On the origin of power laws in equilibrium

A particle in the attractive Coulomb field has an interesting property: its specific heat is constant and negative. We show, both analytically and numerically, that when a classical Hamiltonian system stays in weak contact with one such negative specific heat object, its statistics conforms to a fat-tailed power-law distribution with power index given by C/kB − 1, where kB is the Boltzmann constant and C is the heat capacity. Copyright c © EPLA, 2012 Introduction. – In 1968 Lynden-Bell and Wood [1] pointed out an interesting fact: self-gravitating systems have negative specific heats. As Lynden-Bell later on explained [2] this fact seemed quite natural to astronomers, who know for example that when a star gains energy it expands and cools down [3]; but it appeared incongruous, if not completely wrong, to statistical mechanists, who learn from textbooks that specific heats are necessarily positive. The paradox was resolved by Thirring [4]: While it is true that any system in weak contact with a thermal bath, hence characterized by the canonical distribution, necessarily has a positive specific heat, isolated systems, characterized by the microcanonical distribution, may well have negative specific heats. Since then, many models showing microcanonical negative specific heats have been reported, see, e.g., [5–11], their statistical mechanics has been discussed [12,13], and negative specific heats have been experimentally measured in thermally isolated small atomic clusters [14,15]. Recently it has also been pointed out that if a system is in strong coupling with its environment, likewise it may display negative specific heat [16–23]. Inspired by ref. [24], here we consider an ordinary system S with Hamiltonian HS(x,p) and weakly couple it via an interaction energy h(x,X), to a second system with Hamiltonian HC(X,P), possessing a constant negative microcanonical heat capacity C < 0, i.e., H(x,p,X,P) =HS(x,p)+HC(X,P)+h(x,X) . (1) For the sake of clarity we recall that the heat capacity C is defined as the derivative of a system’s energy with respect to the temperature, C = ∂E/∂T , while the specific heat c is defined as the heat capacity per unit mass. The archetypical example of a system with constant negative microcanonical heat capacity is that of a single particle in the gravitational (or the attractive Coulomb) force field, for which C =−3/2 [2]. Throughout the paper, temperature is expressed in units of energy. In these units kB , Boltzmann’s constant, is dimensionless and equal to 1, and the heat capacity C = ∂E/∂T is dimensionless as well. Our main result is that, provided the total system samples the microcanonical ensemble, the system S samples the power-law distribution p(x,p) = [HS(x,p)−Etot] ∫ dxdp[HS(x,p)−Etot]C−1 , C < 0, (2) where Etot is the (conserved) energy of the total system. To express this result in eq. (2) in the usual set of units, where the temperature is measured in Kelvin, and both the heat capacity and kB have the dimension of Joule/Kelvin, one should replace C in eq. (2) with the dimensionless ratio C/kB . We illustrate this result in fig. 1. A neutral particle, with Hamiltonian HS = p /2m, is confined into a box and makes elastic collisions with a particle carrying a charge q subject to the Coulomb field generated by a fixed charge −q. The charged particle has constant negative heat capacity C =−3/2, and its Hamiltonian reads: HC=−3/2(X,P) =P /2M −α/|X| , (3) where α= q/4πε0 with ε0 the dielectric permittivity of vacuum. According to our main result, eq. (2), since