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.

The interaction between an atom and the quantized electromagnetic field depends on the position of the atom, leading to a force which is the negative gradient of this interaction. At zero temperature, the mean of this force vanishes, but it has a nonzero value for a thermal electromagnetic background. By applying the Heisenberg equations of motion and the Born–Markov approximation, we obtain the mean and correlation of the force, which show that the center-of-mass motion of the atom is damped and diffused. This approach can be easily extended to multi-level atoms, and it is shown that the damping force and diffusion coefficients are invariant under Galilean transformations. In principle, this effect can be utilized to determine the velocity of the laboratory relative to the thermal background.

We study the relation between stock price changes and the difference in the volume of sell and buy orders. Using a soft spin model, we describe the price impact of order imbalances and find an analogy to the fluctuation–dissipation theorem in physical systems. We empirically investigate fluctuations and market friction for a major US stock and find support for our model calculations.

The North Atlantic Oscillation (NAO) monthly index is studied from 1825 till 2002 in order to identify the scaling ranges of its fluctuations upon different delay times and to find out whether or not it can be regarded as a Markov process. A Hurst rescaled range analysis and a detrended fluctuation analysis both indicate the existence of weakly persistent long range time correlations for the whole scaling range and time span. Such correlations are similar to the Brownian fluctuations. The Fokker–Planck equation is derived and Kramers–Moyal coefficients are estimated from the data. They are interpreted in terms of a drift and a diffusion coefficient as in fluid mechanics. All partial distribution functions of the NAO monthly index fluctuations have a form close to a Gaussian, for all time lags, in agreement with the findings of the scaling analyses. This indicates the lack of predictive power of the present NAO monthly index. Yet there are some deviations for large (thus rare) events. Whence suggestions for other measurements are made if some improved predictability of the weather/climate in the North Atlantic is of interest. The subsequent Langevin equation of the NAO signal fluctuations is explicitly written in terms of the diffusion and drift parameters, and a characteristic time scale for these is given in Appendix A.

A mathematical model of tumor growth as a random process is studied, in which there is an interaction between tumor cells and normal cells. A set of two coupled stochastic differential equations defines the dynamics of tumor cells and normal cells. The evolution contains driven growth, therapy, and Gaussian white noises terms. The corresponding Fokker–Planck equation is used to study the large-time behavior of the system. Specifically, for periodic therapy terms, the ratio of the probability of large tumor cells number to the probability of small tumor cells number is investigated. It is seen that increasing the interaction between the tumor cells and normal cells decreases this ratio.

We show how to write a set of brackets for the Langevin equation, describing the dissipative motion of a classical particle, subject to external random forces. The method does not rely on an action principle, and is based solely on the phenomenological description of the dissipative dynamics as given by the Langevin equation. The general expression for the brackets satisfied by the coordinates, as well as by the external random forces, at different times, is determined, and it turns out that they all satisfy the Jacobi identity. Upon quantization, these classical brackets are found to coincide with the commutation rules for the quantum Langevin equation, that have been obtained in the past, by appealing to microscopic conservative quantum models for the friction mechanism.

Monte Carlo simulation based on the three-dimensional Langevin equations and one-body dissipation mechanism has been applied to highly excited compound nucleus to investigate the kinetic energy distribution of fission fragments. The effect of center to center distance between nascent fragments and other dynamical parameters (elongation, neck thickness and mass asymmetry) on average kinetic energy of fragments also studied. Finally, obtained results using this approach are compared with earlier theoretical and experimental data for a few typical systems.

This work is continuation of a stochastic quantization program reported earlier. In this paper, we propose a consistent scheme of doing computations directly in four dimensions using conventional quantum field theory methods.

