Maximum entropy principle Mechanics

Let us now turn to the individual formalism of quantum mechanics again, where thermal states are decomposed in a canonical way according to the maximum entropy principle. We cannot compute yet maximum entropy decompositions and large-deviation entropies for molecular situations. Nevertheless, the (simpler) example of the Curie-Weiss magnet suggests that even in molecular situations ... 

In modem physics, there exist alternative theories for the equilibrium statistical mechanics [1, 2] based on the generalized statistical entropy [3-12]. They are compatible with the second part of the second law of thermodynamics, i.e., the maximum entropy principle [13-14], which leads to uncertainty in the definition of the statistical entropy and consequently the equilibrium probability density functions. This means that the equilibrium statistical mechanics is in a crisis. Thus, the requirements of the equilibrium thermodynamics shall have an exclusive role in selection of the right theory for the equilibrium statistical mechanics. The main difficulty in foundation of the statistical mechanics based on the generalized statistical entropy, i.e., the deformed Boltzmann-Gibbs entropy, is the problem of its connection with the equilibrium thermodynamics. The proof of the zero law of thermodynamics and the principle of additivity... 

Equations (7.100-7.103) form a differential algebraic system of equations where the number of cmistraints used determines the dimensirai of the reduced system. Since the RCCE formulation leads to a general system of ODEs, it is also necessary to select which variables the constraints should be applied to, since different selections may lead to different model reduction errors. Most commonly, constraints are applied to individual species, but this may be more related to practicalities of implementation rather than inherent properties of the system. There is in principle no reason why constraints should not be applied to combinations of species (e.g. lumped variables). To sununarise (Jones and Rigopoulos 2007), an RCCE system comprises a set of ODEs or PDEs that describe the dynamics of the kinetically controlled species taken directly from a detailed mechanism without any approximations and a set of algebraic equations for the computation of the equilibrated species, derived on a physical basis via the maximum entropy principle of thermodynamics. 

In the last section, we used the maximum entropy principle to constract the density operator p characterizing the macroscopic state of some general quantum system for which the information (dy) y = 0,..., is available. Let us turn our attention to the relationship between the statistical mechanics and thermodynamics of this system. Introducing the quantity... 

With the maximum entropy principle and the notion of quasi-static processes at om disposal, we showed that the thermodynamics of a system emerges from the statistical mechanics of that system. Unlike conventional approaches to equilibriitm statistical mechanics, the information-theoretic approach does not require us to appeal to the phenomenological equations of thermodynamics in order to render the statistical mechanical formalism meaningful. 

In this section, we first discuss the basic results from nonequilibrium thermodynamics for continuous and discrete systems. Then we show that all of these results emerge from a quantum statistical mechanical treatment of nonequilibrium systems based on a time-dependent version of the maximum entropy principle. It is shown that this treatment not only includes nonequilibrium thermodynamics, but also provides a more general framework for discussing both equilibriirm and nonequilibrium systems. 

The quantum statistical mechanical theory of relaxation phenomena (QSM theory) is a maximum entropy approach to nonequilibrium processes that is in the same spirit as the statistical mechanical formulations of equilibrium and nonequilibrium thermodynamics given in Sections 3.3 and 5.4. As in the statistical mechanical formulation of nonequilibrium thermodynamics, the maximum entropy principle is assumed to apply to systems out of equilibrium. 

In Sections I.C and YD it was shown that the basic results from equilibrium and nonequilibrium thermodynamics can be established from statistical mechanics by starting from the maximum entropy principle. The success of this approach to the formulation of equilibrium and nonequilibrium thermodynamics suggests that the maximum entropy principle can also be used to formulate a general theory of nonequihbrium processes that automatically includes the thermodynamic description of nonequilibrium systems. In this section, we formulate a theory possessing this character by making use of a time-dependent projection operator P(t) that projects the thermodynamic description p,) of a system out of the global description pt) given by the solution of the Liouville equation. We shall refer to this theory as the maximitm entropy approach to nonequilibrium processes. 

The concept of entropy is present in many disciplines. In statistical mechanics, Boltzmann introduced entropy as a measure of the number of microscopic ways that a given macroscopic state can be realized. A principle of nature is that it prefers systems that have maximum entropy. Shannon has also introduced entropy into communications theory, where entropy serves as a measure of information. The role of entropy in these fields is not disputed in the scientific community. The validity, however, of the Bayesian approach to probability theory and the principle of maximum entropy in this, remains controversial. 

So far we have concentrated on the analysis of detailed rate data, as distinct from their synthesis. It has been implied that vibrational surprisal plots are frequently linear because of some, qualitatively common, dynamic constraint, but this has not been identified. Bernstein and Levine have elegantly reviewed the statistical mechanical basis of the relationship between real distributions and the constraints which lead to them. The general principle is that a system will adopt the distribution with maximum entropy, which is also consistent with all the constraints. Consequently, a complete distribution could be synthesized if the constraints could be independently determined. Alternatively, it should be possible, in principle, to deduce the constraints by observing the distribution. 

Considerable effort has been expended in the attempt to develop a general theory of reaction rates through some extension of thermodynamics or statistical mechanics. Since neither of these sciences can, by themselves, yield any information about rates of reactions, some additional assumptions or postulates must be introduced. An important method of treating systems that are not in equilibrium has acquired the title of irreversible thermodynamics. Irreversible thermodynamics can be applied to those systems that are not too far from equilibrium. The theory is based on the thermodynamic principle that in every irreversible process, that is, in every process proceeding at a finite rate, entropy is created. This principle is used together with the fact that the entropy of an isolated system is a maximum at equilibrium, and with the principle of microscopic reversibility. The additional assumption involved is that systems that are slightly removed from equilibrium may be described statistically in much the same way as systems in equilibrium. 

Derive the criteria of (1) thermal equilibrium (2) mechanical equilibrium and (3) chemical equilibrium from the internal energj minimum principle. These criteria were derived from the entropy maximum principle at the beginning of this chapter. 

Secondly, in the theory of irreversible processes, variation principles may be expected to help establish a general statistical method for a system which is not far from equilibrium, just as the extremal property of entropy is quite important for establishing the statistical mechanics of matter in equilibrium. The distribution functions are determined so as to make thermod5mamic probability, the logarithm of which is the entropy, be a maximum under the imposed constraints. However, such methods for determining the statistical distribution of the s retem are confined to the case of a system in thermodynamic equilibrium. To deal with a system out of equilibrium, we must use a different device for each case, in contrast to the method of statistical thermodynamics, which is based on the general relation between the Helmholtz free energy and the partition function of the system. 

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