×

zbMATH — the first resource for mathematics

The macrodynamics of open systems and the variational principle of the local potential. I. (English) Zbl 0554.93004
The first part of this paper, which consists of two parts, presents a general phenomenological approach for a macroscopic description of the dynamics of open systems far from the equilibrium state. A class of ordinary differential macrodynamic equations is introduced and theorems of existence, uniqueness and stability of stationary states are proved. Singularly perturbed macrodynamic equations are considered. The procedure of simplification for these equations is defined and a theorem on proximity of solutions of the initial and the simplified systems is proved. The variational principle of the local potential for macrodynamic equations is formulated and studied. Iterative procedures for computing stationary states based on this principle are constructed. The second part of the paper is devoted to applications.
Reviewer: M.Rijckaert

MSC:
93A10 General systems
49S05 Variational principles of physics (should also be assigned at least one other classification number in Section 49-XX)
80A30 Chemical kinetics in thermodynamics and heat transfer
34D15 Singular perturbations of ordinary differential equations
37-XX Dynamical systems and ergodic theory
92Cxx Physiological, cellular and medical topics
82B35 Irreversible thermodynamics, including Onsager-Machlup theory
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Callen, H.B., Thermodynamics, (1962), Wiley New York
[2] Groot, S.R.; Mazur, P., Nonequilibrium thermodynamics, (1969), North-Holland Amsterdam
[3] Glansdorff, P.; Prigogine, J., Thermodynamic theory of structure, stability and fluctuations, (1971), Wiley London · Zbl 0246.73005
[4] Nicolis, G.; Prigogine, J., Self-organization in nonequilibrium systems, (1977), Wiley New York · Zbl 0363.93005
[5] Haken, H., Synergetics, (1978), Springer Berlin
[6] Ebeling, V., Structurbildung bei irreversiblen prozessen, (1976), Teubner Rostock
[7] Rozonoer, L.I., Irreversible thermodynamics far from equilibrium, (), 219-238
[8] Vasil’ev, V.M.; Volpert, A.I.; Hudjaev, S.I., On the method of quasistationary concentrations for chemical kinetic equations, J. computation. math. mathematical phys., 687-697, (1973), (in Russian).
[9] Horn, F.; Jackson, R., General mass action kinetics, Archive for rational mechanic and analysis, Vol. 47, No. 2, 81-116, (1972)
[10] Volter, B.V.; Salnikov, I.E., Stability of working regimes of chemical reactors, (1972), Himija Moscow, (in Russian).
[11] Coleman, B.D., On the stability of equilibrium states of general fluids, Archive for rational mechanic and analysis, Vol. 36, No. 2, 1-33, (1970) · Zbl 0211.28802
[12] Rockafellar, R.T., Convex analysis, (1970), Princeton University Press Princeton · Zbl 0202.14303
[13] Kantorovich, L.V.; Akilov, G.P., Functional analysis, (1977), Nauka Moscow, (in Russian). · Zbl 0555.46001
[14] Pontrjagin, L.S., Ordinary differential equations, (1970), Nauka Moscow, (in Russian).
[15] Gelfand, I.M., Lectures on linear algebra, (1971), Nauka Moscow, (in Russian).
[16] Alekseev, V.M.; Tihomirov, V.M.; Fomin, S.V., Optimal control, (1979), Nauka Moscow, (in Russian). · Zbl 0516.49002
[17] Vasil’eva, A.B.; Butuzov, V.F., Asymptotic expansions of solutions of singularly perturbed equations, (1973), Nauka Moscow, (in Russian). · Zbl 0364.34028
[18] Tikhonov, A.M., Systems of differential equations with small parameters, Mathematicheskii sbornik, Vol. 31/73, No. 3, 575-586, (1952), (in Russian).
[19] Hasegava, H., On the relation between the H-theorem and the principle of minimum entropy production, (), 55-71
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.