Onsager Reciprocal Relations - Abstract Formulation

Abstract Formulation

Let denote fluctuations from equilibrium values in several thermodynamic quantities, and let be the entropy. Then, Boltzmann's entropy formula gives for the probability distribution function, A=const. Assuming the fluctuations are small, the probability distribution function can be expressed through the second differential of the entropy

where we are using Einstein summation convention and is a positive definite symmetric matrix.

Using the quasi-stationary equilibrium approximation, that is, Assuming that the system is only slightly non-equilibrium, we have

Suppose we define thermodynamic conjugate quantities as, which can also be expressed as linear functions (for small fluctuations):

Thus, we can write where are called kinetic coefficients

The principle of symmetry of kinetic coefficients or the Onsager's principle states that is a symmetric matrix, that is