**Ergodicity of non-Hamiltonian Equilibrium Systems**

Evans Denis 1,2, Williams Stephen 1,3, Rondoni Lamberto 4,5,6, Searles Debra 1,6,7

1 Australian Institute for Bioengineering & Nanotechnology The University of Queensland, Brisbane QLD, 4072, Australia

2 Department of Applied Mathematics, Research School of Physics and Engineering Australian National University, Canberra ACT, 0200, Australia

3 Research School of Chemistry, Australian National University Canberra ACT, 0200 Australia

4 Dipartimento di Scienze Matematiche and Graphene@Polito Lab, Politecnico di Torino Corso Duca degli Abruzzi 24, 10129 Torino, Italy

5 INFN, Sezione di Torino, Via P. Giuria 1, 10125 Torino, Italy

6 Kavli Institute for Theoretical Physics, Chinese Academy of Sciences, China

7 School of Chemistry and Molecular Biosciences

The University of Queensland, Brisbane QLD, 4072, Australia

### Received:

Received: 30 December 2016; revised: 06 April 2017; accepted: 09 April 2017; published online: 25 September 2017

### DOI: 10.12921/cmst.2016.0000068

### Abstract:

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.

### Key words:

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

### References:

[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 non-equilibrium 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 H. Feshback, Methods of theoretical physics, McGraw-Hill, New York 1953.

[15] D.J. Evans, E.D. G. Cohen, G.P. Morriss, Viscosity of a simple fluid from its maximal lyapunov exponents, Phys. Rev. A 42, 5990 (1990).

[16] J.L. Kaplan J.A. Yorke, Chaotic behavior of multidimensional difference equations in: H.O. Peitgen, H.O. Walther (ed.) Functional differential equations and approximation of fixed points, Springer, Heidelberg, p. 204-227, 1979.