There is a growing interest in the stochastic processes of nonequilibrium systems subject to non-conserved forces, such as the magnetic forces acting on charged particles and the chiral self-propelled force acting on active particles. In this paper, we consider the stationary transport of non-interacting Brownian particles under a constant magnetic field in a position-dependent temperature background. We demonstrate the existence of the Nernst-like stationary density current perpendicular to both the temperature gradient and magnetic field, induced by the intricate coupling between the non-conserved force and the multiplicative noises due to the position-dependent temperature.
Anomalous diffusion phenomenon is an intriguing process that tracer diffusion presents in numerous complex systems. Current experimental and theoretical investigations have reported the emergence of random diffusivity scenarios accompanied by the heterogeneity of the $α$-anomalous diffusion exponents. In this framework, we investigate a heterogeneous ensemble of tracers governed by scaled Brownian motion (sBm). The heterogeneous features are considered on anomalous diffusion exponent and diffusivity. To analyse such systems, we introduce the truncated-Gaussian and truncated chi-squared distrib…
Einstein's theory of Brownian motion is revisited in order to formulate generalized kinetic theory of anomalous diffusion. It is shown that if the assumptions of analyticity and the existence of the second moment of the displacement distribution are relaxed, the fractional derivative naturally appears in the diffusion equation. This is the first demonstration of the physical origin of the fractional derivative, in marked contrast to the usual phenomenological introduction of it. Furthermore, Einstein's approach is generalized to nonlinear kinetic theory to derive the porous-medium-type equatio…
It is shown that in systems with time-dependent and/or spatially nonuniform temperature $T(t,x)$, (i) most of the transport processes is weakly non-ergodic, and (ii) the diffusion (Brownian motion, BM) is anomalous. A few examples of simple arrangements, easy for experimental realization, are discussed in detail. Proposed measurements will enable also the observation of transitions from ergodic to weakly non-ergodic and from normal to anomalous diffusion. New effects are predicted: (i) zero-mean oscillations of $T(t)$ accelerate BM (pumping effect), (ii) the combination of temporal and spatial…
Strong anomalous diffusion phenomena are often observed in complex physical and biological systems, which are characterized by the nonlinear spectrum of exponents $qν(q)$ by measuring the absolute $q$-th moment $\langle |x|^q\rangle$. This paper investigates the strong anomalous diffusion behavior of a two-state process with Lévy walk and Brownian motion, which usually serves as an intermittent search process. The sojourn times in Lévy walk and Brownian phases are taken as power law distributions with exponents $α_+$ and $α_-$, respectively. Detailed scaling analyses are performed for the coex…
It is quite generally assumed that the overdamped Langevin equation provides a quantitative description of the dynamics of a classical Brownian particle in the long time limit. We establish and investigate a paradigm anomalous diffusion process governed by an underdamped Langevin equation with an explicit time dependence of the system temperature and thus the diffusion and damping coefficients. We show that for this underdamped scaled Brownian motion (UDSBM) the overdamped limit fails to describe the long time behaviour of the system and may practically even not exist at all for a certain rang…
We address a recently introduced model describing a system of periodically coupled nonlinear phase oscillators submitted to multiplicative white noises, wherein a ratchet-like transport mechanism arises through a symmetry-breaking noise-induced nonequilibrium phase transition. Numerical simulations of this system reveal amazing novel features such as negative zero-bias conductance and anomalous hysteresis, explained resorting to a strong-coupling analysis in the thermodynamic limit. Using an explicit mean-field approximation we explore the whole ordered phase finding a transition from anomalou…
A system of periodically coupled nonlinear phase oscillators submitted to both additive and multiplicative white noises has been recently shown to exhibit ratchetlike transport, negative zero-bias conductance, and anomalous hysteresis. These features stem from the "asymmetry" of the stationary probability distribution function, arising through a noise-induced nonequilibrium phase transition which is "reentrant" as a function of the multiplicative noise intensity. Using an explicit mean-field approximation we analyze the effect of the multiplicative noises being coloured, finding a contraction …
It has been shown that the weak-interacting limit of the metric-skew-tensor-gravity (MSTG) can explain the anomalous rotation of galaxies without non-baryonic dark matter. We show that MSTG is related to the equilibrium-state of ordinary Brownian motion. We also explore if other stochastic processes can model anomalous rotation. Furthermore, we analyze phase-diagrams that elucidate the condensation of a gravitating cloud towards a Kepler-Newton system and illustrate regions of existence of rotating objects.
