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.

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Starting from the geometrical interpretation of the Rényi entropy, we introduce further extensive generalizations and study their properties. In particular, we found the probability distribution function obtained by the MaxEnt principle with generalized entropies. We prove that for a large class of dynamical systems subject to random perturbations, including particle transport in random media, these entropies play the role of Liapunov functionals. Some physical examples, which can be treated by the generalized Rényi entropies are also illustrated.

Thermodynamic field theory (TFT) allows us to deal with thermodynamic systems submitted even to strong nonequilibrium conditions. The theory formulated in this article enables us to find field equations whose solutions give the generalized relations between the thermodynamic forces and their conjugate flows. It will be shown that evolution of thermodynamic systems is well described in Weyl's space. In the particular case in which the thermodynamic forces and conjugate flows are linked only through a symmetric tensor (the metric tensor), the resulting geometry is Riemannian geometry. When Weyl's space is even-dimensional, the thermodynamic space introduced in this study becomes a differentiable symplectic manifold. As an example of application, the thermoelectric effect and the unimolecular triangular chemical reaction are analyzed in great detail. The Field-Körö s-Noyes model shows the theoretical treatment of a more complex chemical example. Theoretical analysis of materials simultaneously submitted to magnetic fields and electric currents can be found in previous articles. In this case, TFT foresees a new effect: the nonlinear Hall effect. The agreement between the theoretical predictions and experiments is discussed.