Non-Hydrodynamic Limit Dynamics

Abstract

There are many ways to scale such that in the limit the dynamics becomes deterministic. The hydrodynamic limit is singled out from all other possibilities by considering large space-time only. The interaction between particles remains the same and unchanged at every stage of the scaling limit The relative scale between space and time depends on the physical problem under consideration. Mostly we have considered diffusive scaling, space as ε^-1, time as ε^-2. We learned that close to a point of second order phase transition (liquid-gas transition), if space is scaled as ε^-1, then time should be scaled as ε^-z with z = 4 − η with η the anomalous dimension. [The static structure function at the critical point diverges for small k as Ŝ(k) ≅ |k| ^-2+η.] If we would want to study droplet formation, to see droplets of size ε^-1 we have to wait a time of the order ε^-3, which fixes then the relative space-time scale.