Entropy production time variation

Figure 1.18 Kinetics of the reaction A B. Concentrations, rates of entropy production, and variation of the entropy as a function of time. Upper row weak coupling (A = 1,...

It should be clear that the most likely or physical rate of first entropy production is neither minimal nor maximal these would correspond to values of the heat flux of oc. The conventional first entropy does not provide any variational principle for heat flow, or for nonequilibrium dynamics more generally. This is consistent with the introductory remarks about the second law of equilibrium thermodynamics, Eq. (1), namely, that this law and the first entropy that in invokes are independent of time. In the literature one finds claims for both extreme theorems some claim that the rate of entropy production is... 

In an important paper (TNC.l), they offered for the first time an extension of nonequilibrium thermodynamics to nonlinear transport laws. As could be expected, the situation was by no means as simple as in the linear domain. The authors were hoping to find a variational principle generalizing the principle of minimum entropy production. It soon became obvious that such a principle cannot exist in the nonlinear domain. They succeeded, however, to derive a half-principle They decomposed the differential of the entropy production (1) as follows ... 

In this case dxP/dt cannot be reduced to an exact differential and the criterion simply expresses that in the course of time the variation of the thermodynamic forces tends to diminish the entropy production. 

Equation (3.314) shows the volumetric rate of entropy production. Both the flows and the forces may change with time, while they remain constant at the system boundaries at stationary state only. The time variation of P is... 

The time variation of the rate of entropy production with respect to the variation of the thermodynamic force (dJJ) is... 

Example 8.11 Time variation of entropy production in simultaneous chemical reactions Consider two simultaneous reactions and derive relations for the time variation of the entropy production. Assuming that linear laws hold for a two-reaction system, we have... 

We also assume that the phenomenological coefficients are constant. The time variation of the entropy production is... 

Because terms BA/Be, belong to a definite quadratic form, the time variation of entropy production can only decrease with time. 

Example 8.9 Time variation of entropy production in simultaneous chemical reactions... 

The direct representation of the Fourier equation is not possible in a Formal Graph because heat is energy (of the thermal variety) and not a state variable, and also because the heat flux does not correspond to a variable that can be represented in a Formal Graph. The solution consists of taking recourse to state variables of the thermal variety that are the entropy S (basic quantity), the temperature T (effort), and the entropicflow fs (also called entropy production rate). The variation of entropy is linked to the variation of heat according to the energy-per-entity times the variation of entity number... 

Figure 9.2 Racemization of enantiomers as an example of a chemical reaction. The associated entropy production and the time variation of A are shown in (a) and (b). State functions A and G as functions of are shown in (c) and (d)...

Time Variation of Entropy Production and the Stability of Stationary States... 

Let us look at the time variation of the entropy production due to chemical reactions in an open system in the linear regime. As before, we assume homogeneity and unit volume. The entropy production is ... 

Figure 17.5 The time variation of the entropy production P = diS/dt— Y i FkJk for equilibrium and near equilibrium states, (a) For a lluctpation from the equilibrium state, the initial nonzero value of P decreases to its equilibrium value of zero, (b) In the Unear regime, a fluctuation from a nonequilibrium steady state can only increase the value of P above the stationary value Pgu irreversible processes drive P back to its minimum value Pst...

Using equations (17.1.28) and (17.1.29) obtain the time variation of I(t) and Q(t) in a real capacitor and a real inductance. Using these expressions in (17.1.25) and (17.1.26) obtain the entropy production at any time t in these circuit elements with initial current Iq and initial charge Qo-... 

In nonequihbiium thermodynamics, it is assumed that the rate of entropy production inside the system due to the time variation of the driving forces Xy(r, t) decreases in the course of time, i.e.. 

The contribution to the entropy production due to the time variation of the driving forces Xj t) is assumed to conform to the following version of the Glansdorlf-Prigogine evolution theorem ... 

This last point suggests an alternative interpretation of the transport coefficient as the one corresponding to the correlation function evaluated at the point of maximum flux. The second entropy is maximized to find the optimum flux at each x. Since the maximum value of the second entropy is the first entropy sM(x), which is independent of x, one has no further variational principle to invoke based on the second entropy. However, one may assert that the optimal time interval is the one that maximizes the rate of production of the otherwise unconstrained first entropy, 5(x (x, x), x) = x (x,x) Xs(x), since the latter is a function of the optimized fluxes that depend on x. 

---