State variables, nonequilibrium states

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)... 

Simplification of the solution or complete exclusion of the problem of dividing the variables into fast and slow is a great computational advantage of MEIS in comparison with the models of kinetics and nonequilibrium thermodynamics. The problem is eliminated, if there are no constraints in the equilibrium models on macroscopic kinetics. Indeed, the searches for the states corresponding to final equilibrium of only fast variables and states including final equilibrium coordinates of both types of variables with the help of these models do not differ from one another algorithmically. With kinetic constraints the division problem is solved by one of the three methods presented in Section 3.4, which are applied in the majority of cases to slow variables limiting the results of the main studied process. 

The state variables are (41). The time evolution (63) does not involve any nondissipative part and consequently the operator L, in which the Hamiltonian kinematics of (41) is expressed, is absent (i.e., L = 0). Time evolution will be discussed in Section 3.1.3. We now continue to specify the dissipation potential 5. Following the classical nonequilibrium thermodynamics, we introduce first the so-called thermodynamic forces (X 1-.. X k) Jdriving the chemically reacting system to the chemical equilibrium. As argued in nonequilibrium thermodynamics, they are linear functions of (nj,..., nk,) (we recall that n = (p i = 1,2,..., k on the Gibbs-Legendre manifold) with the coefficients... 

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. 

In this chapter we shall discuss the consequences of perturbing a system, which is initially in stable equilibrium, to a neighbouring nonequilibrium state. Since the initial equilibrium is supposed to be stable then the system will tend to return to an equilibrium state. For the moment we shall only concern ourselves with the way in which the thermodynamic variables change as the perturbed system moves back to equilibrium. The characteristics of the final equilibrium state, which is in general different from the initial state, will be discussed in the next chapter. 

F(, is reduced to the ordinary phase rule if we put S = R = 0. Langmuir pointed out that nonequilibrium states are of two kinds steady states, in which the intrinsic properties of all phases and the relative amounts of the phases do not vary with time, and transient states in which at least one of these variables change with time. 

The next step of importance is the description of inhomogeneous systems in terms of local equilibria. This process involves the division of the system to be described into subsystems, as discussed in Sect. 2.2.1, and shown in the upper half of Fig. 2.80. This is to be followed by the description of the system as a function of time as an additional variable. Following the system as a function of time allows the study of the kinetics of the processes seen when approaching equilibrium. Finally, it may be necessary to identify additional, internal variables to describe the nonequiUbrium states as a function of time. The kinetics of polymers may be sufficiently slow to decouple consecutive steps. For very slow responses, it may lead to arrested equilibria, such as seen in materials below their glass transitions as described in Sect. 2.4.4. The following Section makes use of the just summarized concepts in an attempt to achieve a depiction of the nonequilibrium state of macromolecules. 

Physical systems identified by permanently stable and reversible behavior are rare. Unstable phenomena result from inherent fluctuations of the respective state variables. Near a global equilibrium, the fluctuations do not disturb the equilibrium the trend toward equilibrium is distinguished by asymptotically vanishing dissipative contributions. In contrast, nonequilibrium states can amplify the fluctuations, and any local disturbances can even move the whole system into an unstable or metastable state. This feature is an important indication of the qualitative difference between equilibrium and nonequilibrium states. 

Now, the macroscopic behavior of all systems, whether in equilibrium or nonequilibrium states, is classically described in terms of the thermodynamic variables pressure P, temperature T, specific volume V (volume per mole), specific internal energy U (energy per mole), specific entropy S (entropy per mole), concentration or chemical potential /r, and velocity v. In nonequilibrium states, these variables change with respect to space and/or time, and the subject matter is called nonequilibrium thermodynamics. When these variables do not change with respect to space or time, their prediction falls vmder the subject matter of equilibrium thermodynamics. As a matter of notation, we would indicate a nonequilibrium variable such as entropy hy (r,t), where r is a vector that locates a particular region in space (locator vector) and t is the time, whereas the equilibrium notation would simply be . 

These are only associated with unsteady-state systems. An unsteady-state system is a system in which its state variables are changing with time. Unsteady-state open systems will change with time, tending toward the stationary nonequilibrium state , usually called the steady state in the chemical engineering literature. On the other hand, for closed and isolated systems, the unsteady-state behavior tends toward thermodynamic equilibrium. 

In nonequilibrium states and in irreversible processes, temperature - as with all intensive variables - is generally not defined. 

Thermodynamics of irreversible processes delivers a complete description of nonequilibrium states by extending the system of coordinates with a set of hidden variables. Following this route an extended version of the law of mass action can be formulated which holds true for stationary states even in open systems. It identifies stationary nonequilibrium states of cell ensembles as optimised patterns with universal features The reduced version of the cell size distributions of bacteria or yeast belong to the p = 3 class, while the length distribution of the intracellular proteins is described asp>= 1 type. 

As Einstein noted (see the introduction to Chapter 1), it is remarkable that the two laws of thermodynamics are simple to state but they relate so many different quantities and have a wide range of applicability. Thermodynamics gives us many general relations between state variables which are valid for any system in equilibrium. In this section we shall present a few important general relations. We will apply them to particular systems in later Chapters. As we shall see in Chapters 15-17, some of these relations can also be extended to nonequilibrium systems that are locally in equilibrium. 

Once time has been introduced as a variable, one may as well attempt to describe the time dependence of the processes which go from a nonequilibrium state towards equilibrium. The basic assumption is listed in Fig. 2.6 The driving force is proportional to the free energy change for the process, the proportionality constant being available from measurements. 

Space and time-correlation functions of macrovariables around nonequilibrium steady states have recently been calculated from nonequilibrium statistical mechanics [2]. Despite this progress, fluctuation theory for nonequilibrium states remains based, in its essential aspects, on stochastic theory. Specifically, it is assumed that one can define an appropriate set of discrete variables... 

The macroscopic description of nonequilibrium states of fluid systems requires independent variables to specify the extent to which the system deviates from equilibrium and dependent variables to express the rates of processes. 

In order to discuss the macroscopic nonequilibrium state of a fluid system, we will first assume that we can use intensive thermodynamic variables such as the temperature, pressure, density, concentrations, and chemical potentials. In order to justify this assumption we visualize the following process A small portion of the system is suddenly removed from the system and allowed to relax adiabatically to equilibrium at fixed volume. Once equilibrium is reached, intensive thermodynamic variables are well defined and can be measured. The measured values are assigned to a point inside the volume originally occupied by this portion of the system and to the time at which the subsystem was removed. We imagine that this procedure is performed repeatedly at different times and different locations in the system. Interpolation procedures are carried out to obtain smooth functions of position and time to represent the temperature, pressure, and concentrations ... 

The values of these intensive variables at a point are not sufficient for a complete description of the nonequilibrium state at that point. We also need a measure of how strongly the variables depend on position. We use the gradients of these variables for this purpose. The gradient of a scalar function /(x, y, z) is defined by Eq. (B-43) of Appendix B. It is a vector derivative that points in the direction of the most rapid increase of the function and has a magnitude equal to the derivative with respect to distance in that direction. The gradient of the temperature is denoted by VT ... 

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