Equation for entropy production

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... 

Step 1. Insert the molecular flux of entropy given by (25-37) into the defining equation for entropy production [i.e., (25-38)] ... 

On combining the balance equation for energy assuming the absence of velocity gradients, the first law of thermodynamics and assuming Gibbs equation for entropy production for the case of local equilibrium, a the entropy production per unit volume per unit time due to the occurrence of irreversible processes in the system is given by... 

Thermodynamic Governing Equations. Derivation of the expression for entropy production arising from mess transfer requires application of the fundamental balance equations. Potential and kinetic energy effects as well as momentum effects are neglected. With these assumptions the governing equations are given as follows ... 

Two important points need to be mentioned. First, as the mole fraction, y, approaches the equilibrium mole fraction, y g, the integrand approaches zero. Thus the point of minimum entropy production coincides with that of minimum reflux. Second, y > y guarantees that the argument of the logarithm cannot be less than unity, which means that Sp > 0. Finally Equation 24 is only applicable to cases wherein the diffusion processes represent the dominant mechanism for entropy production. 

Engineering systems mainly involve a single-phase fluid mixture with n components, subject to fluid friction, heat transfer, mass transfer, and a number of / chemical reactions. A local thermodynamic state of the fluid is specified by two intensive parameters, for example, velocity of the fluid and the chemical composition in terms of component mass fractions wr For a unique description of the system, balance equations must be derived for the mass, momentum, energy, and entropy. The balance equations, considered on a per unit volume basis, can be written in terms of the partial time derivative with an observer at rest, and in terms of the substantial derivative with an observer moving along with the fluid. Later, the balance equations are used in the Gibbs relation to determine the rate of entropy production. The balance equations allow us to clearly identify the importance of the local thermodynamic equilibrium postulate in deriving the relationships for entropy production. 

In fluid d mamics there is no specific use of the transport equation for entropy other than being a physical condition indicating whether a constitutive relation proposed has a sound physical basis or not (nevertheless, this may be a constraint of great importance in many situations). In this connexion we usually think of the second law of thermodynamics as providing an inequality, expressing the observation that irreversible phenomena lead to entropy production. 

The second statement of the third law (which bears Planck s name) is that as the temperature goes to zero, AS goes to zero for any process for which a reversible path could be imagined, provided the reactants and products are perfect crystals. Here, perfect crystals are defined as those which are non-degenerate, that is, they have only a single quantum state in which they can exist at absolute zero. This statement follows rigorously from Boltzmann s equation for entropy,... 

Theories based on non-equilibrium thermodynamics [3-8] have been applied extensively to elucidate the phenomenon of thermo-osmosis. The methodology of nonequilibrium thermodynamics essentially involves the evaluation of entropy production by the application of the laws of conservation of mass and energy and Gibbs equation. Appropriate fluxes and forces are chosen by suitably splitting the expression for entropy production and subsequently, thermodynamic transport equations are written. The theory of thermo-osmosis based on non-equilibrium thermodynamics is discussed below. 

The balance equation for entropy can be derived using the conservation of energy and the balance equation for the concentrations. This gives us an explicit expression for entropy production or— which can be related to irreversible processes such as heat conduction, diffusion and chemical reactions—and the entropy current is- The formal entropy balance equation is... 

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]. 

Strategy Start by writing a balanced chemical equation for the reaction involved. Then use Equation 17.1 in combination with Table 17.1 to calculate the difference in entropy between products and reactants. For (b) note that you are asked to calculate AS° for one gram of methane. 

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. 

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. 

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 internal entropy production this represents the time-related entropy growth generated within the system (djS/df). The internal entropy production is the most important quantity in the thermodynamics of irreversible systems and reaches its maximum when the system is in a stationary state. The equation for the entropy production is then ... 

Even if one of the processes is not chemical but is categorized as a phase change, for example, evaporation, the extended De Donder s equation (Equation 13.17), is known to be valid. Any large magnitude of entropy production rate (diS/dt) due to evaporation might give a correlation such as... 

Diffusion of the electroactive species within the electrode toward or away from the interface with the electrolyte is an irreversible process. The sum of the products of the forces and fluxes corresponds to the entropy production. In order to avoid space charge accumulation, the motion of at least two types of charged species has to be considered for charge compensation. Onsager s equations read in the isothermal case (neglecting energy fluxes)... 

The plan of this chapter is the following. Section II gives a summary of the phenomenology of irreversible processes and set up the stage for the results of nonequilibrium statistical mechanics to follow. In Section III, it is explained that time asymmetry is compatible with microreversibility. In Section IV, the concept of Pollicott-Ruelle resonance is presented and shown to break the time-reversal symmetry in the statistical description of the time evolution of nonequilibrium relaxation toward the state of thermodynamic equilibrium. This concept is applied in Section V to the construction of the hydrodynamic modes of diffusion at the microscopic level of description in the phase space of Newton s equations. This framework allows us to derive ab initio entropy production as shown in Section VI. In Section VII, the concept of Pollicott-Ruelle resonance is also used to obtain the different transport coefficients, as well as the rates of various kinetic processes in the framework of the escape-rate theory. The time asymmetry in the dynamical randomness of nonequilibrium systems and the fluctuation theorem for the currents are presented in Section VIII. Conclusions and perspectives in biology are discussed in Section IX. 

The eigenmode expansion (13.30) for the local entropy production rate can be expressed in terms of usual laboratory variables Rh Rt (13.14a, b) using transformation equations analogous to those of Section 11.6. In the present Abased framework, the expansion of dEt in intensities [cf. (11.89)] becomes... 

The fundamental question in transport theory is Can one describe processes in nonequilibrium systems with the help of (local) thermodynamic functions of state (thermodynamic variables) This question can only be checked experimentally. On an atomic level, statistical mechanics is the appropriate theory. Since the entropy, 5, is the characteristic function for the formulation of equilibria (in a closed system), the deviation, SS, from the equilibrium value, S0, is the function which we need to use for the description of non-equilibria. Since we are interested in processes (i.e., changes in a system over time), the entropy production rate a = SS is the relevant function in irreversible thermodynamics. Irreversible processes involve linear reactions (rates 55) as well as nonlinear ones. We will be mainly concerned with processes that occur near equilibrium and so we can linearize the kinetic equations. The early development of this theory was mainly due to the Norwegian Lars Onsager. Let us regard the entropy S(a,/3,. ..) as a function of the (extensive) state variables a,/ ,. .. .which are either constant (fi,.. .) or can be controlled and measured (a). In terms of the entropy production rate, we have (9a/0f=a)... 

Equations (8.30) and (8.32) constitute 27 relations for the transport coefficients lik. Further relations can be obtained by eliminating the three fluxes j5, j7, and j9 from the entropy production equation (4.10) by using Eqns. (8.29) and (8.31)... 

If the boundary conditions are zero flux or fixed composition, the last term vanishes. Comparison with the L2 inner product reveals that for evolution according to the diffusion equation, c(x,t) changes so that 5tot (total entropy acceleration ) is its most negative. Thus, entropy production, which is always positive, decreases in time as rapidly as possible when dc/dt [Pg.80]

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