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Isaac Newton Institute for Mathematical Sciences

Picocanonical Ensembles: A theoretical description of metastable states

15th May 2006

Author: Professor D Dhar (Tata Institute of Fundamental Research)


The standard Boltzmann-Gibbs prescription for calculating macroscopic properties of matter in statistical physics involves first calculating the free energy from the partition function in the mico-canonical or canonical ensemble, and then taking suitable derivatives of free energy to determine quantities like specific heat or the pressure. This prescription fails completely to describe the properties of the window glass, or metastable phases like diamonds. The partition function, if correctly calculated, would only give the properties of quartz, or graphite. This is because glasses and diamond are only metastable phases, not "equilibrium states of matter", and the theoretically calculated probability that the equilibrium system will be found in such a state is very very small.

I will describe an alternate description of such states, as systems where the dynamics is not ergodic, and the system samples only a very small part of the full phase space, but can be said to be in thermal equilibrium over this set of states. These ensembles, much smaller than micro-canonical ensembles, will be called pico-canonical ensembles. The general idea of restricting the sum to only over states in the partition function is not new, but has been difficult of make precise in the past, as it is not easy to specify the set of accessible states over which the summation should be restricted. I illustrate the idea by showing how partition functions in these ensembles can be calculated explicitly in a simple one-dimensional model.