The magnetic susceptibilities of NO and OH

A non-isothermal kinetic equation for the distribution function of a sub-system weakly coupled to a bath is derived by modification of the analysis and assumptions of a previous paper [1]. The equation has the form of a generalized non-isothermal Fokker-Planck equation when it is linearized in thermodynamic gradients and only terms through second order in the coupling parameters are retained. Higher order terms in the coupling parameter do not diverge with time. The equation is compared with certain ‘exact’ model results of Ullersma and with the coherence time method. The equation is used to calculate a jump rate for diffusion of a harmonic particle weakly coupled to a lattice and it is found that the jump rate becomes independent of the mass of the particle for a heavy enough particle. The source of the discrepancy of this result with a similar calculation of Prigogine and Bak is indicated. The model of the jump rate is inappropriate for diffusion in a thermal gradient and more appropriate models of the jump are briefly considered. A brief comparison of the derivation of the kinetic equation with the Fano coherence time approximation is made and a difference is noted.     

Pricing on electricity market based on coupled-continuous-time-random-walk concept

In this paper we propose a model of electricity market based on the forward rate dynamics described by a diffusion with jumps as a generalization of the classical diffusion approach. We consider jump components resulting from a coupled continuous-time random walk (CTRW) with jump lengths proportional to the corresponding inter-jump time intervals. In the framework of the model we derive a formula for the EURO-price of a standard European call option, showing applicability of CTRW processes for pricing of financial instruments. The result, obtained by an advance theory of semimartingales, is an essential extension of the pricing formula derived in the classical diffusion model of the forward rate dynamics. It indicates an influence of both, the continuous and the jump parts of the forward rate process on the option price.     

A Simplified “Ratchet” Model of Molecular Motors

A limiting case of one of the ratchet models of Ajdari, Prost, et al. is analyzed. An explicit solution is obtained for the probability distribution as a function of the time for any initial distribution with all the transients included. In the long-time limit the drift velocity and diffusion coefficient are obtained in terms of the microscopic transition rates that are the parameters in the model. In spite of its extreme simplicity, with realistic values of its kinetic parameters the model yields values of the drift velocity and effective force that are of the right magnitude for a molecular motor. The model proves to be a simple special case of Derrida's periodic one-dimensional hopping model, for which he found a solution in the long-time limit.     

Phenomenological coefficients in a concentrated alloy for the dumbbell mechanism

We present an adaptation of the self-consistent mean field (SCMF) theory to calculate the transport coefficients in a concentrated alloy for diffusion by the dumbbell mechanism. In this theory, kinetic correlations are accounted for through a set of effective interactions within a non-equilibrium distribution function of the system. Transport coefficients are calculated for the FCC and BCC multicomponent concentrated alloys for simple sets of jump frequencies, including different stabilities of the different defects. A first approximation leads to an analytical expression of the Onsager coefficients in a binary alloy, and a second approximation provides a more accurate prediction. The results of the SCMF theory are compared with existing models and available Monte Carlo simulations, and an interpretation of the set of effective interactions in terms of a competition between jump frequencies is proposed.     

Modeling and Computation of CO2 Allowance Derivatives Under Jump-Diffusion Processes

In this paper, we study carbon emission trading whose market is gaining popularity as a policy instrument for global climate change. The mathematical model is presented for pricing options on $CO_2$ emission allowance futures with jump diffusion processes, and a so-called fitted finite volume method is proposed to solve the pricing model for the spatial discretization, in which the Crank-Nicolson is employed for time stepping. In addition, the stability and the convergence of the fully discrete scheme are given, and some numerical results, which are compared with the closed form solution and the Monte Carlo simulation solution, are provided to demonstrate the rates of convergence and the robustness of the numerical method.     

Passive scalar in a random wave field: the weak turbulence approach

We consider the evolution of a passive scalar advected by a velocity field which is a superposition of random linear waves. An equation for the average concentration of the passive scalar is derived (in the limit of small molecular diffusion) using the weak turbulence methodology. In addition to the enhanced diffusion, this equation contains the correction to the (Stokes) drift. Both of these terms have the fourth order with respect to wave amplitudes. The formulas for the coefficients of turbulent diffusion and turbulent drift are derived.     

On the theory of surface diffusion: kinetic versus lattice gas approach

The present paper extends the results of a recent analytic kinetic theory of particle-on-substrate diffusion. The approach treats explicitly the molecule–surface interaction and takes into account inter-molecular interaction within the hard particle approximation. The physics influencing the diffusion pre-exponential factor and mechanisms determining the density dependence of collective diffusivity are discussed. The kinetic results are compared with those of the traditional lattice gas hopping models. Analytical expressions for jump rates in the low density limit are derived, and the density dependence of effective jump rates at finite occupancy is discussed. It is shown how the traditional hopping model oversimplifies the picture of diffusion by neglecting the collision part of the hopping process.     

