The Second Law of Nonequilibrium Thermodynamics: How Fast Time Flies

The Second Entropy: A Variational Principle for Time-dependent Systems

“…Onsager invoked the regression hypothesis, which is to say that the flux that develops in an isolated system is the same as that that develops in a system acted upon by an external mechanical or thermodynamic force that is equal to the internal force [6]. In the present second entropy approach it can be shown that in the optimum state the internal force is indeed equal to the external force, and so the present expression holds for the flux in a driven steady state system [1,2].…”

“…The second entropy theory can be briefly illustrated by applying it to the steady state in the linear regime. The Principle is of course much more general than this, and more detailed analyses and broader applications are given elsewhere [1][2][3]31].…”

“…This is the most general quadratic form that can be written consistent with the symmetry of the problem. Denoting the most likely state with an over-line, maximizing the second entropy, ∂S (2)…”

“…The second entropy must reduce to the first entropy upon integration over x . Again for Gaussean statistics, this is the same as evaluating it at x (x, τ ), so that one has the reduction condition S (2)…”

“…The concept has emerged from a theory for non-equilibrium thermodynamics and statistical mechanics that has been developed over the last few years. For reviews see Refs [1,2]. Amongst other things, the theory has lead for the first time to the generalization of the Boltzmann distribution to non-equilibrium systems, and to the development of a Monte Carlo computer algorithm for such time-dependent systems [3].…”