Thermodynamic variation principle

Gage, D. H., Schiffer, M., Kline, S. J. and Reynolds, W. C., The non-existence of a general thermodynamic variational principle, in "Non-Equilibrium Thermodynamics, Variational Techniques and Stability" (R. J. Donnelly, R. Herman, I. Prigogine, Eds.), p. 286. University of Chicago Press, Chicago (1966). 

Just as the perturbation theory described in the previous section, the self-consistent phonon (SCP) method applies only in the case of small oscillations around some equilibrium configuration. The SCP method was originally formulated (Werthamer, 1976) for atomic, rare gas, crystals. It can be directly applied to the translational vibrations in molecular crystals and, with some modification, to the librations. The essential idea is to look for an effective harmonic Hamiltonian H0, which approximates the exact crystal Hamiltonian as closely as possible, in the sense that it minimizes the free energy Avar. This minimization rests on the thermodynamic variation principle ... 

Just as the self-consistent phonon method, the mean field approximation (Kirkwood, 1940 James and Keenan, 1959) is based on the thermodynamic variation principle for the Helmholtz free energy ... 

In this article we have used some of the concepts of quantum-statistical mechanics. These concepts can, of course, be found in the textbooks (Ter Haar, 1966 Feynman, 1972 McQuarrie, 1976), but the ideas that are most relevant to this paper are summarized in this appendix. In particular, we prove the thermodynamic variation principle, which has been applied several times. 

Taking the logarithm of this inequality and multiplying by -/3-1 yields the thermodynamic variation principle, Eq. (A.4). 

Another technique to obtain the effects of the anharmonic terms on the excitation frequencies and the properties of molecular crystals is the Self-Consistent Phonon (SCP) method [71]. This method is based on the thermodynamic variation principle, Eq. (14), for the exact Hamiltonian given in Eq. (10), with the internal coordinates not explicitly considered. As the approximate Hamiltonian one takes the harmonic Hamiltonian of Eq. (18). The force constants in Eq. (18) are not calculated at the equilibrium positions and orientations of the molecules as in Eq. (19), however. Instead, they are considered as variational parameters, to be optimized by minimization of the Helmholtz free energy according to Eq. (14). The optimized force constants are found to be the thermodynamic (and thus temperature dependent) averages of the second derivatives of the potential over the (harmonic) lattice vibrations ... 

Just as the SCP method, the Mean Field (MF) method [74, 75] is based on the thermodynamic variation principle, Eq. (14). This time, however, the approximate Hamiltonian //q is chosen to be the sum of effective single-particle Hamiltonians ... 

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... 

I. Gyarmati, Nonequilibrium Thermodynamics, Field Theory, and Variational Principles, Springer, Berlin, 1970. 

Statistical Mechanics and Thermodynamics of Irreversible Processes, Variational Principles in (Ono). ... 

In an important paper (TNC.l), they offered for the first time an extension of nonequilibrium thermodynamics to nonlinear transport laws. As could be expected, the situation was by no means as simple as in the linear domain. The authors were hoping to find a variational principle generalizing the principle of minimum entropy production. It soon became obvious that such a principle cannot exist in the nonlinear domain. They succeeded, however, to derive a half-principle They decomposed the differential of the entropy production (1) as follows ... 

TNC.15. I. Prigogine, Evolution Criteria, Variational principles and fluctuations, in Nonequilibrium Thermodynamics, Variational Techniques and Stability, University of Chicago, 1966, pp. 3-16. 

The second law of thermodynamics states that an isolated system in equilibrium has maximum entropy. This is the basis for a variational principle often used in determining the equilibrium state of a system. When the system contains several elements which are allowed to exchange mass with each other, the variational principle yields the condition that all elements must have equal chemical potential once equilibrium is established. 

In the 19th century the variational principles of mechanics that allow one to determine the extreme equilibrium (passing through the continuous sequence of equilibrium states) trajectories, as was noted in the introduction, were extended to the description of nonconservative systems (Polak, 1960), i.e., the systems in which irreversibility of the processes occurs. However, the analysis of interrelations between the notions of "equilibrium" and "reversibility," "equilibrium processes" and "reversible processes" started only during the period when the classical equilibrium thermodynamics was created by Clausius, Helmholtz, Maxwell, Boltzmann, and Gibbs. Boltzmann (1878) and Gibbs (1876, 1878, 1902) started to use the terms of equilibria to describe the processes that satisfy the entropy increase principle and follow the "time arrow."... 

To coordinate components, the generalized flows and the thermodynamic forces can be used to define the trajectories of the evolution of nonequilibriun systems in time. A trajectory specifies the curve represented by the flow and force components as a function of time in the flow-force space. A useful trajectory can be found and analyzed by a variation principle. In thermodynamics, the variation principles lead to the least energy dissipation and minimum entropy generation at steady states. According to the most general evolutionary criterion, open chemical reaction systems are dissipative, and evolve toward an asymptotic state in time. 

The ultimate goal of a thermodynamic description of molecular systems, however, is to determine the horizontal displacements of the electronic structure (see Section 7), i.e., transitions from one v-representable molecular density to another. In order to relate the information entropy H[p], possibly involving also the reference densities (equation (92)), to the system energetic parameters one uses the generalized variational principle in the entropy representation [108] ... 

The equalization principle for the electronic chemical potential (equivalently, the electronegativity equalization principle) may be couched in a form reminiscent of the argument from classical thermodynamics [4], However, the chemical potential equalization principle follows most directly from the variational principle and, in particular, Eq. (32). First, define the local chemical potential by... 

The fundamental stability conditions in thermodynamics are formulated as variational principles. Within the zero-temperature canonical ensemble, the quantities n and V are used to specify the state of interest. Suppose one chooses a nonoptimum pressure, P(r). For example, we can divide the system with a partition and place Maxwell s demon at the door between the partitions to ensure that the pressure on one side of the partition is greater than that on the other side of the partition. Then, we have that ... 

I. Gyamati, Non-Equilibrium Thermodynamics Field Hieory and Variational Principles , Springer-Verlag, Beilin, 1970. 

Tolmachev, V. V. 1960. Relationship between the statistic variation principle and the method of forming partial sums for diagrams in the thermodynamic perturbation theory for a modified statement of the problem of Bose-Einstein non-ideal system. Doklady Akademii Nauk USSR. 134, 1324. 

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