Specific heat and entropy of N-body nonextensive systems

Tsallis power laws and finite baths with negative heat capacity

It is often stated that heat baths with finite degrees of freedom i.e., finite baths, are sources of Tsallis distributions for classical Hamiltonian systems. By using well-known fundamental statistical mechanics expressions, we rigorously show that Tsallis distributions with fat tails are possible only for finite baths with constant negative heat capacity, while constant positive heat capacity finite baths yield decays with sharp cutoff with no fat tails. However, the correspondence between Tsallis distributions and finite baths holds at the expense of violating the equipartition theorem for finite classical systems at equilibrium. We comment on the implications of the finite bath for the recent attempts towards a q-generalized central limit theorem.

Classical small systems coupled to finite baths

We have studied the properties of a classical N(S)-body system coupled to a bath containing N(B)-body harmonic oscillators, employing an (N(S)+N(B)) model that is different from most of the existing models with N(S)=1. We have performed simulations for N(S)-oscillator systems, solving 2(N(S)+N(B)) first-order differential equations with N(S)≃1-10 and N(B)≃10-1000, in order to calculate the time-dependent energy exchange between the system and the bath. The calculated energy in the system rapidly changes while its envelope has a much slower time dependence. Detailed calculations of the stationary energy distribution of the system f(S)(u) (u: an energy per particle in the system) have shown that its properties are mainly determined by N(S) but weakly depend on N(B). The calculated f(S)(u) is analyzed with the use of the Γ and q-Γ distributions: the latter is derived with the superstatistical approach (SSA) and microcanonical approach (MCA) to the nonextensive statistics, where q stands for the entropic index. Based on analyses of our simulation results, a critical comparison is made between the SSA and MCA. Simulations have been performed also for the N(S)-body ideal-gas system. The effect of the coupling between oscillators in the bath has been examined by additional (N(S)+N(B)) models that include baths consisting of coupled linear chains with periodic and fixed-end boundary conditions.