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
Read more about this topic: Onsager Reciprocal Relations
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The formulation was used as a motto by the English Nonconformist clergyman Richard Baxter (1615-1691)