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

We discuss a nonlinear model for the relaxation by energy redistribution within an isolated, closed system composed of non-interacting 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 dpi/dt = −pi ln pi − α(t) pi − β(t) eipi where α(t) and β(t) are functionals of the pi(t)’s that maintain invariant the mean energy E = ∑ N i=1 ei pi(t) and the normalization condition 1 = ∑ N i=1 pi(t). The entropy S(t) = −kB ∑ N i=1 pi(t) ln pi(t) is a non-decreasing 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 welldefined for arbitrary initial conditions pi(0) and for all times. 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 not in terms 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 of the nonlinear dynamical equation proposed in a series of papers ended with G.P. Beretta, Found. Phys. 17, 365 (1987) and recently rediscovered in S. Gheorghiu-Svirschevski, Phys. Rev. A 63, 022105 and 054102 (2001). Numerical results illustrate the features of the dynamics and the differences with the rate equations recently considered for the same problem in 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.