A number of random processes in various fields of science is described by phenomenological equations containing a stochastic force, the best known example being the Langevin equation (LE) for the Brownian motion (BM) of particles. Long ago Vladimirsky (1942) proposed a simple method for solving such equations. The method, based on the classical Gibbs statistics, consists in converting the stochastic LE into a deterministic one, and is applicable to linear equations with any kind of memory. When the memory effects are taken into account in the description of the BM, the mean square displacement…
We study fluctuating tilt Brownian ratchets based on fractional subdiffusion in sticky viscoelastic media characterized by a power law memory kernel. Unlike the normal diffusion case the rectification effect vanishes in the adiabatically slow modulation limit and optimizes in a driving frequency range. It is shown also that anomalous rectification effect is maximal (stochastic resonance effect) at optimal temperature and can exhibit a surprisingly good quality. Moreover, subdiffusive current can flow in the counter-intuitive direction upon a change of temperature or driving frequency. The depe…
Brownian motion is a ubiquitous physical phenomenon across the sciences. After its discovery by Brown and intensive study since the first half of the 20th century, many different aspects of Brownian motion and stochastic processes in general have been addressed in Statistical Physics. In particular, there now exist a very large range of applications of stochastic processes in various disciplines. Here, we highlight some of the advances in stochastic processes prompted by novel experimental methods such as superresolution microscopy. Here we provide a summary of some of the recent developments …
We analyze the translational and rotational motion of an ellipsoidal Brownian particle from the viewpoint of stochastic thermodynamics. The particle's Brownian motion is driven by external forces and torques and takes place in an heterogeneous thermal environment where friction coefficients and (local) temperature depend on space and time. Our analysis of the particle's stochastic thermodynamics is based on the entropy production associated with single particle trajectories. It is motivated by the recent discovery that the overdamped limit of vanishing inertia effects (as compared to viscous f…
In this work, we analyze the relaxation of a classical spin interacting with a heat bath, starting from the fully dynamical Hamiltonian description. An analogous problem in the framework of generalized Langevin equation (GLE) with anomalous dissipation is analyzed in details. The Fokker-Planck equation corresponding to GLE is derived and the concept of equilibrium probability distribution is analyzed. In this process we have identified few difficulties to obtain equilibrium distribution for the non-Markovian case and the remedy to overcome this difficulty is also discussed.
Wang et al. [PNAS 106 (2009) 15160] have found that in several systems the linear time dependence of the mean-square displacement (MSD) of diffusing colloidal particles, typical of normal diffusion, is accompanied by a non-Gaussian displacement distribution (DisD), with roughly exponential tails at short times, a situation they termed "anomalous yet Brownian" diffusion. The diversity of systems in which this is observed calls for a generic model. We present such a model where there is "diffusivity memory" but no "direction memory" in the particle trajectory, and we show that it leads to both a…
Based on Brownian ratchets, a counter-intuitive phenomenon has recently emerged -- namely, that two losing games can yield, when combined, a paradoxical tendency to win. A restriction of this phenomenon is that the rules depend on the current capital of the player. Here we present new games where all the rules depend only on the history of the game and not on the capital. This new history-dependent structure significantly increases the parameter space for which the effect operates.
