Work fluctuations in the active Ornstein-Uhlenbeck particle model

We study the large deviations of the power injected by the active force for an active Ornstein-Uhlenbeck particle (AOUP), free or in a confining potential. For the free-particle case, we compute the rate function analytically in d-dimensions from a saddle-point expansion, and numerically in two dimensions by (a) direct sampling of the active work in numerical solutions of the AOUP equations and (b) Legendre-Fenchel transform of the scaled cumulant generating function obtained via a cloning algorithm. The rate function presents asymptotically linear branches on both sides and it is independent of the system's dimensionality, apart from a multiplicative factor. For the confining potential case, we focus on two-dimensional systems and obtain the rate function numerically using both methods (a) and (b). We find a different scenario for harmonic and anharmonic potentials: in the former case, the phenomenology of fluctuations is analogous to that of a free particle, but the rate function might be non-analytic; in the latter case the rate functions are analytic, but fluctuations are realised by entirely different means, which rely strongly on the particle-potential interaction. Finally, we check the validity of a fluctuation relation for the active work distribution. In the free-particle case, the relation is satisfied with a slope proportional to the bath temperature. The same slope is found for the harmonic potential, regardless of activity, and for an anharmonic potential with low activity. In the anharmonic case with high activity, instead, we find a different slope which is equal to an effective temperature obtained from the fluctuation-dissipation theorem.

Fluid and glassy phases of vibrated granular matter studied with a torsion oscillator

Patrick Mayor

Granular matter submitted to external perturbations exhibits various behaviors depending on the vibration intensity: when strongly vibrated, the granular system has a fluid aspect, whereas under low intensity perturbations, it is in a quasi-solid phase. In this work, we clarify these analogies: we discuss to what extent a "granular fluid" is close to a standard liquid and, on the other hand, investigate the similarities between weakly perturbed granular materials and supercooled liquids undergoing a glass transition. We first consider the case where a granular medium undergoes vibrations of high intensity (with accelerations of about 1 to 10 times that of gravity). This vibrated system is investigated using an immersed torsion oscillator which is sensitive to the granular agitation and which, as a result, exhibits irregular angular deflections that can be analyzed to give information on the grains' motion. This oscillator can also be used in forced mode: applying a torque allows measures of the mechanical susceptibility, which displays a resonance peak similar to that of a damped oscillator. It is thus possible to introduce a viscosity parameter for the system, which is found to be inversely proportional to the vibration acceleration imposed to the container. Moreover, by considering both the fluctuations (given by the diffusive noise, with the oscillator in free mode) and the susceptibility (forced mode), the validity of the fluctuation-dissipation theorem can be tested. Surprisingly, it turns out that very complicated system, even though far from equilibrium, satisfies this basic law of equilibrium statistical mechanics in first approximation, and therefore behaves very much like a plain liquid. The immersed oscillator can thus be compared to a pollen particle exhibiting Brownian motion due to the continuous molecular – here, the granular – agitation. We can thus also introduce an "effective temperature" parameter for vibration-fluidized granular matter and discuss its properties. In particular, we observe that the temperature defined is inhomogeneous and anisotropic, contrary to usual liquids. An interesting issue is also studied carefully: when the damping becomes large (for when the imposed vibrations are lower, or when the oscillator is deeply immersed), a stiffening phenomenon is observed, in which the apparent elastic constant increases linearly with the friction. An elementary rheological model suggests that this may be caused by the appearance of a force chains fretwork resisting the probe rotation. Various granular materials are analyzed with this met hod. The grain mass seems to be an important parameter, as well as the grain surface state. Experiments in which the grains are etched with acid in order to modify the surface roughness are also discussed. While we know what happens to a liquid when its temperature is decreased one can wonder what happens to a granular system when the perturbations become critically low. By using vibrations of weak intensity (with accelerations below that of gravity), we study how the system reaches a "frozen" static configuration when the perturbation intensity is decreased. The observed diffusive noise appears to approach the final jammed state according to the Vogel-Fulcher-Tammann law, that also describes the temperature dependence of viscosity or diffusivity in supercooled liquids, thus showing strong analogies with the vitrification process. Here, the parameter playing the role of temperature in the equations is found to depend only on the vibration amplitude. Finally, we briefly discuss how a small modification of the experimental setup may allow to create a "Brownian motor" generating useful work out of the random agitation of the grains.

EPFL2005

Fluid and glassy phases of vibrated granular matter studied with a torsion oscillator

Patrick Mayor

EPFL2005