This paper concerns with the time-reversal characteristics of intrinsic normal diffusion in quantum systems. Time-reversible properties are quantified by the time-reversal test; the system evolved in the forward direction for a certain period is time-reversed for the same period after applying a small perturbation at the reversal time, and the separation between the time-reversed perturbed and unperturbed states is measured as a function of perturbation strength, which characterizes sensitivity of the time reversed system to the perturbation and is called the time-reversal characteristic.
Time-reversal characteristics are investigated for various quantum systems, namely, classically chaotic quantum systems and disordered systems including various stochastic diffusion systems. When the system is normally diffusive, there exists a fundamental quantum unit of perturbation, and all the models exhibit a universal scaling behavior in the time-reversal dynamics as well as in the time-reversal characteristics, which leads us to a basic understanding on the nature of quantum irreversibility.
Biological and synthetic microswimmers display a wide range of swimming trajectories depending on driving forces and torques. In this paper we consider a simple overdamped model of self-propelled particles with a constant self-propulsion speed, but an angular velocity that varies in time. Specifically, we consider the case of both deterministic and stochastic angular velocity reversal, mimicking several synthetic active matter systems like propelled droplets. The orientational correlation function and effective diffusivity is studied using Langevin dynamics simulations and perturbative methods…
We study the time-reversal dynamics of a tight-binding electron in the Watts-Strogatz (WS) small-world networks. The localized initial wave packet at time $t=0$ diffuses as time proceeds until the time-reversal operation, together with the momentum perturbation of the strength $η$, is made at the reversal time $T$. The time irreversibility is measured by $I \equiv |Π(t = 2T) - Π(t = 0)|$, where $Π$ is the participation ratio gauging the extendedness of the wavefunction and for convenience, $t$ is measured forward even after the time reversal . When $η= 0$, the time evolution after $T$ makes th…
The magnetization reversal time of ferromagnetic nanoparticles is investigated in the very low damping regime. The energy-controlled diffusion equation rooted in a generalization of the Kramers escape rate theory for point Brownian particles in a potential to the magnetic relaxation of a macrospin, yields the reversal time in closed integral form. The latter is calculated for a nanomagnet with uniaxial anisotropy with a uniform field applied at an angle to the easy axis and for a nanomagnet with biaxial anisotropy with the field along the easy axis. The results completely agree with those yiel…
We discuss how the first order Langevin equation for the overdamped dynamics of an interacting system has a natural time reversal of simple but surprising form, with consequences for correlation functions. This leads to the correlation of interactions as a strictly restraining term in the time-dependent diffusion tensor of the system, deriving the relation first suggested by Fixman. Applying this to the time-dependent diffusion of dilute polymer coils leads to the quantitative calibration of Kirkwood's approximation for their hydrodynamic radius. We find the generalized ``Fixman Law" for dissi…
A rather simple random walk model on a one-dimensional lattice is put forward. The lattice as a whole switches randomly between two possible states which are spatially symmetric. Both lattice states are identical, but translated by one site with respect to each other, and consist of infinite arrays of absorbing sites separated by two non-absorbing sites. Exact explicit expressions for the long-time velocity and the effective diffusion coefficient are obtained and discussed. In particular, it is shown that the direction of the steady motion can be reversed by conveniently varying the values of …
A closed set of \textit{exact} equations describing statistical theory of turbulent self-diffusion by multivariate-normal turbulent velocity field is derived. In doing so, we first suggest exact formulas for correlations $<f_i(p)f_j(p')R[{\bf f}]>$, $<g(p)R[{\bf f}]>$ and $<g(p)f_j(p')R[{\bf f}]>$ when the functional $R[{\bf f}]$ is functional of functions ${f_i}$'s having multivariate-normal distribution, $g$ and $f_i$'s have joint normal distribution and zero mean values.
