Non-monotonic Relaxation in a Harmonic Well
School of Natural Sciences, Griffith University, Brisbane 4111 QLD, Australia and Queensland Micro- and Nanotechnology Centre, Griffith University, Brisbane 4111 QLD, Australia
* E-mail:o.jepps@griffith.edu.au
Received:
Received: 01 January 2017; revised: 12 March 2017; accepted: 15 March 2017; published online: 30 September 2017
DOI: 10.12921/cmst.2017.0000001
Abstract:
The dissipation function of Evans and Searles has its origins in describing entropy production, yet it has a straightforward dynamical interpretation as well. The ability to consider either dynamical or thermodynamical contexts deepens our understanding of the dissipation function as a concept, and of numerical results involving the dissipation function. One recent, important application of the dissipation function is in relaxation to equilibrium. Here we look at relaxation in a system of interacting molecules that are confined within a harmonic potential, undergoing Hamiltonian dynamics. We note some similarities, but also important differences, to previous studies. The dissipation function sheds light on the periodic return of our system towards its initial state. We find that intermolecular interactions play a much more significant role in the relaxation toward a non-uniform spatial distribution (induced by a conservative background field) than they do toward a uniform distribution, which is reflected in the strongly non-monotonic relaxation we observe. We also find that the maximum dissipation does not occur in the long-time limit, as one might expect of a relaxation process, but shortly after relaxation begins, beyond which a significant net overall decrease in the dissipation function is observed.
Key words:
dissipation function, equilibrium statistical mechanics, microcanonical ensemble, relaxation to equilibrium
References:
[1] Lawrence Sklar, Physics and Chance: Philosophical issues in the foundations of statistical mechanics, Cambridge University Press, Cambridge, 1993.
[2] Roman Frigg, Why Typicality Does Not Explain the Approach to Equilibrium, In Mauricio Suárez, editor, Probabilities, Causes and Propensities in Physics, volume 347 of Studies in Epistemology, Logic, Methodology, and Philosophy of Science, pages 77–93 Springer Dordrecht, 2011.
[3] Stewart Harris, An Introduction to the Theory of the Boltzmann Equation, Holt, Rinehart and Winston Inc., New York, 1971.
[4] J.L. Lebowitz, Boltzmann’s entropy and time’s arrow, Physics Today 46, 32 (1993).
[5] C. Gruber, S Pache, and A Lesne, On the Second Law of Thermodynamics and the Piston Problem, Journal of Statistical Physics 117(3), 739–772 (2004).
[6] Giancarlo Benettin, Ergodicity and time-scales for statistical equilibrium in classical dynamical systems, Progress of Theoretical Physics Supplements 116, 207–234, (1994).
[7] D.J. Searles and D.J. Evans, Physical Review E 50, 1645 (1994).
[8] D.J. Evans, D.J. Searles, and Stephen R. Williams, Journal of Chemical Physics 128, 014504 (2008).
[9] D.J. Evans, D.J. Searles, and Stephen R. Williams, Journal of Chemical Physics 128, 249901 (2008).
[10] D.J. Evans, D.J. Searles, and Stephen R. Williams, Journal of Statistical Mechanics: Theory and Experiment 2009, P07029 (2009).
[11] Owen G. Jepps and Lamberto Rondoni, A dynamical-systems interpretation of the dissipation function, T-mixing and their relation to thermodynamic relaxation, Journal of Physics A: Mathematical and Theoretical 49(15), 154002 (2016).
[12] James C. Reid, Denis J. Evans, and Debra J. Searles, Communication: Beyond Boltzmann’s H-theorem: Demonstration of the relaxation theorem for a non-monotonic approach to equilibrium, Journal of Chemical Physics 136(2), 021101 (2012).
[13] Charlotte F. Petersen, Denis J. Evans, and Stephen R. Williams, Dissipation in monotonic and non-monotonic relaxation to equilibrium, Journal of Chemical Physics 144(7), 074107 (2016).
[14] Owen G. Jepps and Lamberto Rondoni, Deterministic thermostats, theories of nonequilibrium systems and parallels with the ergodic condition, Journal of Physics A: Mathematical and Theoretical 43(13), 133001 (2010).
[15] D.J. Searles and D.J. Evans, Australian Journal of Chemistry 57, 1119 (2004).
[16] D.J. Evans, Stephen R. Williams, D.J. Searles, and Lamberto Rondoni, On Typicality in Nonequilibrium Steady States, Journal of Statistical Physics 164, 842–85 (2016).
[17] Sergio Chibbaro, Lamberto Rondoni, and Angelo Vulpiani, Reductionism, Emergence and Levels of Reality: The Importance of Being Borderline, Springer International Publishing Switzerland, 2014.
[18] D.M. Carberry, J.C. Reid, G.M. Wang, E.M. Sevick, D.J. Searles, and D.J. Evans, Physical Review Letters 92(14), 140601 (2004).
