Adrian Baule

Joint probability distributions for a class of non-Markovian processes.

We consider joint probability distributions for the class of coupled Langevin equations introduced by Fogedby [H. C. Fogedby, Phys. Rev. E 50, 1657 (1994)]. We generalize well-known results for the single-time probability distributions to the case of N -time joint probability distributions. It is shown that these probability distribution functions can be obtained by an integral transform from distributions of a Markovian process. The integral kernel obeys a partial differential equation with fractional time derivatives reflecting the non-Markovian character of the process.

A fractional diffusion equation for two-point probability distributions of a continuous-time random walk

Continuous-time random walks are non-Markovian stochastic processes, which are only partly characterized by single-time probability distributions. We derive a closed evolution equation for joint two-point probability density functions of a subdiffusive continuous-time random walk, which can be considered as a generalization of the known single-time fractional diffusion equation to two-time probability distributions. The solution of this generalized diffusion equation is given as an integral transformation of the probability distribution of an ordinary diffusion process, where the integral kernel is generated by an inverse Levy stable process. Explicit expressions for the two time moments of a diffusion process are given, which could be readily compared with the ones determined from experiments.

Stick?slip motion of solids with dry friction subject to random vibrations and an external field

We investigate a model for the dynamics of a solid object, which moves over a randomly vibrating solid surface and is subject to a constant external force. The dry friction between the two solids is modelled phenomenologically as being proportional to the sign of the objects velocity relative to the surface, and therefore shows a discontinuity at zero velocity. Using a path integral approach, we derive analytical expressions for the transition probability of the objects velocity and the stationary distribution of the work done on the object due to the external force. From the latter distribution, we also derive a fluctuation relation for the mechanical work fluctuations, which incorporates the effect of the dry friction.

Singular features in noise-induced transport with dry friction

We present an exactly solvable nonlinear model for the directed motion of an object due to zero-mean fluctuations on a uniform featureless surface. Directed motion results from the effect of dry (Coulombic) friction coupled to asymmetric surface vibrations with Poissonian shot noise statistics. We find that the transport of the object exhibits striking non-monotonic and singular features: transport actually improves for increasing dry friction up to a critical dry-friction strength Δ* and undergoes a transition to a unidirectional mode of motion at Δ*. This transition is indicated by a cusp singularity in the mean velocity of the object. Moreover, the stationary velocity distribution also contains singular features, such as a discontinuity and a delta peak at zero velocity. Our results highlight that dissipation can in fact enhance transport, which might be exploited in artificial small-scale systems.

Invariant Quantities in Shear Flow

The dynamics of systems out of thermal equilibrium is usually treated on a case-by-case basis without knowledge of fundamental and universal principles. We address this problem for a class of driven steady states, namely, those mechanically driven at the boundaries such as complex fluids under shear. From a nonequilibrium counterpart to detailed balance (NCDB) we derive a remarkably simple set of invariant quantities which remain unchanged when the system is driven. These new nonequilibrium relations are both exact and valid arbitrarily far from equilibrium. Furthermore, they enable the systematic calculation of transition rates in driven systems with state spaces of arbitrary connectivity.

Interplay between consensus and coherence in a model of interacting opinions

Abstract The formation of agents’ opinions in a social system is the result of an intricate equilibrium among several driving forces. On the one hand, the social pressure exerted by peers favors the emergence of local consensus. On the other hand, the concurrent participation of agents to discussions on different topics induces each agent to develop a coherent set of opinions across all the topics in which he/she is active. Moreover, the pervasive action of external stimuli, such as mass media, pulls the entire population towards a specific configuration of opinions on different topics. Here we propose a model in which agents with interrelated opinions, interacting on several layers representing different topics, tend to spread their own ideas to their neighborhood, strive to maintain internal coherence, due to the fact that each agent identifies meaningful relationships among its opinions on the different topics, and are at the same time subject to external fields, resembling the pressure of mass media. We show that the presence of heterogeneity in the internal coupling assigned by agents to their different opinions allows to obtain states with mixed levels of consensus, still ensuring that all the agents attain a coherent set of opinions. Furthermore, we show that all the observed features of the model are preserved in the presence of thermal noise up to a critical temperature, after which global consensus is no longer attainable. This suggests the relevance of our results for real social systems, where noise is inevitably present in the form of information uncertainty and misunderstandings. The model also demonstrates how mass media can be effectively used to favor the propagation of a chosen set of opinions, thus polarizing the consensus of an entire population.

Fluctuation properties of an effective nonlinear system subject to Poisson noise.

We study the work fluctuations of a particle, confined to a moving harmonic potential, under the influence of friction and external asymmetric Poissonian shot noise. This type of noise generalizes the usual Gaussian noise and induces an effective interaction between the noise and the potential, leading to an effectively nonlinear system with singular features. On the basis of an analytic solution we investigate the roles of time scales, symmetries, and singularities in the context of nonequilibrium fluctuations. Our results highlight the nonuniversality of the steady-state fluctuation theorem in stochastic systems.

Two-point correlation function of the fractional Ornstein-Uhlenbeck process

We calculate the two-point correlation function x(t2)x(t1) for a subdiffusive continuous time random walk in a parabolic potential, generalizing well-known results for the single-time statistics to two times. A closed analytical expression is found for initial equilibrium, revealing non-stationarity and a clear deviation from a Mittag-Leffler decay. Our result thus provides a new criterion to assess whether a given stochastic process can be identified as a continuous time random walk.

Rectification of asymmetric surface vibrations with dry friction

of dynamic friction and singular dry friction. We derive an exact solution of the stationary Kolmogorov-Feller (KF) equation in the case of two-sided exponentially distributed amplitudes. The stationary density of the velocity exhibits singular features such as a discontinuity and a delta-peak singularity at zero velocity, and also contains contributions from non-integrable solutions of the KF equation. The mean velocity in our model generally varies non-monotonically as the strength of the dry friction is increased, indicating that transport improves for increased dissipation.

Weak-noise limit of a piecewise-smooth stochastic differential equation

We investigate the validity and accuracy of weak-noise (saddle-point or instanton) approximations for piecewise-smooth stochastic differential equations (SDEs), taking as an illustrative example a piecewise-constant SDE, which serves as a simple model of Brownian motion with solid friction. For this model, we show that the weak-noise approximation of the path integral correctly reproduces the known propagator of the SDE at lowest order in the noise power, as well as the main features of the exact propagator with higher-order corrections, provided the singularity of the path integral associated with the nonsmooth SDE is treated with some heuristics. We also show that, as in the case of smooth SDEs, the deterministic paths of the noiseless system correctly describe the behavior of the nonsmooth SDE in the low-noise limit. Finally, we consider a smooth regularization of the piecewise-constant SDE and study to what extent this regularization can rectify some of the problems encountered when dealing with discontinuous drifts and singularities in SDEs.