By suitable reformulations, we cast the mathematical frameworks of several well-known different approaches to the description of non-equilibrium dynamics into a unified formulation valid in all these contexts, which extends to such frameworks the concept of Steepest Entropy Ascent (SEA) dynamics introduced by the present author in previous works on quantum thermodynamics. Actually, the present formulation constitutes a generalization also for the quantum thermodynamics framework. The analysis emphasizes that in the SEA modeling principle a key role is played by the geometrical metric with respect to which to measure the length of a trajectory in state space. In the near thermodynamic equilibrium limit, the metric tensor turns is directly related to the Onsager's generalized resistivity tensor. Therefore, through the identification of a suitable metric field which generalizes the Onsager generalized resistance to the arbitrarily far non-equilibrium domain, most of the existing theories of non-equilibrium thermodynamics can be cast in such a way that the state exhibits the spontaneous tendency to evolve in state space along the path of SEA compatible with the conservation constraints and the boundary conditions. The resulting unified family of SEA dynamical models are all intrinsically and strongly consistent with the second law of thermodynamics. The nonnegativity of the entropy production is a general and readily proved feature of SEA dynamics. In several of the different approaches to non-equilibrium description we consider here, the SEA concept has not been investigated before. We believe it defines the precise meaning and the domain of general validity of the so-called Maximum Entropy Production principle. Therefore, it is hoped that the present unifying approach may prove useful in providing a fresh basis for effective, thermodynamically consistent, numerical models and theoretical treatments of irreversible conservative relaxation towards equilibrium from far non-equilibrium states. The mathematical frameworks are: A) Statistical or Information Theoretic Models of Relaxation; B) Small-Scale and Rarefied Gases Dynamics (i.e., kinetic models for the Boltzmann equation

We first discuss the geometrical construction and the main mathematical features of the maximum-entropy-production/steepest-entropyascent nonlinear evolution equation proposed long ago by this author in the framework of a fully quantum theory of irreversibility and thermodynamics for a single isolated or adiabatic particle, qubit, or qudit, and recently rediscovered by other authors. The nonlinear equation generates a dynamical group, not just a semigroup, providing a deterministic description of irreversible conservative relaxation towards equilibrium from any non-equilibrium density operator. It satisfies a very restrictive stability requirement equivalent to the Hatsopoulos-Keenan statement of the second law of thermodynamics. We then examine the form of the evolution equation we proposed to describe multipartite isolated or adiabatic systems. This hinges on novel nonlinear projections defining local operators that we interpret as "local perceptions" of the overall system's energy and entropy. Each component particle contributes an independent local tendency along the direction of steepest increase of the locally perceived entropy at constant locally perceived energy. It conserves both the locallyperceived energies and the overall energy, and meets strong separability and non-signaling conditions, even though the local evolutions are not independent of existing correlations. We finally show how the geometrical construction can readily lead to other thermodynamically relevant models, such as of the nonunitary isoentropic evolution needed for full extraction of a system's adiabatic availability.

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 meanei pi(t) and the normalization condition 1 =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.

A novel nonlinear equation of motion is proposed for quantum systems consisting of a single elementary constituent of matter. It is satisfied by pure states and by a special class of mixed states evolving unitarily. But, in general, it generates a nonunitary evolution of the state operator. It keeps the energy invariant and causes the entropy to increase with time until the system reaches a state of equilibrium or a limit cycle. © 1984 Società Italiana di Fisica

A novel nonlinear equation of motion is proposed for a general quantum system consisting of more than one distinguishable elementary constituent of matter. In the domain of idempotent quantummechanical state operators, it is satisfied by all unitary evolutions generated by the Schrödinger equation. But, in the broader domain of nonidempotent state operators not contemplated by conventional quantum mechanics, it generates a generally nonunitary evolution, it keeps the energy invariant and causes the entropy to increase with time until the system reaches a state of equilibrium or a limit cycle. © 1985 Società Italiana di Fisica