We report in this paper a thorough study on the the dynamical mechanics of the fractional Brownian motion systems. Where several non-trivial properties are revealed such as the abundant non-Markovian effects resulted from the fractional characters of the system. In general, the dynamics of the fBm system is found to be of a purely Newton's type, despite of the anomalous fractional properties of the system.
In this study, we analytically formulated the path integral representation of the conditional probabilities for non-Markovian kinetic processes in terms of the free energy of the thermodynamic system. We carry out analytically the time-fractional kinetic equations for these processes. Thus, in a simple way, we generalize path integral solutions of the Markovian to the non-Markovian cases. We conclude that these pedagogical results can be applied to some physical problems such as the deformed ion channels, internet networks and non-equilibrium phase transition problems.
The Fractional Langevin Equation (FLE) describes a non-Markovian Generalized Brownian Motion with long time persistence (superdiffusion), or anti-persistence (subdiffusion) of both velocity-velocity correlations, and position increments. It presents a case of the Generalized Langevin Equation (GLE) with a singular power law memory kernel. We propose and numerically realize a numerically efficient and reliable Markovian embedding of this superdiffusive GLE, which accurately approximates the FLE over many, about r=N lg b-2, time decades, where N denotes the number of exponentials used to approxi…
We consider a Markovian jumping process which is defined in terms of the jump-size distribution and the waiting-time distribution with a position-dependent frequency, in the diffusion limit. We assume the power-law form for the frequency. For small steps, we derive the Fokker-Planck equation and show the presence of the normal diffusion, subdiffusion and superdiffusion. For the Lévy distribution of the step-size, we construct a fractional equation, which possesses a variable coefficient, and solve it in the diffusion limit. Then we calculate fractional moments and define fractional diffusion c…
A Gauss-Newton method for iterative optimization of memory kernels for generalized Langevin thermostats in coarse-grained molecular dynamics simulations
We study the diffusion equation with a position-dependent, power-law diffusion coefficient. The equation possesses the Riesz-Weyl fractional operator and includes a memory kernel. It is solved in the diffusion limit of small wave numbers. Two kernels are considered in detail: the exponential kernel, for which the problem resolves itself to the telegrapher's equation, and the power-law one. The resulting distributions have the form of the Lévy process for any kernel. The renormalized fractional moment is introduced to compare different cases with respect to the diffusion properties of the syste…
We study, using exact numerical simulations, the statistics of the longest excursion l_{\max}(t) up to time t for the fractional Brownian motion with Hurst exponent 0<H<1. We show that in the large t limit, < l_{\max}(t) > \propto Q_\infty t where Q_\infty \equiv Q_\infty(H) depends continuously on H, and in a non trivial way. These results are compared with exact analytical results obtained recently for a renewal process with an associated persistence exponent θ= 1-H. This comparison shows that Q_\infty(H) carries the clear signature of non-Markovian effects for H\neq 1/2. The pre-asymptotic …
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…
We derive a functional equation for the mean first-passage time (MFPT) of a generic self-similar Markovian continuous process to a target in a one-dimensional domain and obtain its exact solution. We show that the obtained expression of the MFPT for continuous processes is actually different from the large system size limit of the MFPT for discrete jump processes allowing leapovers. In the case considered here, the asymptotic MFPT admits non-vanishing corrections, which we call residual MFPT. The case of L/'evy flights with diverging variance of jump lengths is investigated in detail, in parti…
In the present Short Note an idea is proposed to explain the emergence and the observation of processes in complex media that are driven by fractional non-Markovian master equations. Particle trajectories are assumed to be solely Markovian and described by the Continuous Time Random Walk model. But, as a consequence of the complexity of the medium, each trajectory is supposed to scale in time according to a particular random timescale. The link from this framework to microscopic dynamics is discussed and the distribution of timescales is computed. In particular, when a stationary distribution …
Brownian motion is the only random process which is Gaussian, stationary and Markovian. Dropping the Markovian property, i.e. allowing for memory, one obtains a class of processes called fractional Brownian motion, indexed by the Hurst exponent $H$. For $H=1/2$, Brownian motion is recovered. We develop a perturbative approach to treat the non-locality in time in an expansion in $\varepsilon = H-1/2$. This allows us to derive analytic results beyond scaling exponents for various observables related to extreme value statistics: The maximum $m$ of the process and the time $t_{\text{max}}$ at whic…
We investigate fractional Brownian motion with a microscopic random-matrix model and introduce a fractional Langevin equation. We use the latter to study both sub- and superdiffusion of a free particle coupled to a fractal heat bath. We further compare fractional Brownian motion with the fractal time process. The respective mean-square displacements of these two forms of anomalous diffusion exhibit the same power-law behavior. Here we show that their lowest moments are actually all identical, except the second moment of the velocity. This provides a simple criterion which enables to distinguis…
We study the stochastic motion of particles driven by long-range correlated fractional Gaussian noise in a superharmonic external potential of the form $U(x)\propto x^{2n}$ ($n\in\mathbb{N}$). When the noise is considered to be external, the resulting overdamped motion is described by the non-Markovian Langevin equation for fractional Brownian motion. For this case we show the existence of long time, stationary probability density functions (PDFs) the shape of which strongly deviates from the naively expected Boltzmann PDF in the confining potential $U(x)$. We analyse in detail the temporal ap…
First passage time plays a fundamental role in dynamical characterization of stochastic processes. Crucially, our current understanding on the problem is almost entirely relies on the theoretical formulations, which assume the processes under consideration are Markovian, despite abundant non-Markovian dynamics found in complex systems. Here we introduce a simple and physically appealing analytical framework to calculate the first passage time statistics of non-Markovian walkers grounded in a fundamental principle of nonequilibrium statistical physics that connects the fluctuations in stochasti…
This paper is concerned with the stochastic thermodynamics of non-equilibrium Gaussian processes that can exhibit anomalous diffusion. In the systems considered, the noise correlation function is not necessarily related to friction. Thus, there is no conventional fluctuation-dissipation relation (FDR) of the second kind and no unique way to define a temperature. We start from a Markovian process with time-dependent diffusivity (an example being scaled Brownian motion). It turns out that standard stochastic thermodynamic notions can be applied rather straightforwardly by introducing a time-depe…
We analyze the statistical properties of a temporal point process driven by a confined fractional Brownian motion. The event count distribution and power spectral density of this non--Markovian point process exhibit power--law scaling. We show that a nonlinear Markovian point process can reproduce the same scaling behavior. This result indicates a possible link between nonlinearity and apparent non--Markovian behavior.