We derive backward and forward fractional Schrödinger type of equations for the distribution of functionals of the path of a particle undergoing anomalous diffusion. Fractional substantial derivatives introduced by Friedrich and co-workers [PRL {\bf 96}, 230601 (2006)] provide the correct fractional framework for the problem at hand. In the limit of normal diffusion we recover the Feynman-Kac treatment of Brownian functionals. For applications, we calculate the distribution of occupation times in half space and show how statistics of anomalous functionals is related to weak ergodicity breaking…
We present a detailed study of a Brownian particle driven by Carnot-type refrigerating protocol operating between two thermal baths. Both the underdamped as well as the overdamped limits are investigated. The particle is in a harmonic potential with time-periodic strength that drives the particle cyclically between the baths. Each cycle consists of two isothermal steps at different temperatures and two adiabatic steps connecting them. Besides working as a stochastic refrigerator, it is shown analytically that in the quasistatic regime the system can also act as stochastic heater, depending on …
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…
The transport phenomenon (movement and diffusion) of inertia Brownian particles in a periodic potential with non-Gaussian noise is investigated. It is found that proper noise intensity Q will promote particles directional movement(or diffusion), but large Q will inhibit this phenomenon. For large value of Q, the average velocity V (or the diffusion coefficient D) has a maximum with increasing correlation time tau. But for small value of Q, V (or D) decreases with increasing tau. In some cases, for the same value of Q and the same value of tau, non-Gaussian noise can induce particles directiona…
Giant number fluctuations (GNF) are an anomaly universally observed in active fluids with polar or nematic order. In this paper, we show that GNF arise in the fluid phase of active Brownian particles (ABP), where the polar order is absent. GNF in ABP extends over a large but finite length which characterizes the growing velocity correlations. To suppress unwanted phase separation and allow ones to explore the disordered fluid phase at large activities, we impart the inertia, or the mass, to the ABP. A linearized hydrodynamic theory captures our findings, but only qualitatively. We find numeric…
Anomalous diffusion is frequently described by scaled Brownian motion (SBM), a Gaussian process with a power-law time dependent diffusion coefficient. Its mean squared displacement is $\langle x^2(t)\rangle\simeq\mathscr{K}(t)t$ with $\mathscr{K}(t)\simeq t^{α-1}$ for $0<α<2$. SBM may provide a seemingly adequate description in the case of unbounded diffusion, for which its probability density function coincides with that of fractional Brownian motion. Here we show that free SBM is weakly non-ergodic but does not exhibit a significant amplitude scatter of the time averaged mean squared displac…
Generalizing Brownian motion (BM), fractional Brownian motion (FBM) is a paradigmatic selfsimilar model for anomalous diffusion. Specifically, varying its Hurst exponent, FBM spans: sub-diffusion, regular diffusion, and super-diffusion. As BM, also FBM is a symmetric and Gaussian process, with a continuous trajectory, and with a stationary velocity. In contrast to BM, FBM is neither a Markov process nor a martingale, and its velocity is correlated. Based on a recent study of selfsimilar Ito diffusions, we explore an alternative selfsimilar model for anomalous diffusion: fractional Ito motion (…
We present a model of anomalous diffusion consisting of an ensemble of particles undergoing homogeneous Brownian motion except for confinement by randomly placed reflecting boundaries. For power-law distributed compartment sizes, we calculate exact and asymptotic values of the ensemble averaged mean squared displacement and find that it increases subdiffusively, as either a power or the logarithm of time. Numerical simulations show that the probability density function of the displacement is non-Gaussian. We discuss the relevance of the model for the analysis of single-particle tracking experi…
Anomalous statistics in the Langevin equation with fluctuating diffusivity: from Brownian yet non-Gaussian diffusion to anomalous diffusion and ergodicity breaking
Tokyo University of Science（东京科学大学）；Pohang University of Science and Technology（浦项科技大学）；Asia Pacific Centre for Theoretical Physics（亚太理论物理中心）；University of Potsdam（波茨坦大学）；Wakayama University（和歌山大学）；Nagoya University（名古屋大学）；Keio University（庆应义塾大学）
Large scale simulations and analytical theory have been combined to obtain the non-equilibrium velocity distribution, $f(v)$, of randomly accelerated particles in suspension. The simulations are based on an event-driven algorithm, generalised to include friction. They reveal strongly anomalous but largely universal distributions which are independent of volume fraction and collision processes, which suggests a one-particle model should capture all the essential features. We have formulated this one-particle model and solved it analytically in the limit of strong damping, where we find that $f(…
