Generalized Thermodynamics: The Thermodynamics of Irreversible Processes and Generalized Hydrodynami

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Having thus briefly reviewed the fundamentals of the method we proceed next to its application to the derivation of a hydrodynamic formalism. Let us consider a fluid of N interacting particles whose Hamiltonian is of the form of Eq. Further, n 1 and n 2 are the one-particle and two-particles dynamical reduced density functions, namely.

The set of Eq. Therefore, in terms of this basic set of variables the auxiliary coarse-grained nonequilibrium statistical distribution for the system [cf. These parameters are to be identified in the context of hydrodynamics later on.

The basic set of macrovariables extensive nonequilibrium thermodynamic variables are. Let us next consider the equations of evolution for the basic variables, that is Eqs. First it should be noticed that h o is simply proportional to f 1 and so we need to derive the equations for f 1 and f 2. Before proceeding further in the direction of building a hydrodynamic approach three facts concerning the equations of evolution 47 and 48 must be emphasized.

First they are not closed since they involve the three-particle distribution function on the right hand side of Eq. If we incorporate n 3 as a basic variable it leads to an equation of evolution containing n 4 , which in turn if incorporated as a basic variable implies another equation for n 5 and so on. This question will be presented in a future article.

Second, if in Eq.

Third, once the basic set of variables has been chosen, in this case, n 1 and n 2 , to close the system of coupled equations 47 and 48 the method requires to express f 3 and 2 in terms of them. This can be done resorting, for example, to a perturbation method for averages [22], as it was illustrated in previous articles [40, 42]. Let us next consider how to construct a hydrodynamic formalism from Eqs. Let us consider now the steps leading to the equations of hydrodynamics to compare them with the phenomenological classical ones.

For this purpose we introduce: Resorting to the use of Eqs. Moreover, we introduce the quantity. We call the attention to the fact that the quantities above depend on f 1 and f 2 , or, alternatively, on the thermodynamic variables Lagrange multipliers j 1 and j 2. The results embedded in eqs. They were derived first by Kirkwood in [62] using a time smoothing approximation and later on, in by Irving and Zwanzig [63]. Here we have followed very closely the procedure of ref. The main accomplishment of this treatment lies in the fact that our formalism not only allows for the derivation of the hydrodynamic, or better, the conservation equations, but also yields the thermodynamical aspects consistent with then.

In fact we have an explicit form both for an entropy and an entropy production provided by the method itself so we can now proceed to study these quantities. For this purpose we introduce a NSOM entropy density of the form,.

This result is important and interesting enough by itself. Indeed it shows how one can obtain an explicit expression for the entropy consistent with the conservation equations expressed in terms of the first two distribution functions. We are now able to show that the NSOM conservation equations so far developed contain as a limiting case the usual ones arising from the local equilibrium assumption. For that purpose we compare the entropy density, given in the local equilibrium approximation of CIT, namely [18]. The lower index naught indicates the local equilibrium approximation and m o is the chemical potential.

We can thus see that if we write. Since classical hydrodynamics follows from these conservation equations when coupled to the well known constitutive equations, we may conclude that the correction terms involved in D f 1 , D f 2 and similar quantities will lead to a more general version of hydrodynamics which will demand constitutive laws different from the linear ones.

This subject is discussed in a forthcoming paper. Returning to the original expression for the auxiliary coarse grained NSD of Eq. In fact, it expresses the general entropy production that in our approach is consistent with the conservation equations, Eqs. This is a new and interesting result. For if we want to compute hydrodynamic equations from Eqs. In this sense this paper sets the microscopic basis for hydrodynamic equations that are valid beyond LIT. The explicit form for these equations and their generating constitutive laws is the subject of a forthcoming paper.

On the basis of the above paragraph, if in all terms in Eq. Here, we recall, index naught means averages performed with the use of the NSD of Eq. It should be noticed that the expressions derived so far depend on both the extensive and intensive macrovariables, P j and F j respectively, which are related by the nonequilibrium equations of state, Eqs. We proceed next to make explicit that dependence. For that purpose we note that the exponent in the NSD of Eq.

On the basis of this separation we can apply the Heims-Jaynes perturbation expansion method for averages [64], to find that.

Next, we introduce a Boltzmann-like approach, as the one described in reference [42], namely, we first put into evidence in the form of a series expansion the effect of two-particle correlations, namely the part in the exponent of the CHD-NSD that contains n 2. Again we resort to Heims-Jaynes' method [64] to obtain that. Performing the calculation in Eq. As a final note in this section we stress that, according to Eqs. In the present paper we have derived a hydrodynamic description of a fluid of particles interacting through a central force while under, in principle, general nonequilibrium conditions.

