Exact propagator of the Fokker-Planck equation with a space-dependent diffusion coefficient and a time-dependent mean-reverting force

We have investigated the algebraic structure of the Fokker-Planck equation with a variable diffusion coefficient and a time-dependent mean-reverting force. Such a model could be useful to study the general problem of a Brownian walker with a space-dependent diffusion coefficient. We also show that this model is related to the Fokker-Planck equation with a constant diffusion coefficient and a time-dependent anharmonic potential of the form V(x, t) = ?a(t)x ² + b ln x, which has been widely applied to model different physical and biological phenomena, e.g. the study of neuron models and stochastic resonance in monostable nonlinear oscillators. Using the Lie algebraic approach we have derived the exact diffusion propagators for the Fokker-Planck equations associated with different boundary conditions, namely (i) the case of a single absorbing barrier, and (ii) the case of two absorbing barriers. These exact diffusion propagators enable us to study the time evolution of the corresponding stochastic systems. Received 23 October 2001 and Received in final form 24 December 2001     

Nonlinear diffusion under a time dependent external force: q-maximum entropy solutions

Nonlinear diffusion equations provide useful models for a number of interesting phenomena, such as diffusion processes in porous media. We study here a family of nonlinear Fokker-Planck equations endowed both with a power-law nonlinear diffusion term and a drift term with a time dependent force linear in the spatial variable. We show that these partial differential equations exhibit exact time dependent particular solutions of the Tsallis maximum entropy (q-MaxEnt) form. These results constitute generalizations of previous ones recently discussed in the literature [C. Tsallis, D.J. Bukman, Phys. Rev. E 54, R2197 (1996)], concerning q-MaxEnt solutions to nonlinear Fokker-Planck equations with linear, time independent drift forces. We also show that the present formalism can be used to generate approximate q-MaxEnt solutions for nonlinear Fokker-Planck equations with time independent drift forces characterized by a general spatial dependence. Received 25 April 2001 and Received in final form 6 June 2001     

Beliefs and stochastic modelling of interest rate scenario risk

We present a framework that allows for a systematic assessment of risk given a specific model and belief on the market. Within this framework the time evolution of risk is modeled in a twofold way. On the one hand, risk is modeled by the time discrete and nonlinear garch(1,1) process, which allows for a (time-)local understanding of its level, together with a short term forecast. On the other hand, via a diffusion approximation, the time evolution of the probability density of risk is modeled by a Fokker-Planck equation. Then, as a final step, using Bayes theorem, beliefs are conditioned on the stationary probability density function as obtained from the Fokker-Planck equation. We believe this to be a highly rigorous framework to integrate subjective judgments of future market behavior and underlying models. In order to demonstrate the approach, we apply it to risk assessment of empirical interest rate scenario methodologies, i.e. the application of Principal Component Analysis to the the dynamics of bonds. Received 1st August 2000     

Algebraic methods in quantum mechanics: from molecules to polymers

We present a brief review of algebraic techniques developed and applied in molecular spectroscopy in the last five years. We also outline perspectives for new applications of the Lie algebraic method in the first decade of the new century. Received 21 November 2001     

Exact solutions and geometric phase factor of time-dependent three-generator quantum systems

There exist a number of typical and interesting systems and/or models, which possess three-generator Lie-algebraic structure, in atomic physics, quantum optics, nuclear physics and laser physics. The well-known fact that all simple 3-generator algebras are either isomorphic to the algebra sl (2, C) or to one of its real forms enables us to treat these time-dependent quantum systems in a unified way. By making use of both the Lewis-Riesenfeld invariant theory and the invariant-related unitary transformation formulation, the present paper obtains exact solutions of the time-dependent Schr?dinger equations governing various three-generator Lie-algebraic quantum systems. For some quantum systems whose time-dependent Hamiltonians have no quasialgebraic structures, it is shown that the exact solutions can also be obtained by working in a sub-Hilbert-space corresponding to a particular eigenvalue of the conserved generator (i.e., the time-independent invariant that commutes with the time-dependent Hamiltonian). The topological property of geometric phase factors and its adiabatic limit in time-dependent systems is briefly discussed. Received 6 July 2002 / Received in final form 21 October 2002 Published online 11 February 2003     

