Statistical mechanical theory for steady state systems. VII. Nonlinear theory

Abstract

The second entropy theory for nonequilibrium thermodynamics is extended to the nonlinear regime and to systems of mixed parity (even and odd functions of molecular velocities). The steady state phase space probability density is given for systems of mixed parity. The nonlinear transport matrix is obtained and it is shown to yield the analog of the linear Onsager-Casimir reciprocal relations. Its asymmetric part contributes to the flux and to the production of second entropy. The nonlinear transport matrix is not simply expressible as a Green-Kubo fluctuation equilibrium time correlation function. However, here the first nonlinear correction to the transport coefficient is given explicitly as a type of the Green-Kubo equilibrium time correlation function. The theory is illustrated by application to chemical kinetics.

The Second Law of Nonequilibrium Thermodynamics: How Fast Time Flies

“…The asymmetric part of the transport matrix gives zero contribution to the scalar product and so does not contribute to the steady-state rate of first entropy production [7]. This was also observed by Casimir [24] and by Grabert et al [25], Eq.…”

“…The two theories span the very large and the very small. The aim of this chapter is to present a coherent and selfcontained account of these theories, which have been developed by the author and presented in a series of papers [1][2][3][4][5][6][7]. The theory up to the fifth paper has been reviewed previously [8], and the present chapter consolidates some of this material and adds the more recent developments.…”

Time correlations and the second entropy

“…Below we discuss also the behaviour of moments of certain variables. Moments are used in references 11,12 to develop the second-entropy theory. Consider the behaviour of a Fourier mode of the energy density as the box size is increased and k → 0,…”

“…It turns out that for flows the steady state regime holds over intermediate values of the time interval, and that in this regime the second entropy becomes extensive in τ . 7,12 This may be seen starting from the general expression for the most likely terminal vector, Eq. (17), whose time derivative yieldṡ…”

Emerging of Stochastic Dynamical Equalities and Steady State Thermodynamics from Darwinian Dynamics

“…There have renewed interests in stochastic phenomena, ranging from physics, [41,[43][44][45][46][47][48][49][50][51] chemistry, [52][53][54][55] material science, [56][57][58] biology, [19,[59][60][61] and to other fields. [24] These works demonstrate the strong on-going exchange of ideas between physical and biological sciences.…”