The Soret and Dufour Effects in Statistical Dynamics

by R. F. Streater

Proc. Roy. Soc. A, 456, 205-221, 2000.

ABSTRACT

We obtain coupled nonlinear reaction-diffusion equations for a dense fluid moving in a potential, in the spirit of Smoluchowski. The equations obey the first and second laws of thermodynamics. From a discrete model on a lattice, furnished with transition probabilities that grow with energy, we construct the equations for the up-dating of the state; we use the methods of statistical dynamics to reduce the description to self-contained equations for the density and temperature, as functions of space and time. The continuum limit is taken with the help of MAPLE, giving a coupled nonlinear parabolic system, which is not uniformly elliptic. We find that the coupled system obeys the Onsager symmetry, and predicts the Soret effect (particle current in the presence of a temperature gradient) and the Dufour effect (heat flow in the presence of a density gradient). We obtain the effects without the need to take into account the interparticle energy, contrary to the view expressed in the book by Chapman and Cowling, "The mathematical theory of non-uniform gases", Cambridge Univ. Press, (1970), p. 103.

The full paper, including the MAPLE programme, is available on the archive math-ph/9910043.

In recent work by Esteban, Karch, Biler and Nadzieja, a modification of this model is studied, in which a further averaged interparticle Coulomb potential is added. Their model does not include the anomalous convection found here, however.

In a paper "Stability of a hot Smoluchowski fluid", published in Open Systems and Information Dynamics, I study the stability of a static solution of the equations of this paper in one dimension, by the usual method of small perturbations about the solution, and the neglect of terms smaller than linear. The generator of the semigroup of solutions to the coupled linear system is similar (in the technical sense of similarity transform by a bounded operator with bounded inverse) to a direct sum of second space derivatives. Since the spectrum of an operator is unchanged by a similarity transform, our generator is a closed operator with compact resolvent and positive spectrum. The new feature is that the operator is not sectorial, so that the methods of sectorial forms cannot be applied. In fact, the numerical range of the generator of the semigroup is the whole complex plane. The stability of these stationary solutions therefore remains an open question.

The paper, "The Soret and Dufour Effects..." was criticised by one of the referees as being limited to one dimension. This is true only in as much as the multiplicity of states is that appropriate to one dimension. A more stinging criticism was given by Frank Leppington after a lecture on the model at KCL. "Without a velocity field, your theory has not got off the ground", he said. These two points provoked me into writing [137],"Corrections to Fluid Dynamics" and (with Matheus Grasselli) [139], Hydrodynamics in an external field". These provide a hierarchy of equations; if the external potential is put to zero in [139] and the diffusion term omitted, we get the equations of [137]; if the diffusion constant is put equal to zero in [139], we get the Euler equations. Due to the insistence of the referees, the diffusion term in [137] was removed. Not that I agreed with the claim that a diffusion term in the mass-conservation law violated Galilean invariance: it does not. Even with this concession, the paper was rejected, and submitted to a more friendly journal. If the velocity in [137] is put equal to zero, and the diffusion term restored, we get what we would have got for the Soret and Dufour effects if we had put in the correct quantum mechanical multiplicity of states for a three-dimensional system. These differ from the equations of the present paper in that the dependence of the diffusion coefficient on the temperature differs by geometric factors involving square-roots of the temperature. In that sense, the present model is designed for one-dimensional diffusion (in its prediction of the temperature dependence of the Soret coefficient). Some experiments which do not fit the Navier-Stokes equations, but which might need a Soret effect in one-dimensional flow of a one-component gas, are described by Howard Brenner, in "Fluid Mechanics Revisited", to appear in Phys Rev E. This phenomenon was discovered by Reynolds in 1879, and is known as thermal transpiration.

In spite of some adverse referees' comment, the editors agreed to publish the paper, because it predicted the Soret and Dufour effects in a single-component fluid. Chapman had predicted that these effects would only be found in gas mixtures. This is indeed the current view, but thermal transpiration seems to show otherwise. Thermal transpiration was explained by Reynolds using N-s theory by making use of Maxwell's "slip" boundary conditions. These have been criticised by Brenner as not conforming to the first law of thermodynamics.