A geometric, dynamical approach to thermodynamics

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Abstract
We present a geometric and dynamical approach to the micro-canonical ensemble of classical Hamiltonian systems. We generalize the arguments in Rugh (1997 Phys. Rev. Lett. 78 772-4) and show that the energy-derivative of a micro-canonical average is itself micro-canonically observable. In particular, temperature, specific heat and higher-order derivatives of the entropy can be observed dynamically. We give perturbative, asymptotic formulae by which the canonical ensemble itself can be reconstructed from micro-canonical measurements only. Using our geometrical tools we rederive formulae by Lebowitz et al and Pearson et al, relating, for example, specific heat to fluctuations in the kinetic energy. We show that under natural assumptions on the fluctuations in the kinetic energy the micro-canonical temperature is asymptotically equivalent to the standard canonical definition using the kinetic energy.