Entropy production

The rate of entropy production per unit volume, a, by the reaction as s/s = 

It might be thought that some principle concerning entropy production, such as the minimum principle invoked in irreversible thermodynamics, might distinguish between equivalent sets of reactions, but the next theorem shows that this is not so. 

Theorem 16. If 5 r and 0lf = y r are two equivalent sets of reactions over the same species, their rates of entropy production are the same. 

In a set of coupled reactions %, the signs of all the terms in the sum far need not be the same although the sum, a, will be positive. A particular reaction can be called positive or negative according as its contribution to a is positive or negative. Hooyman [5] has defended the use of this terminology in the context of Rem. 9 above.  

If the foregoing observations are indeed prolegomena, it is desirable to indicate how far they are related to what has been said elsewhere and what may be said after them. 

Let us reconsider the Second Law of Thermodynamics (cf. Kondepudi and Prigogine 1998). We partition the entropy change into dS due to the exchange of energy and/or material with the surroundings and due to the irreversible process in the system itself i.e. 

The result (D.57) suggests (whether it is correct or not) that the system, even if it is isolated, involves an internal change. Considering the equilibrium state of an isolated system in which no internal heat source d Q exists, we have 

in the irreversible process approaching the equilibrium state, the entropy is increasing dS 0), while at the equilibrium state, it reaches a maximum. Note that for a closed or open system entropy can decrease in general if some energy flux is applied to the system. 

Combining the First Law of Thermodynamics (3.40) and the Second Law of Thermodynamics discussed above, we can summarize the laws governing nonequilibrium thermodynamics as follows  

If Stokes power formula is satisfied (cf. Sect. 3.2.4), the above formulae can then be modified as follows  

Consider a flat interface, without internal properties, separating two pure metals. The entropy production at the interface per vmit area and time is written as  

This formulation enables us to directly obtain the entropy production at the interface without performing a passage to the boundary on an elementary volume surroimding the interface. 

Z can be deconstructed into a symmetrical tensor and a antisymmetrical tensor Z Z = Z +Z. The tensor Z is associated with the axial vector Z. The Onsager symmetry relations are written thus, in the presence of a magnetic field Z (H) = Z, (-H). From this, we can deduce that Z(H) and 

Z (H)are, respectively, an odd-numbered and even-numbered fimction of the field H. 

For weak magnetic fields, a Taylor expansion can be written  

Due to the local nature of the fluctuation approach, we did not pay much attention to the origin of the currents. We want to be more precise in this respect. Therefore we separate the total entropy change, dS, into two distinctly different contributions, i.e. 

While di S can never be negative, deS does not have a definite sign and can be positive or negative. 

As an example consider a closed system at constant temperature. For a reversible process we have learned that 

Including irreversible processes inside the system this becomes 

Below we shall show that this relation remains valid outside the linear regime. Here we briefly mention a consequence of this for the Onsager coefficients. Inserting Eq. (7.8) into Eq. (7.60) yields 

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From ( A3.2.12). it is seen that the rate of entropy production is given by... 

Consider an isotropic fluid in which viscous phenomena are neglected. Concentrations and temperature are non-unifonn in this system. The rate of entropy production may be written... 

Chemical reactions may be added to the situation giving an entropy production of... 

The lack of correlation between the flucUiating stress tensor and the flucUiating heat flux in the third expression is an example of the Curie principle for the fluctuations. These equations for flucUiating hydrodynamics are arrived at by a procedure very similar to that exliibited in the preceding section for difllisioii. A crucial ingredient is the equation for entropy production in a fluid... 

In other words, if we look at any phase-space volume element, the rate of incoming state points should equal the rate of outflow. This requires that be a fiinction of the constants of the motion, and especially Q=Q i). Equilibrium also implies d(/)/dt = 0 for any /. The extension of the above equations to nonequilibriiim ensembles requires a consideration of entropy production, the method of controlling energy dissipation (diennostatting) and the consequent non-Liouville nature of the time evolution [35]. 

When a process is completely reversible, the equahty holds, and the lost work is zero. For irreversible processes the inequality holds, and the lost work, that is, the energy that becomes unavailable for work, is positive. The engineering significance of this result is clear The greater the irreversibility of a process, the greater the rate of entropy production and the greater the amount of energy that becomes unavailable for work. Thus, every irreversibility carries with it a price. 

The entropy production terms in (7.4) and (7.5) come from plastic deforma-tional heating. Following Wallace [15]... 

Units of Energy 209. The First Law of Thermodynamics 210, Entropy Production Flow Systems 214. Application of (he Second Law 216. Summary of Thermodynamic Equations 223. 

In Equation 2-114, is the rate of entropy production within the control volume symbols with dots refer to the time rate of change of the quantity in question. The second law requires that the rate of entropy production be positive. 

Because the rate of entropy production is negative, the device violates the second law and is therefore impossible. Note that the device would be theoretically possible if the final pressure were specified as 400 psia or less by the inventor. That is, at = 400 psia, T = 40°F, h = 290 Btu/lb, and S, = 1.25 Btu/lb °R, the entropy production rate would be... 

