Large-time dynamics and aging of a polymer chain in a random potential

We study the out-of-equilibrium large-time dynamics of a Gaussian polymer chain in a quenched random potential. The dynamics studied is a simple Langevin dynamics commonly referred to as the Rouse model. The equations for the two-time correlation and response functions are derived within the Gaussian variational approximation. In order to implement this approximation faithfully, we employ the supersymmetric representation of the Martin-Siggia-Rose dynamical action. For a short-ranged correlated random potential the equations are solved analytically in the limit of large times using certain assumptions concerning the asymptotic behavior. Two possible dynamical behaviors are identified depending upon the time separation: a stationary regime and an aging regime. In the stationary regime time translation invariance holds and so does the fluctuation dissipation theorem. The aging regime which occurs for large time separations of the two-time correlation functions is characterized by a history dependence and the breakdown of certain equilibrium relations. The large-time limit of the equations yields equations among the order parameters that are similar to the equations obtained in statics using replicas. In particular the aging solution corresponds to the broken replica solution. But there is a difference in one equation that leads to important consequences for the solution. The stationary regime corresponds to the motion of the polymer inside a local minimum of the random potential, whereas in the aging regime the polymer hops between different minima. As a by-product we also solve exactly the dynamics of a chain in a random potential with quadratic correlations.