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Domain wall motion in magnetic materials

Consider the boundary between two magnetic domains of opposite orientations, in a ferroelectric material. At the mesoscopic level, the smooth transition from one domain to the other is described by a soliton of zero velocity. As an external field is applied (or a global current is imposed), one domain is favored with respect to the other and the soliton moves with an asymptotically fixed velocity.

Such materials may present disordered features, for instance due to sparse magnetic impurities or to local variation of couplings. In that case, as observed in experiments, the soliton really starts to move when the field is larger than a given critical value. Before this threshold, the soliton moves slowly, in a thermally activated ``creep'' regime. This situation is similar to the depining of an elastic line in a random potential, the role of the line being played by the position of the soliton.

However, the corresponding predicted creep exponents are not compatible with those measured in experiments. This discrepancy may arise from an incomplete description of the soliton. We investigate the role of the spin phase at the location of the soliton. It represents a supplementary degree of freedom, coupled to the position of the soliton, and prevents the use of a simple elastic depining approach. We also examine how one can account for the dissipation of magnetic energy in a reliable way (there's presently no common agreement on it, though it's an essential ingredient for the motion of the soliton).

Collaboration: S. Barnes, T. Giamarchi, and A. Rosso

Fluctuations in transport models: mapping nonequilibrium onto equilibrium

Microscopic models of lattice gas with hard-core interactions, or more generally exclusion rules, such as (a)symmetric exclusion processes, are tackled either at the microscopic level (using available exact solutions) or directly at the macroscopic level (relying on an effective fluctuating hydrodynamics). We formulated a coherent-state approach which provides a field-theoretic description of such systems in a systematic way, both at micro- and macroscopic levels. In the macroscopic limit, one recovers the usual hydrodynamic limit of exclusion processes (yielding for instance EW or KPZ fluctuations, depending on microscopic rates).

One quantity of interest in such models is the probability of observing a large deviation of the density profile, especially when driven out of equilibrium. We focused on one-dimensional systems in contact with reservoirs of different densities at boundaries. At the level of fluctuating hydrodynamics, we showed that the system can be mapped onto a (dual) equilibrium isolated system, for which all properties are known. The difference between densities is mapped into the fixed mean density of the dual model. The non-equilibrium long range correlation are accounted by the non-locality of the mapping. The large deviation function of the density profiled is further directly recovered using the mapping.

We subsequently showed that such duality can be extended to other non-equilibrium diffusive processes, even at the microscopic level, yielding microscopic exact solutions otherwise difficulty obtained.

Collaboration: J. Tailleur, J. Kurchan

Fluctuation of current in equilibrium and non-equilibrium systems

Non-equilibrium phenomena are ubiquitous in Nature, though there exist no such general approach as the Boltzmann-Gibbs theory for equilibrium system. The celebrated fluctuation theorem is one result valid in general in non-equilibrium. It expresses a symmetry of the large deviation function (ldf) associated to the current Q flowing through the system on a large window of time. Although this symmetry is valid in general, the determination of the ldf reveals often strenuous, though it bears its physical interpretation.

We focused on subdiffusive systems. The ldf associated to Q is singular and characterized by a non-trivial exponent, related to the anomalously large fluctuations of the current in the steady state (quantitatively, the variance of the current Q grows more than linearly with respect to the system size). We used dynamical renormalization group techniques to determine this exponent for the KPZ universality class, a driven diffusive system exhibiting a continuous non-equilibrium phase transition, and for a disordered system. In all cases, we could relate this exponent to other dynamical exponents, thus emphasizing that ldf's reveal worth tools to classify non-equilibrium system.

Collaboration: U. Täuber, Z. Rácz, and F. van Wijland

Fluctuations of the current Q (or of other dynamical observables, such a the number of events K in a history) present striking features also in equilibrium. We determined the large deviations of these two observables for the symmetric exclusion process with periodic boundaries conditions. We compared the results obtained from two techniques, namely (i) fluctuating hydrodynamics (valid in the macroscopic limit) and (ii) Bethe Ansatz, which is in principle exact, but more intricate. Both methods yield the same result in the large size limit, which shows that fluctuating hydrodynamics is reliable when determining large fluctuations of K or Q. Moreover, we found that the fluctuations of K and Q are governed by the same (new) universal scaling function, thus pertaining to different observables in the Edwards-Anderson universality class.

