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 of the canonical ensemble with an exponential factor involving a continuous function of the Hamiltonian . 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 , the generalized canonical ensemble reproduces, in the thermodynamic limit, all the microcanonical equilibrium properties of the many-body system represented by even if this system has a nonconcave microcanonical entropy function. This is something that in general the standard 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 functions is discussed in detail; it leads to the so-called Gaussian ensemble.

Statistical equilibrium predictions of jets and spots on Jupiter.

An equilibrium statistical theory of coherent structures is applied to midlatitude bands in the northern and southern hemispheres of Jupiter. The theory imposes energy and circulation constraints on the large-scale motion and uses a prior distribution on potential vorticity fluctuations to parameterize the small-scale turbulent eddies. Nonlinearly stable coherent structures are computed by solving the constrained maximum entropy principle governing the equilibrium states of the statistical theory. The theoretical predictions are consistent with the observed large-scale features of the weather layer if and only if the prior distribution has anticyclonic skewness, meaning that intense anticyclones predominate at small scales. Then the computations show that anticyclonic vortices emerge at the latitudes of the Great Red Spot and the White Ovals in the southern band, whereas in the northern band no vortices form within the zonal jets. Recent observational data from the Galileo mission support the occurrence of intense small-scale anticyclonic forcing. The results suggest the possibility of using equilibrium statistical theory for inverse modeling of the small-scale characteristics of the Jovian atmosphere from observed features.