Nonequilibrium partition identity
(Carberry et al. 2004.) Thus in spite of the Second Law Inequality which might lead you to expect that the average would decay exponentially with time, the exponential probability ratio given by the FT exactly cancels the negative exponential in the average above leading to an average which is unity for all time!
The first derivation of the NonEquilibrium Partition Identity for Hamiltonian systems was by Yamada and Kawaski in 1967. For thermostatted deterministic systems the first derivation was by Morriss and Evans in 1985.
- K. Kawasaki and J.D. Gunton, Phys. Rev. A, 8, 2048 (1973)
- T. Yamada and K. Kawasaki, Prog. Theo. Phys., 38, 1031 (1967)
- G.P. Morriss and D.J. Evans,"Isothermal response theory", Molecular Physics, '54, 629 (1985).
- D. M. Carberry, S. R. Williams, G. M. Wang, E. M. Sevick and D. J. Evans, "The Kawasaki identity and the fluctuation theorem", Journal of Chemical Physics, 121, 8179 – 8182 (2004).
- Fluctuation theorem - Provides an equality that quantifies fluctuations in time averaged entropy production in a wide variety of nonequilibrium systems.
- Crooks fluctuation theorem -Provides a Fluctuation Theorem between two equilibrium states. ImpliesJarzynski equality.
- Jarzynski Equality on arxiv.org
- "Fluctuation-Dissipation: Response Theory in Statistical Physics" by Umberto Marini Bettolo Marconi, Andrea Puglisi, Lamberto Rondoni, Angelo Vulpiani, 
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