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1.
Using the Lie algebraic approach we have derived the exact diffusion propagator of the Fokker-Planck equation with a time-dependent
variable diffusion coefficient and a time-dependent mean-reverting force between two absorbing boundaries. The exact diffusion
propagator not only enables us to study the time evolution of the corresponding stochastic system, but the knowledge of the
propagator can also provide a benchmark for testing approximate numerical or analytical procedures. Furthermore, the Lie algebraic
method is very simple and could be easily extended to the more general Fokker-Planck equations with well-defined algebraic
structures.
Received 18 December 2002 / Received in final form 3 March 2003 Published online 24 April 2003 相似文献
2.
C. Giordano A.R. Plastino M. Casas A. Plastino 《The European Physical Journal B - Condensed Matter and Complex Systems》2001,22(3):361-368
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 相似文献
3.
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. 相似文献
4.
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 相似文献
5.
S. I. Denisov W. Horsthemke P. H?nggi 《The European Physical Journal B - Condensed Matter and Complex Systems》2009,68(4):567-575
We derive the generalized Fokker-Planck equation associated with the
Langevin equation (in the Ito sense) for an overdamped particle in an external
potential driven by multiplicative noise with an arbitrary distribution of the
increments of the noise generating process. We explicitly consider this
equation for various specific types of noises, including Poisson white noise
and Lévy stable noise, and show that it reproduces all Fokker-Planck
equations that are known for these noises. Exact analytical, time-dependent and
stationary solutions of the generalized Fokker-Planck equation are derived and
analyzed in detail for the cases of a linear, a quadratic, and a tailored
potential. 相似文献
6.
A new way of constructing the steady state solutions of the Fokker-Planck Equations(FPE's) is described in which the concept of a vortex field h plays an important role.The steady state solutions can be classified according to the type of h.In particular, one easily obtains almost all the exact global solutions known so far for the stationary FPE's, as well as the exact local solutions near the fixed points of the corres-ponding deterministic equations.In this way some new properties of FPE are explored. 相似文献
7.
The exact solutions to the Cauchy problem for two equations, which are slight generalizations of the so-called linear vectorial Fokker-Planck equation, are found using Feynman’s disentangling techniques and algebraic (operational) methods. This approach may be considered as a generalization of the Suzuki method for solving the one-dimensional linear Fokker-Planck equation. 相似文献
8.
《Journal of Nonlinear Mathematical Physics》2013,20(1-2):132-136
Abstract Symmetry properties of some Fokker-Planck equations are studied. In the one-dimensional case, when symmetry groups turn out to be six-parameter ones, this allows to find changes of variables to reduce such Fokker-Planck equations to the one-dimensional heat equation. The symmetry and the family of exact solutions of the Kramers equation are obtained. 相似文献
9.
LIUHong-Zhun PANZu-Liang 《理论物理通讯》2005,44(1):15-18
By a known transformation, (2 1)-dimensional Brioer Kaup equations are turned to a single equation.The classical Lie symmetry analysis and similarity reductions are performed for this single equation. From some of reduction equations, new exact solutions are obtained, which contain previous results, and more exact solutions can be created directly by abundant known solutions of the Burgers equations and the heat equations. 相似文献
10.
LIU Hong-Zhun PAN Zu-Liang 《理论物理通讯》2005,44(7)
By a known transformation, (2 1)-dimensional Brioer-Kaup equations are turned to a single equation.The classical Lie symmetry analysis and similarity reductions axe performed for this single equation. From some of reduction equations, new exact solutions are obtained, which contain previous results, and more exact solutions can be created directly by abundant known solutions of the Burgers equations and the heat equations. 相似文献
11.
U. Al Khawaja 《Physics letters. A》2009,373(31):2710-2716
We consider a general form of the Gross-Pitaevskii equation with time- and space-dependent effective mass, external potential and strength of interatomic interaction. Using the inverse-scattering method, we derive the integrability condition of this equation within a general scheme that can be used to find exact solutions of a wide range of linear and nonlinear partial differential equations. We use this condition to derive exact solitonic solutions of the one-dimensional time-dependent Gross-Pitaevskii equation corresponding to a Bose-Einstein condensate trapped by a periodic potential. Both attractive and repulsive interatomic interactions are considered. The values of the parameters of the potential can be chosen such that the periodic potential becomes almost identical to that of an optical lattice. 相似文献
12.
