Projection operator formalism and entropy.pdf

a r X i v : c o n d - m a t / 0 7 0 3 6 7 2 v 1 [ c o n d - m a t .s t a t - m e c h ] 2 6 M a r 2 0 0 7 Projection operator formalism and entropy E.A.J.F. Peters? Dept. of Chemical Engineering Technische Universiteit Eindhoven P.O. Box 513 5600 MB Eindhoven The Netherlands (Dated: February 4, 2008) The entropy definition is deduced by means of (re)deriving the generalized non-linear Langevin equation using Zwanzig projector operator formalism. It is shown to be necessarily related to an invariant measure which, in classical mechanics, can always be taken to be the Liouville measure. It is not true that one is free to choose a “relevant” probability density independently as is done in other flavors of projection operator formalism. This observation induces an entropy expression which is valid also outside the thermodynamic limit and in far from equilibrium situations. The Zwanzig projection operator formalism therefore gives a deductive derivation of non-equilibrium, and equilibrium, thermodynamics. The entropy definition found is closely related to the (generalized) microcanonical Boltzmann-Planck definition but with some subtle differences. No “shell thickness” arguments are needed, nor desirable, for a rigorous definition. The entropy expression depends on the choice of macroscopic variables and does not exactly transform as a scalar quantity. The relation with expressions used in the GENERIC formalism are discussed. PACS numbers: 05.70.Ln, 05.40.-a, 05.20.Gg, 05.10.Gg I. INTRODUCTION The classical, Boltzmann-Planck, definition of entropy is the logarithm of the number of microstates correspond- ing to a macroscopic state of a system (times the Boltz- mann constant kB). Within classical mechanics this defi- nition causes some fundamental problems since the num- ber of microstates is not countable. The common res- olution is to define a unit volume of microscopic phase space. Within classical mechanics the motivation why this is reasonable is found in Liou