There are three different approaches to a thermodynamic theory of continuum that can be distinguished. These approaches differ from each other by the fundamental postulates on which the theory is based. All of them are characterized by the same fundamental requirement that the results should be obtained without having recourse to statistical or kinetic theories. None of these approaches is concerned with the atomic structure of the material. Therefore, they represent a pure phenomenological approach. The principal postulates of the first approach, usually called the classical thermodynamics of irreversible processes, are documented. The principle of local state is assumed to be valid. The equation of entropy balance is assumed to involve a term expressing the entropy production which can be represented as a sum of products of fluxes and forces. This term is zero for a state of equilibrium and positive for an irreversible process. The fluxes are function of forces, not necessarily linear. However, the reciprocity relations concern only coefficients of the linear terms of the series expansions. Using methods of this approach, a thermodynamic description of elastic, rheologic and plastic materials was obtained. 

Entropy, Entropy Production, Auto Catalysis and Oscillating Reactions 69... 

A reaction at steady state is not in equilibrium. Nor is it a closed system, as it is continuously fed by fresh reactants, which keep the entropy lower than it would be at equilibrium. In this case the deviation from equilibrium is described by the rate of entropy increase, dS/dt, also referred to as entropy production. It can be shown that a reaction at steady state possesses a minimum rate of entropy production, and, when perturbed, it will return to this state, which is dictated by the rate at which reactants are fed to the system [R.A. van Santen and J.W. Niemantsverdriet, Chemical Kinetics and Catalysis (1995), Plenum, New York]. Hence, steady states settle for the smallest deviation from equilibrium possible under the given conditions. Steady state reactions in industry satisfy these conditions and are operated in a regime where linear non-equilibrium thermodynamics holds. Nonlinear non-equilibrium thermodynamics, however, represents a regime where explosions and uncontrolled oscillations may arise. Obviously, industry wants to avoid such situations ... 

According to irreversible thermodynamics, the entropy production per unit volume S for an isothermal system can be written... 

Self-organization seems to be counterintuitive, since the order that is generated challenges the paradigm of increasing disorder based on the second law of thermodynamics. In statistical thermodynamics, entropy is the number of possible microstates for a macroscopic state. Since, in an ordered state, the number of possible microstates is smaller than for a more disordered state, it follows that a self-organized system has a lower entropy. However, the two need not contradict each other it is possible to reduce the entropy in a part of a system while it increases in another. A few of the system s macroscopic degrees of freedom can become more ordered at the expense of microscopic disorder. This is valid even for isolated, closed systems. Eurthermore, in an open system, the entropy production can be transferred to the environment, so that here even the overall entropy in the entire system can be reduced. 

On a related point, there have been other variational principles enunciated as a basis for nonequilibrium thermodynamics. Hashitsume [47], Gyarmati [48, 49], and Bochkov and Kuzovlev [50] all assert that in the steady state the rate of first entropy production is an extremum, and all invoke a function identical to that underlying the Onsager-Machlup functional [32]. As mentioned earlier, Prigogine [11] (and workers in the broader sciences) [13-18] variously asserts that the rate of first entropy production is a maximum or a minimum and invokes the same two functions for the optimum rate of first entropy production that were used by Onsager and Machlup [32] (see Section HE). 

Evans and Baranyai [51, 52] have explored what they describe as a nonlinear generalization of Prigogine s principle of minimum entropy production. In their theory the rate of (first) entropy production is equated to the rate of phase space compression. Since phase space is incompressible under Hamilton s equations of motion, which all real systems obey, the compression of phase space that occurs in nonequilibrium molecular dynamics (NEMD) simulations is purely an artifact of the non-Hamiltonian equations of motion that arise in implementing the Evans-Hoover thermostat [53, 54]. (See Section VIIIC for a critical discussion of the NEMD method.) While the NEMD method is a valid simulation approach in the linear regime, the phase space compression induced by the thermostat awaits physical interpretation even if it does turn out to be related to the rate of first entropy production, then the hurdle posed by Question (3) remains to be surmounted. 

A significant question is whether the asymmetric contribution to the transport matrix is zero or nonzero. That is, is there any coupling between the transport of variables of opposite parity The question will recur in the discussion of the rate of entropy production later. The earlier analysis cannot decide the issue, since can be zero or nonzero in the earlier results. But some insight can be gained into the possible behavior of the system from the following analysis. 

The asymmetric part of the transport matrix gives zero contribution to the scalar product and so does not contribute to the steady-state rate of first entropy production [7]. This was also observed by Casimir [24] and by Grabert et al. [25], Eq. (17). 

As stressed at the end of the preceding section, there is no proof that the asymmetric part of the transport matrix vanishes. Casimir [24], no doubt motivated by his observation about the rate of entropy production, on p. 348 asserted that the antisymmetric component of the transport matrix had no observable physical consequence and could be set to zero. However, the present results show that the function makes an important and generally nonnegligible contribution to the dynamics of the steady state even if it does not contribute to the rate of first entropy production. 

The optimum rate of first entropy production may also be written in terms of the fluxes,... 

The first question posed in the introduction, Question (3), makes the point that one cannot have a theory for the nonequilibrium state based on the first entropy or its rate of production. It ought to be clear that the steady state, which corresponds to the most likely flux, x(x, i), gives neither the maximum nor the minimum of Eq. (61), the rate of first entropy production. From that equation, the extreme rates of first entropy production occur when x = oo. Theories that invoke the Principle of Minimum Dissipation, [10-12, 32] or the Principle of... 

The constrained rate of first entropy production for the total system is... 

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