For a system which may be partitioned into M subsystems A1, A2, …, AM, such that configurations of subsystems Ai, Aj correspond to realizations of discrete, random variables si, sj, not necessarily isomorphic, and such that the probability for a configuration of the total system is p(s1, s2, …, sM) = f1(s1) … fM(sM)g1(s1, s2) × g2(s2, s3) … gM−1(sM−1, sM), we prove that p(s1, s2, …, sM) = p1(s1 | s2)p2(s2 | s3) … pM−1(sM−1 | sM)pM(sM), where pi(si | si+1) is the conditional probability for si given si+1, and pi(si) is the reduced probability for si. This result yields a decomposition of the total entropy into ``single‐subsystem entropies'' and ``nearest‐neighbor'' cumulant‐like terms only. This Markovian decomposition applies to systems with short‐range interactions for which transfer‐matrix methods are introduced.
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Research Article|
October 31 2003
Entropy Decomposition and Transfer‐Matrix Problems
H. Falk
H. Falk
Department of Physics, City College of the City University of New York, New York, New York 10031
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J. Math. Phys. 13, 608–609 (1972)
Article history
Received:
November 11 1971
Citation
H. Falk; Entropy Decomposition and Transfer‐Matrix Problems. J. Math. Phys. 1 May 1972; 13 (5): 608–609. https://doi.org/10.1063/1.1666023
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