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On an operator identity central to projection operator methodology

Lamb, Wilson and Murdoch, Ian and Stewart, John (2001) On an operator identity central to projection operator methodology. Physica A: Statistical Mechanics and its Applications, 298 (1). pp. 121-139. ISSN 0378-4371

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Derivations of master equations using projection operator methodology are based upon an identity whose validity has been established in only limited contexts. Its proof requires precise definitions of eLt and , where L is the Liouville operator and P the projection operator associated with the limited system information of interest. Here, for interacting particles confined to a box, the existence and uniqueness of system dynamics is demonstrated. A distributional extension of L defined in an L1 space is derived for which is the corresponding updating operator. Attempts to define within current semigroup theory are outlined, and a possible future approach indicated.