  We introduce a dispersion approximation of weak, entropy solutions of
multidimensional scalar conservation laws using variational kinetic
representation, where equilibrium densities satisfy the Gibb's entropy
minimization principle for a piecewise linear, convex entropy. For such
solutions, we show that small scale discontinuities, measured by the entropy
increments, propagate with characteristic velocities, while the large scale,
shock-type discontinuities propagate with speeds close to the speeds of
classical shock waves. In the zero-limit of the scale parameter, approximate
solutions converge to a unique, entropy solution of a scalar conservation law.
