Lévy processes and Schrödinger equation

We analyze the extension of the well known relation between Brownian motion and the Schrödinger equation to the family of the Lévy processes. We consider a Lévy-Schrödinger equation where the usual kinetic energy operator-the Laplacian-is generalized by means of a selfadjoint, pseudodifferential operator whose symbol is the logarithmic characteristic of an infinitely divisible law. The Lévy-Khintchin formula shows then how to write down this operator in an integro-differential form. When the underlying Lévy process is stable we recover as a particular case the fractional Schrödinger equation. A few examples are finally given and we find that there are physically relevant models-such as a form of the relativistic Schrödinger equation-that are in the domain of the non stable Lévy-Schrödinger equations.     

Fluctuation-driven directed transport in the presence of Lévy flights

The role of Lévy flights on fluctuation-driven transport in time independent periodic potentials with broken spatial symmetry is studied. Two complementary approaches are followed. The first one is based on a generalized Langevin model describing overdamped dynamics in a ratchet type external potential driven by Lévy white noise with stability index α in the range 1<α<2. The second approach is based on the space fractional Fokker-Planck equation describing the corresponding probability density function (PDF) of particle displacements. It is observed that, even in the absence of an external tilting force or a bias in the noise, the Lévy flights drive the system out of the thermodynamic equilibrium and generate an up-hill current (i.e., a current in the direction of the steeper side of the asymmetric potential). For small values of the noise intensity there is an optimal value of α yielding the maximum current. The direction and magnitude of the current can be manipulated by changing the Lévy noise asymmetry and the potential asymmetry. For a sharply localized initial condition, the PDF of staying at the minimum of the potential exhibits scaling behavior in time with an exponent bigger than the −1/α exponent corresponding to the force free case.     

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→2. 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.     

Noise-induced collective regimes of complex system in contact with a random reservoir

Based on a system-reservoir model, where the reservoir is driven by an external white Gaussian noise, we study the behavior of system components in Weiss mean-field approach and Gaussian approximation for moments. Crossover from individual to cooperative dynamics of the system components is due to noise. The system displays a transition similar to diversity-induced phase transition. The analytical results are compared with the numerical simulations.     

Can spontaneous cell movements be modelled as Lévy walks?

Spontaneous cell movement is a random motion that takes place in the absence of external guiding stimuli. The spontaneous movements of HaCaT and NHDF cells (cells of the epidermis) are well represented as continuous Markovian processes driven by multiplicative noise [D. Selmeczi, S. Mosler, P.H. Hagedorn, N.B. Larsen, H. Flyvbjerg, Biophysical Journal 89 (2005) 912]. Model components are, however, ad hoc as they are inspired by fits to experimental data. As a consequence, model agreement with experimental data does not add much to our understanding of spontaneous movements of these cells beyond demonstrating that they can be modelled phenomenologically. Here it is noted that a slight re-parameterization and re-interpretation of the driving noise leads to the model of Lubashevsky et al. (2009) [I. Lubashevsky, R. Friedrich, A. Heuer, Physical Review E 79 (2009) 011110] that realises Lévy walks as Markovian stochastic processes. This brings forth new biological insight as Lévy walks are advantageous when searching in the absence of external stimuli and without knowledge of the target distribution, as may be the case with cells of the epidermis that form new tissue by locating and then attaching on to one another. The Hänggi-Klimontovich interpretation of the driving noise in the model of Lubashevsky et al. (2009) and Cauchy distributions of predicted velocities do, however, appear problematic, even unphysical. Here it is shown that these are perceived rather than actual difficulties. Intermittent stop-start motions of the kind displayed by some cells and protozoan are found to underlie the formulation of the model of Lubashevsky et al. (2009) and the velocities of starved Dictyostelium discoideum (a unicellular organism) are found to be Cauchy distributed to a good approximation. It is therefore suggested that the model of Lubashevsky et al. (2009) can describe the spontaneous movements of some cells, and that some cells have spontaneous movement patterns that can be approximated by Lévy walks, as first proposed by Schuster and Levandowsky (1996) [F.L. Schuster, M. Levandowsky, Journal of Eukaryotic Microbiology 43 (1996) 150].     

Langevin equation with two fractional orders

A new type of fractional Langevin equation of two different orders is introduced. The solutions for this equation, known as the fractional Ornstein-Uhlenbeck processes, based on Weyl and Riemann-Liouville fractional derivatives are obtained. The basic properties of these processes are studied. An example of the spectral density of ocean wind speed which has similar spectral density as that of Weyl fractional Ornstein-Uhlenbeck process is given.     

Correlation in the velocity of a Brownian particle induced by frictional anisotropy and magnetic field

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 by the simultaneous influence of magnetic field and anisotropic dissipation.     

Detecting unknown paths on complex networks through random walks

In this article, we investigate the problem of detecting unknown paths on complex networks through random walks. To detect a given path on a network a random walker should pass through the path from its initial node to its terminal node in turn. We calculate probability ?(t) that a random walker detects a given path on a connected network in t steps when it starts out from source node s. We propose an iteration formula for calculating ?(t). Generating function of ?(t) is also derived. Major factors affecting ?(t), such as walking time t, path length l, starting point of the walker, structure of the path, and topological structure of the underlying network are further discussed. Among these factors, two most outstanding ones are walking time t and path length l. On the one hand, ?(t) increases as t increases, and ?(∞)=1, which shows that the longer the walking time, the higher the chance of detecting a given path, and the walker will discover the path sooner or later so long as it keeps wandering on the network. On the other hand, ?(t) drops substantially as path length l increases, which shows that the longer the path, the lower the chance for the walker to find it in a given time. Apart from path length, path structure also has obvious effect on ?(t). Starting point of the walker has only minor influence on ?(t), but topological structure of the underlying network has strong influence on ?(t). Simulations confirm our analytic results.     

