Stepping molecular motor amid Lévy white noise

Directed transport of particles in coupled fractional-order systems excited by Lévy noise

This paper investigates the directed transport of particles in a coupled fractional-order system excited by Lévy noise. Numerical simulations reveal the effects of fractional order, Lévy noise and coupling coefficients on the directed transport. It is found that there exists an optimal fractional order, which maximizes the directed transport of particles. The optimal fractional order for the directed transport shifts to the left or right with different noise parameters, which means that the appropriate fractional order and noise parameters should be taken into account to maximize the directed transport. Meanwhile, the increase of the scale and symmetry parameters intensifies the directed transport of the particles, while the increase of the stability index suppresses the directed transport, so appropriate Lévy noise parameters will effectively amplify the directed transport. In addition, strong coupling can also effectively promote the directed transport of particles. These studies may provide a theoretical basis for the design of nanomachines, improving drug delivery across cell membranes and treating diseases of the nervous system.

Path integration method for stochastic responses of differential equations under Lévy white noise

A path integration (PI) approach that is progressive for studying the stochastic response driven by Lévy white noise is presented. First, a probability mapping is constructed, which decouples the domain of interest for the system state and the probability space derived from the randomness of Lévy white noise within a short time interval. Then, solving the probability mapping yields the short-time response of the system. Finally, the stochastic evolution of the system can be grasped in a stepwise manner based on the fundamental concept of the PI method. The applicability and effectiveness of our approach in addressing the transient and stationary responses under Lévy white noises are verified by Monte Carlo simulation results. Moreover, the advances in utilization of this method are that it eliminates the restriction of the previous PI method on the controlling parameter of Lévy white noises, and it is highly efficient for solving responses of systems under Lévy white noises.

A thermodynamical model of non-deterministic computation in cortical neural networks

Neuronal populations in the cerebral cortex engage in probabilistic coding, effectively encoding the state of the surrounding environment with high accuracy and extraordinary energy efficiency. A new approach models the inherently probabilistic nature of cortical neuron signaling outcomes as a thermodynamic process of non-deterministic computation. A mean field approach is used, with the trial Hamiltonian maximizing available free energy and minimizing the net quantity of entropy, compared with a reference Hamiltonian. Thermodynamic quantities are always conserved during the computation; free energy must be expended to produce information, and free energy is released during information compression, as correlations are identified between the encoding system and its surrounding environment. Due to the relationship between the Gibbs free energy equation and the Nernst equation, any increase in free energy is paired with a local decrease in membrane potential. As a result, this process of thermodynamic computation adjusts the likelihood of each neuron firing an action potential. This model shows that non-deterministic signaling outcomes can be achieved by noisy cortical neurons, through an energy-efficient computational process that involves optimally redistributing a Hamiltonian over some time evolution. Calculations demonstrate that the energy efficiency of the human brain is consistent with this model of non-deterministic computation, with net entropy production far too low to retain the assumptions of a classical system.

Quantum Bohmian-Inspired Potential to Model Non-Gaussian Time Series and Its Application in Financial Markets

We have implemented quantum modeling mainly based on Bohmian mechanics to study time series that contain strong coupling between their events. Compared to time series with normal densities, such time series are associated with rare events. Hence, employing Gaussian statistics drastically underestimates the occurrence of their rare events. The central objective of this study was to investigate the effects of rare events in the probability densities of time series from the point of view of quantum measurements. For this purpose, we first model the non-Gaussian behavior of time series using the multifractal random walk (MRW) approach. Then, we examine the role of the key parameter of MRW, λ, which controls the degree of non-Gaussianity, in quantum potentials derived for time series. Our Bohmian quantum analysis shows that the derived potential takes some negative values in high frequencies (its mean values), then substantially increases, and the value drops again for rare events. Thus, rare events can generate a potential barrier in the high-frequency region of the quantum potential, and the effect of such a barrier becomes prominent when the system transverses it. Finally, as an example of applying the quantum potential beyond the microscopic world, we compute quantum potentials for the S&P financial market time series to verify the presence of rare events in the non-Gaussian densities and demonstrate deviation from the Gaussian case.

