Generalized canonical ensembles and ensemble equivalence

This paper is a companion piece to our previous work [J. Stat. Phys. 119, 1283 (2005)], which introduced a generalized canonical ensemble obtained by multiplying the usual Boltzmann weight factor $e^{-\beta H}$ of the canonical ensemble with an exponential factor involving a continuous function $g$ of the Hamiltonian $H$. We provide here a simplified introduction to our previous work, focusing now on a number of physical rather than mathematical aspects of the generalized canonical ensemble. The main result discussed is that, for suitable choices of $g$, the generalized canonical ensemble reproduces, in the thermodynamic limit, all the microcanonical equilibrium properties of the many-body system represented by $H$ even if this system has a nonconcave microcanonical entropy function. This is something that in general the standard $(g=0)$ canonical ensemble cannot achieve. Thus a virtue of the generalized canonical ensemble is that it can often be made equivalent to the microcanonical ensemble in cases in which the canonical ensemble cannot. The case of quadratic $g$ functions is discussed in detail; it leads to the so-called Gaussian ensemble.

Figure 1

Geometric interpretation of supporting lines in relation to the graph of the microcanonical entropy function $s\left(u\right)$ (full line) and its concave envelope or concave hull $s^{**}\left(u\right)$ (dashed line). The point $a$ in the figure has the property that $s\left(u\right)$ admits a supporting line at $a$; i.e., there exists a line passing through $(a,s\left(a\right))$ that lies above the graph of $s\left(u\right)$. In this case, $s\left(a\right)=s^{**}\left(a\right)$. The point $b$ in the figure has the property that $s\left(u\right)$ admits no supporting line at $b$. In this case $s\left(b\right)\ne s^{**}\left(b\right)$.

Figure 2

Free-energy function $\phi \left(\beta \right)$ associated with the nonconcave entropy function $s\left(u\right)$ shown in Fig. 1. The region of nonconcavity of $s\left(u\right)$ is signaled at the level of $\phi \left(\beta \right)$ by the appearance of a point ${\beta }_c$ where $\phi \left(\beta \right)$ is nondifferentiable. ${\beta }_c$ equals the slope of the affine part of $s^{**}\left(u\right)$, while the left and right derivatives of $\phi$ at ${\beta }_c$ equal $u_h$ and $u_l$, respectively.

Figure 3

Schematic illustration of the effect of $g$ on the entropy and free-energy functions. (a) Initial entropy $s\left(u\right)$ and its corresponding free energy $\phi \left(\beta \right)$ (see Figs. 1, 2). (b) Modified entropy $s_g\left(u\right)$ having a smaller region of nonconcavity than $s\left(u\right)$, and its corresponding generalized free energy ${\phi }_g\left(\alpha \right)$. (c) A modified entropy $s_g\left(u\right)$ rendered fully concave by $g$; its corresponding generalized free energy ${\phi }_g\left(\alpha \right)$ is everywhere differentiable.

Figure 4

Example of a point of $s\left(u\right)$ which does not admit a supporting line but admits a supporting parabola. Such a point is accessible to the Gaussian ensemble but not to the canonical ensemble.