  We consider a simple one dimensional stochastic model of heat transport which
locally conserves both energy and momentum and which is coupled to heat
reservoirs with different temperatures at its two ends. The steady state is
analyzed and the model is found to obey the Fourier law with finite heat
conductivity. In the infinite length limit, the steady state is described
locally by an equilibrium Gibbs state. However finite size corrections to this
local equilibrium state are present. We analyze these finite size corrections
by calculating the on-site fluctuations of the momentum and the two point
correlation of the momentum and energy. These correlations are long ranged and
have scaling forms which are computed explicitly. We also introduce a
multi-lane variant of the model in which correlations vanish in the steady
state. The deviation from local equilibrium in this model as expressed in terms
of the on-site momentum fluctuations is calculated in the large length limit.

Statistical Mechanics