Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userBrownian regime of finite-N corrections to particle motion in the XY hamiltonian mean field model
119
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We study the dynamics of the N-particle system evolving in the XY hamiltonian mean field (HMF) model for a repulsive potential, when no phase transition occurs. Starting from a homogeneous distribution, particles evolve in a mean field created by the interaction with all others. This interaction does not change the homogeneous state of the system, and particle motion is approximately ballistic with small corrections. For initial particle data approaching a waterbag, it is explicitly proved that corrections to the ballistic velocities are in the form of independent brownian noises over a time scale diverging not slower than $N^{2/5}$ as $N to infty$, which proves the propagation of molecular chaos. Molecular dynamics simulations of the XY-HMF model confirm our analytical findings.
rate research
Read More
We consider the spherical model on a spider-web graph. This graph is effectively infinite-dimensional, similar to the Bethe lattice, but has loops. We show that these lead to non-trivial corrections to the simple mean-field behavior. We first determine all normal modes of the coupled springs problem on this graph, using its large symmetry group. In the thermodynamic limit, the spectrum is a set of $delta$-functions, and all the modes are localized. The fractional number of modes with frequency less than $omega$ varies as $exp (-C/omega)$ for $omega$ tending to zero, where $C$ is a constant. For an unbiased random walk on the vertices of this graph, this implies that the probability of return to the origin at time $t$ varies as $exp(- C t^{1/3})$, for large $t$, where $C$ is a constant. For the spherical model, we show that while the critical exponents take the values expected from the mean-field theory, the free-energy per site at temperature $T$, near and above the critical temperature $T_c$, also has an essential singularity of the type $exp[ -K {(T - T_c)}^{-1/2}]$.
Complex systems display anomalous diffusion, whose signature is a space/time scaling $xsim t^delta$ with $delta e 1/2$ in the Probability Density Function (PDF). Anomalous diffusion can emerge jointly with both Gaussian, e.g., fractional Brownian motion, and power-law decaying distributions, e.g., Levy Flights (LFs) or Levy Walks (LWs). LFs get anomalous scaling, but also infinite position variance and also infinite energy and discontinuous velocity. LWs are based on random trapping events, resemble a Levy-type power-law distribution that is truncated in the large displacement range and have finite moments, finite energy and discontinuous velocity. However, both LFs and LWs cannot describe friction-diffusion processes. We propose and discuss a model describing a Heterogeneous Ensemble of Brownian Particles (HEBP) based on a linear Langevin equation. We show that, for proper distributions of relaxation time and velocity diffusivity, the HEBP displays features similar to LWs, in particular power-law decaying PDF, long-range correlations and anomalous diffusion, at the same time keeping finite position moments and finite energy. The main differences between the HEBP model and two LWs are investigated, finding that, even if the PDFs are similar, they differ in three main aspects: (i) LWs are biscaling, while HEBP is monoscaling; (ii) a transition from anomalous ($delta e 1/2$) to normal ($delta = 1/2$) diffusion in the long-time regime; (iii) the power-law index of the position PDF and the space/time diffusion scaling are independent in the HEBP, while they both depend on the scaling of the inter-event time PDF in LWs. The HEBP model is derived from a friction-diffusion process, it has finite energy and it satisfies the fluctuation-dissipation theorem.
Fractional Brownian motion is a non-Markovian Gaussian process indexed by the Hurst exponent $Hin [0,1]$, generalising standard Brownian motion to account for anomalous diffusion. Functionals of this process are important for practical applications as a standard reference point for non-equilibrium dynamics. We describe a perturbation expansion allowing us to evaluate many non-trivial observables analytically: We generalize the celebrated three arcsine-laws of standard Brownian motion. The functionals are: (i) the fraction of time the process remains positive, (ii) the time when the process last visits the origin, and (iii) the time when it achieves its maximum (or minimum). We derive expressions for the probability of these three functionals as an expansion in $epsilon = H-tfrac{1}{2}$, up to second order. We find that the three probabilities are different, except for $H=tfrac{1}{2}$ where they coincide. Our results are confirmed to high precision by numerical simulations.
We study the dynamics of the center of mass of a Brownian particle levitated in a Paul trap. We focus on the overdamped regime in the context of levitodynamics, comparing theory with our numerical simulations and experimental data from a nanoparticle in a Paul trap. We provide an exact analytical solution to the stochastic equation of motion, expressions for the standard deviation of the motion, and thermalization times by using the WKB method under two different limits. Finally, we prove the power spectral density of the motion can be approximated by that of an Ornstein-Uhlenbeck process and use the found expression to calibrate the motion of a trapped particle.
We present the analysis of the first passage time problem on a finite interval for the generalized Wiener process that is driven by Levy stable noises. The complexity of the first passage time statistics (mean first passage time, cumulative first passage time distribution) is elucidated together with a discussion of the proper setup of corresponding boundary conditions that correctly yield the statistics of first passages for these non-Gaussian noises. The validity of the method is tested numerically and compared against analytical formulae when the stability index $alpha$ approaches 2, recovering in this limit the standard results for the Fokker-Planck dynamics driven by Gaussian white noise.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
