György Steinbrecher

Symmetry group and group representations associated with the thermodynamic covariance principle

The main objective of this work [previously appeared in literature, the thermodynamical field theory (TFT)] is to determine the nonlinear closure equations (i.e., the flux-force relations) valid for thermodynamic systems out of Onsagers region. The TFT rests upon the concept of equivalence between thermodynamic systems. More precisely, the equivalent character of two alternative descriptions of a thermodynamic system is ensured if, and only if, the two sets of thermodynamic forces are linked with each other by the so-called thermodynamic coordinate transformations (TCT). In this work, we describe the Lie group and the group representations associated to the TCT. The TCT guarantee the validity of the so-called thermodynamic covariance principle (TCP): The nonlinear closure equations, i.e., the flux-force relations, everywhere and in particular outside the Onsager region, must be covariant under TCT. In other terms, the fundamental laws of thermodynamics should be manifestly covariant under transformations between the admissible thermodynamic forces, i.e., under TCT. The TCP ensures the validity of the fundamental theorems for systems far from equilibrium. The symmetry properties of a physical system are intimately related to the conservation laws characterizing that system. Noethers theorem gives a precise description of this relation. We derive the conserved (thermodynamic) currents and, as an example of calculation, a system out of equilibrium (tokamak plasmas) where the validity of TCP imposed at the level of the kinetic equations is also analyzed.

Family of probability distributions derived from maximal entropy principle with scale invariant restrictions

Using statistical thermodynamics, we derive a general expression of the stationary probability distribution for thermodynamic systems driven out of equilibrium by several thermodynamic forces. The local equilibrium is defined by imposing the minimum entropy production and the maximum entropy principle under the scale invariance restrictions. The obtained probability distribution presents a singularity that has immediate physical interpretation in terms of the intermittency models. The derived reference probability distribution function is interpreted as time and ensemble average of the real physical one. A generic family of stochastic processes describing noise-driven intermittency, where the stationary density distribution coincides exactly with the one resulted from entropy maximization, is presented.

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

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.

Derivation of reference distribution functions for Tokamak-plasmas by statistical thermodynamics

A general approach for deriving the expression of reference distribution functions by statistical thermodynamics is illustrated, and applied to the case of a magnetically confined plasma. The local equilibrium is defined by imposing the minimum entropy production, which applies only to the linear regime near a stationary thermodynamically non-equilibrium state and the maximum entropy principle under the scale invariance restrictions. This procedure may be adopted for a system subject to an arbitrary number of thermodynamic forces, however, for concreteness, we analyze, afterwords, a magnetically confined plasma subject to three thermodynamic forces, and three energy sources: (i) the total Ohmic heat, supplied by the transformer coil; (ii) the energy supplied by neutral beam injection (NBI); and (iii) the RF energy supplied by ion cyclotron resonant heating (ICRH) system which heats the minority population. In this limit case, we show that the derived expression of the distribution function is more general than that one, which is currently used for fitting the numerical steady-state solutions obtained by simulating the plasma by gyro-kinetic codes. An application to a simple model of fully ionized plasmas submitted to an external source is discussed. Through kinetic theory, we fixed the values of the free parameters linking them with the external power supplies. The singularity at low energy in the proposed distribution function is related to the intermittency in the turbulent plasma.

Reference distribution functions for magnetically confined plasmas from the minimum entropy production theorem and the MaxEnt principle, subject to the scale-invariant restrictions

We derive the expression of the reference distribution function for magnetically confined plasmas far from the thermodynamic equilibrium. The local equilibrium state is fixed by imposing the minimum entropy production theorem and the maximum entropy (MaxEnt) principle, subject to scale invariance restrictions. After a short time, the plasma reaches a state close to the local equilibrium. This state is referred to as the reference state. The aim of this Letter is to determine the reference distribution function (RDF) when the local equilibrium state is defined by the above mentioned principles. We prove that the RDF is the stationary solution of a generic family of stochastic processes corresponding to an universal Landau-type equation with white parametric noise. As an example of application, we consider a simple, fully ionized, magnetically confined plasmas, with auxiliary Ohmic heating. The free parameters are linked to the transport coefficients of the magnetically confined plasmas, by the kinetic theory.

