Thermodynamics entropy production

Linear momentum (L) operator, time reversal symmetry and, 243-244 Linear scaling, multiparticle collision dynamics, nonideal fluids, 137 Linear thermodynamics entropy production, 20-23 formalities, 8-11... 

Maximum Dissipation Principle, linear thermodynamics, entropy production, 21—23... 

The entropies per unit time as well as the thermodynamic entropy production entering in the formula (101) can be interpreted in terms of the numbers of paths satisfying different conditions. In this regard, important connections exist between information theory and the second law of thermodynamics. 

We notice that the only difference between both dynamical entropies is the exchange of oa and oa in the transition probabilities appearing in the logarithm. According to Eq. (101), the thermodynamic entropy production of this process would be equal to... 

The number of typical paths generated by the random process increases as exp(/ifl). In this regard, the Kolmogorov-Sinai entropy per unit time is the rate of production of information by the random process. On the other hand, the time-reversed entropy per unit time is the rate of production of information by the time reversals of the typical paths. The thermodynamic entropy production is the difference between these two rates of information production. With the formula (101), we can recover a result by Landauer [50] and Bennett [51] that erasing information in the memory of a computer is an irreversible process of... 

In irreversible thermod3mamics, the second law of thermodynamics dictates that entropy of an isolated system can only increase. From the second law of thermodynamics, entropy production in a system must be positive. When this is applied to diffusion, it means that binary diffusivities as well as eigenvalues of diffusion matrix are real and positive if the phase is stable. This section shows the derivation (De Groot and Mazur, 1962). 

Principles of thermodynamics find applications in all branches of engineering and the sciences. Besides that, thermodynamics may present methods and generalized correlations for the estimation of physical and chemical properties when there are no experimental data available. Such estimations are often necessary in the simulation and design of various processes. This chapter briefly covers some of the basic definitions, principles of thermodynamics, entropy production, the Gibbs equation, phase equilibria, equations of state, and thermodynamic potentials. 

A statistical mechanical model of thermodynamic entropy production in a sequence-structure system suggests that the shared thermodynamic entropy is the probability function that weighs any sequence average. The sequence information is defined as the length of the shortest string that encodes the sequence. The connection between sequence evolution and nonequilibrium thermodynamics is that the minimal length encoding of specific amino acids will have the same dependence on sequence as the shared thermodynamic entropy. 

The Lyapunov function resembles the thermodynamic entropy production function and the asymptotic stability principle. If the eigenvalues of the coefficient matrix of the quadratic form of the entropy production are very large, then the convergence to equilibrium state will be rapid. 

When we consider the dependence of excess heat production on an external transformation, we can connect thermodynamic quantities and underlying dynamics. We have derived the theorem similar to the fluctuation-dissipation theorem [10]. The theorem shows that thermodynamic entropy production such as excess heat can be written as a correlation function between Einstein-Shanon entropy functions. Through the correlation function the thermodynamic entropy production is related to the underlying dynamics. 

We have obtained several interesting results from the theorem If the period of the external transformation is much longer than the relaxation time, then thermodynamic entropy production is proportional to the ratio of the period and relaxation time. The relaxation time is proportional to the inverse of the Kolmogorov-Sinai entropy for small strongly chaotic systems. Thermodynamic entropy production is proportional to the inverse of the dynamical entropy [11]. On the other hand, thermodynamic entropy production is proportional to the dynamical entropy when the period of the external transformation is much shorter than the relaxation time. Furthermore, we found fractional scaling of the excess heat for long-period external transformations, when the system has longtime correlation such as 1 /fa noise. Since excess heat is measured as the area of a hysteresis loop [12], these properties can be confirmed in experiments. 

The area of the hysteresis loop is proportional to jT, so it increases for greater T. The thermodynamic entropy production is proportional to the dynamical entropy production in this case [22, 23]. In a large chaotic system, we can expect less thermodynamic entropy production for weak diffusion. 

According to the second law of thermodynamics, entropy production must be positive. Consequently, the graph of the constitutive relation must be within the first and third quadrant. 

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. 

There are three different approaches to a thermodynamic theory of continuum that can be distinguished. These approaches differ from each other by the fundamental postulates on which the theory is based. All of them are characterized by the same fundamental requirement that the results should be obtained without having recourse to statistical or kinetic theories. None of these approaches is concerned with the atomic structure of the material. Therefore, they represent a pure phenomenological approach. The principal postulates of the first approach, usually called the classical thermodynamics of irreversible processes, are documented. The principle of local state is assumed to be valid. The equation of entropy balance is assumed to involve a term expressing the entropy production which can be represented as a sum of products of fluxes and forces. This term is zero for a state of equilibrium and positive for an irreversible process. The fluxes are function of forces, not necessarily linear. However, the reciprocity relations concern only coefficients of the linear terms of the series expansions. Using methods of this approach, a thermodynamic description of elastic, rheologic and plastic materials was obtained. 

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. 

On a related point, there have been other variational principles enunciated as a basis for nonequilibrium thermodynamics. Hashitsume [47], Gyarmati [48, 49], and Bochkov and Kuzovlev [50] all assert that in the steady state the rate of first entropy production is an extremum, and all invoke a function identical to that underlying the Onsager-Machlup functional [32]. As mentioned earlier, Prigogine [11] (and workers in the broader sciences) [13-18] variously asserts that the rate of first entropy production is a maximum or a minimum and invokes the same two functions for the optimum rate of first entropy production that were used by Onsager and Machlup [32] (see Section HE). 

It should be clear that the most likely or physical rate of first entropy production is neither minimal nor maximal these would correspond to values of the heat flux of oc. The conventional first entropy does not provide any variational principle for heat flow, or for nonequilibrium dynamics more generally. This is consistent with the introductory remarks about the second law of equilibrium thermodynamics, Eq. (1), namely, that this law and the first entropy that in invokes are independent of time. In the literature one finds claims for both extreme theorems some claim that the rate of entropy production is... 

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). 

Fig. 5.65 Dependence of the solar conversion efficiency (CE) on the threshold wavelength (Ag) for a quantum converter at AM 1.2. Curve 1 Fraction of the total solar power convertible by an ideal equilibrium converter with no thermodynamic and kinetic losses. Curve 2 As 1 but the inherent thermodynamic losses (detailed balance and entropy production) are considered. Continuous line Efficiency of a regenerative photovoltaic cell, where the thermodynamic and kinetic losses are considered. The values of Ag for some semiconductors are also shown (according to J. R. Bolton et al.)... 

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. 

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