Maximum of Entropy

We may point out a special case in that the energy of the spring does not change with length at all. Then X = 0 from the first term on the right-hand side of Eq. (2.58) and the gases will expand to zero pressure. 

We examine now the entropy of two subsystems under the constraint of constant volume and constant energy. We have two systems with the energies Ui(Si, Vi) and 112(82, V2). These energies are functions of their entropies denoted as i, S2 and their volumes Vi,V2. The total energy and the total volume should be constant  

We form now the total entropy and add the constraint equations (2.59) multiplied with undetermined coefficients Xu, kv - 

In this procedure we do not search for an extremum for the total energy. Instead, we assume the total energy to be constant. We do not use a condition of constant entropy but we search an extremum of the entropy under the constraint of constant energy and constant volume. Nevertheless, it turns out that under these constraints not only the pressures of the two systems are equal but also the temperatures. 

However, if we search for an extremum of the energy under the constraint of constant total volume V at — Vi — V2 = 0 and of constant total entropy Stot — Si — S2 = 0, the same result would be obtained. In fact, we would have to evaluate an expression like 

---

This corresponds de facto to a rotational gas with the maximum of entropy, because in this idealized model it is supposed that rotation covers isoenergetic angular positions with equal probability. 

Maximum of entropy in linear steady-state system... 

In such system the rate of increase of entropy Prigogine theorem. Therefore, the entropy of the system is maximum otherwise, an eventual rising of entropy would cause new fluxes increasing system entropy So is maximum, and the Prigogine theorem is equivalent to the principle of maximum of entropy for the system. 

We assume that the complex system in start-up regime is described statistically and comprises a number (assemblage) of microsystems in nonsteady-state, while the system is approximately characterized by the limited maximum of entropy. 

Systems that do not tend to approach a maximum of entropy might be basically of interest in cosmology, if the big bang stops to bang. However, we should annotate that at the time when Clausius formulated the second law, nothing was known about the big bang. 

If the subsystems are connected via a diathermic wall, heat may be exchanged. However, the individual mol numbers and the volumes are fixed. So we are left over to vary S and S" with the constraint Utot = C. A further constraint is needed, which turns out to be the maximum of entropy principle. 

These equations indicate that a maximum of entropy is achieved at equilibrium. On the other hand, if we demand an isolated system that is even not transparent for entropy, then we must conclude that under such circumstances entropy will be generated, if such as system is moving toward equilibrium. We have used here the principle of maximum entropy under the constraints of constant energy and constant volume. 

This admissibihty is criticized by Rajagopal and his school using instead the assumption of maximum of entropy production rate, their further restrictions may be obtained, see, e.g., [37-39]. However, there are also reservations to the principle of maximum entropy production [40]. 

Strictly speaking the composition law in Equations 7 is not restricted only by addition. But finding the maximum of entropy 8... 

Relationship 1 reflects a universal principle of entropy increase in systems seeking an equilibrium. Consequently, the equilibrium itself is characterized by a maximum of entropy. 

There may be situations when pjquations 7 and 9 are realized while Equations 8 and 10 are not realized, i.c. the system is stable toward infinitely small perturbations while being unstable toward finite ones (a local maximum of entropy or a minimum of internal energy with at least one additional extremum). In such cases it is generally agreed to speak of a metastable equilibrium (metastable state) of the system. Conditions 8 and 10 define a stable equilibrium. When simultaneously breaking conditions 7 10, the system proves to be absolutely unstable. 

Both the entropy production and the divergence of entropy fluxes are functions of two variables (say, interlameUar distance and I). Each of these functions independently has no maximum. Yet, at the subset of variables corresponding to the constraint of steady-state balance of entropy, the maximum of entropy production does exist and should correspond to the most probable evolution path. 

It is an inference naturally suggested by the general increase of entropy which accompanies the changes occurring in any isolated material system that when the entropy of the system has reached a maximum, the system will be in a state of equilibrium. Although this principle has by no means escaped the attention of physicists, its importance does not seem to have been duly appreciated. Little has been done to develop the principle as a foundation for the general theory of thermodynamic equilibrium (my italics). ... 

