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Micro-canonical thermodynamics: Why does heat flow from hot to cold

We show how to use a central limit approximation for additive co-cycles to describe non-equilibrium and far from equilibrium thermodynamic behavior. We consider first two weakly coupled Hamiltonian dynamical systems initially at different micro-canonical temperatures. We describe a stochastic model where the energy-transfer between the two systems is considered as a random variable satisfying a central limit approximation. We show that fluctuations in energy observables are linearly related to the heat-transfer (dissipation). As a result, on average, heat flows from hot to cold. We also consider the far from equilibrium situation of a non-Hamiltonian thermostatted system as in Evans et al. {\em Phys.\ Rev.\ Lett.} {\bf 71}, 2401 (1993). Applying the same central limit approximation we re-derive their relation for the violation of the 2nd law of thermodynamics. We note that time-reversal symmetry is not used in our derivation.