[19] J.D. Weeks, D.Chandler, and H.C. Andersen, Journal of Chemical Physics 54, 5237 (1971).
[20] H.Risken, The Fokker-Planck equation: methods of solution and applications, Springer-Verlag, New York, 1996.
The dissipation function of Evans and Searles has its origins in describing entropy production, yet it has a straightforward dynamical interpretation as well. The ability to consider either dynamical or thermodynamical contexts deepens our understanding of the dissipation function as a concept, and of numerical results involving the dissipation function. One recent, important application of the dissipation function is in relaxation to equilibrium. Here we look at relaxation in a system of interacting molecules that are confined within a harmonic potential, undergoing Hamiltonian dynamics. We note some similarities, but also important differences, to previous studies. The dissipation function sheds light on the periodic return of our system towards its initial state. We find that intermolecular interactions play a much more significant role in the relaxation toward a non-uniform spatial distribution (induced by a conservative background field) than they do toward a uniform distribution, which is reflected in the strongly non-monotonic relaxation we observe. We also find that the maximum dissipation does not occur in the long-time limit, as one might expect of a relaxation process, but shortly after relaxation begins, beyond which a significant net overall decrease in the dissipation function is observed.
Key words:
dissipation function, equilibrium statistical mechanics, microcanonical ensemble, relaxation to equilibrium
References:
[1] Lawrence Sklar, Physics and Chance: Philosophical issues in the foundations of statistical mechanics, Cambridge University Press, Cambridge, 1993.
[2] Roman Frigg, Why Typicality Does Not Explain the Approach to Equilibrium, In Mauricio Suárez, editor, Probabilities, Causes and Propensities in Physics, volume 347 of Studies in Epistemology, Logic, Methodology, and Philosophy of Science, pages 77–93 Springer Dordrecht, 2011.
[3] Stewart Harris, An Introduction to the Theory of the Boltzmann Equation, Holt, Rinehart and Winston Inc., New York, 1971.
[4] J.L. Lebowitz, Boltzmann’s entropy and time’s arrow, Physics Today 46, 32 (1993).
[5] C. Gruber, S Pache, and A Lesne, On the Second Law of Thermodynamics and the Piston Problem, Journal of Statistical Physics 117(3), 739–772 (2004).
[6] Giancarlo Benettin, Ergodicity and time-scales for statistical equilibrium in classical dynamical systems, Progress of Theoretical Physics Supplements 116, 207–234, (1994).
[7] D.J. Searles and D.J. Evans, Physical Review E 50, 1645 (1994).
[8] D.J. Evans, D.J. Searles, and Stephen R. Williams, Journal of Chemical Physics 128, 014504 (2008).
[9] D.J. Evans, D.J. Searles, and Stephen R. Williams, Journal of Chemical Physics 128, 249901 (2008).
[10] D.J. Evans, D.J. Searles, and Stephen R. Williams, Journal of Statistical Mechanics: Theory and Experiment 2009, P07029 (2009).
[11] Owen G. Jepps and Lamberto Rondoni, A dynamical-systems interpretation of the dissipation function, T-mixing and their relation to thermodynamic relaxation, Journal of Physics A: Mathematical and Theoretical 49(15), 154002 (2016).
[12] James C. Reid, Denis J. Evans, and Debra J. Searles, Communication: Beyond Boltzmann’s H-theorem: Demonstration of the relaxation theorem for a non-monotonic approach to equilibrium, Journal of Chemical Physics 136(2), 021101 (2012).
[13] Charlotte F. Petersen, Denis J. Evans, and Stephen R. Williams, Dissipation in monotonic and non-monotonic relaxation to equilibrium, Journal of Chemical Physics 144(7), 074107 (2016).
[14] Owen G. Jepps and Lamberto Rondoni, Deterministic thermostats, theories of nonequilibrium systems and parallels with the ergodic condition, Journal of Physics A: Mathematical and Theoretical 43(13), 133001 (2010).
[15] D.J. Searles and D.J. Evans, Australian Journal of Chemistry 57, 1119 (2004).
[16] D.J. Evans, Stephen R. Williams, D.J. Searles, and Lamberto Rondoni, On Typicality in Nonequilibrium Steady States, Journal of Statistical Physics 164, 842–85 (2016).
[17] Sergio Chibbaro, Lamberto Rondoni, and Angelo Vulpiani, Reductionism, Emergence and Levels of Reality: The Importance of Being Borderline, Springer International Publishing Switzerland, 2014.
[18] D.M. Carberry, J.C. Reid, G.M. Wang, E.M. Sevick, D.J. Searles, and D.J. Evans, Physical Review Letters 92(14), 140601 (2004).
[19] J.D. Weeks, D.Chandler, and H.C. Andersen, Journal of Chemical Physics 54, 5237 (1971).
[20] H.Risken, The Fokker-Planck equation: methods of solution and applications, Springer-Verlag, New York, 1996.