We study the Langevin equation with both a white noise and a colored noise. We construct the Lagrangian as well as the Hamiltonian for the generalized Langevin equation which leads naturally to a path integral description from first principles. This derivation clarifies the meaning of the additional fields introduced by Martin, Siggia and Rose in their functional formalism. We show that the transition amplitude, in this case, is the generating functional for correlation functions. We work out explicitly the correlation functions for the Markovian process of the Brownian motion of a free particle as well as for that of the non-Markovian process of the Brownian motion of a harmonic oscillator (Uhlenbeck–Ornstein model). The path integral description also leads to a simple derivation of the Fokker–Planck equation for the generalized Langevin equation.

In an earlier paper, it has been shown that the ultra violet divergence structure of anomalous $U(1)$ axial vector gauge model in the stochastic quantization scheme is different from that in the conventional quantum field theory. Also, it has been shown that the model is expected to be renormalizable. Based on the operator formalism of the stochastic quantization, a new approach to anomalous $U(1)$ axial vector gauge model is proposed. The operator formalism provides a convenient framework for analysis of ultra violet divergences, but the computations in a realistic model become complicated. In this paper a new approach to do computations in the model is formulated directly in four dimensions. The suggestions put forward here will lead to simplification in the study of applications of the axial vector gauge theory, as well as those of other similar models.

We study the transport of heat in a three-dimensional, harmonic crystal of slab geometry whose boundaries and the intermediate surfaces are connected to stochastic, white noise heat baths at different temperatures. Heat baths at the intermediate surfaces are required to fix the initial state of the slab in respect of its surroundings. We allow the flow of energy fluxes between the intermediate surfaces and the attached baths and impose conditions that relate the widths of Gaussian noises of the intermediate baths. The radiated heat obeys Newton's law of cooling when intermediate baths collectively constitute the environment surrounding the slab. We show that Fourier's law holds in the continuum limit. We obtain an exponentially falling temperature profile from high to low temperature end of the slab and this very nature of the profile was already confirmed by Ingen-Hausz's experiment. Temperature profile of similar nature is also obtained in the one-dimensional version of this model.

In this paper, a Langevin equation for predator population with multiplicative correlated noise is analyzed. The noise source, which is a nonnegative random pulse noise with regulated periodicity, corresponds to the prey population cycling. The increase of periodicity of noise affects the average predator density at the stationary state.

The steady-state work fluctuations for an overdamped Brownian particle driven by an external stochastic force are analyzed by using the Langevin approach. The first two moments of the work distribution are given analytically, both are proportional to the time duration of the work injection. The first two moments determine a unique Gaussian function, and the deviations of the probability density functions (PDFs) of work from the Gaussian functions are measured. The tails of the PDFs are shown to approximate the Gaussian functions. In the limit of weak external force and large time interval, the PDF can be approximated by the Gaussian form and the steady-state fluctuation theorem is held.

We study transport of heat in an ordered, anharmonic crystal in the form of slab geometry in three dimensions. Apart from attaching baths of Langevin type to two extreme surfaces, we also attach baths of same type to the intermediate surfaces of the slab. Since the crystal is uninsulated, it exchanges energy with the intermediate heat baths. We find that both Fourier’s law of heat conduction and the Newton’s law of cooling hold to leading order in anharmonic coupling. The leading behavior of the temperature profile is exponentially falling from high to low temperature surface of the slab. As the anharmonicity increases, profiles fall more below the harmonic one in the log plot. In the thermodynamic limit thermal conductivity remains independent of the environment temperature and its leading order anharmonic contribution is linearly proportional to the temperature change between the two extreme surfaces of the slab. A fast crossover from one-dimensional (1D) to three-dimensional (3D) behavior of the thermal conductivity is observed in the system.

The relativistic collective diffusion in one-dimensional molecular system is investigated through nonequilibrium molecular dynamics with Monte Carlo methods. We have proposed the relationship among the speed, the temperature, the density distribution and the collective diffusion coefficient of particles in a relativistic moving system. It is found that the relativistic speed of the system has no effect on the temperature, but the collective diffusion coefficient decreases to zero as the velocity of the system approaches to the speed of light. The collective diffusion coefficient is modified as $D^{\prime }=D{(1-\frac{w^2}{c^2})}^{\frac{3}{2}}$ for satisfying the relativistic circumstances. The present results may contribute to the understanding of the behavior of the particles transport diffusion in a high speed system, as well as enlighten the study of biological metabolism at relativistic high speed situation.

