Condensed Matter Seminar: Tempered diffusion and some other joyful flux-limited processes

Abstract:

The classical transport theory as expressed by, say, the Fokker-Planck equation, lives in an analytical paradise, but in sin. Not only its response to initial datum spreads at once everywhere, oblivious of the basic tenets of physics, but it also induces an infinite flux across a sharp interface. Attempting to overcome these difficulties one notices that the moment expansion of any of the micro ensembles of the kind that beget the equations of the classical mathematical physics, say the Chapman-Enskog expansion of Boltzmann Eq., if extended beyond the second moment, yields an ill posed PDE (the Pawula Paradox)!  

We shall describe a number of mathematical strategies to overcome these generic difficulties. The resulting flux-limited transport equations are well posed and capture some of the crucial effects of the original ensemble lost in moment expansion. For instance, initial discontinuities do not dissolve at once but persist for a while. Landau-Ginzburg Free Energy functional may sustain finite jumps. There is a critical transition from analytical to discontinuous states with embedded sub-shock(s). Moreover, both convection-diffusion and reaction-diffusion processes admit finite sub-shocks which emerge whenever upstream downstream disparity crosses a critical threshold. 

Event Organizer: Prof. Sasha Gerber