This was done in the spirit of the IST based on the nonequilibrium statistical operator method, a formalism which appears as belonging to the realm of Jaynes' Predictive Statistical Mechanics involving Bayesian and maximum informational-entropy methods. The description of the macroscopic state of the system is done in terms of the single- and two-particle reduced dynamic distribution functions, which, in principle, allow to express any dynamic observable of relevance in terms of them. The equations of evolution for the nonequilibrium averages of those quantities, namely, the density distribution functions, were obtained within the nonlinear generalized transport theory that the MaxEnt-NSOM provides.

These equations were next used to obtain the equations of evolution for the usual hydrodynamic variables, namely the particle density, momentum density, and internal energy density. In that way there followed their balance equations including dissipative sources. We call the attention to the fact that these equations involve the three-particle distribution function, which if it is incorporated as a basic variable would lead us to a generalization to arbitrary nonequilibrium condition of the BBGKY hierarchy.

To introduce only n 1 and n 2 implies the need of a truncation procedure necessary for the practical use of the method as discussed elsewhere [40] in a way reminiscent of a truncation of Grad's moments method. At this point, once in possession of the general expressions we have derived, within the framework of IST, we have made contact with classical conservation equations based on CIT. We showed that in fact a particular choice of the MaxEnt-NSOM Lagrange multipliers, what is equivalent to a NSOM approach that includes as basic variables only the particle density, momentum density, and energy density, leads to the well known results of classical hydrodynamics, mainly to Gibbs relation defining local equilibrium [Cf.

Hence, on the one hand, this proves that essentially classical hydrodynamics is contained in IST as a limiting case, and on the other that the method provides a way to go beyond the limitations of CIT.

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The procedure developed in section III evidences the contributions that are beyond the local equilibrium and other restrictions imposed in CIT [Cf. Of course in Eqs. One way to do that is to introduce the fluxes, that is, all higher order fluxes are to be incorporated to the basic set of variables to be used to describe the macroscopic state of the system and its evolution in arbitrarily far away from equilibrium conditions. This includes nonlocality and retro-effects, thus allowing to deal with short wavelength and high frequency phenomena in a generalized extended hydrodynamics in, eventually, far-from-equilibrium conditions.

Also, the constitutive equations of classical hydrodynamics that lead to the uncomfortable question of propagation of thermal perturbations at infinite speed become generalized equations which in restricted conditions first order extension of CIT are of the hyperbolic type in the form of the so called telegraphist equation [40, 41].

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Next step is of course a full development of these subject, presently under way and to be reported in future articles. To obtain a general relation [except the implicit ones given by Eqs.

Generalized Thermodynamics: The Thermodynamics of by Byung Chan Eu PDF

We used a quite simple one consisting in, as already noted, a truncated description that would produce classical conservation equation [Cf. Next, neglecting two-particle correlation [Cf. Local and time-dependent temperature, chemical potential, drift velocity, and pressure, can be defined and the average kinetic energy satisfies, locally, the equipartition theorem.

As noted before, the extension of these concepts, mainly that of nonequilibrium temperature, entropy and entropy production under quite general dissipative conditions is an open question. As final words, we comment that the MaxEnt-NSOM, which possesses a remarkable compactness, having by far a most appealing structure, and based on reasonable sound principles [14,,] offers a very effective method for dealing with nonlinear and nonlocal in space and time irreversible processes in far-from-equilibrium conditions in many-body systems. As noticed, it provides microscopic foundations to IST, and in this paper we have stressed the fact that it provides a seemingly far-reaching-generalized hydrodynamics.

We have shown how classical conservation equations are recovered when imposing very limiting restrictive conditions, bringing into evidence - at least in a general way - the contributions to be expected to allow for going outside the domain of validity of CIT. In a sense this anticipates that besides the quasi-conserved quantities basically density and energy density one needs to introduce, to an appropriate degree depending on the problem in hands, relaxing fluxes of ever increasing order. Since the seventies we are having a proficous colaboration, that, we hope, will extend into next coming century and millenium.

Anderson, Science , ; Physics Today, July , pp. Addison Wesley, Readings, MA, North Holland, Amsterdam, Today 25 11 , 23 ; ibid. Stengers, Order out of the Chaos: Prigogine, From Being to Becoming: Ziman, Electrons and Phonons: Nauka, Moscow, ]. Also the recently published, D. Dorffman, in XIV Int. Bogoliubov, Lectures in Quantum Statistics , Vols. Japan 33 , ; M. Japan 18 , North Holland, Amsterdam, , pp.

JETP 26 , Garcia-Colin, Physica A , Operational Approaches , H. Grandy, Foundations of Statistical Mechanics , Vol. Kluwer, Dordrecht, , pp. Luzzi, Physica A , Fluids 2 , ; Trans. Mexico 34 , ; Acta Phys. Hungarica 66 , 79 A 21 , A 42 , ; S. A 43 , ; ibid. Abstract In this article the thermodynamically consistent formulation of generalized hydrodynamics is reviewed and applications to shock—wave structures, ultrasonic wave absorption and dispersion and microchannel flows of the generalized hydrodynamics so formulated are discussed. Mathematical, Physical and Engineering Sciences username.

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