Nonequilibrium systems entropy

The generic case is a subsystem with phase function x(T) that can be exchanged with a reservoir that imposes a thermodynamic force Xr. (The circumflex denoting a function of phase space will usually be dropped, since the argument T distinguishes the function from the macrostate label x.) This case includes the standard equilibrium systems as well as nonequilibrium systems in steady flux. The probability of a state T is the exponential of the associated entropy, which is the total entropy. However, as usual it is assumed (it can be shown) [9] that the... 

GEN. 128. 1. Prigogine, The message of entropy, in Workshop, Patterns, Defects and Microstructures in Nonequilibrium Systems, Austin, Texas 1986. 

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. 

We tend to identify 5p(F) as the entropy production in a nonequilibrium system, whereas B(F) is a term that contributes just at the beginning and end of the nonequilibrium process. Note that the entropy production 5p(F) is antisymmetric under time reversal, 5p(F ) = -5p(F), expressing the fact that the entropy production is a quantity related to irreversible motion. According to Eq. (21) paths that produce a given amount of entropy are much more probable than those... 

Beyond the domain of validity of the minimum entropy production theorem (i.e., far from equilibrium), a new type of order may arise. The stability of the thermodynamic branch is no longer automatically ensured by the relations (8). Nevertheless it can be shown that even then, with fixed boundary conditions, nonequilibrium systems always obey to the inequality1... 

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)... 

The foundation of irreversible thermodynamics is the concept of entropy production. The consequences of entropy production in a dynamic system lead to a natural and general coupling of the driving forces and corresponding fluxes that are present in a nonequilibrium system. 

System with random fluxes is defined as the nonequilibrium system where the fluxes of substance, heat, etc. change randomly. One can cite numerous examples of such systems turbulent gas-liquid systems with intensive heat/mass transfer, turbulent fluids containing dispersed solids, etc. In the case of pore formation, such situation is realized when the heat fluxes change randomly because of air fluidization or mechanical mixing. All macroscopic measured parameters of stationary turbulent flows, like their pressure, temperature, excess (free) energy, entropy, etc. do not change with time, while their values and directions in different spots of the flows can vary significantly. 

In nonequilibrium systems, the intensive properties of temperature, pressure, and chemical potential are not uniform. However, they all are defined locally in an elemental volume with a sufficient number of molecules for the principles of thermodynamics to be applicable. For example, in a region A , we can define the densities of thermodynamic properties such as energy and entropy at local temperature. The energy density, the entropy density, and the amount of matter are expressed by uk(T, Nk), s T, Nk), and Nk, respectively. The total energy U, the total entropy S, and the total number of moles N of the system are determined by the following volume integrals ... 

Since the temperature is not uniform for the whole system, the total entropy is not a function of the other extensive properties of U, V, and N. However, with the local temperature, the entropy of a nonequilibrium system is defined in terms of an entropy density, sk. 

The reversibility of molecular behavior gives rise to a kind of symmetry in which the transport processes are coupled to each other. Although a thermodynamic system as a whole may not be in equilibrium, the local states may be in local thermodynamic equilibrium all intensive thermodynamic variables become functions of position and time. The definition of energy and entropy in nonequilibrium systems can be expressed in terms of energy and entropy densities u(T,Nk) and s(T,Nk), which are the functions of the temperature field T(x) and the mole number density Y(x) these densities can be measured. The total energy and entropy of the system is obtained by the following integrations... 

In every nonequilibrium system, an entropy effect exists either within the system or through the boundary of the system. Entropy is an extensive property, and if a system consists of several parts, the total entropy is equal to the sum of the entropies of each part. Entropy balance is... 

Nonisothermal reaction-diffusion systems represent open, nonequilibrium systems with thermodynamic forces of temperature gradient, chemical potential gradient, and affinity. The dissipation function or the rate of entropy production can be used to identify the conjugate forces and flows to establish linear phenomenological equations. For a multicomponent fluid system under mechanical equilibrium with n species and A r number of chemical reactions, the dissipation function 1 is... 

The Gibbs stability theory condition may be restrictive for nonequilibrium systems. For example, the differential form of Fourier s law together with the boundary conditions describe the evolution of heat conduction, and the stability theory at equilibrium refers to the asymptotic state reached after a sufficiently long time however, there exists no thermodynamic potential with a minimum at steady state. Therefore, a stability theory based on the entropy production is more general. 

