Nonequilibrium stationary state

In their subsequent works, the authors treated directly the nonlinear equations of evolution (e.g., the equations of chemical kinetics). Even though these equations cannot be solved explicitly, some powerful mathematical methods can be used to determine the nature of their solutions (rather than their analytical form). In these equations, one can generally identify a certain parameter k, which measures the strength of the external constraints that prevent the system from reaching thermodynamic equilibrium. The system then tends to a nonequilibrium stationary state. Near equilibrium, the latter state is unique and close to the former its characteristics, plotted against k, lie on a continuous curve (the thermodynamic branch). It may happen, however, that on increasing k, one reaches a critical bifurcation value k, beyond which the appearance of the... 

Intensive properties that specify the state of a substance are time independent in equilibrium systems and in nonequilibrium stationary states. Extensive properties specifying the state of a system with boundaries are also independent of time, and the boundaries are stationary in a particular coordinate system. Therefore, the stationary state of a substance at ary point is related to the stationary state of the system. 

Therefore, the total entropy produced within the system must be discharged across the boundary at stationary state. For a system at stationary state, boundary conditions do not change with time. Consequently, a nonequilibrium stationary state is not possible for an isolated system for which deS/dt = 0. Also, a steady state cannot be maintained in an adiabatic system in which irreversible processes are occurring, since the entropy produced cannot be discharged, as an adiabatic system cannot exchange heat with its surroundings. In equilibrium, all the terms in Eq. (3.48) vanish because of the absence of both entropy flow across the system boundaries and entropy production due to irreversible processes, and we have dJS/dt = d dt = dS/dt = 0. 

Potassium leaves the cell, while the net flow of sodium is inward. A nonequilibrium stationary state for the cell at rest is maintained by the sodium and potassium pumps, which pump out the entering sodium ions and pump the leaking potassium ions back into the cell interior, using a certain metabolic output. The sodium transfer is coupled with the chemical reaction. The electrochemical potential difference for sodium ions is expressed as... 

The term to the right of the equal sign in Eq. (12.32) is the excess entropy production. Equations (12.31) and (12.32) describe the stability of equilibrium and nonequilibrium stationary states. The term 82S is a Lyapunov functional for a stationary state. 

If P is the entropy production in a nonequilibrium stationary state, the change of P due to small changes in the forces 8Xt and in the flows 8Jt is... 

This inequality is identical to the Le Chatelier principle for nonequilibrium stationary states, since the disturbance 8Xk has the same sign as the flow Jk, indicating a decrease in the disturbance. For example, an increase in the gradient of the chemical potential will cause the mass flow and diminish the gradient. Hence, the stationary state will return to its original status. 

Equations (12.27) and (12.64) show the stability of the nonequilibrium stationary states in light of the fluctuations Sev The linear regime requires P > 0 and dP/dt < 0, which are Lyapunov conditions, as the matrix (dAJdej) is negative definite at near equilibrium. 

There are two types of macroscopic structures equilibrium and dissipative ones. A perfect crystal, for example, represents an equilibrium structure, which is stable and does not exchange matter and energy with the environment. On the other hand, dissipative structures maintain their state by exchanging energy and matter constantly with environment. This continuous interaction enables the system to establish an ordered structure with lower entropy than that of equilibrium structure. For some time, it is believed that thermodynamics precludes the appearance of dissipative structures, such as spontaneous rhythms. However, thermodynamics can describe the possible state of a structure through the study of instabilities in nonequilibrium stationary states. 

The experiments were run in a continuous-flow stirred tank reactor (CSTR) (fig. 6.2) with the reaction system at a nonequilibrium stationary state, such that the reactions run spontaneously from glucose to G3P and 3PG. The concentrations of the species at this state are close to those of physiological conditions. The metabolites G6P, F6P, F1,6BP, DHAP, G3P, and 3PG were detected and analyzed by capillary electrophoresis. Typical relative errors were 4% for G6P, 11% for F6P, 15% for F1,6BP, 9% for DHAP, 6% for 3PG and 3% for G3P. 

The equation above proves that with the linear reaction flows, the entropy production is minimized at nonequilibrium stationary state where the reaction velocities are equal to each other 7ri = Ji2-... 

A function L satisfying the equation above is called a Lyapunov function. The second variation of entropy L = —d S may be used as a Lyapunov functional if the stationary state satisfies dKidf > 0. A functional is a set of functions that are mapped to a real or complex value. Hence, a nonequilibrium stationary state is stable if... 

The entropy for a stochastic macromolecular mechanics becomes 5 = —ks Jpln pdB. In a nonequilibrium stationary states where the probabilities are time-independent (dPjdt = 0), the entropy production is... 

NONEQUILIBRIUM STATIONARY STATES AND THEIR STABILITY LINEAR REGIME... 

In the previous section we have seen some examples of nonequilibrium stationary states in which one or more thermodynamic forces were maintained at... 

In the linear regime, the total entropy production in a system subject to flow of energy and matter, diS/dt = odV, reaches a minimum value at the nonequilibrium stationary state. ... 

That is, P = diS/dt is minimized when the flow J2 corresponding to the unconstrained force F2 vanishes. This result can easily be generalized to an arbitrary number of forces and flows. The stationary state is the state of minimum entropy production in which the flows Jk, corresponding to the unconstrained forces, are zero. Although nonequilibrium stationary states are generally obtained through kinetic considerations, minimization of entropy production provides an alternative way. 

In the previous section, by using kinetics, we have already seen that the nonequilibrium stationary state is completely specified by (17.1.18) ... 

In the linear regime, the entropy production is minimized at the nonequilibrium stationary state. 

These conditions ensure the stability of the nonequilibrium stationary states in... 

Let P be the entropy production in a nonequilibrium stationary state. Since P = y dV = lyY FkJkdV, the rate of change in P can be written as... 

A function L that satisfies (18.3.3) is called a Lyapunov function. If the variables are functions of position (as concentrations in a nonequilibrium system can be), L is called a Lyapunov functional—a functional is a mapping of a set of functions to a number, real or complex. The notion of stability is not restricted to stationary states it can also be extended to periodic states [4]. However, since we are interested in the stability of nonequilibrium stationary states, we shall not deal with the stability of periodic states at this point. 

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