Potential symmetry and invariant solutions of Fokker-Planck equation in cylindrical coordinates related to magnetic field diffusion in magnetohydrodynamics including the Hall current

Lie groups involving potential symmetries are applied in connection with the system of magnetohydrodynamic equations for incompressible matter with Ohm's law for finite resistivity and Hall current in cylindrical geometry. Some simplifications allow to obtain a Fokker-Planck type equation. Invariant solutions are obtained involving the effects of time-dependent flow and the Hall-current. Some interesting side results of this approach are new exact solutions that do not seem to have been reported in the literature.     

Non-homogeneous random walks, generalised master equations, fractional Fokker-Planck equations, and the generalised Kramers-Moyal expansion

A generalised random walk scheme for random walks in an arbitrary external potential field is investigated. From this concept which accounts for the symmetry breaking of homogeneity through the external field, a generalised master equation is constructed. For long-tailed transfer distance or waiting time distributions we show that this generalised master equation is the genesis of apparently different fractional Fokker-Planck equations discussed in literature. On this basis, we introduce a generalisation of the Kramers-Moyal expansion for broad jump length distributions that combines multiples of both ordinary and fractional spatial derivatives. However, it is shown that the nature of the drift term is not changed through the existence of anomalous transport statistics, and thus to first order, an external potential Φ(x) feeds back on the probability density function W through the classical term ∝/ x (x)W(x, t), i.e., even for Lévy flights, there exists a linear infinitesimal generator that accounts for the response to an external field. Received 30 June 2000 and Received in final form 12 November 2000     

Time-dependent formulation of the linear unitary transformation and the time evolution of general time-dependent quadratic Hamiltonian systems

A time-dependent closed-form formulation of the linear unitary transformation for harmonic-oscillator annihilation and creation operators is presented in the Schrödinger picture using the Lie algebraic approach. The time evolution of the quantum mechanical system described by a general time-dependent quadratic Hamiltonian is investigated by combining this formulation with the time evolution equation of the system. The analytic expressions of the evolution operator and propagator are found. The motion of a charged particle with variable mass in the time-dependent electric field is considered as an illustrative example of the formalism. The exact time evolution wave function starting from a Gaussian wave packet and the operator expectation values with respect to the complicated evolution wave function are obtained readily.     

Scaling behavior of a nonlinear oscillator with additive noise,white and colored

We study analytically and numerically the problem of a nonlinear mechanical oscillator with additive noise in the absence of damping. We show that the amplitude, the velocity and the energy of the oscillator grow algebraically with time. For Gaussian white noise, an analytical expression for the probability distribution function of the energy is obtained in the long-time limit. In the case of colored, Ornstein-Uhlenbeck noise, a self-consistent calculation leads to (different) anomalous diffusion exponents. Dimensional analysis yields the qualitative behavior of the prefactors (generalized diffusion constants) as a function of the correlation time. Received 10 October 2002 Published online 6 March 2003 RID="a" ID="a"e-mail: mallick@spht.saclay.cea.fr     

Fluctuations of the quark densities in nuclei

We study the static scalar susceptibility of the nuclear medium, i.e., the change of the quark condensate for a small modification of the quark mass. In the linear sigma model it is linked to the in-medium sigma propagator and its magnitude increases due to the mixing with the softer modes of the nucleon-hole excitations. We show that the pseudoscalar susceptibility, which is large in the vacuum, owing to the smallness of the pion mass, follows the density evolution of the quark condensate and thus decreases. At normal nuclear matter density the two susceptibilities become much closer, a partial chiral symmetry restoration effect as they become equal when the full restoration is achieved. Received: 20 July 2002 / Accepted: 14 September 2002 / Published online: 21 January 2003 RID="a" ID="a"e-mail: chanfray@ipnl.in2p3.fr Communicated by A. Molinari     

