Ergodicity of non-Hamiltonian Equilibrium Systems

• Autorzy: Evans D. J. , Williams S. R. , Rondoni L. , Searles D. J.

Identyfikatory

DOI

10.12921/cmst.2016.0000068

• Języki publikacji: EN

Abstrakty

EN

It is well known that ergodic theory can be used to formally prove a form of relaxation to microcanonical equilibrium for finite, mixing Hamiltonian systems. In this manuscript we substantially modify this proof using an approach similar to that used in umbrella sampling, and use this approach to consider relaxation in both Hamiltonian and non- Hamiltonian systems. In doing so, we demonstrate the need for a form of ergodic consistency of the initial and final distribution. The approach only applies to relaxation of averages of physical properties and low order probability distribution functions. It does not provide any information about whether the full 6N-dimensional phase space distribution relaxes towards the equilibrium distribution or how long the relaxation of physical averages takes.

Słowa kluczowe

EN

equilibrium; ergodic theory; relaxation to equilibrium; distribution function; statistical mechanics

• Wydawca: Institute of Bioorganic Chemistry Scientific Publishers OWN, Polish Academy of Sciences
• Czasopismo: Computational Methods in Science and Technology
• Rocznik: 2017
• Tom: Vol. 23, No. 3
• Strony: 175--184
• Opis fizyczny: Bibliogr. 16 poz.

Twórcy

autor

Evans D. J.

• Australian Institute for Bioengineering & Nanotechnology The University of Queensland, Brisbane QLD, 4072, Australia
• Department of Applied Mathematics, Research School of Physics and Engineering Australian National University, Canberra ACT, 0200, Australia

autor

Williams S. R.

• Australian Institute for Bioengineering & Nanotechnology The University of Queensland, Brisbane QLD, 4072, Australia
• Research School of Chemistry, Australian National University Canberra ACT, 0200 Australia

autor

Rondoni L.

• Dipartimento di Scienze Matematiche and Graphene@Polito Lab, Politecnico di Torino Corso Duca degli Abruzzi 24, 10129 Torino, Italy
• INFN, Sezione di Torino, Via P. Giuria 1, 10125 Torino, Italy
• Kavli Institute for Theoretical Physics, Chinese Academy of Sciences, China

autor

Searles D. J.

• Australian Institute for Bioengineering & Nanotechnology The University of Queensland, Brisbane QLD, 4072, Australia
• Kavli Institute for Theoretical Physics, Chinese Academy of Sciences, China
• School of Chemistry and Molecular Biosciences The University of Queensland, Brisbane QLD, 4072, Australia

Bibliografia

• [1] D.J. Evans, S.R. Williams, L. Rondoni, A mathematical proof of the zeroth "law" of thermodynamics and the nonlinear Fourier "law" for heat flow, J. Chem. Phys., 137, 194109 (2012).
• [2] D.J. Evans, D.J. Searles, S.R. Williams, Fundamentals of classical statistical thermodynamics, dissipation, relaxation and fluctuation theorems, Wiley-VCH Verlag, Weinheim, Germany 2016.
• [3] Y.G. Sinai, Introduction to ergodic theory, Princeton University Press, Princeton, NJ, USA 1976.
• [4] D.J. Evans, D.J. Searles, S.R. Williams, Dissipation and the relaxation to equilibrium, J. Stat. Mech. Theor. Exp. P07029 (2009).
• [5] D.J. Evans, D.J. Searles, S.R.Williams, A simple mathematical proof of Boltzmann’s equal a priori probability hypothesis in: C. Chmelik, N. Kanellopoulos, J. Karger, T. Doros (ed.) Diffusion fundamentals III, Leipziger Universitatsverlag, Leipzig, p. 367- 374, 2009.
• [6] D.J. Evans G.P. Morriss, Statistical mechanics of nonequilibrium liquids, Academic, London 1990.
• [7] H. Green, The molecular theory of fluids, North-Holland, Amsterdam 1952.
• [8] R. Balescu, Equilibrium and non equilibrium statistical mechanics, Wiley Interscience, New York 1975.
• [9] J.D. Crawford J.R. Cary, Decay of correlations in a chaotic measure-preserving transformation, Physica D 6, 223 (1983).
• [10] D.J. Evans, D.J. Searles, S.R. Williams, On the fluctuation theorem for the dissipation function and its connection with response theory, J. Chem. Phys. 128, 014504 (2008).
• [11] D.J. Evans, D.J. Searles, S.R. Williams, Erratum: On the fluctuation theorem for the dissipation function and its connection with response theory (vol 128, artn 014504, 2008), J. Chem. Phys. 128, 249901 (2008).
• [12] D.J. Evans, D.J. Searles, S.R. Williams, Musings on thermostats, J. Chem. Phys. 133, 104106 (2010).
• [13] C. Thompson, Mathematical statistical mechanics, Collier Macmillan Ltd, London 1972.
• [14] P.M. Morse and H. Feshback, Methods of theoretical physics, McGraw-Hill, New York 1953.
• [15] D.J. Evans, E.D. G. Cohen, and G.P. Morriss, Viscosity of a simple fluid from its maximal lyapunov exponents, Phys. Rev. A 42, 5990 (1990).
• [16] J.L. Kaplan and J.A. Yorke, Chaotic behavior of multidimensional difference equations in: H.O. Peitgen, and H.O. Walther (ed.) Functional differential equations and approximation of fixed points, Springer, Heidelberg, p. 204-227, 1979.

Uwagi

PL

Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę (zadania 2017).