Coupled continuous-time random walk approach to the Rachev-Rüschendorf model for financial data

In this paper we expand the Rachev-Rüschendorf asset-pricing model introducing a coupled continuous-time-random-walk-(CTRW)-like form of the random number of price changes. Such a form results from the concept of the random clustering procedure (that resembles the coarse-graining methods of statistical physics) and, on the other hand, indicates applicability of the CTRW idea, widely used in physics to model anomalous diffusion, for describing financial markets. In the framework of the proposed model we derive the limiting distributions of log-returns and the corresponding pricing formulas for European call option. In order to illustrate the obtained theoretical results we present their fitting with several sets of financial data.     

Chaos-induced diffusion in nonlinear discrete dynamics

The large-scale motion of one-dimensional discrete systemX _t+1=X _t+f _a(X _t), (t=0, 1,2,...;f _a(X+1)=f _a(X)) is studied. This motion can be asymptotically decoposed into a drift and a diffusion (chaos-induced diffustion). We derive the formulae for the drift velocityv as the average jump number per step and for the diffusion coefficientD in terms of the jump number's time correlation function. It is shown that the coarsegrained probability distribution is asymptotically gaussian. Considering piecewise linear models and the sinusoidal one, we study the onset behavior of diffusion and its gross behavior. It is found thatD is proportional to the length of the region with a non-zero jump number, if the critical dynamics is well-defined. Otherwise we have logarithmic or inverse power corrections to the simple law originating from an intermittency type behavior or from band splitting phenomena. Maps with a smooth maximum (as e.g. the sinusoidal) exhibit several additional types of trajectories: running modes with broken symmetry, localized trajectories, regular periodic or chaotic, non-diffusive or diffusive ones. These dynamical states appear in a nested window structure, which is described.     

Macroscopic description of a subdiffusion-controlled bimolecular reaction

The random trap model is used to derive equations describing reaction-subdiffusion systems with diffusion-controlled (infinitely fast) bimolecular reaction. A hierarchy of equations in terms of distribution functions is closed by using a quasi-equilibrium condition in the equation for the two-particle distribution function. The reaction terms in the resulting equations contain products between concentrations and diffusion jump rates, rather than products of concentrations as dictated by the law of mass action. The same equations are also derived in the framework of a nonlinear continuous-time random walk model. The equations are used to show that inhomogeneity of the medium may manifest itself by fractional-order reaction terms.     

On drift,diffusion and geometry

We present some reflections on the links between drift, diffusion and geometry. For this purpose, we examine different sources of “diffusion models”, in physics and in mathematics. We observe that diffusion processes may arise from original models either deterministic, or random but where dynamics and noise are clearly delineated. In the end, we get a diffusion process where noise and dynamics (“drift”) are generally intimately entangled in a second-order partial differential operator. We focus on the following questions. Are there implicit geometric structures to properly define a diffusion? How are drift/dynamics and diffusion mixed? Are there geometric structures needed to separate drift and diffusion? We stress the importance of recurrent differential geometric structures – connections and Riemannian metrics – needed to properly define a “diffusion term” and also to separate drift from diffusion.     

Two-state random walk model of lattice diffusion. 1. Self-correlation function

Diffusion with interruptions (arising from localized oscillations, or traps, or mixing between jump diffusion and fluid-like diffusion, etc.) is a very general phenomenon. Its manifestations range from superionic conductance to the behaviour of hydrogen in metals. Based on a continuous-time random walk approach, we present a comprehensive two-state random walk model for the diffusion of a particle on a lattice, incorporating arbitrary holding-time distributions for both localized residence at the sites and inter-site flights, and also the correct first-waiting-time distributions. A synthesis is thus achieved of the two extremes of jump diffusion (zero flight time) and fluid-like diffusion (zero residence time). Various earlier models emerge as special cases of our theory. Among the noteworthy results obtained are: closed-form solutions (ind dimensions, and with arbitrary directional bias) for temporally uncorrelated jump diffusion and for the ‘fluid diffusion’ counterpart; a compact, general formula for the mean square displacement; the effects of a continuous spectrum of time scales in the holding-time distributions, etc. The dynamic mobility and the structure factor for ‘oscillatory diffusion’ are taken up in part 2.     

Continuous time random walk with jump length correlated with waiting time

A coupled continuous time random walk (CTRW) model is proposed, in which the jump length of a walker is correlated with waiting time. The power law distribution is chosen as the probability density function of waiting time and the Gaussian-like distribution as the probability density function of jump length. Normal diffusion, subdiffusion and superdiffusion can be realized within the present model. It is shown that the competition between long-tailed distribution and correlation of jump length and waiting time will lead to different diffusive behavior.     

Limit theorems in financial market models

Invariance principle states that a scaled simple random walk converges to the standard Brownian motion.In this article, we present a discrete time stochastic process, which reflects a microstructure of market dynamics, and prove a convergence to a scaling limit process with a drift term and a jump term. These terms are derived from a macroscopic condition on volumes traded in some time intervals. The mathematical tools for obtaining our results are Dobrushin-Hryniv theory and the method of cluster expansion developed in mathematical studies of statistical mechanics.     

