Non-equilibrium thermodynamics

A profound difference separates equilibrium from non-equilibrium thermodynamics. Equilibrium thermodynamics ignores the time-courses of physical processes. In contrast, non-equilibrium thermodynamics attempts to describe their time-courses in continuous detail.

Equilibrium thermodynamics restricts its considerations to processes that have initial and final states of thermodynamic equilibrium; the time-courses of processes are deliberately ignored. Consequently, equilibrium thermodynamics allows processes that pass through states far from thermodynamic equilibrium, that cannot be described even by the variables admitted for non-equilibrium thermodynamics,[5] such as time rates of change of temperature and pressure.[6] For example, in equilibrium thermodynamics, a process is allowed to include even a violent explosion that cannot be described by non-equilibrium thermodynamics.[5] Equilibrium thermodynamics does, however, for theoretical development, use the idealized concept of the "quasi-static process". A quasi-static process is a conceptual (timeless and physically impossible) smooth mathematical passage along a continuous path of states of thermodynamic equilibrium.[7] It is an exercise in differential geometry rather than a process that could occur in actuality.

Non-equilibrium thermodynamics, on the other hand, attempting to describe continuous time-courses, needs its state variables to have a very close connection with those of equilibrium thermodynamics.[8] This profoundly restricts the scope of non-equilibrium thermodynamics, and places heavy demands on its conceptual framework.

The suitable relationship that defines non-equilibrium thermodynamic state variables is as follows. On occasions when the system happens to be in states that are sufficiently close to thermodynamic equilibrium, non-equilibrium state variables are such that they can be measured locally with sufficient accuracy by the same techniques as are used to measure thermodynamic state variables, or by corresponding time and space derivatives, including fluxes of matter and energy. In general, non-equilibrium thermodynamic systems are spatially and temporally non-uniform, but their non-uniformity still has a sufficient degree of smoothness to support the existence of suitable time and space derivatives of non-equilibrium state variables.

Because of the spatial non-uniformity, non-equilibrium state variables that correspond to extensive thermodynamic state variables have to be defined as spatial densities of the corresponding extensive equilibrium state variables. On occasions when the system is sufficiently close to thermodynamic equilibrium, intensive non-equilibrium state variables, for example temperature and pressure, correspond closely with equilibrium state variables. It is necessary that measuring probes be small enough, and rapidly enough responding, to capture relevant non-uniformity. Further, the non-equilibrium state variables are required to be mathematically functionally related to one another in ways that suitably resemble corresponding relations between equilibrium thermodynamic state variables.[9] In reality, these requirements are very demanding, and it may be difficult or practically, or even theoretically, impossible to satisfy them. This is part of why non-equilibrium thermodynamics is a work in progress.

According to Wildt[19] (see also Essex[20][21][22]), current versions of non-equilibrium thermodynamics ignore radiant heat; they can do so because they refer to laboratory quantities of matter under laboratory conditions with temperatures well below those of stars. At laboratory temperatures, in laboratory quantities of matter, thermal radiation is weak and can be practically nearly ignored. 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.

The terms 'classical irreversible thermodynamics'[4] and 'local equilibrium thermodynamics' are sometimes used to refer to a version of non-equilibrium thermodynamics that demands certain simplifying assumptions, as follows. The assumptions have the effect of making each very small volume element of the system effectively homogeneous, or well-mixed, or without an effective spatial structure, and without kinetic energy of bulk flow or of diffusive flux. Even within the thought-frame of classical irreversible thermodynamics, care[12] is needed in choosing the independent variables[23] for systems. In some writings, it is assumed that the intensive variables of equilibrium thermodynamics are sufficient as the independent variables for the task (such variables are considered to have no 'memory', and do not show hysteresis); in particular, local flow intensive variables are not admitted as independent variables; local flows are considered as dependent on quasi-static local intensive variables.

Also it is assumed that the local entropy density is the same function of the other local intensive variables as in equilibrium; this is called the local thermodynamic equilibrium assumption[9][12][16][17][24][25][26][27] (see also Keizer (1987)[28]). Radiation is ignored because it is transfer of energy between regions, which can be remote from one another. In the classical irreversible thermodynamic approach, there is allowed very small spatial variation, from very small volume element to adjacent very small 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 means that spatial structure cannot contribute as it properly should to the global entropy assessment for the system. This approach assumes spatial and temporal continuity and even differentiability of locally defined intensive variables such as temperature and internal energy density. All of these are very stringent demands. Consequently, this approach can deal with only a very limited range of phenomena. This approach is nevertheless valuable because it can deal well with some macroscopically observable phenomena.

In other writings, local flow variables are considered; these might be considered as classical by analogy with the time-invariant long-term time-averages of flows produced by endlessly repeated cyclic processes; examples with flows are in the thermoelectric phenomena known as the Seebeck and the Peltier effects, considered by Kelvin in the nineteenth century and by Lars Onsager in the twentieth.[24][29] These effects occur at metal junctions, which were originally effectively treated as two-dimensional surfaces, with no spatial volume, and no spatial variation.