For nonequilibrium systems far from global equilibrium, the second law does not impose the sign of entropy variation due to the terms djS and d S, as illustrated in Figure 12.2. Therefore, there is no universal Lyapunov function. For a multicomponent fluid system with n components, entropy production in terms of conjugate forces Xu flows Jj, and / number of chemical reactions is... 

The theory treating near-equilibrium phenomena is called the linear nonequilibrium thermodynamics. It is based on the local equilibrium assumption in the system and phenomenological equations that linearly relate forces and flows of the processes of interest. Application of classical thermodynamics to nonequilibrium systems is valid for systems not too far from equilibrium. This condition does not prove excessively restrictive as many systems and phenomena can be found within the vicinity of equilibrium. Therefore equations for property changes between equilibrium states, such as the Gibbs relationship, can be utilized to express the entropy generation in nonequilibrium systems in terms of variables that are used in the transport and rate processes. The second law analysis determines the thermodynamic optimality of a physical process by determining the rate of entropy generation due to the irreversible process in the system for a required task. 

Equations (A.23) and (A.25) pertain to equilibrium conditions of homogeneous systems. Such systems have constant properties over space and time and there is no entropy production. We shall now be interested in systems, away from equilibrium where properties vary as functions of location as well as time. Tb apply the results of thermodynamics to nonequilibrium systems., the principle of local (microscopic) equilibrium is invoked. For that reason it is useful to work with the thermodynamic variables on a unit volume basis. Equation (A.25) then becomes... 

Inequalities (3.2) and (3.3) are generalizations of the principle of the minimal entropy production rate in the course of spontaneous evolution of its system to the stationary state. They are independent of any assump tions on the nature of interrelations of fluxes and forces under the condi tions of the local equilibrium. Expression (3.2), due to its very general nature, is referred to as the Qlansdorf-Prigogine universal criterion of evolution. The criterion implies that in any nonequilibrium system with the fixed boundary conditions, the spontaneous processes lead to a decrease in the rate of changes of the entropy production rate induced by spontaneous variations in thermodynamic forces due to processes inside the system (i.e., due to the changes in internal variables). The equals sign in expres sion (3.2) refers to the stationary state. 

Such a generalization is consistent with the Second Law of Thermod5mamics, since the 77 theorem and the generalized definition of entropy together lead to the conclusion that the entropy of an isolated nonequilibrium system increases monotonically, as it approaches equilibrium. 

In 1902, T. W. Richards found experimentally that the free-energy increment of a reaction approached the enthalpy change asymptotically as the temperature was decreased. From a study of Richards data, Nernst suggested that at absolute zero the entropy increment of reversible reactions among perfect crystalline solids is zero. This heat theorem was restated by Planck in 1912 in the form The entropy of all perfect crystalline solids is zero at absolute zero.f This postulate is the third law of thermodynamics. A perfect crystal is one in true thermodynamic equilibrium. Apparent deviations from the third law are attributed to the fact that measurements have been made on nonequilibrium systems. 

Summary. The Second Law was postulated as a simple general statement on heat exchange in cyclic processes. It was demonstrated that when this statement is combined with the properties of thermodynamic systems and universe introduced in Sect. 1.2 the existence of the absolute temperature and entropy follows, even out of equilibrium. The entropy should satisfy an inequality (1.21) which can be viewed as an alternative form of the Second Law and is called the entropy inequality. However, enttopy need not be unique especially in complex (nonequilibrium) systems or processes and even the ttansferability of the proof of its existence at such conditions remains unclear. Even in such cases the supposed existence of entropy can give important information on possible behavior which can be subjected to experimental testing. 

Thermodynamics plays an important role in the stability analysis of transport and rate processes, and the nonequilibrium thermodynamics approach in particular may enhance and broaden this role. This chapter reviews stability analysis based on the conventional Gibbs approach and tbe nonequilibrium thermodynamics theory. It considers the stability of equilibrium, near-equilibrium, and far-from-equilibrium states with some case studies. The entropy production approach for nonequilibrium systems appears to be more general for stability analysis. One major implication of the nonequilibrium thermodynamics theory is the introduction of distance from global equilibrium as a constraint for determining the stability of nonequilibrium systems. When a system is far from global equilibrium, the possibility of new organized structures of matter arise beyond an instability point. 

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