Nonlinear model dynamics for closed-system, constrained, maximal-entropy-generation relaxation by energy redistribution

We discuss a nonlinear model for relaxation by energy redistribution within an isolated, closed system composed of noninteracting identical particles with energy levels eᵢ with i=1,2,,N. The time-dependent occupation probabilities pᵢ(t) are assumed to obey the nonlinear rate equations ${τ}{{0.2em}{0ex}}d{p}ᵢ∕dt={-}{p}ᵢ{{0.2em}{0ex}}ln{{0.2em}{0ex}}{p}ᵢ{-}{α}(t){p}ᵢ{-}{β}(t){e}ᵢ{p}ᵢ$ where ${α}(t)$ and ${β}(t)$ are functionals of the ${p}ᵢ(t)$'s that maintain invariant the mean energy $E={{∑}}ᵢ₌₁N{e}ᵢ{p}ᵢ(t)$ and the normalization condition $1={{∑}}ᵢ₌₁N{p}ᵢ(t)$. The entropy $S(t)={-}{k}B{{∑}}ᵢ₌₁N{p}ᵢ(t)ln{{0.2em}{0ex}}{p}ᵢ(t)$ is a nondecreasing function of time until the initially nonzero occupation probabilities reach a Boltzmann-like canonical distribution over the occupied energy eigenstates. Initially zero occupation probabilities, instead, remain zero at all times. The solutions pᵢ(t) of the rate equations are unique and well defined for arbitrary initial conditions pᵢ(0) and for all times. The existence and uniqueness both forward and backward in time allows the reconstruction of the ancestral or primordial lowest entropy state. By casting the rate equations in terms not of the pᵢ's but of their positive square roots √pᵢ, they unfold from the assumption that time evolution is at all times along the local direction of steepest entropy ascent or, equivalently, of maximal entropy generation. These rate equations have the same mathematical structure and basic features as the nonlinear dynamical equation proposed in a series of papers ending with G. P. Beretta, Found. Phys. 17, 365 (1987) and recently rediscovered by S. Gheorghiu-Svirschevski [Phys. Rev. A 63, 022105 (2001);63, 054102 (2001)]. Numerical results illustrate the features of the dynamics and the differences from the rate equations recently considered for the same problem by M. Lemanska and Z. Jaeger [Physica D 170, 72 (2002)]. We also interpret the functionals kBα(t) and kBβ(t) as nonequilibrium generalizations of the thermodynamic-equilibrium Massieu characteristic function and inverse temperature, respectively.