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Dynamics of Macrosystems: Proceedings of a Workshop on the by Elie Bienenstock (auth.), Prof. Dr. Jean-Pierre Aubin, Prof.

By Elie Bienenstock (auth.), Prof. Dr. Jean-Pierre Aubin, Prof. Dr. Donald Saari, Prof. Dr. Karl Sigmund (eds.)

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Read or Download Dynamics of Macrosystems: Proceedings of a Workshop on the Dynamics of Macrosystems Held at the International Institute for Applied Systems Analysis (IIASA), Laxenburg, Austria, September 3–7, 1984 PDF

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Extra info for Dynamics of Macrosystems: Proceedings of a Workshop on the Dynamics of Macrosystems Held at the International Institute for Applied Systems Analysis (IIASA), Laxenburg, Austria, September 3–7, 1984

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B) F/2 < TE q < TE P < F and hence E q < E P < 2E q for all (E,A,~) E (0,00)3 with A > ~ This inequality may be interpretated as qualitative equipartition of the "energy" = Ep + E up to the q factor of 2 even in situations where the parameters A and (c) ~ E differ by orders of magnitude. F(E,~,~) < F(E,A,~) < F(E,A1/3~2/3,A1/3~2/3) for all (E,A,~) E' (0,00)3 with A > ~ 6. THE CANONICAL ENSEMBLE The canonical ensemble is defined as the Boltzmann distribution in the (p,q)-phase space, which is g(p,q) = Z(BA)-1z(B~)-1 exp[-BH(p,q)] Table 2 summarizes the calculations for the symmetric case A = ~ 1, from which the general case is straightforward to obtain.

In case of (12) one gets an integral and a sum of residua. Furtheron,. as shown in a forthcoming paper [6], these two parts coincide with ,+(E) and ,_(E), respectively. Hence one gets oo , , + (E) Jo (E) 1 + e -sE (sz(s» -1 (14) ds 00 L e - kE (k Z (k) ) -1 (15 ) k=1 We turn to expansions in two limiting cases. Small values of E correspond to low temperature and large s. Large values of E correspond to high temperature and small s. First turn to the low temperature case. g. 3) one can transform back this series term by term and

16: 103-112. Okubo, A. (1980). Diffusion and Ecological Problems: Mathematical Models. Springer-Verlag, Berlin. , and Wolff, R. (1974). Dynamical systems under constant organization. III. Cooperative and competitive behaviour of hypercycles. J. Differential Equations. 32: 357-368. Sigmund, K. and Schuster, P. (1984). Permanence and uninvadibility for deterministic population models. In Stochastic Phenomena and Chaotic Behaviour in Complex Systems. Springer-Verlag, Berlin. Smoller, J. (1982). Shock Waves and React{on-Diffusion Equations.

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