On completeness of description of an equilibrium canonical ensemble by reduced s-particle distribution function

In this article it is shown that in a classical equilibrium canonical ensemble of molecules with $s$-body interaction full Gibbs distribution can be uniquely expressed in terms of a reduced s-particle distribution function. This means that whenever a number of particles $N$ and a volume $V$ are fixed the reduced $s$-particle distribution function contains as much information about the equilibrium system as the whole canonical Gibbs distribution. The latter is represented as an absolutely convergent power series relative to the reduced $s$-particle distribution function. As an example a linear term of this expansion is calculated. It is also shown that reduced distribution functions of order less than $s$ don't possess such property and, to all appearance, contain not all information about the system under consideration.Comment: This work was reported on the International conference on statistical physics "SigmaPhi2008", Crete, Greece, 14-19 July 200

Best approximation by downward sets with applications

We develop a theory of downward sets for a class of normed ordered spaces. We study best approximation in a normed ordered space X by elements of downward sets, and give necessary and sufficient conditions for any element of best approximation by a closed downward subset of X. We also characterize strictly downward subsets of X, and prove that a downward subset of X is strictly downward if and only if each its boundary point is Chebyshev. The results obtained are used for examination of some Chebyshev pairs (W,x), where x E X and W is a closed downward subset of X.C

Markovian Equilibrium in Infinite Horizon Economies with Incomplete Markets and Public Policy

We develop an isotone recursive approach to the problem of existence, computation, and characterization of nonsymmetric locally Lipschitz continuous (and, therefore, Clarke-differentiable) Markovian equilibrium for a class of infinite horizon multiagent competitive equilibrium models with capital, aggregate risk, public policy, externalities, one sector production, and incomplete markets. The class of models we consider is large, and examples have been studied extensively in the applied literature in public economics, macroeconomics, and financial economics. We provide sufficient conditions that distinguish between economies with isotone Lipschitizian Markov equilibrium decision processes (MEDPs) and those that have only locally Lipschitzian (but not necessarily isotone) MEDPs. As our fixed point operators are based upon order continuous and compact non-linear operators, we are able to provide sufficient conditions under which isotone iterative fixed point constructions converge to extremal MEDPs via successive approximation. We develop a first application of a new method for computing MEDPs in a system of Euler inequalities using isotone fixed point theory even when MEDPs are not necessarily isotone. The method is a special case of a more general mixed monotone recursive approach. We show MEDPs are unique only under very restrictive conditions. Finally, we prove monotone comparison theorems in Veinott's strong set order on the space of public policy parameters and distorted production functions