Entropy production systems

Consider an isotropic fluid in which viscous phenomena are neglected. Concentrations and temperature are non-unifonn in this system. The rate of entropy production may be written... 

Units of Energy 209. The First Law of Thermodynamics 210, Entropy Production Flow Systems 214. Application of (he Second Law 216. Summary of Thermodynamic Equations 223. 

A reaction at steady state is not in equilibrium. Nor is it a closed system, as it is continuously fed by fresh reactants, which keep the entropy lower than it would be at equilibrium. In this case the deviation from equilibrium is described by the rate of entropy increase, dS/dt, also referred to as entropy production. It can be shown that a reaction at steady state possesses a minimum rate of entropy production, and, when perturbed, it will return to this state, which is dictated by the rate at which reactants are fed to the system [R.A. van Santen and J.W. Niemantsverdriet, Chemical Kinetics and Catalysis (1995), Plenum, New York]. Hence, steady states settle for the smallest deviation from equilibrium possible under the given conditions. Steady state reactions in industry satisfy these conditions and are operated in a regime where linear non-equilibrium thermodynamics holds. Nonlinear non-equilibrium thermodynamics, however, represents a regime where explosions and uncontrolled oscillations may arise. Obviously, industry wants to avoid such situations ... 

According to irreversible thermodynamics, the entropy production per unit volume S for an isothermal system can be written... 

Self-organization seems to be counterintuitive, since the order that is generated challenges the paradigm of increasing disorder based on the second law of thermodynamics. In statistical thermodynamics, entropy is the number of possible microstates for a macroscopic state. Since, in an ordered state, the number of possible microstates is smaller than for a more disordered state, it follows that a self-organized system has a lower entropy. However, the two need not contradict each other it is possible to reduce the entropy in a part of a system while it increases in another. A few of the system s macroscopic degrees of freedom can become more ordered at the expense of microscopic disorder. This is valid even for isolated, closed systems. Eurthermore, in an open system, the entropy production can be transferred to the environment, so that here even the overall entropy in the entire system can be reduced. 

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. 

A significant question is whether the asymmetric contribution to the transport matrix is zero or nonzero. That is, is there any coupling between the transport of variables of opposite parity The question will recur in the discussion of the rate of entropy production later. The earlier analysis cannot decide the issue, since can be zero or nonzero in the earlier results. But some insight can be gained into the possible behavior of the system from the following analysis. 

The constrained rate of first entropy production for the total system is... 

As in the case of the isolated system, the asymmetric part of the transport matrix does not contribute to the scalar product or to the steady-state rate of first entropy production. All of the first entropy produced comes from the reservoirs, as it must since in the steady state the structure of the subsystem and hence its first entropy doesn t change. The rate of first entropy production is of course positive. 

Nonlinear Hamiltonian system, geometric transition state theory, 200-201 Nonlinear thermodynamics coefficients linear limit, 36 entropy production rate, 39 parity, 28-29... 

The plot of CE = Pout/Ps (from Eqs (5.10.33) and (5.10.37)) versus Ag for AM 1.2 is shown in Fig. 5.65 (curve 1). It has a maximum of 47 per cent at 1100 nm. Thermodynamic considerations, however, show that there are additional energy losses following from the fact that the system is in a thermal equilibrium with the surroundings and also with the radiation of a black body at the same temperature. This causes partial re-emission of the absorbed radiation (principle of detailed balance). If we take into account the equilibrium conditions and also the unavoidable entropy production, the maximum CE drops to 33 per cent at 840 nm (curve 2, Fig. 5.65). 

As already mentioned, a continual inflow of energy is necessary to maintain the stationary state of a living system. It is mostly chemical energy which is injected into the system, for example by activated amino acids in protein biosynthesis (see Sect. 5.3) or by nucleoside triphosphates in nucleic acid synthesis. Energy flow is always accompanied by entropy production (dS/dt), which is composed of two contributions ... 

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 ... 

Starting from the second law of thermodynamics, it is possible to derive a principle according to which the change of entropy production in the neighbourhood of a stationary state is always negative if the flows in the system are kept constant and only the forces varied. As already mentioned, the entropy production reaches a minimum value in the stationary state of the system. If it is at a minimum, and a positive fluctuation occurs, the system reverts to the minimum, and a stable state is again reached. 

If the entropy production in the stationary state falls, a negative fluctuation occurs, and the system becomes unstable the stable stationary state of the system is disturbed or destroyed. The system reacts by changing its composition until a new stable state is reached. In this new state, the system is characterised by a lower entropy content than that present prior to the fluctuation, since only a negative entropy change can occur. However, the lower entropy corresponds to a higher degree of order in the system. 

Fig. 9.3 Entropy-time diagram of an evolution process. If a negative fluctuation of the internal entropy production o occurs in a system, the controlling stationary state is terminated. An instability occurs, starting from which a new stable state is taken up. The change in the internal entropy is always negative (5Si < 0). The new stationary state has a lower entropy, i.e., the order of the system is increased (Eigen, 1971a, b)...

The affinities that define the rate of entropy production in continuous systems are therefore gradients of intensive parameters (in entropy representation) rather than discrete differences. For instance, the affinities associated with the z-components of energy and matter flow for constituent k, in this notation would be... 

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 second law of thermodynamics asserts that the total entropy 5 of a system may change in time because of exchanges with its environment and internal entropy production which is vanishing at equilibrium and positive out of equilibrium [5]... 

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