Nonlinear closure relations theory for transport processes in nonequilibrium systems

Abstract

A decade ago, a macroscopic theory for closure relations has been proposed for systems out of Onsager's region. This theory is referred to as the thermodynamic field theory (TFT). The aim of this work was to determine the nonlinear flux-force relations that respect the thermodynamic theorems for systems far from equilibrium. We propose a formulation of the TFT where one of the basic restrictions, namely, the closed-form solution for the skew-symmetric piece of the transport coefficients, has been removed. In addition, the general covariance principle is replaced by the De Donder-Prigogine thermodynamic covariance principle (TCP). The introduction of TCP requires the application of an appropriate mathematical formalism, which is referred to as the entropy-covariant formalism. By geometrical arguments, we prove the validity of the Glansdorff-Prigogine universal criterion of evolution. A new set of closure equations determining the nonlinear corrections to the linear ("Onsager") transport coefficients is also derived. The geometry of the thermodynamic space is non-Riemannian. However, it tends to be Riemannian for high values of the entropy production. In this limit, we recover the transport equations found by the old theory. Applications of our approach to transport in magnetically confined plasmas, materials submitted to temperature, and electric potential gradients or to unimolecular triangular chemical reactions can be found at references cited herein. Transport processes in tokamak plasmas are of particular interest. In this case, even in the absence of turbulence, the state of the plasma remains close to (but, it is not in) a state of local equilibrium. This prevents the transport relations from being linear.

Generalized extensive entropies for studying dynamical systems in highly anisotropic phase spaces

“…This work gives several perspectives. Through the thermodynamical field theory (TFT) [35], it is possible to estimate the PDF when the nonlinear contributions cannot be neglected [36]. The next task should be to establish the relation between the reference, stationary PDF, derived by the MaxEnt principle applied to GRE, subject to scale-invariant restrictions, with the ones found by the TFT.…”

Symmetry group and group representations associated with the thermodynamic covariance principle

“…In a previous work, one of us introduced a macroscopic theory for closure relations for systems out of Onsager s region [1]- [2]. The most important closure relations are the so-called transport equations, relating the dissipative fluxes to the thermodynamic forces that produce them.…”

“…The coordinates of the Thermodynamic Space are the thermodynamic forces, the metric is identified with the (positive-definite) piece of the transport coefficients and the parallel transport of a vector is made by the affine connection constructed in such a way that the Universal Criterion of Evolution (UCE) is automatically satisfied. We recall that the UCE is valid for systems out of equilibrium and subject to time-independent boundary conditions [1].…”

“…Constraint A) allows introducing the so called Space of the Thermodynamic Forces (or, simply, the Thermodynamic Space) in the following manner (see Fig. (1) and Ref. [1]). The coordinates of the Thermodynamic Space are the thermodynamic forces, the metric is identified with the (positive-definite) piece of the transport coefficients and the parallel transport of a vector is made by the affine connection constructed in such a way that the Universal Criterion of Evolution (UCE) is automatically satisfied.…”

“…[1] is to determine the nonlinear flux-force relations such that A) respect the thermodynamic theorems for systems far from equilibrium; B) are covariant under the Thermodynamic Coordinate Transformations (TCT) i.e., the closure relations should be covariant under the transformations of the thermodynamic forces leaving unaltered both the entropy production, σ, and the Glansdorff-Prigogine dissipative quantity, P [for the definition of P , see the forthcoming Eq. (3)].…”