Quantum confinement and negative heat capacity

Thermodynamics dictates that the specific heat of a system is strictly non-negative. However, in finite classical systems there are well-known theoretical and experimental cases where this rule is violated, in particular finite atomic clusters. Here, we show for the first time that negative heat capacity can also occur in finite quantum systems. The physical scenario on which this effect might be experimentally observed is discussed. Copyright c © EPLA, 2013 Thermodynamics dictates that the specific heat of a system is strictly non-negative, implying that the addition (subtraction) of energy cannot result in a decrease (increase) of the system’s temperature [1]. Nevertheless there are well-known cases where this rule is apparently violated [2,3]. Systems in weak contact with a thermal bath, which are described by the Canonical ensemble, may not have a negative heat capacity. Systems that are not included in this condition, such as isolated systems that have to be described by the Microcanonical ensemble, may display negative heat capacity. For example, Schmidt et al. measured, in an elegant series of experiments, the negative heat capacity of a Na cluster with 147 atoms for temperatures neighboring the melting temperature of the cluster [4–6]. The origin of the negative heat capacity in clusters has been extensively examined by Berry and coworkers [7–12], and other isolated classical systems have been studied by Campisi et al. [13] and by Dunkel and Hilbert [14,15]. Other examples of negative heat capacity can be found in the literature, such as strongly coupled open quantum systems [16,17] and dissipative quantum systems [18,19]. At the astronomical scale negative heat capacities have been known for years [20], where it is observed that stars and star clusters increase their temperature as they age while losing energy by radiation [21]. Therefore, invoking the thermodynamic limit is not sufficient to guarantee the equivalence of Canonical and Microcanonical ensembles. The key to theoretically reconcile these results with the thermodynamics was addressed by Thirring and coworkers: A system may display negative heat capacity, even in the thermodynamics limit, provided that it is not ergodic [22–24]. In this letter we investigate whether this phenomenon could also be observed in the quantum domain, in particular, for a small isolated system described within the Microcanonical formalism. Following previous ideas [25,26] on a minimal model having negative specific heat for classical systems we study the effect of the delocalization of the wave function on the average kinetic energy of the system, and also on several definitions of temperature corresponding to the Canonical and Microcanonical statistics. Let us consider a 1d potential well trapped between impenetrable walls: V (x) = ⎪⎨ ⎪⎩ −U0, if |x| < a, 0, if a ≤ |x| < L, ∞, if |x| ≤ L. (1) This potential represents a simple example that suffices to show how a negative heat capacity emerges. The solution of the Schrödinger equation for one particle in V (x)