Phase-sensitive detection of anomalous diffusion dynamics in the neuronal membrane induced by ion channel gating

Non-ergodicity of neuronal dynamics from rapid ion channel gating through the membrane induces membrane displacement statistics that deviate from Brownian motion. The membrane dynamics from ion channel gating were imaged by phase-sensitive optical coherence microscopy. The distribution of optical displacements of the neuronal membrane showed a Lévy-like distribution and the memory effect of the membrane dynamics by the ionic gating was estimated. The alternation of the correlation time was observed when neurons were exposed to channel-blocking molecules. Non-invasive optophysiology by detecting the anomalous diffusion characteristics of dynamic images is demonstrated.

Optimal harvesting strategy for stochastic hybrid delay Lotka-Volterra systems with Lévy noise in a polluted environment

This paper concerns the dynamics of two stochastic hybrid delay Lotka-Volterra systems with harvesting and Lévy noise in a polluted environment (i.e., predator-prey system and competitive system). For every system, sufficient and necessary conditions for persistence in mean and extinction of each species are established. Then, sufficient conditions for global attractivity of the systems are obtained. Finally, sufficient and necessary conditions for the existence of optimal harvesting strategy are provided. The accurate expressions for the optimal harvesting effort (OHE) and the maximum of expectation of sustainable yield (MESY) are given. Our results show that the dynamic behaviors and optimal harvesting strategy are closely correlated with both time delays and three types of environmental noises (namely white Gaussian noises, telephone noises and Lévy noises).

Dynamical behavior of a nonlocal Fokker-Planck equation for a stochastic system with tempered stable noise

We characterize a stochastic dynamical system with tempered stable noise, by examining its probability density evolution. This probability density function satisfies a nonlocal Fokker-Planck equation. First, we prove a superposition principle that the probability measure-valued solution to this nonlocal Fokker-Planck equation is equivalent to the martingale solution composed with the inverse stochastic flow. This result together with a Schauder estimate leads to the existence and uniqueness of strong solution for the nonlocal Fokker-Planck equation. Second, we devise a convergent finite difference method to simulate the probability density function by solving the nonlocal Fokker-Planck equation. Finally, we apply our aforementioned theoretical and numerical results to a nonlinear filtering system by simulating a nonlocal Zakai equation.

Transport of coupled particles in rough ratchet driven by Lévy noise

This paper studies the transport of coupled particles in a tilted rough ratchet potential. The relationship between particles transport and roughness, noise intensity, external force, coupling strength, and free length is explored numerically by calculating the average velocity of coupled particles. Related investigations have found that rough potential can accelerate the process of crossing the barrier by increasing the particles velocity compared with smooth potential. It is based on the fact that the roughness on the potential surface is like a "ladder," which helps particles climb up and blocks them from sliding down. Moreover, superimposing an appropriate external force on the coupled particles or strengthening the Lévy noise leads to the particles velocity to increase. It is worth emphasizing that when the external force is selected properly, an optimal roughness can be found to maximize the particles velocity. For a given roughness, an optimal coupling coefficient is discovered to match the maximum velocity. And once the coupling coefficient is greater than the optimal value, the particles velocity drops sharply to zero. Furthermore, our results also indicate that choosing an appropriate free length between particles can also speed up transport.

Thermodynamics of Superdiffusion Generated by Lévy-Wiener Fluctuating Forces

Scale free Lévy motion is a generalized analogue of the Wiener process. Its time derivative extends the notion of “white noise” to non-Gaussian noise sources, and as such, it has been widely used to model natural signal variations described by an overdamped Langevin stochastic differential equation. Here, we consider the dynamics of an archetypal model: a Brownian-like particle is driven by external forces, and noise is represented by uncorrelated Lévy fluctuations. An unperturbed system of that form eventually attains a steady state which is uniquely determined by the set of parameter values. We show that the analyzed Markov process with the stability index α<2 violates the detailed balance, i.e., its stationary state is quantified by a stationary probability density and nonvanishing current. We discuss consequences of the non-Gibbsian character of the stationary state of the system and its impact on the general form of the fluctuation–dissipation theorem derived for weak external forcing.