Optical tweezers setup is often used to probe the motion of individual tracer particle, which promotes the study of relaxation dynamics of a generic process confined in a harmonic potential. We uncover the dependence of ensemble- and time-averaged mean square displacements of confined processes on the velocity correlation function $C(t,t+τ)$ of the original process. With two different scaling forms of $C(t,t+τ)$ for small $τ$ and large $τ$, the stationary value and the relaxation behaviors can be obtained immediately. The gotten results are valid for a large amount of anomalous diffusion proce…
Einstein's explanation of Brownian motion provided one of the cornerstones which underlie the modern approaches to stochastic processes. His approach is based on a random walk picture and is valid for Markovian processes lacking long-term memory. The coarse-grained behavior of such processes is described by the diffusion equation. However, many natural processes do not possess the Markovian property and exhibit to anomalous diffusion. We consider here the case of subdiffusive processes, which are semi-Markovian and correspond to continuous-time random walks in which the waiting time for a step…
The motion of particles in random potentials occurs in several natural phenomena ranging from the mobility of organelles within a biological cell to the diffusion of stars within a galaxy. A Brownian particle moving in the random optical potential associated to a speckle, i.e., a complex interference pattern generated by the scattering of coherent light by a random medium, provides an ideal mesoscopic model system to study such phenomena. Here, we derive a theory for the motion of a Brownian particle in a speckle and, in particular, we identify its universal characteristic timescale levering o…
Recent studies on Brownian motors driven by colored non Gaussian noises have shown that the departure of the noise distribution from Gaussian behavior induces an enhancement of its current and efficiency. Here we discuss some new aspects of this phenomenon focusing in some analytical results based in an adiabatic approximation, and in the analysis of the long probability distribution tails' role.
Based on the dynamical quantization method we derive a quantum phase-space non-Markovian Smoluchowski equation describing the non-inertial Brownian motion of a harmonic oscillator immersed in a generic environment. In the long-time regime we investigate the tunneling phenomenon by evaluating the quantum Kramers escape rate of a Brownian particle over a potential barrier. As far as a quantum thermal reservoir is concerned, it is found that our steady quantum Kramers rate may depend upon no friction constant in the strong friction domain. This preposterous feature may suggest the existence of no…
This work puts forward a generalization of the well-known rocking Markovian Brownian ratchets to the realm of antipersistent non-Markovian subdiffusion in viscoelastic media. A periodically forced subdiffusion in a parity-broken ratchet potential is considered within the non-Markovian generalized Langevin equation (GLE) description with a power-law memory kernel $η(t)\propto t^{-α}$ ($0<α<1$). It is shown that subdiffusive rectification currents, defined through the mean displacement and subvelocity $v_α$, $<δx(t)>\sim v_α t^α/ Γ(1+α)$, emerge asymptotically due to the breaking of the detailed…
Three-dimensional Monte Carlo simulations provide a striking confirmation to a recent theoretical prediction: the Brownian non-Gaussian diffusion of critical self-avoiding walks. Although the mean square displacement of the polymer center of mass grows linearly with time (Brownian behavior), the initial probability density function is strongly non-Gaussian and crosses over to Gaussianity only at large time. Full agreement between theory and simulations is achieved without the employment of fitting parameters. We discuss simulation techniques potentially capable of addressing the study of anoma…
Fractional Brownian motion and the fractional Langevin equation are models of anomalous diffusion processes characterized by long-range power-law correlations in time. We employ large-scale computer simulations to study these models in two geometries, (i) the spreading of particles on a semi-infinite domain with an absorbing wall at one end and (ii) the stationary state on a finite interval with absorbing boundaries at both ends and a source in the center. We demonstrate that the probability density and other properties of the fractional Langevin equation can be mapped onto the corresponding q…
A dynamics between Newton and Langevin formalisms is elucidated within the framework of the generalized Langevin equation. For thermal noise yielding a vanishing zero-frequency friction the corresponding non-Markovian Brownian dynamics exhibits anomalous behavior which is characterized by ballistic diffusion and accelerated transport. We also investigate the role of a possible initial correlation between the system degrees of freedom and the heat-bath degrees of freedom for the asymptotic long-time behavior of the system dynamics. As two test beds we investigate (i) the anomalous energy relaxa…
