Non-equilibrium thermodynamics
[1] Nevertheless, some natural systems and processes remain beyond the scope of equilibrium thermodynamic methods due to the existence of non variational dynamics, where the concept of free energy is lost.Another fundamental and very important difference is the difficulty, in defining entropy at an instant of time in macroscopic terms for systems not in thermodynamic equilibrium.Together with the concept of entropy production, this provides a powerful tool in process optimisation, and provides a theoretical foundation for exergy analysis.[10] In reality, these requirements, although strict, have been shown to be fulfilled even under extreme conditions, such as during phase transitions, at reacting interfaces, and in plasma droplets surrounded by ambient air.[14][15] Some concepts of particular importance for non-equilibrium thermodynamics include time rate of dissipation of energy (Rayleigh 1873,[16] Onsager 1931,[17] also[10][18]), time rate of entropy production (Onsager 1931),[17] thermodynamic fields,[19][20][21] dissipative structure,[5] and non-linear dynamical structure.[18] One problem of interest is the thermodynamic study of non-equilibrium steady states, in which entropy production and some flows are non-zero, but there is no time variation of physical variables.But, for example, atmospheric physics is concerned with large amounts of matter, occupying cubic kilometers, that, taken as a whole, are not within the range of laboratory quantities; then thermal radiation cannot be ignored.Even within the thought-frame of classical irreversible thermodynamics, care[18] is needed in choosing the independent variables[29] for systems.In the classical irreversible thermodynamic approach, there is allowed spatial variation from infinitesimal volume element to adjacent infinitesimal volume element, but it is assumed that the global entropy of the system can be found by simple spatial integration of the local entropy density.This approach assumes spatial and temporal continuity and even differentiability of locally defined intensive variables such as temperature and internal energy density.Thus time comes into the picture more deeply than for time-dependent local equilibrium thermodynamics with memoryless materials, but fluxes are not independent variables of state.There are many examples of stationary non-equilibrium systems, some very simple, like a system confined between two thermostats at different temperatures or the ordinary Couette flow, a fluid enclosed between two flat walls moving in opposite directions and defining non-equilibrium conditions at the walls.Laser action is also a non-equilibrium process, but it depends on departure from local thermodynamic equilibrium and is thus beyond the scope of classical irreversible thermodynamics; here a strong temperature difference is maintained between two molecular degrees of freedom (with molecular laser, vibrational and rotational molecular motion), the requirement for two component 'temperatures' in the one small region of space, precluding local thermodynamic equilibrium, which demands that only one temperature be needed.If the only extensive quantity that is allowed to fluctuate is the internal energy, all the other ones being kept strictly constant, the temperature of the system is measurable and meaningful.If the stationary state of the process is stable, then the unreproducible fluctuations involve local transient decreases of entropy.Fluctuations about stable stationary states are extremely small except near critical points (Kondepudi and Prigogine 1998, page 323).Local thermodynamic equilibrium of matter[10][5][30][31][32] (see also Keizer (1987)[33] 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.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 above equation is valid for small deviations from equilibrium; The dynamics of internal variables in general case is considered by Pokrovskii.In section 8 of the third chapter of his book,[49] Prigogine has specified three contributions to the variation of entropy of the considered system at the given volume and constant temperatureThe middle term in (1) depicts energy dissipation (entropy production) due to the relaxation of internal variablesThe article on Onsager reciprocal relations considers the stable near-steady thermodynamically non-equilibrium regime, which has dynamics linear in the forces and flux densities.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.Nicolis (1999)[52] 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 (2008)[53] 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 1931[17] origination of this subject.There is good experimental evidence that heat convection does not obey extremal principles for time rate of entropy production.[54] Theoretical analysis shows that chemical reactions do not obey extremal principles for the second differential of time rate of entropy production.[58] Also, ideas from non-equilibrium thermodynamics and the informatic theory of entropy have been adapted to describe general economic systems.