A further extension of local equilibrium thermodynamics is to allow that materials may have "memory", so that their constitutive equations depend not only on present values but also on past values of local equilibrium variables. 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.[30]

Extended irreversible thermodynamics is a branch of non-equilibrium thermodynamics that goes outside the restriction to the local equilibrium hypothesis. The space of state variables is enlarged by including the fluxes of mass, momentum and energy and eventually higher order fluxes. The formalism is well-suited for describing high-frequency processes and small-length scales materials.

Local thermodynamic equilibrium of matter[9][16][25][26][27] (see also Keizer (1987)[28] 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.[32] The longer relaxation time is of the order of magnitude of times taken for the macroscopic dynamical structure of the system to change. 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[32] and other approaches have to be proposed, see for instance Extended irreversible thermodynamics. For example, in the atmosphere, the speed of sound is much greater than the wind speed; this favours the idea of local thermodynamic equilibrium of matter for atmospheric heat transfer studies at altitudes below about 60 km where sound propagates, but not above 100 km, where, because of the paucity of intermolecular collisions, sound does not propagate.

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'.[33] He defined 'local thermodynamic equilibrium' in a 'cell' by requiring that it macroscopically absorb and spontaneously emit radiation as if it were in radiative equilibrium in a cavity at the temperature of the matter of the '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.

To describe deviation of the thermodynamic system from equilibrium, in addition to constitutive variables ${\displaystyle x_{1},x_{2},...,x_{n}}$ that are used to fix the equilibrium state, as was described above, a set of variables ${\displaystyle \xi _{1},\xi _{2},\ldots }$ that are called internal variables have been introduced. 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 tending to disappear; the local law of disappearing can be written as relaxation equation for each internal variable

${\displaystyle {\frac {d\xi _{i}}{dt}}=-{\frac {1}{\tau _{i}}}\,\left(\xi _{i}-\xi _{i}^{(0)}\right),\quad i=1,\,2,\ldots ,}$

(1)

where ${\displaystyle \tau _{i}=\tau _{i}(T,x_{1},x_{2},\ldots ,x_{n})}$ is a relaxation time of a corresponding variables. It is convenient to consider the initial value ${\displaystyle \xi _{i}^{0}}$ are equal to zero. The above equation is valid for small deviations from equilibrium; The dynamics of internal variables in general case is considered by Pokrovskii.[42]

Entropy of the system in non-equilibrium is a function of the total set of variables

${\displaystyle S=S(T,x_{1},x_{2},,x_{n};\xi _{1},\xi _{2},\ldots )}$

(1)

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 section 8 of the third chapter of his book,[43] Prigogine has specified three contributions to the variation of entropy of the considered system at the given volume and constant temperature ${\displaystyle T}$ . The increment of entropy ${\displaystyle S}$ can be calculated according to the formula

${\displaystyle T\,dS=\Delta Q-\sum _{j}\,\Xi _{j}\,\Delta \xi _{j}+\sum _{\alpha =1}^{k}\,\eta _{\alpha }\,\Delta N_{\alpha }.}$

(1)

The first term on the right hand side of the equation presents a stream of thermal energy into the system; the last term—a part of a stream of energy ${\displaystyle h_{\alpha }}$ coming into the system with the stream of particles of substances ${\displaystyle \Delta N_{\alpha }}$ that can be positive or negative, ${\displaystyle \eta _{\alpha }=h_{\alpha }-\mu _{\alpha }}$, where ${\displaystyle \mu _{\alpha }}$ is chemical potential of substance ${\displaystyle \alpha }$. The middle term in (1) depicts energy dissipation (entropy production) due to the relaxation of internal variables ${\displaystyle \xi _{j}}$. 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 theory can be generalised,[44][42] to consider any deviation from the equilibrium state as an internal variable, so that we consider the set of internal variables ${\displaystyle \xi _{j}}$ in equation (1) to consist of the quantities defining not only degrees of completeness of all chemical reactions occurring in the system, but also the structure of the system, gradients of temperature, difference of concentrations of substances and so on.

Following Section III of Rayleigh (1873),[10] Onsager (1931, I)[11] showed that in the regime where both the flows (${\displaystyle J_{i}}$) are small and the thermodynamic forces (${\displaystyle F_{i}}$) vary slowly, the rate of creation of entropy ${\displaystyle (\sigma )}$ is linearly related to the flows:

${\displaystyle \sigma =\sum _{i}J_{i}{\frac {\partial F_{i}}{\partial x_{i}}}}$

and the flows are related to the gradient of the forces, parametrized by a matrix of coefficients conventionally denoted ${\displaystyle L}$:

${\displaystyle J_{i}=\sum _{j}L_{ij}{\frac {\partial F_{j}}{\partial x_{j}}}}$

from which it follows that:

${\displaystyle \sigma =\sum _{i,j}L_{ij}{\frac {\partial F_{i}}{\partial x_{i}}}{\frac {\partial F_{j}}{\partial x_{j}}}}$

The second law of thermodynamics requires that the matrix ${\displaystyle L}$ be positive definite. Statistical mechanics considerations involving microscopic reversibility of dynamics imply that the matrix ${\displaystyle L}$ is symmetric. 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.[42]