A Brownian particle floating in a narrow corrugated (sinusoidal) channel with fluctuating cross section exhibits non-Gaussian normal diffusion. Its displacements are distributed according to a Gaussian law for very short and asymptotically large observation times, whereas a robust exponential distribution emerges for intermediate observation times of the order of the channel fluctuation correlation time. For intermediate to large observation times the particle undergoes normal diffusion with one and the same effective diffusion constant. These results are analytically interpreted without havin…
There is a well-known mapping between energy normal (super-) diffusion and normal (anomalous) heat conduction in one-dimensional (1D) nonlinear lattices. The momentum conserving nonlinear lattices exhibit energy super-diffusion behavior with the only exception of coupled rotator model. Yet, for all other 1D momentum nonconserving nonlinear lattices studied so far, the energy diffusion or heat conduction is normal. Here we propose a 1D nonlinear lattice model with negative couplings, which is momentum non-conserving due to the translational symmetry breaking. Our numerical results show that ene…
Normal diffusion in corrugated potentials with spatially uncorrelated Gaussian energy disorder famously explains the origin of non-Arrhenius $\exp[-σ^2/(k_BT^2)]$ temperature-dependence in disordered systems. Here we show that unbiased diffusion remains asymptotically normal also in the presence of spatial correlations decaying to zero. However, due to a temporal lack of self-averaging transient subdiffusion emerges on mesoscale, and it can readily reach macroscale even for moderately strong disorder fluctuations of $σ\sim 4-5\, k_BT$. Due to its nonergodic origin such subdiffusion exhibits a …
We have studied the reversal time of the magnetisation in two dimensional Ising ferromagnet in the presence of externally applied uniform magnetic field using Monte Carlo simulation based on Metropolis single spin flip algorithm. Then we have investigated the change in reversal time due to the presence of quenched random field in addition to the uniform magnetic field. We report the results of statistical distribution of reversal times in the presence of three different types of the distributions (namely, uniform, bimodal and normal) of random fields and compared the results with those obtaine…
A relation between the effective diffusion coefficient in a lattice with random site energies and random trasition rates and the macroscopic conductivity in a random resistor network allows for elucidating possible sources of anomalous diffusion in random potential models. We show that subdiffusion is only possible either if the mean Boltzmann factor in the corresponding potential diverges or if the percolation concentration in the system is equal to unity (or both), and that superdiffusion is impossible in our system under any condition. We show also other useful applications of this relation…
We consider the out-of-equilibrium dynamics of the Hamiltonian Mean Field (HMF) model, by focusing in particular on the properties of single-particle diffusion. As we shall here demonstrate analytically, if the autocorrelation of momenta in the so-called quasi-stationary states can be fitted by a q-exponential, then diffusion ought to be normal for q<2, at variance with the interpretation of the numerical experiments proposed in Refs. A. Pluchino, A. Rapisarda, Progress in Theoretical Physics Supplement 162, 18 (2006); A. Pluchino, V. Latora, A. Rapisarda, Physica A 338, 60 (2004); A. Rapisard…
A comment on the Letter by A. Rebenshtok, S. Denisov, P. Hänggi, and E. Barkai, Phys. Rev. Lett., vol. 112, 110601 (2014). It is shown that the recent claims that the particle distributions or densities can become non-normalizable in the case of anomalous Levy walk diffusion cannot be justified. In particular, one can define a normalizable finite-time density which become infinite covariant density (ICD) in the limit of infinite time. The former one can be used to find any generalized moment, while the latter one does not. In this respect, we point out that any decent real or numerical experim…
A paradigmatic nonhyperbolic dynamical system exhibiting deterministic diffusion is the smooth nonlinear climbing sine map. We find that this map generates fractal hierarchies of normal and anomalous diffusive regions as functions of the control parameter. The measure of these self-similar sets is positive, parameter-dependent, and in case of normal diffusion it shows a fractal diffusion coefficient. By using a Green-Kubo formula we link these fractal structures to the nonlinear microscopic dynamics in terms of fractal Takagi-like functions.
A theoretical framework is developed for the phenomenon of non-Gaussian normal diffusion that has experimentally been observed in several heterogeneous systems. From the Fokker-Planck equation with the dynamical structure with largely separated time scales, a set of three equations are derived for the fast degree of freedom, the slow degree of freedom and the coupling between these two hierarchies. It is shown that this approach consistently describes "diffusing diffusivity" and non-Gaussian normal diffusion.