Collaboration: C. Appert-Rolland, B. Derrida, and F. van Wijland

Numerical approach to large deviations in continuous time

In contrast to the equilibrium case, where Boltzmann-Gibbs theory is available, fluctuations in nonequilibrium phenomena can't be understood with tools such general as free-energies or entropies. One has instead to rely on a direct analysis of the distribution of dynamical observables, like the particle or energy current Q flowing through a system in a large window of time t. The complete (i.e. beyond Gaussian) fluctuations of Q/t are encoded into a large deviation function (ldf).

However, the very definition of ldf's renders their direct numerical observation almost impossible, as the probability of observing a value of Q/t far from its average typically decreases exponentially with time. Giardinŕ, Kurchan and Peliti introduced a numerical procedure which overcomes this difficulty for discrete time Markov chains. Nevertheless, most physical processes evolve continuously in time, and one thus has to choose an arbitrary time step dt to discretize the dynamics, balancing between algorithm efficiency and errors arising from the approximation. Systems featuring a single time scale in the steady state present in general different time scales in their large deviations, depending on the kind of histories probed, which makes the choice of dt strenuous. A simple example is given by traffic flow models, in which the typical time scale is fixed on average, but vary by a factor equal to the number N of ``cars'' when comparing jammed histories (where O(1) cars move) and free flowing histories (where O(N) cars move).

We provided a direct continuous-time algorithm, and shown on specific examples that the method can be used successfully in systems where the presence of different time scales renders the discrete-time approach difficult. We also applied the method to probe ldf's in systems with a dynamical phase transition –- revealed in our context through the appearance of a non-analyticity in the ldf's -- whose origin takes place in the coexistence of ``active'' and ``inactive'' histories (see also next point).

Collaboration: J. Tailleur

Statistics over histories in stochastic systems and glassy phenomena

Equilibrium thermodynamics relies on the statistical analysis of configurations taken by systems at fixed time.~This approach is not well suited to situations where dynamical aspects are significant (non-equilibrium steady states, aging phenomena). To overcome this problem, we imported concepts of the theory of dynamical systems into the description of systems with Markov dynamics. These consist in focusing on the various histories (and their fluctuations) that the systems may follow. The construction relies on the definition of a dynamical partition function and a dynamical free energy, in analogy with Boltzmann thermodynamics.

We showed that the dynamical free energy, a central quantity in dynamical system theory, is endowed with a natural physical interpretation. It appears as the large deviation function of a time-integrated observable, which represents the dynamical complexity of an history. It thus directly probes the nature of histories that a system may follow, distinguishing for instance between ``active'' and ``inactive'' histories.

We applied these tools to glassy phenomena, for which dynamical aspects predominate. We showed that the steady state of some glass formers is characterized by a first-order coexistence between active and inactive dynamical phases. A singularity in the dynamical free-energy reflects the corresponding first-order (dynamical) transition. We formulated a local mean-field approach, based on a Landau-like dynamical free energy, which provides a quantitative description of the coexistence of these two dynamical phases.

Collaboration: C. Appert-Rolland, J.P. Garrahan, R. Jack, E. Pitard, K. van Duijvendijk, and F. van Wijland

Collective motion of animals

The collective motion of a large number of animals can be tackled from simple local rules of interaction among individuals. However, only few aspects about the nature of such interaction are known, and models and theories mainly rely on untested assumptions. Recent experiments on starlings assert that the interaction does not depend on the metric distance, as most current models and theories assume, but rather on the topological distance: each bird interacts on average with a fixed number of neighbors, irrespective of the metric distance (within physiological limitations).

To probe the evolutionary advantages of a topological interaction in Nature, we devised numerical experiments to compare both kinds of interaction, in situations where birds flocks are subjected to an external perturbation. We performed simulations of predator attack, and obstacle bypass. The robustness of flocks was tested by measuring the cohesion (number of sub-flocks after interaction) and health (related to the number of isolated birds) of the modeled flocks.

It turns out that topological interaction grants significantly higher cohesion of the aggregation compared to a standard metric one. We concluded that topological interaction constitute a key point to maintain flock's cohesion against the large density changes caused by external perturbations.

Collaboration: A. Cavagna, I. Giardina, and G. Parisi