In this paper we are going to derive the Fokker–Planck equation (FPE) which is based on the Langevin equation for the stochastic model of a turbulent cascade and then we exactly solve the FPE for a certain case and find an explicit form of the probability distribution function (PDF) by the operator method. Finally, by using this PDF, we calculate the velocity moments for this system.

The effects of carrying capacity of environment $K$ for degradation (the $K$ effect for short) on the constitutive gene expression and a simple genetic regulation system are investigated by employing a stochastic Langevin equation combined with the corresponding Fokker–Planck equation for the two stochastic systems subjected to internal and external noises. This $K$ effect characterizes the limited degradation ability of the environment for RNA or proteins, such as insufficient catabolic enzymes. The $K$ effect could significantly change the distribution of mRNA copy-number in constitutive gene expression, and interestingly, it leads to the Fano factor slightly larger than one if only the internal noise exists. Therefore, that the recent experimental measurements suggest the Fano factor deviates from one slightly [D. L. Jones, R. C. Brewster and R. Phillips, Science346 (2014) 1533], probably originates from the $K$ effect. The $K$ effects on the steady and transient properties of genetic regulation system, have been investigated in detail. It could enhance the mean first passage time significantly especially when the noises are weak and reduce the signal-to-noise ratio in stochastic resonance substantially.

We illustrate two distinct approaches to the Mittag–Leffler relaxation, as a mathematical expression suitable for the interpretation of real data produced by complex systems, and especially those of physiological interest. The first approach is based on interpreting the fluctuation–dissipation process under study as obtained via Subordination to the Ordinary Fluctuation–Dissipation (SOFD) process. The second approach rests on the Generalized Langevin Equation (GLE). We prove that in the real cases of truncated time series the two theories generate a survival probability in the form of a stretched exponential, and that this property makes it hard to assess if a given time series obeys the GLE or the SOFD prescription. Some conjectures are made on the possibility of distinguishing the GLE from the SOFD predictions through the analysis of a single time series.

In the Langevin or Ornstein–Uhlenbeck approach to diffusion, stochastic increments are applied to the velocity rather than to the space variable. The density of this process satisfies a linear partial differential equation of the general form of a transport equation which is hyperbolic with respect to the space variable but parabolic with respect to the velocity variable, the Klein–Kramers or simply Kramers equation. This modeling approach allows for a more detailed description of individual movement and orientation dependent interaction than the frequently used reaction diffusion framework.

For the Kramers equation, moments are computed, the infinite system of moment equations is closed at several levels, and telegraph and diffusion equations are derived as approximations. Then nonlinearities are introduced such that the semi-linear reaction Kramers equation describes particles which move and interact on the same time-scale. Also for these nonlinear problems a moment approach is feasible and yields nonlinear damped wave equations as limiting cases.

We apply the moment method to the Kramers equation for chemotactic movement and obtain the classical Patlak–Keller–Segel model. We discuss similarities between chemotactic movement of bacteria and gravitational movement of physical particles.

Stochastic Langevin molecular dynamics for nuclei is derived from the Ehrenfest Hamiltonian system (also called quantum classical molecular dynamics) in a Kac–Zwanzig setting, with the initial data for the electrons stochastically perturbed from the ground state and the ratio M of nuclei and electron mass tending to infinity. The Ehrenfest nuclei dynamics is approximated by the Langevin dynamics with accuracy o(M^-1/2) on bounded time intervals and by o(1) on unbounded time intervals, which makes the small friction and o(M^-1/2) diffusion terms visible. The initial electron probability distribution is a Gibbs density at low temperature, motivated by a stability and consistency argument. The diffusion and friction coefficients in the Langevin equation satisfy the Einstein's fluctuation–dissipation relation.