When boundaries impose to the system different local conditions, e. The latter are the thermodynamic forces driving fluxes of extensive properties through the system. It may be shown that the Legendre transformation changes the maximum condition of the entropy valid at equilibrium in a minimum condition of the extended Massieu function for stationary states, no matter whether at equilibrium or not. In thermodynamics one is often interested in a stationary state of a process, allowing that the stationary state include the occurrence of unpredictable and experimentally unreproducible fluctuations in the state of the system.
The fluctuations are due to the system's internal sub-processes and to exchange of matter or energy with the system's surroundings that create the constraints that define the process. If the stationary state of the process is stable, then the unreproducible fluctuations involve local transient decreases of entropy. The reproducible response of the system is then to increase the entropy back to its maximum by irreversible processes: the fluctuation cannot be reproduced with a significant level of probability.
Fluctuations about stable stationary states are extremely small except near critical points Kondepudi and Prigogine , page There are theorems about the irreversible dissipation of fluctuations. Here 'local' means local with respect to the abstract space of thermodynamic coordinates of state of the system. If the stationary state is unstable, then any fluctuation will almost surely trigger the virtually explosive departure of the system from the unstable stationary state.
This can be accompanied by increased export of entropy. The scope of present-day non-equilibrium thermodynamics does not cover all physical processes. A condition for the validity of many studies in non-equilibrium thermodynamics of matter is that they deal with what is known as local thermodynamic equilibrium. Local thermodynamic equilibrium of matter      see also Keizer  means that conceptually, for study and analysis, the system can be spatially and temporally divided into 'cells' or 'micro-phases' of small infinitesimal size, in which classical thermodynamical equilibrium conditions for matter are fulfilled to good approximation.
These conditions are unfulfilled, for example, in very rarefied gases, in which molecular collisions are infrequent; and in the boundary layers of a star, where radiation is passing energy to space; and for interacting fermions at very low temperature, where dissipative processes become ineffective.
When these 'cells' are defined, one admits that matter and energy may pass freely between contiguous 'cells', slowly enough to leave the 'cells' in their respective individual local thermodynamic equilibria with respect to intensive variables. One can think here of two 'relaxation times' separated by order of magnitude.
The shorter is of the order of magnitude of times taken for a single 'cell' to reach local thermodynamic equilibrium. If these two relaxation times are not well separated, then the classical non-equilibrium thermodynamical concept of local thermodynamic equilibrium loses its meaning  and other approaches have to be proposed, see for instance Extended irreversible thermodynamics.
Edward A. Milne , thinking about stars, gave a definition of 'local thermodynamic equilibrium' in terms of the thermal radiation of the matter in each small local 'cell'. Then it strictly obeys Kirchhoff's law of equality of radiative emissivity and absorptivity, with a black body source function. The key to local thermodynamic equilibrium here is that the rate of collisions of ponderable matter particles such as molecules should far exceed the rates of creation and annihilation of photons.
It is pointed out by W. Grandy Jr,     , that entropy, though it may be defined for a non-equilibrium system, is when strictly considered, only a macroscopic quantity that refers to the whole system, and is not a dynamical variable and in general does not act as a local potential that describes local physical forces.
Under special circumstances, however, one can metaphorically think as if the thermal variables behaved like local physical forces. The approximation that constitutes classical irreversible thermodynamics is built on this metaphoric thinking. This point of view shares many points in common with the concept and the use of entropy in continuum thermomechanics,     which evolved completely independently of statistical mechanics and maximum-entropy principles.
To describe deviation of the thermodynamic system from equilibrium, in addition to constitutive variables x 1 , x 2 ,.
Non-Equilibrium Entropy and Irreversibility
The equilibrium state is considered to be stable and the main property of the internal variables, as measures of non-equilibrium of the system, is their trending to disappear; the local law of disappearing can be written as relaxation equation for each internal variable. The above equation is valid for small deviations from equilibrium; The dynamics of internal variables in general case is considered by Pokrovskii.
The essential contribution to the thermodynamics of the non-equilibrium systems was brought by Prigogine , when he and his collaborators investigated the systems of chemically reacting substances.
The stationary states of such systems exists due to exchange both particles and energy with the environment. In the case of chemically reacting substances, which was investigated by Prigogine, the internal variables appear to be measures of incompleteness of chemical reactions, that is measures of how much the considered system with chemical reactions is out of equilibrium.
The fundamental relation of classical equilibrium thermodynamics . Following Onsager ,I ,  let us extend our considerations to thermodynamically non-equilibrium systems. For classical non-equilibrium studies, we will consider some new locally defined intensive macroscopic variables. We can, under suitable conditions, derive these new variables by locally defining the gradients and flux densities of the basic locally defined macroscopic quantities.
Such locally defined gradients of intensive macroscopic variables are called 'thermodynamic forces'. They 'drive' flux densities, perhaps misleadingly often called 'fluxes', which are dual to the forces. These quantities are defined in the article on Onsager reciprocal relations. Establishing the relation between such forces and flux densities is a problem in statistical mechanics. The article on Onsager reciprocal relations considers the stable near-steady thermodynamically non-equilibrium regime, which has dynamics linear in the forces and flux densities.
In stationary conditions, such forces and associated flux densities are by definition time invariant, as also are the system's locally defined entropy and rate of entropy production. Notably, according to Ilya Prigogine and others, when an open system is in conditions that allow it to reach a stable stationary thermodynamically non-equilibrium state, it organizes itself so as to minimize total entropy production defined locally.
Non-equilibrium thermodynamics - Wikipedia
This is considered further below. One wants to take the analysis to the further stage of describing the behaviour of surface and volume integrals of non-stationary local quantities; these integrals are macroscopic fluxes and production rates. In general the dynamics of these integrals are not adequately described by linear equations, though in special cases they can be so described. This fact is called the Onsager reciprocal relations.
The generalization of the above equations for the rate of creation of entropy was given by Pokrovskii.
13.4: Entropy Changes in Reversible Processes
Until recently, prospects for useful extremal principles in this area have seemed clouded. Nicolis  concludes that one model of atmospheric dynamics has an attractor which is not a regime of maximum or minimum dissipation; she says this seems to rule out the existence of a global organizing principle, and comments that this is to some extent disappointing; she also points to the difficulty of finding a thermodynamically consistent form of entropy production.
Another top expert offers an extensive discussion of the possibilities for principles of extrema of entropy production and of dissipation of energy: Chapter 12 of Grandy  is very cautious, and finds difficulty in defining the 'rate of internal entropy production' in many cases, and finds that sometimes for the prediction of the course of a process, an extremum of the quantity called the rate of dissipation of energy may be more useful than that of the rate of entropy production; this quantity appeared in Onsager's  origination of this subject.
Other writers have also felt that prospects for general global extremal principles are clouded. A recent proposal may perhaps by-pass those clouded prospects. It is also used to give a description of the dynamics of nanoparticles, which can be out of equilibrium in systems where catalysis and electrochemical conversion is involved. From Wikipedia, the free encyclopedia. This article may need to be rewritten to comply with Wikipedia's quality standards. You can help. The discussion page may contain suggestions.
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