arXiv subjects

Tomasz Srokowski

Publications and source records attributed to Tomasz Srokowski.

16 recordsLinked to original sources

arXiv2017

Random walk in nonhomogeneous environments: A possible approach to human and animal mobility

The random walk process in a nonhomogeneous medium, characterised by a Lévy stable distribution of jump length, is discussed. The width depends on a position: either before the jump or after that. In the latter case, the density slope is affected by the variable width and the variance may be finite; then all kinds of the anomalous diffusion are predicted. In the former case, only the time characteristics are sensitive to the variable width. %while the former case resolves itself to a problem with a variable jumping rate, The corresponding Langevin equation with different interpretations of the multiplicative noise is discussed. The dependence of the distribution width on position after jump is interpreted in terms of cognitive abilities and related to such problems as migration in a human population and foraging habits of animals.

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arXiv2016

Escape process in systems characterised by stable noises and position-dependent resting times

Stochastic systems characterised by a random driving in a form of the general stable noise are considered. The particle experiences long rests due to the traps the density of which is position-dependent and obeys a power-law form attributed to the underlying self-similar structure. Both one and two dimensional case are analysed. The random walk description involves a position-dependent waiting time distribution. On the other hand, the stochastic dynamics is formulated in terms of the subordination technique where the random time generator is position-dependent. The first passage time problem is addressed by evaluating a first passage time density distribution and an escape rate. The influence of the medium nonhomogeneity on those quantities is demonstrated; moreover, the dependence of the escape rate on the stability index and the memory parameter is evaluated. Results indicate essential differences between the Gaussian case and the case involving Levy flights.

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arXiv2015

Levy flights and nonhomogenous memory effects: relaxation to a stationary state

The non-Markovian stochastic dynamics involving Levy flights and a potential in the form of a harmonic and non-linear oscillator is discussed. The subordination technique is applied and the memory effects, which are nonhomogeneous, are taken into account by a position-dependent subordinator. In the non-linear case, the asymptotic stationary states are found. The relaxation pattern to the stationary state is derived for the quadratic potential: the density decays like a linear combination of the Mittag-Leffler functions. It is demonstrated that in the latter case the density distribution satisfies a fractional Fokker-Planck equation. The densities for the non-linear oscillator reveal a complex picture, qualitatively dependent on the potential strength, and the relaxation pattern is exponential at large time.

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arXiv2015

Anomalous diffusion in stochastic systems with nonhomogeneously distributed traps

The stochastic motion in a nonhomogeneous medium with traps is studied and diffusion properties of that system are discussed. The particle is subjected to a stochastic stimulation obeying a general Lévy stable statistics and experiences long rests due to traps the density of which depends on the position. The memory is taken into account by subordination of that process to a random time; then the subordination equation is position-dependent. The problem is approximated by means of a decoupling of the trap geometry and memory and exactly solved for a power-law trap density, corresponding to a fractal medium structure, in the case of the Gaussian statistics: the density distribution and moments are derived. Depending on geometry and memory parameters, the system may reveal both the subdiffusion and enhanced diffusion. A similar analysis is performed for the Lévy flights where the finiteness of the variance follows from a multiplicative noise, as a result of impurities and defects at the boundary. Two diffusion regimes are found: in the bulk and near the surface. The anomalous diffusion exponent as a function of the system parameters is derived.

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arXiv2014

Asymmetric Levy flights in nonhomogeneous environments

We consider stochastic systems involving general -- non-Gaussian and asymmetric -- stable processes. The random quantities, either a stochastic force or a waiting time in a random walk process, explicitly depend on the position. A fractional diffusion equation corresponding to a master equation for a jumping process with a variable jumping rate is solved in a diffusion limit. It is demonstrated that for some model parameters the equation is satisfied in that limit by the stable process with the same asymptotics as the driving noise. The Langevin equation containing a multiplicative noise, depending on the position as a power-law, is solved; the existing moments are evaluated. The motion appears subdiffusive and the transport depends on the asymmetry parameter: it is fastest for the symmetric case. A special case of the one-sided distribution is discussed.

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arXiv2014

Anomalous diffusion in nonhomogeneous media: Time-subordinated Langevin equation approach

Diffusion in nonhomogeneous media is described by a dynamical process driven by a general Levy noise and subordinated to a random time; the subordinator depends on the position. This problem is approximated by a multiplicative process subordinated to a random time: it separately takes into account effects related to the medium structure and the memory. Density distributions and moments are derived from the solutions of the corresponding Langevin equation and compared with the numerical calculations for the exact problem. Both subdiffusion and enhanced diffusion are predicted. Distribution of the process satisfies the fractional Fokker-Planck equation.

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arXiv2013

Bistable generalised Langevin dynamics driven by correlated noise possessing a long jump distribution: barrier crossing and stochastic resonance

The generalised Langevin equation with a retarded friction and a double-well potential is solved. The random force is modelled by a multiplicative noise with long jumps. Probability density distributions converge with time to a distribution similar to a Gaussian but tails have a power-law form. Dependence of the mean first passage time on model parameters is discussed. Properties of the stochastic resonance, emerging as a peak in the plot of the spectral amplification against the temperature, are discussed for various sets of the model parameters. The amplification rises with the memory and is largest for the cases corresponding to the large passage time.

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arXiv2013

Fluctuations in multiplicative systems with jumps

Fluctuation properties of the Langevin equation including a multiplicative, power-law noise and a quadratic potential are discussed. The noise has the Levy stable distribution. If this distribution is truncated, the covariance can be derived in the limit of large time; it falls exponentially. Covariance in the stable case, studied for the Cauchy distribution, exhibits a weakly stretched exponential shape and can be approximated by the simple exponential. The dependence of that function on system parameters is determined. Then we consider a dynamics which involves the above process and obey the generalised Langevin equation, the same as for Gaussian case. The resulting distributions possess power-law tails - that fall similarly to those for the driving noise - whereas central parts can assume the Gaussian shape. Moreover, a process with the covariance 1/t at large time is constructed and the corresponding dynamical equation solved. Diffusion properties of systems for both covariances are discussed.

