A geometric, dynamical approach to thermodynamics

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.