The stress-driven migration of point defects to a slowly moving crack

The migration kinetics of point defects near a slowly moving brittle crack are studied under the condition of pure drift. In the pure-drift approximation it is assumed that the point-defect flow in the vicinity of a crack tip is dominated by the elastic interaction between the stress field of the crack and a point defect and that concentration gradient effects can be neglected. While such a pure-drift approach has been shown to be useful to calculate the short-time diffusion kinetics of impurity-induced subcritical crack growth, previous applications are based on the drift solutions for a stationary crack. In the present paper, the first-order drift diffusion equation for a slowly moving crack at uniform velocity is solved. This yields the flow lines of the point defects and the impurity segregation rate directly in terms of the crack growth rate. The flow line patterns reveal important insights with respect to the point-defect migration kinetics near a steadily advancing crack. Although the calculation is entirely elastic, it is shown that the present drift model maintains some relevance also in the presence of a plastic zone ahead of the crack tip.     

Stochastic modeling of experimental chaotic time series

Methods developed recently to obtain stochastic models of low-dimensional chaotic systems are tested in electronic circuit experiments. We demonstrate that reliable drift and diffusion coefficients can be obtained even when no excessive time scale separation occurs. Crisis induced intermittent motion can be described in terms of a stochastic model showing tunneling which is dominated by state space dependent diffusion. Analytical solutions of the corresponding Fokker-Planck equation are in excellent agreement with experimental data.     

Cosmic-Ray Modulation: an Ab Initio Approach

A better understanding of cosmic-ray modulation in the heliosphere can only be gained through a proper understanding of the effects of turbulence on the diffusion and drift of cosmic rays. We present an ab initio model for cosmic-ray modulation, incorporating for the first time the results yielded by a two-component turbulence transport model. This model is solved for periods of minimum solar activity, utilizing boundary values chosen so that model results are in fair to good agreement with spacecraft observations of turbulence quantities, not only in the solar ecliptic plane but also along the out-of-ecliptic trajectory of the Ulysses spacecraft. These results are employed as inputs for modelled slab and 2D turbulence energy spectra. The latter spectrum is chosen based on physical considerations, with a drop-off at the very lowest wavenumbers commencing at the 2D outerscale. There currently exist no models or observations for this quantity, and it is the only free parameter in this study. The modelled turbulence spectra are used as inputs for parallel mean free path expressions based on those derived from quasi-linear theory and perpendicular mean free paths from extended nonlinear guiding center theory. Furthermore, the effects of turbulence on cosmic-ray drifts are modelled in a self-consistent way, employing a recently developed model for drift along the wavy current sheet. The resulting diffusion coefficients and drift expressions are applied to the study of galactic cosmic-ray protons and antiprotons using a three-dimensional, steady-state cosmic-ray modulation code, and sample solutions in fair agreement with multiple spacecraft observations are presented.     

An improved memristor model for brain-inspired computing

Memristors, as memristive devices, have received a great deal of interest since being fabricated by HP labs. The forgetting effect that has significant influences on memristors' performance has to be taken into account when they are employed. It is significant to build a good model that can express the forgetting effect well for application researches due to its promising prospects in brain-inspired computing. Some models are proposed to represent the forgetting effect but do not work well. In this paper, we present a novel window function, which has good performance in a drift model. We analyze the deficiencies of the previous drift diffusion models for the forgetting effect and propose an improved model. Moreover,the improved model is exploited as a synapse model in spiking neural networks to recognize digit images. Simulation results show that the improved model overcomes the defects of the previous models and can be used as a synapse model in brain-inspired computing due to its synaptic characteristics. The results also indicate that the improved model can express the forgetting effect better when it is employed in spiking neural networks, which means that more appropriate evaluations can be obtained in applications.     

Modelling and Numerical Valuation of Power Derivatives in Energy Markets

In this work we investigate the pricing of swing options in a model where the underlying asset follows a jump diffusion process. We focus on the derivation of the partial integro-differential equation (PIDE) which will be applied to swing contracts and construct a novel pay-off function from a tree-based pay-off matrix that can be used as initial condition in the PIDE formulation. For valuing swing type derivatives we develop a theta implicit-explicit finite difference scheme to discretize the PIDE using a Gaussian quadrature method for the integral part. Based on known results for the classical theta-method the existence and uniqueness of solution to the new implicit-explicit finite difference method is proven. Various numerical examples illustrate the usability of the proposed method and allow us to analyse the sensitivity of swing options with respect to model parameters. In particular, the effects of number of exercise rights, jump intensities and dividend yields will be investigated in depth.     

The valuation of equity warrants in a fractional Brownian environment

In this paper, we discuss the valuation of equity warrants in the geometric fractional Brownian environment based on the equilibrium condition. Using the conditional expectation we present a fractional pricing model for equity warrants and analyze the influence of the Hurst parameter. Then we propose an optimization procedure to obtain the valuation of equity warrants. Some numerical examples are given to demonstrate the pricing results by comparing different pricing models. Furthermore, we provide an empirical study to show how to apply our model in realistic contexts, and these comparative results of different pricing models show that the pricing model proposed in this paper matches the actual price quite well.