SCDA for 3D lattice gases with repulsive interaction

The selfconsistent diagram approximation (SCDA) is generalized for three-dimensional lattice gases with nearest neighbor repulsive interactions. The free energy is represented in a closed form through elementary functions. Thermodynamical (phase diagrams, chemical potential and mean square fluctuations), structural (order parameter, distribution functions) as well as diffusional characteristics are investigated. The calculation results are compared with the Monte Carlo simulation data to demonstrate high precision of the SCDA in reproducing the equilibrium lattice gas characteristics. It is shown that similarly to two-dimensional systems the specific statistical memory effects strongly influence the lattice gas diffusion in the ordered states. Received 7 August 2002 / Received in final form 22 January 2003 Published online 24 April 2003     

Random walks with imperfect trapping in the decoupled-ring approximation

We investigate random walks on a lattice with imperfect traps. In one dimension, we perturbatively compute the survival probability by reducing the problem to a particle diffusing on a closed ring containing just one single trap. Numerical simulations reveal this solution, which is exact in the limit of perfect traps, to be remarkably robust with respect to a significant lowering of the trapping probability. We demonstrate that for randomly distributed traps, the long-time asymptotics of our result recovers the known stretched exponential decay. We also study an anisotropic three-dimensional version of our model. We discuss possible applications of some of our findings to the decay of excitons in semiconducting organic polymer materials, and emphasize the crucial influence of the spatial trap distribution on the kinetics. Received 23 July 2001 / Received in final form 14 May 2002 Published online 13 August 2002     

Noise correlations in shear flows

We consider the effects of a shear on velocity fluctuations in a flow. The shear gives rise to a transient amplification that not only influences the amplitude of perturbations but also their time correlations. We show that, in the presence of white noise, time correlations of transversal velocity components are exponential and that correlations of the longitudinal components are exponential with an algebraic prefactor. Cross correlations between transversal and downstream components are strongly asymmetric and provide a clear indication of non-normal amplification. We suggest experimental tests of our predictions. Received 26 February 2002 / Received in final form 11 March 2003 Published online 20 June 2003 RID="a" ID="a"e-mail: bruno.eckhardt@physik.uni-marburg.de RID="b" ID="b"Also at: Jawaharlal Nehru Centre for Advanced Scientific Research, Bangalore, India     

Parity and ruin in a stochastic game

We study an elementary two-player card game where in each round players compare cards and the holder of the card with the smaller value wins. Using the rate equations approach, we treat the stochastic version of the game in which cards are drawn randomly. We obtain an exact solution for arbitrary initial conditions. In general, the game approaches a steady state where the card value densities of the two players are proportional to each other. The leading small value behavior of the initial densities determines the corresponding proportionality constant, while the next correction governs the asymptotic time dependence. The relaxation toward the steady state exhibits a rich behavior, e.g., it may be algebraically slow or exponentially fast. Moreover, in ruin situations where one player eventually wins all cards, the game may even end in a finite time. Received 24 August 2001 and Received in final form 12 November 2001     

Relativistic diffusion model

Evidence is presented that diffusion drives colliding many-particle systems at relativistic energies from the initial δ–functions in rapidity towards the equilibrium distribution. Analytical solutions of a linear Fokker-Planck equation represent rapidity spectra for participant protons in central heavy-ion collisions at SPS-energies accurately. Thermal equilibrium in the interaction region is not attained, nonequilibrium features persist and can account for the broad rapidity spectra. Received: 24 November 1998 / Revised version: 20 February 1999     

Model-dependent nature of blowtorch effect on the escape and equilibration rates

We analyze the relaxation behavior of a bistable system when the background temperature profile is inhomogeneous due to the presence of a localized hot region (blowtorch) on one side of the potential barrier. Since the diffusion equation for inhomogeneous medium is model-dependent, we consider two physical models to study the kinetics of such system. Using a conventional stochastic method, we obtain the escape and equilibration rates of the system for the two physical models. For both models, we find that the hot region enhances the escape rate from the well where it is placed while it retards the escape rate from the other well. However, the value of the escape rate from the well where the hot region is placed differs for the two models while that of the escape rate from the other well is identical for both. This work, for the first time, gives a detailed report of the similarities and differences of the escape rates and, hence, exposes the common and distinct features of the two known physical models in determining the way the bistable system relaxes. Received 25 September 2001     

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.