  In this work we study variational properties of approximate solutions of
scalar conservation laws. Solutions of this type are described by a kinetic
equation which is similar to the kinetic representation of admissible weak
solutions due to Lions-Perthame-Tadmor, but also retain small scale
non-equilibrium behavior. We show that approximate solutions can be obtained
from a BGK-type equation with equilibrium densities satisfying Gibb's entropy
minimization principle.