Anomalous diffusion is predicted for Brownian particles in inhomogeneous viscosity landscapes by means of scaling arguments, which are substantiated through numerical simulations. Analytical solutions of the related Fokker-Planck equation in limiting cases confirm our results. For an ensemble of particles starting at a spatial minimum (maximum) of the viscous damping we find subdiffusive (superdiffusive) motion. Superdiffusion occurs also for a monotonically varying viscosity profile. We suggest different substances for related experimental investigations.
Fractional Brownian motion (FBM), a non-Markovian self-similar Gaussian stochastic process with long-ranged correlations, represents a widely applied, paradigmatic mathematical model of anomalous diffusion. We report the results of large-scale computer simulations of FBM in one, two, and three dimensions in the presence of reflecting boundaries that confine the motion to finite regions in space. Generalizing earlier results for finite and semi-infinite one-dimensional intervals, we observe that the interplay between the long-time correlations of FBM and the reflecting boundaries leads to strik…
Quantitatively understanding of the dynamics of an active Brownian particle (ABP) interacting with a viscoelastic polymer environment is a scientific challenge. It is intimately related to several interdisciplinary topics such as the microrheology of active colloids in a polymer matrix and the athermal dynamics of the in vivo chromosome or cytoskeletal networks. Based on Langevin dynamics simulation and analytic theory, here we explore such a viscoelastic active system in depth using a star polymer of functionality $f$ with the center cross-linker particle being ABP. We observe that the ABP cr…
Functionals of Brownian motion have diverse applications in physics, mathematics, and other fields. The probability density function (PDF) of Brownian functionals satisfies the Feynman-Kac formula, which is a Schrodinger equation in imaginary time. In recent years there is a growing interest in particular functionals of non-Brownian motion, or anomalous diffusion, but no equation existed for their PDF. Here, we derive a fractional generalization of the Feynman-Kac equation for functionals of anomalous paths based on sub-diffusive continuous-time random walk. We also derive a backward equation …
Consider a chaotic dynamical system generating Brownian motion-like diffusion. Consider a second, non-chaotic system in which all particles localize. Let a particle experience a random combination of both systems by sampling between them in time. What type of diffusion is exhibited by this {\em random dynamical system}? We show that the resulting dynamics can generate anomalous diffusion, where in contrast to Brownian normal diffusion the mean square displacement of an ensemble of particles increases nonlinearly in time. Randomly mixing simple deterministic walks on the line we find anomalous …
A standard approach to study time-dependent stochastic processes is the power spectral density (PSD), an ensemble-averaged property defined as the Fourier transform of the autocorrelation function of the process in the asymptotic limit of long observation times, $T\to\infty$. In many experimental situations one is able to garner only relatively few stochastic time series of finite $T$, such that practically neither an ensemble average nor the asymptotic limit $T\to\infty$ can be achieved. To accommodate for a meaningful analysis of such finite-length data we here develop the framework of singl…
In this paper we revisit the Brownian motion on the basis of {the fractional Langevin equation which turns out to be a particular case of the generalized Langevin equation introduced by Kubo in 1966. The importance of our approach is to model the Brownian motion more realistically than the usual one based on the classical Langevin equation, in that it takes into account also the retarding effects due to hydrodynamic back-flow, i.e. the added mass and the Basset memory drag. We provide the analytical expressions of the correlation functions (both for the random force and the particle velocity) …
We study the motion of charged Brownian particles in an external magnetic field. It is found that a correlation appears between the components of particle velocity in the case of anisotropic friction, approaching asymptotically zero in the stationary limit. If magnetic field is smaller compared to the critical value, determined by frictional anisotropy, the relaxation of the correlation is non-oscillating in time. However, in a larger magnetic field this relaxation becomes oscillating. The phenomenon is related to the statistical dependence of the components of transformed random force caused …
Two models for quantum Brownian motion - the Oscillator Bath (OB) model and the Random-Band-Matrix (RBM) model - are compared and a relation between the spectral density function I(w) and the variance (Vab)^2 is established. The extension to a fractal environment is then considered and the microscopic origin of anomalous diffusion is discussed. In particular, it is shown that the asymptotic behavior of the normalized velocity auto-correlation function (VACF) is entirely determined by the band form factor. This allows for a microscopic derivation of the "whip-back" effect.
