Steepest entropy ascent model for far-nonequilibrium thermodynamics: Unified implementation of the maximum entropy production principle

By suitable reformulations, we cast the mathematical frameworks of several well-known different approaches to the description of nonequilibrium 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 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-nonequilibrium domain, most of the existing theories of nonequilibrium 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 is intrinsically and strongly consistent with the second law of thermodynamics. The non-negativity of the entropy production is a general and readily proved feature of SEA dynamics. In several of the different approaches to nonequilibrium 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 nonequilibrium states. The mathematical frameworks we consider are the following: (A) statistical or information-theoretic models of relaxation; (B) small-scale and rarefied gas dynamics (i.e., kinetic models for the Boltzmann equation); (C) rational extended thermodynamics, macroscopic nonequilibrium thermodynamics, and chemical kinetics; (D) mesoscopic nonequilibrium thermodynamics, continuum mechanics with fluctuations; and (E) quantum statistical mechanics, quantum thermodynamics, mesoscopic nonequilibrium quantum thermodynamics, and intrinsic quantum thermodynamics.

Figure 1

Pictorial representation of the linear manifold spanned by the vectors $|{\Psi }_i)$ and the orthogonal projection of $\left|\Phi \right)$ onto this manifold which defines the Lagrange multipliers ${\beta }_i$ in the case of a uniform metric $\widehat{G}=\widehat{I}$. The construction defines also the generalized affinity vector, which in this case is $\left|\Lambda \right)=|\Phi -{\sum }_i{\beta }_i{\Psi }_i)$.

Figure 2

Pictorial representation of the SEA variational construction in the case of a uniform metric $\widehat{G}=\widehat{I}$. The circle represents the condition $\left({\Pi }_{\gamma }\right|{\Pi }_{\gamma })={\dot{\epsilon }}^2$. The vector $|{\Pi }_{\gamma })$ must be orthogonal to the $|{\Psi }_i)$'s in order to satisfy the conservation constraints ${\Pi }_{C_i}=\left({\Psi }_i\right|{\Pi }_{\gamma })=0$. In order to maximize the scalar product $(\Phi -{\sum }_i{\beta }_i{\Psi }_i|{\Pi }_{\gamma })$, $|{\Pi }_{\gamma })$ must have the same direction as $|\Phi -{\sum }_i{\beta }_i{\Psi }_i)$.

Figure 3

Pictorial representation of the linear manifold spanned by the vectors ${\widehat{G}}^{-1/2}{\Psi }_i$ and the orthogonal projection of ${\widehat{G}}^{-1/2}\left|\Phi \right)$ onto this manifold which defines the Lagrange multipliers ${\beta }_i$ in the case of a nonuniform metric $\widehat{G}$. The construction defines also the generalized affinity vector $\left|\Lambda \right)={\widehat{G}}^{-1/2}|\Phi -{\sum }_i{\beta }_i{\Psi }_i)$.

Figure 4

Pictorial representation of the SEA variational construction in the case of a nonuniform metric $\widehat{G}$. The circle represents the condition $({\Pi }_{\gamma }|\widehat{G}\left|{\Pi }_{\gamma }\right)={\dot{\epsilon }}^2$, corresponding to the norm of vector ${\widehat{G}}^{1/2}\left|{\Pi }_{\gamma }\right)$. This vector must be orthogonal to the ${\widehat{G}}^{-1/2}\left|{\Psi }_i\right)$'s in order to satisfy the conservation constraints ${\Pi }_{C_i}=\left({\Psi }_i\right|{\Pi }_{\gamma })=0$. In order to maximize the scalar product ${\Pi }_S=\left(\Phi \right|{\Pi }_{\gamma })=(\Phi -{\sum }_i{\beta }_i{\Psi }_i|{\Pi }_{\gamma })$, vector ${\widehat{G}}^{1/2}\left|{\Pi }_{\gamma }\right)$ must have the same direction as $\left|\Lambda \right)={\widehat{G}}^{-1/2}|\Phi -{\sum }_i{\beta }_i{\Psi }_i)$.