Environmental protection and resource use have to be considered in a comprehensive framework, and all of the relevant economic and natural scientific aspects have to be taken into consideration. The concepts of entropy and sustainability are useful in this regard. The entropy concept says that every system will tend toward maximum disorder if left to itself. In other words, in the absence of sound environmental policy. Earth s energy sources will be converted to heat and pollutants that must be received by Earth. The concept of sustainability has to do with... 

Calculations were usually carried out under the conditions of a pH-maximum of protein bonding. The experimental results show that the interaction of proteins and most other complex organic substances with CP is accompanied by an increase in the entropy of the system. 

The equilibrium is evidently stable when the entropy is a maximum, for then every possible change would diminish the entropy. The equilibrium will be unstable when the entropy is a minimum for a given value of the energy. This implies that if there are several conceivable neighbouring states with the same energy, that with the least entropy will correspond with a state of unstable equilibrium, whilst the others with more entropy will be essentially unstable states, except the one with the greatest amount of entropy, which will be the state of stable... 

The temperature or enthalpy of the gas may then be plotted to a base of entropy to give a Fanno line.iA This line shows the condition of the fluid as it flows along the pipe. If the velocity at entrance is subsonic (the normal condition), then the enthalpy will decrease along the pipe and the velocity will increase until sonic velocity is reached. If the flow is supersonic at the entrance, the velocity will decrease along the duct until it becomes sonic. The entropy has a maximum value corresponding to sonic velocity as shown in Figure 4.11. (Mach number Ma < 1 represents sub-sonic conditions Ma > 1 supersonic.)... 

This is a law about the equilibrium state, when macroscopic change has ceased it is the state, according to the law, of maximum entropy. It is not really a law about nonequilibrium per se, not in any quantitative sense, although the law does introduce the notion of a nonequilibrium state constrained with respect to structure. By implication, entropy is perfectly well defined in such a nonequilibrium macrostate (otherwise, how could it increase ), and this constrained entropy is less than the equilibrium entropy. Entropy itself is left undefined by the Second Law, and it was only later that Boltzmann provided the physical interpretation of entropy as the number of molecular configurations in a macrostate. This gave birth to his probability distribution and hence to equilibrium statistical mechanics. 

The equilibrium state, which is denoted x, is by definition both the most likely state, p(x E) > p(x E), and the state of maximum constrained entropy, iS,(T (x /ij > iS 0(x j. This is the statistical mechanical justification for much of the import of the Second Law of Equilibrium Thermodynamics. The unconstrained entropy, as a sum of positive terms, is strictly greater than the maximal constrained entropy, which is the largest term, S HE) >. S(1 (x j. However, in the thermodynamic limit when fluctuations are relatively negligible, these may be equated with relatively little error, S HE) . S(1 (x j. 

Physically the variational procedure based on the second entropy may be interpreted like this. If the flux E were increased beyond its optimum value, then the rate of entropy consumption by the subsystem would be increased due to its increased dynamic order by a greater amount than the entropy production of the reservoirs would be increased due to the faster transfer of heat. The converse holds for a less than optimum flux. In both cases the total rate of second entropy production would fall from its maximum value. 

According to the latter model, the crystal is described as formed of anumber of equal scatterers, all randomly, identically and independently distributed. This simplified picture and the interpretation of the electron density as a probability distribution to generate a statistical ensemble of structures lead to the selection of the map having maximum relative entropy with respect to some prior-prejudice distribution m(x) [27, 28],... 

In the absence of any data, the maximum of the entropy functional is reached for p(r) = m(r). Any substantial departure from the model, observed in the final map, is really contained in the data as, with the new definition, it costs entropy. This paper presents illustrations of model testing in the case of intermetallic and molecular compounds. 

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). 

The tendency of systems to reach a maximum state of entropy has been applied to the social sciences. Does the second law of thermodynamics help to explain the increase in garbage on the streets Justify your answer. 

---