The functional generalization of the Boltzmann-Vlasov equation and its Gauge-like symmetry

We argue that one can model deviations from the ensemble average in non-equilibrium statistical mechanics by promoting the Boltzmann equation to an equation in terms of {\em functionals} , representing possible candidates for phase space distributions inferred from a finite observed number of degrees of freedom. We find that, provided the collision term and the Vlasov drift term are both included, a gauge-like redundancy arises which does not go away even if the functional is narrow. We argue that this effect is linked to the gauge-like symmetry found in relativistic hydrodynamics \cite{bdnk} and that it could be part of the explanation for the apparent fluid-like behavior in small systems in hadronic collisions and other strongly-coupled small systems\cite{zajc}. When causality is omitted this problem can be look at via random matrix theory show, and we show that in such a case thermalization happens much more quickly than the Boltzmann equation would infer. We also sketch an algorithm to study this problem numerically