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Non-equilibrium statistical mechanics is formulated from the non-equilibrium probability density in phase space. This is the exponential of the associated reservoir entropy, which is shown to consist of a static and a dynamic part. Both the fluctuation form and the explicit formally exact form involving the prior work are given. The second entropy is used to analyse the transition probability, the stochastic dissipative equations of motion, and the time correlation function. The non-equilibrium fluctuation dissipation theorem is established for phase space. The non-equilibrium probability density is shown to obey explicitly the Fokker-Planck equation. Irreversibility and the change in entropy on a trajectory are discussed. The odd projection of the dynamic part of the reservoir entropy is analysed in detail. An expression for the path entropy is obtained, and from this the fluctuation and work theorems are derived.

*Keywords: *
non-equilibrium;
statistical mechanics;
transition probability;
stochastic;
trajectory;
fluctuation dissipation theorem;
work theorem

*Chapter.*
*37117 words.*
*Illustrated.*

*Subjects: *
Condensed Matter Physics

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