Phase space hydrodynamics method for low dimensional systems: evolution of shock fronts and approach to thermalization

 I will describe a new method for treating aspects of time evolution of 1+1 dimensional gas of fermions (or hard-core bosons). The differential equations of conventional hydrodynamics, describing the time evolution of fermion density, fail to work beyond the instant when a shock front develops. It has been known, both from experiments with cold atoms as well as from quantum many-body simulations that these systems show thermalization at long times; because of the singularity mentioned above, conventional hydrodynamics fails to show such long term behaviour. We will present an analytic treatment (originally discovered in a string theory context) in terms of  phase space density which remains smooth even when shocks form (topologically,  the reason is that a shock in real space becomes a fold in phase space which is smooth). Using this method, we are able to give an analytic derivation of the thermalization and derive a set of universal exponents characterising the relaxation to thermal behaviour. We also discuss generalisations to some 2+1 dimensional systems.