Alexandre Lefèvre

Tapping spin glasses and ferromagnets on random graphs.

We consider a tapping dynamics, analogous to that in experiments on granular media, on spin glasses and ferromagnets on random thin graphs. Between taps, zero temperature single spin flip dynamics takes the system to a metastable state. Tapping corresponds to flipping simultaneously any spin with probability p. This dynamics leads to a stationary regime with a steady state energy E(p). We analytically solve this dynamics for the one-dimensional ferromagnet and +/-J spin glass. Numerical simulations for spin glasses and ferromagnets of higher connectivity are carried out; in particular, we find a novel first order transition for the ferromagnetic systems.

Possible Test of the Thermodynamic Approach to Granular Media

We study the steady state distribution of the energy of the Sherrington-Kirkpatrick model driven by a tapping mechanism which mimics the mechanically driven dynamics of granular media. The dynamics consists of two phases: a zero temperature relaxation phase which leads the system to a metastable state, then a tapping which excites the system thus reactivating the relaxational dynamics. Numerically, we investigate whether the distribution of the energies of the blocked states obtained agrees with a simple canonical form of the Edwards measure. It is found that this canonical measure is in good agreement with the dynamically measured energy distribution. A possible experimental test of the Edwards measure based on the study here is proposed.

Dynamics of interacting particle systems: stochastic process and field theory

We present an approach to the dynamics of interacting particle systems, which allows us to derive path integral formulae from purely stochastic considerations. We show that the resulting field theory is a dual version of the standard theory of Doi and Peliti. This clarifies both the origin of the Cole–Hopf map between the two approaches and the occurrence of imaginary noises in effective Langevin equations for reaction–diffusion systems. The advantage of our approach is that it focuses directly on the density field. We show some applications, in particular on the zero range process, hydrodynamic limits and large deviation functional.

Tapping thermodynamics of the one-dimensional Ising model

We analyse the steady-state regime of a one-dimensional Ising model under a tapping dynamics recently introduced by analogy with the dynamics of mechanically perturbed granular media. The idea that the steady-state regime may be described by a flat measure over metastable states of fixed energy is tested by comparing various steady-state time-averaged quantities in extensive numerical simulations with the corresponding ensemble averages computed analytically with this flat measure. The agreement between the two averages is excellent in all the cases examined, showing that a static approach is capable of predicting certain measurable properties of the steady-state regime.

Steady state behavior of mechanically perturbed spin glasses and ferromagnets.

A zero temperature dynamics of Ising spin glasses and ferromagnets on random graphs of finite connectivity is considered. Like granular media, these systems have an extensive entropy of metastable states. We consider the problem of what energy a randomly prepared spin system falls to before becoming stuck in a metastable state. We then introduce a tapping mechanism, analogous to that found in real experiments on granular media. This tapping, corresponding to flipping each spin with probability p simultaneously, leads to a stationary regime with a steady state energy E(p). We explicitly solve this problem for the one-dimensional ferromagnet and the +/-J spin glass, and carry out extensive numerical simulations for spin systems of higher connectivity. In addition our simulations on the ferromagnetic systems reveal a first order transition, whereas the usual thermodynamic transition on these graphs is second order.

Self-diffusion in a system of interacting Langevin particles

The behavior of the self-diffusion constant of Langevin particles interacting via a pairwise interaction is considered. The diffusion constant is calculated approximately within a perturbation theory in the potential strength about the bare diffusion constant. It is shown how this expansion leads to a systematic double expansion in the inverse temperature beta and the particle density rho. The one-loop diagrams in this expansion can be summed exactly and we show that this result is exact in the limit of small beta and rhobeta constants. The one-loop result can also be resummed using a semiphenomenological renormalization group method which has proved useful in the study of diffusion in random media. In certain cases the renormalization group calculation predicts the existence of a diverging relaxation time signaled by the vanishing of the diffusion constant, possible forms of divergence coming from this approximation are discussed. Finally, at a more quantitative level, the results are compared with numerical simulations, in two dimensions, of particles interacting via a soft potential recently used to model the interaction between coiled polymers.

Metastable states of a ferromagnet on random thin graphs

Abstract:We calculate the mean number of metastable states of an Ising ferromagnet on random thin graphs of fixed connectivity c. We find, as for totally connected mean field spin glasses, that this mean increases exponentially with the number of sites, and is the same as that calculated for the ±J spin glass on the same graphs. An annealed calculation of the average number of metastable states of energy E per spin is carried out. For small c, an analytic result is obtained. These results are compared with the one obtained for the corresponding ±J spin glasses, in order to discuss the role played by loops on thin graphs and hence the effect of real frustration on the distribution of metastable states.

Role of the interaction matrix in mean-field spin glass models.

Mean-field models of two-spin Ising spin glasses with interaction matrices taken from ensembles that are invariant under O(N) transformations are studied. A general study shows that the nature of the spin glass transition can be deduced from the eigenvalue spectrum of the interaction matrix. A simple replica approach is derived to carry out the average over the O(N) disorder. The analytic results are confirmed by the extensive Monte Carlo simulations for large system sizes and by the exact enumeration for small system sizes.

The number of metastable states in the generalized random orthogonal model

We calculate the number of metastable states in the generalized random orthogonal model. The results obtained are verified by exact numerical enumeration for small system sizes taking into account finite size effects. These results are compared with those for a Hopfield model in order to examine the effect of strict orthonormality of neural network patterns on the number of metastable states.

Phase transitions in the steady state behavior of mechanically perturbed spin glasses and ferromagnets

A zero temperature dynamics of Ising spin glasses and ferromagnets on random graphs of finite connectivity is considered. Like granular media, these systems have an extensive entropy of metastable states. We consider the problem of what energy a randomly prepared spin system falls to before becoming stuck in a metastable state. We then introduce a tapping mechanism, analogous to that found in real experiments on granular media. This tapping, corresponding to flipping each spin with probability p simultaneously, leads to a stationary regime with a steady state energy E(p). We explicitly solve this problem for the one-dimensional ferromagnet and the +/-J spin glass, and carry out extensive numerical simulations for spin systems of higher connectivity. In addition our simulations on the ferromagnetic systems reveal a first order transition, whereas the usual thermodynamic transition on these graphs is second order.