Informational Extended Thermodynamics of Phonon and Self-Diffusion Components of Heat Conduction in Liquids

Abstract

Heat flow in a liquid is divided into a component carried by longitudinal sound and a component carried by self-diffusing molecules. These components are averages of operators for which expressions are given involving phonon numbers and the probability that a given molecule is free to diffuse out of its cage. On the informationtheoretic model, these operators can be used to calculate extended thermodynamic forces which are Lagrange multipliers used in maximizing the entropy. Following arguments that the two heat flux components are uncorrelated, we find forces and rate equations to be uncoupled in leading order. Analytic expressions for the forces can be used to calculate the non-linear dependence on the temperature gradient of the evolution equations for the heat flux components. Also, it is found that in a steady state the condition of constancy of the structural parameters does not lead to nonanalytic solutions far from equilibrium, and so pressure and viscosity will not depend non-analytically on the gradient of temperature. IntroductionFollowing a model of Debye [1] applied to gases, an early formulation of extended thermodynamics of heat conduction [2] in liquids adopted a phonon picture of transport. In this model, heat is carried by collective longitudinal hydrodynamic modes with wavelengths long in comparison with intermolecular spacing. In addition to the phonon mechanism, we can imagine a self-diffusion component. An atom, confined to a cage formed by neighbours, can occasionally diffuse away when a random superposition of high-frequency hypersound waves, with wavelengths of the order of intermolecular distances, causes a local expansion sufficient for escape from the cage. The diffusing molecule carries heat, producing heat flux J s which is added to the phonon component, Jp, to give the total heat flow, J.In accord with this picture, an extended thermodynamic treatment was proposed [3] based on coupled equations for J s and J p . It was also assumed that the time-rate of Brought to you by | Purdue University Libraries Authenticated Download Date | 6/13/15 2:29 AM 240 R· E. Nettleton change of the density gradient can be expressed in terms of J s and J p . This phenomenology evolved long before the informational approach [4, 5] to thermodynamics of heat conduction which allows us to calculate the entropy as a function of J s and J^, provided we have microscopic operators, J s and J p^ of which the fluxes are ensemble averages. By introducing an explicit function for J s , we can study the coupling of the heat flux components in the thermodynamic forces. The operators can also be used in a statistical derivation of the kinetic equations [6] to show that, in linear approximation, J s and J p obey independent Cattaneo-Vernottetype equations. This simplifies the earlier phenomenology [3], eliminating some of the coupling coefficients.The model for J 5 involves the probability Ë that an atom at r/ can cross the local potential barriers and move into a new cage. The relaxation of J s is coupled nonlinearly ...