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arXiv2012

Anomalous diffusion in systems driven by the stable Levy noise with a finite noise relaxation time and inertia

Dynamical systems driven by a general Lévy stable noise are considered. The inertia is included and the noise, represented by a generalised Ornstein-Uhlenbeck process, has a finite relaxation time. A general linear problem (the additive noise) is solved: the resulting distribution converges with time to the distribution for the white-noise, massless case. Moreover, a multiplicative noise is discussed. It can make the distribution steeper and the variance, which is finite, depends sublinearly on time (subdiffusion). For a small mass, a white-noise limit corresponds to the Stratonovich interpretation. On the other hand, the distribution tails agree with the Ito interpretation if the inertia is very large. An escape time from the potential well is calculated.

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arXiv2012

Multiplicative Levy noise in bistable systems

Stochastic motion in a bistable, periodically modulated potential is discussed. The system is stimulated by a white noise increments of which have a symmetric stable Lévy distribution. The noise is multiplicative: its intensity depends on the process variable like |x|^{-θ}. The Stratonovich and Itô interpretations of the stochastic integral are taken into account. The mean first passage time is calculated as a function of θfor different values of the stability index αand size of the barrier. Dependence of the output amplitude on the noise intensity reveals a pattern typical for the stochastic resonance. Properties of the resonance as a function of α, θ and size of the barrier are discussed. Both height and position of the peak strongly depends on θ and on a specific interpretation of the stochastic integral.

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arXiv2011

Anomalous diffusion for a correlated process with long jumps

We discuss diffusion properties of a dynamical system, which is characterised by long-tail distributions and finite correlations. The particle velocity has the stable Lévy distribution; it is assumed as a jumping process (the kangaroo process) with a variable jumping rate. Both the exponential and the algebraic form of the covariance -- defined for the truncated distribution -- are considered. It is demonstrated by numerical calculations that the stationary solution of the master equation for the case of power-law correlations decays with time, but a simple modification of the process makes the tails stable. The main result of the paper is a finding that -- in contrast to the velocity fluctuations -- the position variance may be finite. It rises with time faster than linearly: the diffusion is anomalously enhanced. On the other hand, a process which follows from a superposition of the Ornstein-Uhlenbeck-Lévy processes always leads to position distributions with a divergent variance which means accelerated diffusion.

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arXiv2011

Correlated Lévy noise in linear dynamical systems

Linear dynamical systems, driven by a non-white noise which has the Levy distribution, are analysed. Noise is modelled by a specific stochastic process which is defined by the Langevin equation with a linear force and the Levy distributed symmetric white noise. Correlation properties of the process are discussed. The Fokker-Planck equation driven by that noise is solved. Distributions have the Levy shape and their width, for a given time, is smaller than for processes in the white noise limit. Applicability of the adiabatic approximation in the case of the linear force is discussed.

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arXiv2010

Nonlinear stochastic equations with multiplicative Lévy noise

The Langevin equation with a multiplicative Lévy white noise is solved. The noise amplitude and the drift coefficient have a power-law form. A validity of ordinary rules of the calculus for the Stratonovich interpretation is discussed. The solution has the algebraic asymptotic form and the variance may assume a finite value for the case of the Stratonovich interpretation. The problem of escaping from a potential well is analysed numerically; predictions of different interpretations of the stochastic integral are compared.

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arXiv2009

Multiplicative Lévy processes: Itô versus Stratonovich interpretation

Langevin equation with a multiplicative stochastic force is considered. That force is uncorrelated, it has the Lévy distribution and the power-law intensity. The Fokker-Planck equations, which correspond both to the Itô and Stratonovich interpretation of the stochastic integral, are presented. They are solved for the case without drift and for the harmonic oscillator potential. The variance is evaluated; it is always infinite for the Itô case whereas for the Stratonovich one it can be finite and rise with time slower that linearly, which indicates subdiffusion. Analytical results are compared with numerical simulations.

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arXiv2009

Stretched-Gaussian asymptotics of the truncated Lévy flights for the diffusion in nonhomogeneous media

The Lévy, jumping process, defined in terms of the jumping size distribution and the waiting time distribution, is considered. The jumping rate depends on the process value. The fractional diffusion equation, which contains the variable diffusion coefficient, is solved in the diffusion limit. That solution resolves itself to the stretched Gaussian when the order parameter $μ\to2$. The truncation of the Lévy flights, in the exponential and power-law form, is introduced and the corresponding random walk process is simulated by the Monte Carlo method. The stretched Gaussian tails are found in both cases. The time which is needed to reach the limiting distribution strongly depends on the jumping rate parameter. When the cutoff function falls slowly, the tail of the distribution appears to be algebraic.

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arXiv2009

Fractional Fokker-Planck equation for Lévy flights in nonhomogeneous environments

The fractional Fokker-Planck equation, which contains a variable diffusion coefficient, is discussed and solved. It corresponds to the Lévy flights in a nonhomogeneous medium. For the case with the linear drift, the solution is stationary in the long-time limit and it represents the Lévy process with a simple scaling. The solution for the drift term in the form $λ\hbox{sgn}(x)$ possesses two different scales which correspond to the Lévy indexes $μ$ and $μ+1$ $(μ<1)$. The former component of the solution prevails at large distances but it diminishes with time for a given $x$. The fractional moments, as a function of time, are calculated. They rise with time and the rate of this growth increases with $λ$.

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