Burgers Turbulence and the Energy Landscape of Randomly Pinned Objects

Abstract

The physics of elastic objects pinned by random impurities is certainly one of the most topical current themes of statistical mechanics [1]. Two different general approaches have been proposed to describe the statics of these pinned manifolds, for which perturbation theory badly fails. The first one is the ‘functional renormalisation group’ (FRG) which aims at constructing the correlation function for the effective pinning potential acting on long wavelengths using renormalisation group. (RG) ideas [3, 4]. The second is the variational replica method which combines a Gaussian trial Hamiltonian with ‘replica symmetry breaking’ (RSB) to obtain results in the low temperature, strongly pinned phase [5, 6, 7]. The aim of our work [8] was to understand the connections between these two approches. We show that both formalisms suggest the same physical picture: the effective, long wavelength pinning potential is a succession of parabolic wells of random depth, matching on singular points where the effective force (i.e. the derivative of the potential) is discontinuous. These discontinuities induce a singularity in the effective potential correlation function, and are encoded in the replica language by the RSB. The replica calculation furthermore provides an explicit construction of this effective (random) potential, and hence, in turn, information on the statistics of the depth of the potential minima. We hope that our construction can also shed light on the dynamics of such objects. For example, their relaxation can be analyzed in terms of hops between the different minima (‘traps’), corresponding to metastable long wavelength configurations. The statistics of barrier heights control the trapping time distribution, and hence the low frequency response and its possible aging behaviour [9, 10].