57, 1119 (2004). [ISI]
  11. D. J. Evans, D. J. Searles, and S. R. Williams, J. Stat. Mech.: Theory Exp. P07029 (2009).
  12. D. J. Evans and D. J. Searles, Adv. Phys. 51, 1529 (2002).
  13. Conformal relaxation occurs when the non-equilibrium distribution is of the form (f(Gamma, t) = exp ( − betaH + lambda(t)g(Gamma))/Z, [for all]t) and the deviation function, g, is a constant over the relaxation.
  14. D. M. Carberry, J. C. Reid, G. M. Wang, E. M. Sevick, D. J. Searles, and D. J. Evans, Phys. Rev. Lett. 92, 140601 (2004). [MEDLINE]
  15. M. Allen and D. Tildesley, Computer Simulation of Liquids (Oxford University Press, Oxford, 1994).
  16. D. Evans and G. Morriss, Statistical Mechanics of Nonequilibrium Liquids (Cambridge Unviersity Press, Cambridge, England, 2008).
  17. S. R. Williams, D. J. Searles, and D. J. Evans, Phys. Rev. E 70, 066113 (2004).
  18. Simulation parameters: 50 fluid particles, 22 wall particles, Deltat = 1 × 10−3, T = 1, rho = 0.3, and 100 000 trajectories.
  19. D. J. Evans, D. J. Searles, and S. R. Williams, J. Chem. Phys. 133, 054507 (2010)JCPSA6000133000005054507000001.

Figures (click on thumbnails to view enlargements)

FIG.1
(a) Plot of ln of the two sides of the ESFT, Eq. ( 4 ), for binned data with lines represent the expected behaviour. The agreement is extremely good. (b) and (c) Plot of the average instantaneous dissipation (Ω(Γ(t)), —) and the dissipation theorem (θ(Ω,t) = 〈Ω(0)〉+ ∫ 0t〈Ω(0)Ω(s)〉ds, - - -) with time (b) and the difference between the two functions bounded by the standard error of the difference (c). Only by plotting the difference, it is possible to observe the difference between the two functions. (d) Plot of the average of the total dissipation with time. In contrast with the instantaneous dissipation function, (b), the average of the total dissipation function is positive definite.

FIG.1 Download High Resolution Image (.zip file) | Export Figure to PowerPoint


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