Dynamical stability of the one-dimensional rigid Brownian rotator: the role of the rotator's spatial size and shape

We investigate dynamical stability of a single propeller-like shaped molecular cogwheel modelled as the fixed-axis rigid rotator. In the realistic situations, rotation of the finite-size cogwheel is subject to the environmentally-induced Brownian-motion effect that we describe by utilizing the quantum Caldeira-Leggett master equation. Assuming the initially narrow (classical-like) standard deviations for the angle and the angular momentum of the rotator, we investigate the dynamics of the first and second moments depending on the size, i.e. on the number of blades of both the free rotator as well as of the rotator in the external harmonic field. The larger the standard deviations, the less stable (i.e. less predictable) rotation. We detect the absence of the simple and straightforward rules for utilizing the rotator's stability. Instead, a number of the size-related criteria appear whose combinations may provide the optimal rules for the rotator dynamical stability and possibly control. In the realistic situations, the quantum-mechanical corrections, albeit individually small, may effectively prove non-negligible, and also revealing subtlety of the transition from the quantum to the classical dynamics of the rotator. As to the latter, we detect a strong size-dependence of the transition to the classical dynamics beyond the quantum decoherence process.

Boltzmann-distribution-equivalent for Lévy noise and how it leads to thermodynamically consistent epicatalysis

Nonequilibrium systems commonly exhibit Lévy noise. This means that the distribution for the size of the Brownian fluctuations has a "fat" power-law tail. Large Brownian kicks are then more common as compared to the ordinary Gaussian distribution. We consider a two-state system, i.e., two wells and a barrier in between. The barrier is sufficiently high for a barrier crossing to be a rare event. When the noise is Lévy, we do not get a Boltzmann distribution between the two wells. Instead we get a situation where the distribution between the two wells also depends on the height of the barrier that is in between. Ordinarily, a catalyst, by lowering the barrier between two states, speeds up the relaxation to an equilibrium, but does not change the equilibrium distribution. In an environment with Lévy noise, on the other hand, we have the possibility of epicatalysis, i.e., a catalyst effectively altering the distribution between two states through the changing of the barrier height. After deriving formulas to quantitatively describe this effect, we discuss how this idea may apply in nuclear reactors and in the biochemistry of a living cell.

Roughness-enhanced transport in a tilted ratchet driven by Lévy noise

The enhanced transport of particles by roughness in a tilted rough ratchet potential subject to a Lévy noise is investigated in this paper. Due to the roughness, the transport process exhibits quite different properties compared to the smooth case. We find that the roughness on the potential wall functions like a ladder to provide the convenience for particles to climb up but hinder them to slide down. The mean first passage time from one well to its right adjacent well and the mean velocity are, respectively, calculated versus the roughness, the external force, and the Lévy stability index. Our results show that the roughness is able to induce an enhancement on the mean velocity of particles and accelerate the barrier crossing process. The general conditions require a small external force and a small Lévy stability index. We find that with increasing external forces, the enhancement areas of roughness and Lévy stability index both shrink. However, for the Lévy stability index within the enhancement area, its increase will enlarge the enhancement area of roughness. On the contrary, under the same conditions we observe that for a Gaussian noise the roughness always reduces the corresponding mean velocity which is very different from the case of Lévy noise.

Transports in a rough ratchet induced by Lévy noises

We study the transport of a particle subjected to a Lévy noise in a rough ratchet potential which is constructed by superimposing a fast oscillating trigonometric function on a common ratchet background. Due to the superposition of roughness, the transport process exhibits significantly different properties under the excitation of Lévy noises compared to smooth cases. The influence of the roughness on the directional motion is explored by calculating the mean velocities with respect to the Lévy stable index α and the spatial asymmetry parameter q of the ratchet. Variations in the splitting probability have been analyzed to illustrate how roughness affects the transport. In addition, we have examined the influences of roughness on the mean first passage time to know when it accelerates or slows down the first passage process. We find that the roughness can lead to a fast reduction of the absolute value of the mean velocity for small α, however the influence is small for large α. We have illustrated that the ladder-like roughness on the potential wall increases the possibility for particles to cross the gentle side of the ratchet, which results in an increase of the splitting probability to right for the right-skewed ratchet potential. Although the roughness increases the corresponding probability, it does not accelerate the mean first passage process to the right adjacent well. Our results show that the influences of roughness on the mean first passage time are sensitive to the combination of q and α. Hence, the proper q and α can speed up the passage process, otherwise it will slow down it.