Proof of the symmetry of the transport matrix for diffusion and heat flow in fluid systems

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Abstract
A phenomenological proof is given of the symmetry of the transport matrix (B i j ) of coefficients in the constitutive laws that describe mass and energytransport in a multicomponent fluid for the following choices of frame, forces, and fluxes: the local center of mass frame; the forces ∇(μi/T), where μ0=−1, T is the temperature, and μ i for i=1,...K is the chemical potential of the ith component; and the fluxes j i , where j0 is the nonconvective internal energy flux and j i for i=1,...K is the mass flux of the ith component. The proof is based on Onsager’s reciprocity theory and proceeds by using the equations of hydrodynamics and the constitutive equations, postulated to describe mass and energytransport in the fluid, to calculate the time development of a specially selected fluctuation, when that fluctuation is considered to be macrovariation. The resulting dynamical equations are cast into the Onsager form by use of the generalized forces obtained from a thermodynamic calculation of the entropy change ΔS associated with the fluctuation; the principle of local equilibrium is used in the latter calculation. Our proof then follows by a demonstration that the symmetry of the matrix (B i j )is equivalent to the symmetry of the matrix (L i j ) established by Onsager’s abstract proof. Special forms of the constitutive equations with symmetric coefficients are given for the case in which the dependence of the driving forces on temperature is isolated. Our proof is based on thermodynamics and phenomenological equations and is therefore independent of any particular statistical model; it does, however, agree with the conclusions of the statistical mechanical analysis of the fluid model studied by Kirkwood and Fitts and with a phenomenological analysis by Onsager for the special case of constant temperature and pressure.

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