We present an English translation of Erwin Schrödinger's paper on "On the Reversal of the Laws of Nature". In this paper Schrödinger analyses the idea of time reversal of a diffusion process. Schrödinger's paper acted as a prominent source of inspiration for the works of Bernstein on reciprocal processes and of Kolmogorov on time reversal properties of Markov processes and detailed balance. The ideas outlined by Schrödinger also inspired the development of probabilistic interpretations of quantum mechanics by Fényes, Nelson and others as well as the notion of "Euclidean Quantum Mechanics" as p…
Based on the generalized Langevin equation for the momentum of a Brownian particle a generalized asymptotic Einstein relation is derived. It agrees with the well-known Einstein relation in the case of normal diffusion but continues to hold for sub- and super-diffusive spreading of the Brownian particle's mean square displacement. The generalized asymptotic Einstein relation is used to analyze data obtained from molecular dynamics simulations of a two-dimensional soft disk fluid. We mainly concentrated on medium densities for which we found super-diffusive behavior of a tagged fluid particle. A…
In this work we study the transition from normal to anomalous diffusion of Brownian particles on disordered potentials. The potential model consists of a series of "potential hills" (defined on unit cell of constant length) whose heights are chosen randomly from a given distribution. We calculate the exact expression for the diffusion coefficient in the case of uncorrelated potentials for arbitrary distributions. We particularly show that when the potential heights have a Gaussian distribution (with zero mean and a finite variance) the diffusion of the particles is always normal. In contrast w…
Motivated by electronic transport in graphene-like structures, we study the diffusion of a classical point particle in Fermi potentials situated on a triangular lattice. We call this system a soft Lorentz gas, as the hard disks in the conventional periodic Lorentz gas are replaced by soft repulsive scatterers. A thorough computational analysis yields both normal and anomalous (super) diffusion with an extreme sensitivity on model parameters. This is due to an intricate interplay between trapped and ballistic periodic orbits, whose existence is characterized by tongue-like structures in paramet…
We consider Langevin equation with dichotomously fluctuating diffusivity, where the diffusion coefficient changes dichotomously in time, in order to study fluctuations of time-averaged observables in temporary heterogeneous diffusion process. We find that occupation time statistics is a powerful tool for calculating the time-averaged mean square displacement in the model. We show that the time-averaged diffusion coefficients are intrinsically random when the mean sojourn time for one of the states diverges. Our model provides anomalous fluctuations of time-averaged diffusivity, which have rele…
Active Brownian motion with intermittent direction reversals are common in a class of bacteria like {\it Myxococcus xanthus} and {\it Pseudomonas putida}. We show that, for such a motion in two dimensions, the presence of the two time scales set by the rotational diffusion constant $D_R$ and the reversal rate $γ$ gives rise to four distinct dynamical regimes: (I) $t\ll \min (γ^{-1}, D_R^{-1}),$ (II) $γ^{-1}\ll t\ll D_R^{-1}$, (III) $D_R^{-1} \ll t \ll γ^{-1}$, and (IV) $t\gg \max (γ^{-1}$, $D_R^{-1})$, showing distinct behaviors. We characterize these behaviors by analytically computing the po…
From extensive numerical simulations, we find that periodic polygonal billiard channels with angles which are irrational multiples of pi generically exhibit normal diffusion (linear growth of the mean squared displacement) when they have a finite horizon, i.e. when no particle can travel arbitrarily far without colliding. For the infinite horizon case we present numerical tests showing that the mean squared displacement instead grows asymptotically as t log t. When the unit cell contains accessible parallel scatterers, however, we always find anomalous super-diffusion, i.e. power-law growth wi…
We study the diffusion of an ensemble of overdamped particles sliding over a tilted random poten- tial (produced by the interaction of a particle with a random polymer) with long-range correlations. We found that the diffusion properties of such a system are closely related to the correlation function of the corresponding potential. We model the substrate as a symbolic trajectory of a shift space which enables us to obtain a general formula for the diffusion coefficient when normal diffusion occurs. The total time that the particle takes to travel through n monomers can be seen as an ergodic s…
The paper presents a unified approach to different fluctuation relations for classical nonequilibrium dynamics described by diffusion processes. Such relations compare the statistics of fluctuations of the entropy production or work in the original process to the similar statistics in the time-reversed process. The origin of a variety of fluctuation relations is traced to the use of different time reversals. It is also shown how the application of the presented approach to the tangent process describing the joint evolution of infinitesimally close trajectories of the original process leads to …
Particles driven through a periodic potential by an external constant force are known to exhibit a pronounced peak of the diffusion around a critical force that defines the transition between locked and running states. It has recently been shown both experimentally and numerically that this peak is greatly enhanced if some amount of spatial disorder is superimposed on the periodic potential. Here we show that beyond a simple enhancement lies a much more interesting phenomenology. For some parameter regimes the system exhibits a rich variety of behaviors from normal diffusion to superdiffusion,…
We consider a particle transport process in a one-dimensional system with a thin membrane, described by a normal diffusion equation. We consider two boundary conditions at the membrane that are linear combinations of integral operators, with time dependent kernels, which act on the functions and their spatial derivatives define on both membrane surfaces. We show how boundary conditions at the membrane change the temporal evolution of the first and second moments of particle position distribution (the Green's function) which is a solution to normal diffusion equation. As these moments define th…
The non-Gaussian normal diffusion, i.e., the probability distribution function (PDF) is non-Gaussian but the mean squared displacement (MSD) depends on time linearly, has been observed in particle motions. Here we show by numerical simulations that this phenomenon may manifest in energy diffusion along lattices at a non-zero, finite temperature. The model we study is one-dimensional disordered lattices with on-site potential. We find that the energy-density fluctuations are spatially localized if the nonlinear interaction is suppressed, but may relax with a non-Gaussian PDF and a linear time-d…
