Irreversible processes, equilibrium nonequilibrium thermodynamics

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 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)... 

In this chapter, we first introduce the principles of irreversible or nonequilibrium thermodynamics as opposed to those of equilibrium thermodynamics. Then, we identify important thermodynamic forces X (the cause) and their associated flow rates / (the effect). We show how these factors are responsible for the rate with which the entropy production increases and available work decreases in a process. This gives an excellent insight into the origin of the incurred losses. We pay attention to the relation between flows and forces and the possibility of coupling of processes and its implications. 

The MEIS developers relying on the capabilities of modem computers and computational mathematics started the work whichresulted in an essential expansion of the application area of "good, old" classical thermodynamics and in the possibility to study (using thermodynamics) any states on all possible motion trajectories of a nonequilibrium system. In other words, they put forward the goal to use the models of equilibrium not only to determine the directions of irreversible processes but to estimate the attainability of desired and undesired states on these directions. 

Development of the "flow" MEIS with the form reminding the models of nonequilibrium thermodynamics seems to be a very promising direction in equilibrium modeling of physical and chemical systems. Application of these models opens prospects for simpler analysis and solution of many complex problems related to the calculations of processes considered to be irreversible in principle. Certainly the flows in MEIS are interpreted statically as the coordinates of states. Thermodynamic interpretations are naturally extended to the kinetic coefficients that relate these flows with forces. Correctness of such interpretations is confirmed by the application of MP, being the theory of equilibrium states, as the terms for MEIS description. 

Feasibility of applying the models of equilibrium thermodynamics to the analysis of nonequilibrium irreversible processes were described in Section 2 of this chapter. This section discusses the comparative efficiency of such application to solve diverse theoretical and applied problems. 

The next sphere of competition between equilibrium and nonequilibrium thermodynamics is the analysis of irreversible trajectories. A popular opinion about the possibility for the equilibrium thermodynamics only to determine admissible directions of motion for nonequilibrium processes was already mentioned in Introduction. However, the more... 

Fortunately, several simplifications can be made (Nye, 1957). Transport phenomena, for example, are processes whereby systems transition from a state of nonequilibrium to a state of equilibrium. Thus, they fall within the realm of irreversible or nonequilibrium thermodynamics. Onsager s theorem, which is central to nonequilibrium thermodynamics, dictates that as a consequence of time-reversible symmetry, the off-diagonal elements of a transport property tensor are symmetrical (i.e., xy = X/,-). This is known as a reciprocal relation. The Norwegian physical chemist Lars Onsager (1903-1976) was awarded the 1968 Nobel Prize in Chemistry for reciprocal relations. Thus, the tensor above can be rewritten as... 

Other examples of transport properties include electrical and thermal conductivity. Transport of a physical quantity along a determined direction due to a gradient is an irreversible process by which a system transitions from a nonequilibrium state to an equilibrium state (e.g., compositional or thermal homogeneity). Therefore, it is outside the realm of equilibrium thermodynamics. (For this reason, equilibrium thermodynamics is more appropriately termed thermostatics.) Transport processes must be studied by irreversible thermodynamics. 

The kinetic theory leads to the definitions of the temperature, pressure, internal energy, heat flow density, diffusion flows, entropy flow, and entropy source in terms of definite integrals of the distribution function with respect to the molecular velocities. The classical phenomenological expressions for the entropy flow and entropy source (the product of flows and forces) follow from the approximate solution of the Boltzmann kinetic equation. This corresponds to the linear nonequilibrium thermodynamics approach of irreversible processes, and to Onsager s symmetry relations with the assumption of local equilibrium. 

The stability of transport and rate systems is studied either by nonequilibrium thermodynamics or by conventional rate theory. In the latter, the analysis is based on Poincare s variational equations and Lyapunov functions. We may investigate the stability of a steady state by analyzing the response of a reaction system to small disturbances around the stationary state variables. The disturbed quantities are replaced by linear combinations of their undisturbed stationary values. In nonequilibrium thermodynamics theory, the stability of stationary states is associated with Progogine s principle of minimum entropy production. Stable states are characterized by the lowest value of the entropy production in irreversible processes. The applicability of Prigogine s principle of minimum entropy production is restricted to stationary states close to global thermodynamic equilibrium. It is not applicable to the stability of continuous reaction systems involving stable and unstable steady states far from global equilibrium. The steady-state deviation of entropy production serves as a Lyapunov function. 

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. 