Fractional Brownian motion is a non-Markovian Gaussian process $X_t$, indexed by the Hurst exponent $H$. It generalises standard Brownian motion (corresponding to $H=1/2$). We study the probability distribution of the maximum $m$ of the process and the time $t_{\rm max}$ at which the maximum is reached. They are encoded in a path integral, which we evaluate perturbatively around a Brownian, setting $H=1/2 + \varepsilon$. This allows us to derive analytic results beyond the scaling exponents. Extensive numerical simulations for different values of $H$ test these analytical predictions and show …
We present a treatment of non-Markovian character of memory by incorporating different forms of Mittag-Leffler (ML) function, which generally arises in the solution of fractional master equation, as different memory functions in the Generalized Kolmogorov-Feller Equation (GKFE). The cross-over from the short time (stretched exponential) to long time (inverse power law) approximations of the ML function incorporated in the GKFE is proven. We have found that the GKFE solutions are the same for negative exponential and for upto frst order expansion of stretched exponential function for very small…
Fractional Brownian motion is a self-affine, non-Markovian and translationally invariant generalization of Brownian motion, depending on the Hurst exponent $H$. Here we investigate fractional Brownian motion where both the starting and the end point are zero, commonly referred to as bridge processes. Observables are the time $t_+$ the process is positive, the maximum $m$ it achieves, and the time $t_{\rm max}$ when this maximum is taken. Using a perturbative expansion around Brownian motion ($H=\frac12$), we give the first-order result for the probability distribution of these three variables,…
The Dicke model famously exhibits a phase transition to a superradiant phase with a macroscopic population of photons and is realized in multiple settings in open quantum systems. In this work, we study a variant of the Dicke model where the cavity mode is lossy due to the coupling to a Markovian environment while the atomic mode is coupled to a colored bath. We analytically investigate this model by inspecting its low-frequency behavior via the Schwinger-Keldysh field theory and carefully examine the nature of the corresponding superradiant phase transition. Integrating out the fast modes, we…
The purpose of this work is to propose a non-Markovian and nonlinear model of subdiffusive transport that involves adhesion affects the cells escape rates form position x, with chemotaxis. This leads the escape rates to be dependent on the particles density at the neighbours as well as the chemotactic gradient. We systematically derive subdiffusive fractional master equation, then we consider the diffusive limit of the fractional master equation. We finally solve the resulted fractional subdiffusive master equation stationery and analyse the role of adhesion in the resulted stationary density.
The Wiener's path integral plays a central role in the studies of Brownian motion. Here we derive exact path-integral representations for the more general \emph{fractional} Brownian motion (fBm) and for its time derivative process -- the fractional Gaussian noise (fGn). These paradigmatic non-Markovian stochastic processes, introduced by Kolmogorov, Mandelbrot and van Ness, found numerous applications across the disciplines, ranging from anomalous diffusion in cellular environments to mathematical finance. Still, their exact path-integral representations were previously unknown. Our formalism …
A living non-Newtonian matter like the cell cortex and tissues are driven out-of-equilibrium at multiple spatial and temporal scales. The stochastic dynamics of a particle embedded in such a medium are non-Markovian, given by a generalized Langevin equation. Due to the non-Markovian nature of the dynamics, the heat dissipation and the entropy production rate cannot be computed using the standard methods for Markovian processes. In this work, to calculate heat dissipation, we use an effective Markov description of the non-Markovian dynamics, which includes the degrees-of-freedom of the medium. …
We study Markovian stochastic motion on a graph with finite number of nodes and adiabatically periodically driven transition rates. We show that, under general conditions, the quantized currents that appear at low temperatures are a manifestation of topological invariants in the counting statistics of currents. This observation provides an approach for classification of topological properties of the counting statistics, as well as for extensions of the phenomenon of the robust quantization of currents at low temperatures to the properties of the counting statistics which persist to finite temp…