Cauchy flights in confining potentials

We analyze confining mechanisms for Lévy flights evolving under an influence of external potentials. Given a stationary probability density function (pdf), we address the reverse engineering problem: design a jump-type stochastic process whose target pdf (eventually asymptotic) equals the preselected one. To this end, dynamically distinct jump-type processes can be employed. We demonstrate that one “targeted stochasticity” scenario involves Langevin systems with a symmetric stable noise. Another derives from the Lévy-Schrödinger semigroup dynamics (closely linked with topologically induced super-diffusions), which has no standard Langevin representation. For computational and visualization purposes, the Cauchy driver is employed to exemplify our considerations.     

On a super-diffusive, nonlinear birth and death process

The explicit and fully analytic transient solution for the transition probability density associated with a nonlinear birth and death process on Z is constructed. The time-dependent variance is proportional to t+Bt² (with B being a constant), thus exhibiting a super-diffusive behavior. The space continuous limit of this process is a well-known diffusion process with nonlinear drift for which the transition probability density is also known explicitly in a very simple way.     

Noise-induced coupling and phase transition in initially homogeneous bistable system

We show that a colored spatial noise induces a heterogeneous behavior and coupling of initially uncoupled single bistable units. A formal approximation reduces a non-Markovian stochastic process described by the initial set of equations into Markovian process in terms of Langevin equation, for which a simple piecewise linear emulation was used to represent the nonlinear deterministic force. It turned out that the coupling leads to a phase transition due to the noise-induced diffusive term. As an example, a typical bistable noisy system with symmetric double-well potential was studied.     

Exact solutions of the Fokker-Planck equations with moving boundaries

By means of time-dependent similarity transformations, we derive exact solutions of the Fokker-Planck equations with moving boundaries in the presence of: (1) a time-dependent linear force and (2) a time-dependent nonlinear force. The method of similarity transformation is simple and can be easily applied to more general Fokker-Planck equations. Furthermore, the knowledge of the exact solutions in closed form can be useful as a benchmark to test approximate numerical or analytical procedures.     

Fluctuation on Brownian motor in the presence of entropic barrier

Transporting velocity of a loaded Brownian motor with entropic barrier is investigated in the presence of an asymmetric unbiased force. It is found that in the presence of the entropic barrier, the stall force of the Brownian motor does not change monotonously with temperature. The average velocity of the Brownian motor is a peaked function of thermal noise and amplitude of the asymmetric unbiased external force, which indicates that a definite fluctuation can facilitate the loaded Brownian motor moving. With the increase of the load, the range of temperature and amplitude of the asymmetric unbiased external force for Brownian motor working become smaller. The limited area for Brownian motor working is given on the load-temperature plane. The threshold of fluctuation for Brownian motor working is found, and the minimum of asymmetric parameter of unbiased external force for Brownian motor working is given.     

A long-range memory stochastic model of the return in financial markets

We present a nonlinear stochastic differential equation (SDE) which mimics the probability density function (PDF) of the return and the power spectrum of the absolute return in financial markets. Absolute return as a measure of market volatility is considered in the proposed model as a long-range memory stochastic variable. The SDE is obtained from the analogy with an earlier proposed model of trading activity in the financial markets and generalized within the nonextensive statistical mechanics framework. The proposed stochastic model generates time series of the return with two power law statistics, i.e., the PDF and the power spectral density, reproducing the empirical data for the one-minute trading return in the NYSE.     

Effective potential for complex Langevin equations

We construct an effective potential for the complex Langevin equation on a lattice. We show that the minimum of this effective potential gives the space–time and Langevin time average of the complex Langevin field. The loop expansion of the effective potential is matched with the derivative expansion of the associated Schwinger–Dyson equation to predict the stationary distribution to which the complex Langevin equation converges.     

The rate matching effect: A hidden property of aperiodic stochastic resonance

We study a system S generating Poisson events, and a corresponding dichotomous signal as well, perturbed by a system P, also generating Poisson events and a corresponding dichotomous signal. The rates of events productions for system and perturbation are gS and gP, respectively. We call S events the events produced by the system S and P events those produced by the perturbation P. We show that this simple model reproduces the essence of recent experimental and theoretical results on aperiodic stochastic resonance. More remarkably, this simplified version of aperiodic stochastic resonance allows us to discover a property that has been overlooked by the earlier research work. The rate matching condition gS=gP is the border between two distinctly different conditions: For gS<gP, the P events are attractors of the S events and for gS>gP they become repellers of the S events. The transition from the former to the latter condition is very marked and takes place in a short region of either gS or gP, depending on which is the parameter changed, thereby resulting in a discontinuous transition.     

Complex Langevin equations and Schwinger–Dyson equations

Stationary distributions of complex Langevin equations are shown to be the complexified path integral solutions of the Schwinger–Dyson equations of the associated quantum field theory. Specific examples in zero dimensions and on a lattice are given. The relevance to the study of quantum field theory solution space is discussed.