Some processes may have forces operating far away from equilibrium where the linear phenomenological equations are no longer applicable. Such a domain of irreversible phenomena, such as some chemical reactions, periodic oscillations, and bifurcation, is examined by extended nonequilibrium thermodynamics. Extending the methods of thermodynamics to treat the linear and nonlinear phenomena, and such dissipative structures are attracting scientists from various disciplines. 

Both diffusion and conduction are nonequilibrium (irreversible) processes and are therefore not amenable to the methods of equilibrium thermodynamics or equilibrium statistical mechanics. In these latter disciplines, the concepts of time and change are absent. It is possible, however, to imagine a situation where the two processes oppose and balance each other and a pseudoequilibrium obtains. This is done as follows (Fig. 4.62). 

While in a reversible process the system passes through a sequence of equilibrium thermodynamic states in an irreversible process it passes through a series of nonequilibrium states. Denote by Tp/(f) the system s actual nonequilibrium state at time t when it is undergoing the irreversible process T —> Vp. Denote by Tpi the equilibrium system state with the same macroscopic parameter values as Tp/(f). (That is, for state Tp> the macroscopic parameters are constrained to have the fixed values N,V,E,A, ...,... 

In equilibrium thermodynamics model A and in model B not far from equilibrium (and with no memory to temperature) the entropy may be calculated up to a constant. Namely, in both cases S = S(V, T) (2.6)2, (2.25) and we can use the equilibrium processes (2.28) in B or arbitrary processes in A for classical calculation of entropy change by integration of dS/dT or dS/dV expressible by Gibbs equations (2.18), (2.19), (2.38) through measurable heat capacity dU/dT or state Eqs.(2.6>, (2.33) (with equilibrium pressure P° in model B). This seems to accord with such a property as in (1.11), (1.40) in Sects. 1.3, 1.4. As we noted above, here the Gibbs equations used were proved to be valid not only in classical equilibrium thermodynamics (2.18), (2.19) but also in the nonequilibrium model B (2.38) and this expresses the local equilibrium hypothesis in model B (it will be proved also in nonuniform models in Chaps.3 (Sect. 3.6), 4, while in classical theories of irreversible processes [12, 16] it must be taken as a postulate). 

Equilibrium is characterized by the equality of the chemical potential. Nonequilibrium is therefore induced by the gradient of the chemical potential. In the thermodynamics of irreversible processes, chemical potential gradient is the fundamentally correct driving force for diffusion. According to the Gibbs-Duhem equation (2.3-5), we have ... 

The work of Carnot, published in 1824, and later the work of Clausius (1850) and Kelvin (1851), advanced the formulation of the properties of entropy, temperature, and the second law. Clausius introduced the word entropy. The second law is a statement of existence of stable equilibrium states and distinguishes thermodynamics from mechanics and other fields of physics. The many stable equilibrium states and various other equilibrium and nonequilibrium states contemplated in thermodynamics are not contemplated in mechanics (Gyftopoulos and Beretta, 2005). The second law is a qualitative statement on the accessibility of energy and the direction of progress of real processes. For example, the efficiency of a reversible engine is a function of temperature only, and efficiency cannot exceed unity. These statements are the results of the first and second laws, and can be used to define an absolute scale of temperature that is independent of any material properties used to measure it. A quantitative description of the second law emerges by determining entropy and entropy production in irreversible processes. 

Before introducing the notion of nonequilibrium thermodynamics we shall first summarize briefly the linear and nonlinear laws between thermodynamic fluxes and forces. A key concept when describing an irreversible process is the macroscopic state parameter of an adiabatically isolated system These parameters are denoted by. At equilibrium the state parameters have values A , while an arbitrary state which is near or far from the equilibrium may be specified by the deviations from the equilibrium state ... 

Thermodynamics can be divided into subjects which deal with 1) equilibrium, (2) nonequilibrium, and (3) irreversible processes. Ail three of these subdivisions are important in hydrocarbon reservoirs and in the interpretation of laboratory experiments for the understanding of hydrocarbon reservoirs. However, equilibrium thermodynamics is by far the most important and the best understood subject. According to Tisza (1966), the subdivision of equilibrium thermodynamics can be carried out further into Gibbsian thermodynamics and the early thermodynamics of Clausius and Kelvin. The latter considered the thermodynamic system as a black box, and all the relevant information was then derived from the energy absorbed and the work done by the system. The concepts of internal energy, U, and entropy, S, from the observable quantities are then established. In Gibbsian thermodynamics, the concepts of internal energy and entropy are assumed to be known and are used to provide a detailed description of the subsystems in equilibrium (we will soon define some of the terms used above). 

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