  We consider a class of stochastic models for energy transport and study
relations between the thermal conductivity and some static observables, such as
the static conductivity, which is defined as the contribution of static
correlations in Green-Kubo formula. The class of models is a generalization of
two specific models derived by Gaspard and Gilbert as mesoscopic dynamics of
energies for two-dimensional and three-dimensional locally confined hard-discs.
They claim some equalities hold between the thermal conductivity and several
static observables and also conjecture that these equations are universal in
the sense that they hold for mesoscopic dynamics of energies for confined
particles interacting through hard-core collisions. In this paper, we give
sufficient and necessary conditions for these equalities to hold in the class
we introduce. In particular, we prove that the equality between the thermal
conductivity and other static observables holds if and only if the model obeys
the gradient condition. Since the gradient condition does not hold for models
derived by Gaspard and Gilbert, our result implies a part of their claim is
incorrect.
