Negative Specific Heat in the Canonical Statistical Ensemble

According to thermodynamics, the specific heat of Boltzmannian short-range interacting systems is a positive quantity. Less intuitive properties are instead displayed by systems characterized by long-range interactions. In that case, the sign of specific heat depends on the considered statistical ensemble: Negative specific heat can be found in isolated systems, which are studied in the framework of the microcanonical ensemble; on the other hand, it is generally recognized that a positive specific heat should always be measured in systems in contact with a thermal bath, for which the canonical ensemble is the appropriate one. We demonstrate that the latter assumption is not generally true: One can, in principle, measure negative specific heat also in the canonical ensemble if the system under scrutiny is non-Boltzmannian and/or out-of-equilibrium.

Figure 1

Lynden-Bell entropic curve, $s=s(e)$, calculated for the initial condition $f_0=0.1096$, in the framework of the microcanonical ensemble. Inset: Difference $\Delta s$ between magnetized and nonmagnetized branches of the entropic curve, as a function of energy. A change of sign in $\Delta s(e)$ is the signal of a phase transition; see also 16. There is a second-order phase transition at $e=0.6123$ and a first-order phase transition at $e=0.5987$. Note that the latter cannot be noticed on the (too large) scale used to plot the entropic curve.

Figure 2

(a) Kinetic temperature $T_{\mathrm{kin}}$ as a function of the Lynden-Bell Lagrange multiplier $\beta$, calculated for the initial condition $f_0=0.1096$. (b) Kinetic caloric curve, $1/T_{\mathrm{kin}}$ vs $e$ (continuous line), and thermodynamic caloric curve, $1/T_{\mathrm{th}}=\beta$ vs $e$ (dotted line), calculated at $f_0=0.1096$.