No production of entropy in the Euler fluid

ABSTRACT

We derive the Euler equations as the hydrodynamic limit of a stochastic model of a hard-sphere gas. We show that the system does not produce entropy.

REMARKS

in Nonlocal elliptic and parabolic problems, Banach Center Publications, 66, Warsaw, 2004.

The entropy in the title refers to the von Neumann entropy of the lattice gas, where the sample space is described by the absence or presence of a particle at each site, and, if a particle is there, by its momentum. The momentum of a particle is taken to lie on a lattice of integers, in units of a smallest quantum of momentum. The stochastic model refers to a hopping model, where the rate of a hop is given by the velocity of the particle, if it is present. After a hop for a random time, the particle thermalizes at the location of its arrival, this being the lattice site nearest its Newtonian path, in the presence of a given external potential energy, Phi. The state of the system is given by a Boltzmann density function,

mu(x,k,t)= N(x,k,t)p(x,k,t),

defining a measure on the sample space at each time. We sketch a derivation of the master equation, which is to replace Boltzmann's equation as the dynamics of the gas. The probability of a hop is given by an exponential law, whose parameters are determined by the present values of mu over the path followed by the particle. Thus, it is a non-linear "Markov process" in the sense of Alicki and Messer. There is thus a survival probability for each hop, less than one if collisions can occur. If we assume that the mean free time is zero, we arrive at the Euler equations for fluid motion, of a compressible gas with temperature. We show that in this limit, there is no entropy production.

The article can be found in the archives nlin.CG.0310008.