How low is the thermodynamical limit?

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We analyze how can some dynamical process lead to (almost) exponential distribution of hadrons without instantaneous equipartition with a heat bath. We present a model for parton dressing whic re-combines the exponential from cut power law minijet distributions as a limiting distribution. Thermal models and power law tails are interpreted within the framework of the more general Tsallis distribution.
Talk presented at NATO School "Structure and Dynamics of Elementary Matter", Kemar, Turkey, Sept.22.-Oct.2, 2003 by T.S.Biro. LateX, special style file, 4 extra .ps files

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