  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 e_i with i=1,2,...,N. The time-dependent
occupation probabilities p_i(t) are assumed to obey the nonlinear rate
equations tau dp_i/dt=-p_i ln p_i+ alpha(t)p_i-beta(t)e_ip_i where alpha(t) and
beta(t) are functionals of the p_i(t)'s that maintain invariant the mean energy
E=sum_i e_ip_i(t) and the normalization condition 1=sum_i p_i(t). The entropy
S(t)=-k sum_i p_i(t) ln p_i(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 p_i(t) of the
rate equations are unique and well-defined for arbitrary initial conditions
p_i(0) and for all times. Existence and uniqueness both forward and backward in
time allows the reconstruction of the primordial lowest entropy state. The 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).