Category Theoretic Properties of the A. Rényi and C. Tsallis Entropies

The problem of embedding the Tsallis, Renyi and generalized Renyi entropies in the framework of category theory and their axiomatic foundation is studied. To this end, we construct a special category MES related to measured spaces. We prove that both of the Renyi and Tsallis entropies can be imbedded in the formalism of category theory by proving that the same basic partition functional that appears in their definitions, as well as in the associated Lebesgue space norms, has good algebraic compatibility properties. We prove that this functional is both additive and multiplicative with respect to the direct product and the disjoint sum (the coproduct) in the category MES, so it is a natural candidate for the measure of information or uncertainty. We prove that the category MES can be extended to monoidal category, both with respect to the direct product as well as to the coproduct. The basic axioms of the original Renyi entropy theory are generalized and reformulated in the framework of category MES and we prove that these axioms foresee the existence of an universal exponent having the same values for all the objects of the category MES. In addition, this universal exponent is the parameter, which appears in the definition of the Tsallis and Renyi entropies. It is proved that in a similar manner, the partition functional that appears in the definition of the Generalized Renyi entropy is a multiplicative functional with respect to direct product and additive with respect to the disjoint sum, but its symmetry group is reduced compared to the case of classical Renyi entropy.

Stationary distribution functions for ohmic Tokamak-plasmas in the weak-collisional transport regime by MaxEnt principle

In previous works, we derived stationary density distribution functions (DDF) where the local equilibrium is determined by imposing the maximum entropy (MaxEnt) principle, under the scale invariance restrictions, and the minimum entropy production theorem. In this paper we demonstrate that it is possible to reobtain these DDF solely from the MaxEnt principle subject to suitable scale invariant restrictions in all the variables. For the sake of concreteness, we analyse the example of ohmic, fully ionized, tokamak-plasmas, in the weak-collisional transport regime. In this case we show that it is possible to reinterpret the stationary distribution function in terms of the Prigogine distribution function where the logarithm of the DDF is directly linked to the entropy production of the plasma. This leads to the suggestive idea that also the stationary neoclassical distribution functions, for magnetically confined plasmas in the collisional transport regimes, may be derived solely by the MaxEnt principle.

A note on the application of the Prigogine theorem to rotation of tokamak-plasmas in absence of external torques.

Rotation of tokamak-plasmas, not at the mechanical equilibrium, is investigated using the Prigogine thermodynamic theorem. This theorem establishes that, for systems confined in rectangular boxes, the global motion of the system with barycentric velocity does not contribute to dissipation. This result, suitably applied to toroidally confined plasmas, suggests that the global barycentric rotations of the plasma, in the toroidal and poloidal directions, are pure reversible processes. In case of negligible viscosity and by supposing the validity of the balance equation for the internal forces, we show that the plasma, even not in the mechanical equilibrium, may freely rotate in the toroidal direction with an angular frequency, which may be higher than the neoclassical estimation. In addition, its toroidal rotation may cause the plasma to rotate globally in the poloidal direction at a speed faster than the expression found by the neoclassical theory. The eventual configuration is attained when the toroidal and poloidal angular frequencies reaches the values that minimize dissipation. The physical interpretation able to explain the reason why some layers of plasma may freely rotate in one direction while, at the same time, others may freely rotate in the opposite direction, is also provided. Invariance properties, herein studied, suggest that the dynamic phase equation might be of the second order in time. We then conclude that a deep and exhaustive study of the invariance properties of the dynamical and thermodynamic equations is the most correct and appropriate way for understanding the triggering mechanism leading to intrinsic plasma-rotation in toroidal magnetic configurations.