Muhittin Mungan, Bogazici University
Frenkel Kontorawa Models, Free Energy Recursions and Burgers Shocks
The Frenkel Kontorowa Model is a classical infinite chain of atoms linked by elastic springs with equilibrium spacing a that is subject to an external periodic potential of period b. The model is characterized by the competition of two different length scales, a and b. The lowest energy configurations of the chain have a very complex depedence on a/b and the relative strength of the underlying potential, giving rise to commensurate or incommensurate configurations, and transitions between these, as the parameters of the model are varied. The ground state configurations are also closely related to the unstable (hyperbolic) orbits of 2 dimensional hamiltonian maps such as the standard map. It is possible to approach the problem from a statistical mechanical point of view, constructing a transfer matrix that captures the evolution of the free-energy, as particles are added to the chain. Recursion equations for the evolution of the free-energy were derived by Griffiths et al. for the zero-temperature case and for the general case by Feigelman et al. One of the major advantages of the transfer matrix description is that besides the inclusion of non-zero temperature, this approach readily accomodates the case of random as opposed to periodic potentials as well, allowing for the treatment of phenomena such as 1d charge density wave systems, pinned polymers, dna-unzipping etc. Over the last 10 years it has been realized that a continuum hydrodynamic type evolution underlies the discrete free energy evolution as captured by the transfer matrix description. For the case of an elastic chain of particles embedded in an external potential, this evolution turns out to be governed by an iterated Burgers Equation, and the emerging shock discontinuities have a natural interpretation in terms of meta-stable states and the pinning of the elastic chain. In this talk, I will present the connections of FK type models and the Burgers-type evolution and illustrate this connection by means of an exactly solvable (non-trivial) FK model that was worked out recently by Cem Yolcu and myself.