Symmetry and conservation laws alone yield nonlinear fluctuating hydrodynamics equations whose sound and heat modes both flow to a KPZ fixed point with dynamical exponent 3/2, confirmed by simulations matching the Prahofer-Spohn function.
Spohn, Generalized Gibbs ensembles of the classical Toda chain, J
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The Δ-model adds a fixed Δ to collision velocities in vibrated granular fluids to enable Enskog kinetic theory predictions of temperature, equation of state, Navier-Stokes coefficients, mixture nonequipartition, and unconditional stability of homogeneous states.