Alberto Petri

Clustering and Non-Gaussian Behavior in Granular Matter

We investigate the properties of a model of granular matter consisting of N Brownian particles on a line, subject to inelastic mutual collisions. This model displays a genuine thermodynamic limit for the mean values of the energy, and the energy dissipation. When the typical relaxation time t associated with the Brownian process is small compared with the mean collision time tc the spatial density is nearly homogeneous and the velocity probability distribution is Gaussian. In the opposite limit t ? tc one has strong spatial clustering, with a fractal distribution of particles, and the velocity probability distribution strongly deviates from the Gaussian one. [S0031-9007(98)07496-1]

Stieltjes moment problem via fractional moments

Stieltjes moment problem is considered to recover a probability density function from the knowledge of its infinite sequence of ordinary moments. The approximate density is obtained through maximum entropy technique, under the constraint of few fractional moments. The latter are numerically obtained from the infinite sequence of ordinary moments and are chosen in such a way as to convey the maximum information content carried by the ordinary moments. As a consequence a model with few parameters is obtained and intrinsic numerical instability is avoided. It is proved that the approximate density is useful for calculating expected values and some other interesting probabilistic quantities.

Hausdorff moment problem via fractional moments

We outline an efficient method for the reconstruction of a probability density function from the knowledge of its infinite sequence of ordinary moments. The approximate density is obtained resorting to maximum entropy technique, under the constraint of some fractional moments. The latter ones are obtained explicitly in terms of the infinite sequence of given ordinary moments. It is proved that the approximate density converges in entropy to underlying density, so that it turns out to be useful for calculating expected values.

Brownian ratchet in a thermal bath driven by Coulomb friction.

The rectification of unbiased fluctuations, also known as the ratchet effect, is normally obtained under statistical nonequilibrium conditions. Here we propose a new ratchet mechanism where a thermal bath solicits the random rotation of an asymmetric wheel, which is also subject to Coulomb friction due to solid-on-solid contacts. Numerical simulations and analytical calculations demonstrate a net drift induced by friction. If the thermal bath is replaced by a granular gas, the well-known granular ratchet effect also intervenes, becoming dominant at high collision rates. For our chosen wheel shape the granular effect acts in the opposite direction with respect to the friction-induced torque, resulting in the inversion of the ratchet direction as the collision rate increases. We have realized a new granular ratchet experiment where both these ratchet effects are observed, as well as the predicted inversion at their crossover. Our discovery paves the way to the realization of micro and submicrometer Brownian motors in an equilibrium fluid, based purely upon nanofriction.

Brownian Forces in Sheared Granular Matter

We present results from a series of experiments on a granular medium sheared in a Couette geometry and show that their statistical properties can be computed in a quantitative way from the assumption that the resultant from the set of forces acting in the system performs a Brownian motion. The same assumption has been utilized, with success, to describe other phenomena, such as the Barkhausen effect in ferromagnets, and so the scheme suggests itself as a more general description of a wider class of driven instabilities.

Stochastic dynamics of a sheared granular medium

We experimentally investigate the response of a sheared granular medium in a Couette geometry. The apparatus exhibits the expected stick-slip motion and we probe it in the very intermittent regime resulting from low driving. Statistical analysis of the dynamic fluctuations reveals notable regularities. We observe a possible stability property for the torque distribution, reminiscent of the stability of Gaussian independent variables. In this case, however, the variables are correlated and the distribution is skewed. Moreover, the whole dynamical intermittent regime can be described with a simple stochastic model, finding good quantitative agreement with the experimental data. Interestingly, a similar model has been previously introduced in the study of magnetic domain wall motion, a source of Barkhausen noise. Our study suggests interesting connections between different complex phenomena and reveals some unexpected features that remain to be explained.

Shear stress fluctuations in the granular liquid and solid phases.

We report on experimentally observed shear stress fluctuations in both granular solid and fluid states, showing that they are non-Gaussian at low shear rates, reflecting the predominance of correlated structures (force chains) in the solidlike phase, which also exhibit finite rigidity to shear. Peaks in the rigidity and the stress distributions skewness indicate that a change to the force-bearing mechanism occurs at the transition to fluid behavior, which, it is shown, can be predicted from the behavior of the stress at lower shear rates. In the fluid state stress is Gaussian distributed, suggesting that the central limit theorem holds. The fiber bundle model with random load sharing effectively reproduces the stress distribution at the yield point and also exhibits the exponential stress distribution anticipated from extant work on stress propagation in granular materials.

Nonequilibrium fluctuations in a frictional granular motor: experiments and kinetic theory.

We report the study of an experimental granular Brownian motor, inspired by the one published in Eshuis et al. [Phys. Rev. Lett. 104, 248001 (2010)], but different in some ingredients. As in that previous work, the motor is constituted by a rotating blade, the surfaces of which break the rotation-inversion symmetry through alternated patches of different inelasticity, immersed in a gas of granular particles. The main difference of our experimental setup is in the orientation of the main axis, which is parallel to the (vertical) direction of shaking of the granular fluid, guaranteeing an isotropic distribution for the velocities of colliding grains, characterized by a variance v(0)(2). We also keep the granular system diluted, in order to compare with Boltzmann-equation-based kinetic theory. In agreement with theory, we observe the crucial role of Coulomb friction which induces two main regimes: (i) rare collisions, with an average angular velocity ~v(0)(3), and (ii) frequent collisions (FC), with ~v(0). We also study the fluctuations of the angle spanned in a large-time interval Δθ, which in the FC regime is proportional to the work done upon the motor. We observe that the fluctuation relation is satisfied with a slope which weakly depends on the relative collision frequency.

Analysis of metal cutting acoustic emissions by time series models.

We analyse some acoustic emission time series obtained from a lathe machining process. Considering the dynamic evolution of the process, we apply two classes of well known stationary stochastic time series models. We apply a preliminary root mean square (RMS) transformation followed by an auto regressive moving average analysis; results thereof are mainly related to the description of the continuous part (plastic deformation) of the signal. An analysis of acoustic emission, as some previous works show, may also be performed with the scope of understanding the evolution of the ageing process that causes the degradation of the working tools. Once the importance of the discrete part of the acoustic emission signals (i.e. isolated amplitude bursts) in the ageing process is understood, we apply a stochastic analysis based on point processes’ waiting times between bursts and to identify a parameter with which to characterise the wear level of the working tool. A Weibull distribution seems to adequately describe the waiting times distribution.

Statistical properties of acoustic emission signals from metal cutting processes

Acoustic emission (AE) data from single point turning machining are analyzed in this paper in order to gain a greater insight of the signal statistical properties for tool condition monitoring applications. A statistical analysis of the time series data amplitude and root mean square (rms) value at various tool wear levels are performed, finding that aging features can be revealed in all cases from the observed experimental histograms. In particular, AE data amplitudes are shown to be distributed with a power-law behavior above a crossover value. An analytic model for the rms values probability density function is obtained resorting to the Jaynes’ maximum entropy principle; novel technique of constraining the modeling function under few fractional moments, instead of a greater amount of ordinary moments, leads to well-tailored functions for experimental histograms.