Narrow Results

The reconstruction of stochastic evolution equations from time-series data in terms of the Langevin equation and the corresponding Fokker–Planck equation is often challenged by the inevitably finite temporal sampling of time-series data. With the aim of improving the reconstruction of stochastic evolution equations from empirical time-series data, one can use a finite sampling $\tau$ expansion of the Kramers–Moyal (KM) conditional moments. Depending on the number of terms used from such expansion one obtains different orders of approximation of the KM coefficients. In this paper, we derive and employ the second-order approximation $\sim {\tau }^2$ of the KM conditional moments for the Langevin dynamics and demonstrate that its use leads to the modification of the diffusion coefficient. To demonstrate the substantial potential for practical applications, we utilize data-driven techniques to estimate fluctuations in synthetic time series generated from both linear and nonlinear diffusive processes. Furthermore, we show that when Langevin dynamics is specifically used to model asset prices (here gold price), the use of the second-order approximation of the KM coefficients leads to more accurate prices simulation and prediction that have not been observed in similar work by others.

An approach to infinite-dimensional integration which unifies the case of oscillatory integrals and the case of probabilistic type integrals is presented. It provides a truly infinite-dimensional construction of integrals as linear functionals, as much as possible independent of the underlying topological and measure theoretical structure. Various applications are given, including, next to Feynman path integrals, Schrödinger and diffusion equations, as well as higher order hyperbolic and parabolic equations.

Human activities can play a crucial role in the statistical properties of observables in many complex systems such as social, technological, and economic systems. We demonstrate this by looking into the heavy-tailed distributions of observables in fatal plane and car accidents. Their origin is examined and can be understood as stochastic processes that are related to human activities. Simple mathematical models are proposed to illustrate such processes and compared with empirical results obtained from existing databanks.

We present a fragmentation model that generates almost any inverse power-law size distribution, including dual-scaled versions, consistent with the underlying dynamics of systems with earthquake-like behavior. We apply the model to explain the dual-scaled power-law statistics observed in an Internet access dataset that covers more than 32 million requests. The non-Poissonian statistics of the requested data sizes $m$ and the amount of time $\tau$ needed for complete processing are consistent with the Gutenberg–Richter–law. Inter-event times $\delta t$ between subsequent requests are also shown to exhibit power-law distributions consistent with the generalized Omori law. Thus, the dataset is similar to the earthquake data except that two power-law regimes are observed. Using the proposed model, we are able to identify underlying dynamics responsible in generating the observed dual power-law distributions. The model is universal enough for its applicability to any physical and human dynamics that is limited by finite resources such as space, energy, time or opportunity.

We point out that in a Brownian dynamics simulation it is justified to use arbitrary distribution functions of random numbers if the moments exhibit the correct limiting behavior prescribed by the Fokker-Planck equation. Our argument is supported by a simple analytical consideration and some numerical examples: We simulate the Wiener process, the Ornstein-Uhlenbeck process and the diffusion in a Φ⁴ potential, using both Gaussian and uniform random numbers. In these examples, the rate of convergence of the mean first exit time is found to be nearly identical for both types of random numbers.

In this study, we investigate a single-item, periodic-review inventory problem where the production capacity is limited and unmet demand is backordered. We assume that customer demand in each period is a stationary, discrete random variable. Linear holding and backorder cost are charged per unit at the end of a period. In addition to the variable cost charged per unit ordered, a positive fixed ordering cost is incurred with each order given. The optimization criterion is the minimization of the expected cost per period over a planning horizon. We investigate the infinite horizon problem by modeling the problem as a discrete-time Markov chain. We propose a heuristic for the problem based on a particular solution of this stationary model, and conduct a computational study on a set of instances, providing insight on the performance of the heuristic.

This paper deals with a precise description of the region of zitterbewegung below the Compton scale and the stochastic nature associated with it. We endeavor to delineate this particular region by means of Ito’s calculus and instigate certain features that are in sharp contrast with conventional physics. Interestingly, our work substantiates that the zitterbewegung region represents a pre-space–time region and from therein emerges the notion of our conventional space–time. Interestingly, this unique region engenders the relativistic and quantum mechanical aspects of space–time.

We reconsider the problem of diffusion of particles at constant speed and present a generalization of the Telegrapher process to higher-dimensional stochastic media (d > 1) where the particle can move along 2^d directions. We derive the equations for probability density function using the "formulae of differentiation" of Shapiro and Loginov. The model is an advancement over similiar models of photon migration in multiply scattering media for it results in a true diffusion at constant speed in the limit of large dimensions.

A driven diffusive system (DDS) is a lattice-gas in contact with a thermal bath in the presence of an external field. Such DDS constantly gains (losses) energy from (to) the driving field (thermal bath) and therefore, for long enough time, it reaches a non-equilibrium steady-state (NESS) with a generally unknown statistical distribution. It is found that if the constant driving is replaced by an oscillatory field of magnitude E and period τ, the system exhibits a crossover from NESS to a quasi-equilibrium state (QES) driven by τ. The crossover behavior is characterized by a typical crossover time which is proportional to the lattice side and consequently relevant to confined systems.

Sound happens in time. Human ears detect more objective regularities of a sonic flow, than those we are in general able to describe, yet. We review some studies of sound in the time-domain. Sound streams to be detected by human ears have a hierarchical organization. This can also be seen on the time-organization of entropic flow of the signal, that we choose as variable. We describe an experiment where ability to synchronize by ear a spoken signal with another signal of disturbs is matched to the entropic content of the spoken signal; i.e. the disturb is placed more precisely if it happens in areas of high entropy. This study is phenomenological.

