Revenons a nos moutons and keep in mind Eq. (5.13) of Chapter 2 written for the density of entropy production, s:

If the particles of mth sort under the action of internal force Fm move with the current density Jm, then, in accordance with Ohm's rule, dAirrev = V^mFmJmdt. By substituting it into Eq. (3.1), we get

Formally, the right side of Eq. (3.2) can be written in a bilinear form with (3M + K) items:

3M+K

where XA and J A are the generalised thermodynamic forces and fluxes defined as

Xa = Flm, J a = Jlm (l = 1,2; 3; m = 1,..., M; a = 1, ..., 3M),

Xa = Ak, Ja = vt (k = 1, ...,K; a = 1, ..., 3M + K).

The index l = 1,2, 3 appears here since the forces and fluxes are three-dimensional vectors.

The product sT is a function of dissipation C (see Section 2.5). At thermodynamic equilibrium, all the thermodynamic forces and fluxes simultaneously vanish; therefore, in the vicinity of the equilibrium, the fluxes can be considered as linear functions of forces:

This is the so-called Onsager's linear relation, and the Lap is named Onsager's coefficients. It is obvious that in this case the expression for C = ST can be represented as a quadratic form:

3M+K

From the Second Law, it follows that s $ 0, where s = 0 only for Xa = 0. This means that the matrix IlL^H must be positive definite that, in turn, imposes certain constraints on Onsager's coefficients. For instance, for a, ¡3 = 1.2, these constraints have the form:

Other constraints appear from temporal and spatial symmetries of the system. Since the equations of microphysics are symmetrical with respect to time, then La3 = L^a (Onsager, 1931).

Let us consider some spatial-isotropic medium, where all processes are divided into two classes: scalar (chemical reactions) and vector (particles flows). By virtue of isotropy, scalar causes cannot induce anisotropic effects, and vice versa (Curie-Prigogine's principle). Thus, Onsager's coefficients corresponding to coupling between forces and fluxes, described by tensors of different ranges, are equal to zero.

Let us calculate the variations of entropy production, forces and fluxes at T = const in the vicinity of some steady state (dynamic equilibrium) s(0), Xa0), Ja0':

Since at the dynamic equilibrium, da/dt = 0, then the term within brackets also vanishes. Therefore

3M+K

Since the matrix \\Lap\\ is positive definite then 8a $ 0, 5(di5/dt) = V8a $ 0, where 8a = 8(di5/dt) = 0 only if 8X« = 0. This is the main statement of Prigogine's theorem: the entropy production of linear irreversible process reaches a maximum in a dynamic equilibrium. When the system approaches the equilibrium, the forces and fluxes change in such a way that the entropy production is always decreasing (Prigogine, 1955).

The theorem can be considered as a criterion of the system evolution in a linear region; a criterion for non-linear domain has been formulated by Glansdorff and Prigogine (1971). We shall formulate the criterion for a partial case of chemical reactions, which takes place within a system. Since in this case, d'Airrev/dt = 0, then we have from Eq. (3.1):

By varying the chemical affinities (see Eq. (5.12) in Chapter 2), 8Ak = — vmk8/xk, and substituting these variations into Eq. (3.8), we obtain

V dt dt k1

/ >cl then for isotherm processes we get finally from Eq. (3.9)

dxa 1

Gibbs has shown the matrix Wd^m/d^W to be positive definite, therefore

This is the Glansdorff-Prigogine evolutionary criterion, or Glansdorff-Prigogine's theorem: the change of entropy production caused by variations of forces is either negative or equal to zero. As a partial case of the theorem for linear process, we get the following

form of Prigogine's criterion:

There are a lot of investigations where the rate of entropy change in biological systems is estimated by means of experimental measurement of their heat production. These experiments could be correct if the entropy of reactants does not depend on the degree of completeness of the chemical reactions. In this case, the affinity of process is directly connected with its thermal effect, A = —(dH/dg)T p where the H is enthalpy, and the entropy production must be proportional to the effect, so that dS < _ ifd^

A typical temporal pattern of heat production by the fertilised amphibian egg is shown in Fig. 3.1 (Lurie and Wagensberg, 1979).

Fig. 3.1 is a good illustration to Prigogine's theorem: one can see that the entropy production for a rather large period of time is a monotonous decrease with time tending to a certain positive value.

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