The comb model is a simplified description for anomalous diffusion under geometric constraints. It represents particles spreading out in a two-dimensional space where the motions in the x-direction are allowed only when the y coordinate of the particle is zero. Here, we propose an extension for the comb model via Langevin-like equations driven by fractional Gaussian noises (long-range correlated). By carrying out computer simulations, we show that the correlations in the y-direction affect the diffusive behavior in the x-direction in a non-trivial fashion, resulting in a quite rich diffusive s…
We introduce a fractional Klein-Kramers equation which describes sub-ballistic superdiffusion in phase space in the presence of a space-dependent external force field. This equation defines the differential L{é}vy walk model whose solution is shown to be non-negative. In the velocity coordinate, the probability density relaxes in Mittag-Leffler fashion towards the Maxwell distribution whereas in the space coordinate, no stationary solution exists and the temporal evolution of moments exhibits a competition between Brownian and anomalous contributions.
Anomalous-diffusion, the departure of the spreading dynamics of diffusing particles from the traditional law of Brownian-motion, is a signature feature of a large number of complex soft-matter and biological systems. Anomalous-diffusion emerges due to a variety of physical mechanisms, e.g., trapping interactions or the viscoelasticity of the environment. However, sometimes systems dynamics are erroneously claimed to be anomalous, despite the fact that the true motion is Brownian -- or vice versa. This ambiguity in establishing whether the dynamics as normal or anomalous can have far-reaching c…
Two losing games, when alternated in a periodic or random fashion, can produce a winning game. This paradox occurs in a family of stochastic processes: if one combines two or more dynamics where a given quantity decreases, the result can be a dynamic system where this quantity increases. The paradox could be applied to a number of stochastic systems and has drawn the attention of researchers from different areas. In this paper we show how the phenomenon can be used to design Brownian or molecular motors, i.e., thermal engines that operate by rectifying fluctuations. We briefly review the liter…
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…
Fractional Brownian motion is a Gaussian stochastic process with stationary, long-time correlated increments and is frequently used to model anomalous diffusion processes. We study numerically fractional Brownian motion confined to a finite interval with reflecting boundary conditions. The probability density function of this reflected fractional Brownian motion at long times converges to a stationary distribution showing distinct deviations from the fully flat distribution of amplitude $1/L$ in an interval of length $L$ found for reflected normal Brownian motion. While for superdiffusion, cor…
We analyze the scaled voter model, which is a generalization of the noisy voter model with time-dependent herding behavior. We consider the case when the intensity of herding behavior grows as a power-law function of time. In this case, the scaled voter model reduces to the usual noisy voter model, but it is driven by the scaled Brownian motion. We derive analytical expressions for the time evolution of the first and second moments of the scaled voter model. In addition, we have derived an analytical approximation of the first passage time distribution. By numerical simulation, we confirm our …