Nonequilibrium systems driven by additive or multiplicative dichotomous Markov noise appear in a wide variety of physical and mathematical models. We review here some prototypical examples, with an emphasis on analytically-solvable situations. In particular, it has escaped attention till recently that the standard results for the long-time properties of such systems cannot be applied when unstable fixed points are crossed in the asymptotic regime. We show how calculations have to be modified to deal with these cases and present a few relevant applications — the hypersensitive transport, the rocking ratchet, and the stochastic Stokes' drift. These results reinforce the impression that dichotomous noise can be put on par with Gaussian white noise as far as obtaining analytical results is concerned. They convincingly illustrate the interplay between noise and nonlinearity in generating nontrivial behaviors of nonequilibrium systems and point to various practical applications.

We review a novel approach to evaluate the ground-state properties of many-body lattice systems based on an exact probabilistic representation of the dynamics and its long time approximation via a central limit theorem. The choice of the asymptotic density probability used in the calculation is discussed in detail.

The continuous-time random walk (CTRW) model is a useful tool for the description of diffusion in nonequilibrium systems, which is broadly applied in nature and life sciences, e.g., from biophysics to geosciences. In particular, the integro-differential equations for diffusion and diffusion-advection are derived asymptotically from the decoupled CTRW model and a generalized Chapmann–Kolmogorov equation, with generic waiting time probability density function (PDF) and external force. The advantage of the integro-differential equations is that they can be used to investigate the entire diffusion process i.e., covering initial-, intermediate- and long-time ranges of the process. Therefore, this method can distinguish the evolution detail for a system having the same behavior in the long-time limit but with different initial- and intermediate-time behaviors. An integro-differential equation for diffusion-advection is also presented for the description of the subdiffusive and superdiffusive regime. Moreover, the methods of solving the integro-differential equations are developed, and the analytic solutions for PDFs are obtained for the cases of force-free and linear force.

The energetics of a Brownian heat engine and heat pump driven by position dependent temperature, known as the Büttiker–Landauer heat engine and heat pump, is investigated by numerical simulations of the inertial Langevin equation. We identify parameter values for optimal performance of the heat engine and heat pump. Our results qualitatively differ from approaches based on the overdamped model. The behavior of the heat engine and heat pump, in the linear response regime is examined under finite time conditions and we find that the efficiency is lower than that of an endoreversible engine working under the same condition. Finally, we investigate the role of different potential and temperature profiles to enhance the efficiency of the system. Our simulations show that optimizing the potential and temperature profile leads only to a marginal enhancement of the system performance due to the large entropy production via the Brownian particle's kinetic energy.

We have studied the single-particle heat engine and refrigerator driven by time-asymmetric protocol of finite duration. Our system consists of a particle in a harmonic trap with time-periodic strength that drives the particle cyclically between two baths. Each cycle consists of two isothermal steps at different temperatures and two adiabatic steps connecting them. The system works in irreversible mode of operation even in the quasistatic regime. This is indicated by finite entropy production even in the large cycle time limit. Consequently, Carnot efficiency for heat engine or Carnot coefficient of performance (COP) for refrigerators is not achievable. We further analyzed the phase diagram of heat engines and refrigerators. They are sensitive to time-asymmetry of the protocol. Phase diagram shows several interesting features, often counterintuitive. The distribution of stochastic efficiency and COP is broad and exhibits power-law tails.

The Tokuyama–Mori projection operator method for a reduced time-convolutionless description of a local temporal behavior of an open quantum system interacting with the weakly dissipative and fluctuating pervasive environment is applied to a Markov chain subject to random transition probabilities. The solution to the problem of the multimodal dynamics of a two-stage absorbing Markov chain with the fluctuating forward rate constant augmented by a symmetric dichotomous stochastic process is found exactly and compared with that of the problem for the same Markov chain with the fluctuating backward rate constant. It is shown that these two different tetramodal solutions cannot generally be reduced to but be complementary to each other. In the limit of very frequent fluctuations in forward/backward rate constants of a two-stage absorbing Markov chain, as well as in the case of a one-stage recurrent Markov chain, both solutions become bimodal and superimposed to one another. However, there is a distinction between using of those solutions for the dynamics of a two-stage absorbing Markov chain in the limit of very rare fluctuations at the critical point, in which the former solution shows the resonance effect exhibiting itself as the stochastic immobilization in an initial state, while the latter demonstrates the deterministic decay to the other state.

New stochastic models of thermodiffusion are constructed and their hydrodynamical limits are studied through a first-order Chapman–Enskog expansion. These models differ from earlier ones by taking into account all first-order contributions proportional to the temperature gradient and, thus, allow for both positive and negative Soret coefficients, in accordance with observations.

In this paper, we model the tick-by-tick dynamics of markets by using the continuous-time random walk (CTRW) model. We employ a sum of products of power law and stretched exponential functions for the waiting time probability distribution function; this function can fit well the waiting time distribution for BUND futures traded at LIFFE in 1997.

We show that based on the mathematics of Markov processes and particularly based on the definition of Kramers–Moyal coefficients, it is possible to estimate the deterministic part of the dynamics for a broad class of nonlinear noisy systems. In particular, we show that for different kinds of noise perturbations, including non-Langevin force with finite correlation time and independent measurement noise, the deterministic part can be reconstructed.

A two-variable bistable system modeling the selection between competing options is considered in the presence of additive white noise, representing environmental or internal variability. It is shown that when the strengths of the noises acting on the evolution equations of the two variables are different a symmetry-breaking process takes place, whereby the state in which the excess variable is perturbed by the strongest noise becomes predominant.