L. Borland F. Pennini A.R. Plastino A. Plastino 《The European Physical Journal B - Condensed Matter and Complex Systems》1999,12(2):285-297
Nonlinear Fokker-Planck equations (e.g., the diffusion equation for porous medium) are important candidates for describing anomalous diffusion in a variety of systems.
In this paper we introduce such nonlinear Fokker-Planck equations with general state-dependent diffusion, thus significantly
generalizing the case of constant diffusion which has been discussed previously. An approximate maximum entropy (MaxEnt) approach
based on the Tsallis nonextensive entropy is developed for the study of these equations. The MaxEnt solutions are shown to
preserve the functional relation between the time derivative of the entropy and the time dependent solution. In some particular
important cases of diffusion with power-law multiplicative noise, our MaxEnt scheme provides exact time dependent solutions. We also prove that the stationary solutions of the nonlinear Fokker-Planck equation with diffusion
of the (generalized) Stratonovich type exhibit the Tsallis MaxEnt form.
Received 26 February 1999 相似文献
13.
Marius Grigorescu 《Physica A》2008,387(26):6497-6504
Probability waves in the configuration space are associated with coherent solutions of the classical Liouville or Fokker-Planck equations. Distributions localized in the momentum space provide action waves, described by the probability density and the generating function of the Hamilton-Jacobi theory. It is shown that by introducing a minimum distance in the coordinate space, the action distributions aquire the phase-space dispersion specific to the quantum objects. At finite temperature, probability density waves propagating with the sound velocity can arise as nonstationary solutions of the classical Fokker-Planck equation. The results suggest that in a system of quantum Brownian particles, a transition from complex to real probability waves could be observed. 相似文献
14.
In this paper we find the exact solutions to a class of pluridimensional Fokker-Planck equations with constant diffusion and non linear drifts and discuss the solutions found. We also write the non linear drifts used in an intrinsic form. 相似文献
15.
The time-dependent analytic solutions of three classes of multidimensional Fokker-Planck equations with nonlinear drift are presented together with eigenvalues which are complex and depend essentially on the correlation functions of the fluctuations. 相似文献
16.
We investigate a subdiffusive, fractional Fokker-Planck dynamics occurring in time-varying potential landscapes and thereby disclose the failure of the fractional Fokker-Planck equation (FFPE) in its commonly used form when generalized in an ad hoc manner to time-dependent forces. A modified FFPE (MFFPE) is rigorously derived, being valid for a family of dichotomously alternating force fields. This MFFPE is numerically validated for a rectangular time-dependent force with zero average bias. For this case, subdiffusion is shown to become enhanced as compared to the force free case. We question, however, the existence of any physically valid FFPE for arbitrary varying time-dependent fields that differ from this dichotomous varying family. 相似文献
17.
In this paper we have analytically solved the Fokker-Planck equation (FPE) associated with a fairly large class of multiplicative
stochastic processes with time-varying nonliner drift and diffusion coefficients, which has wide applicability in various
areas of physics, e.g. nonlinear optics and chemical reaction dynamics. By exploiting the dynamical symmetry of the FPE, we
apply the Lie-algebraic approach to derive the time-dependent analytical closed-form solutions. The derived solutions fall
into two different categories, namely (i) one with a moving absorbing boundary, and (ii) one with a fixed absorbing boundary
at the origin, depending upon the model parameters. The corresponding escape (or survival) probabilities are also evaluated
analytically. We believe that not only our analytically exact results can serve as standard models upon which the discussion
of more complicated problems can be based, but they can also be useful as a benchmark to test approximate numerical or analytical
procedures. 相似文献
18.
A system of nonlocal electron transport equations for electrostatic perturbations in (omega,k) space in a high-Z plasma is derived from the Fokker-Planck equation for arbitrary relations between the time, space, and collisionality scales. The closed scheme for obtaining the longitudinal plasma susceptibility epsilon(omega,k) in the entire (omega,k) plane is proposed. Regions in the (omega,k) plane have been mapped for problems such as the relaxation of the local temperature enhancement with a time-dependent heat conductivity. The electron dielectric permittivity has been calculated over the entire range of parameters, including the transition region between Vlasov and Fokker-Planck equation solutions. 相似文献
19.
In this study, the generalized \(\tan (\phi /2)\)-expansion method and He’s semi-inverse variational method (HSIVM) are applied to seek the exact solitary wave solutions for the resonant nonlinear Schrödinger equation with time-dependent coefficients. Using these methods, we investigate exact solutions for the nonlinear resonant Schrödinger equation with time-dependent coefficients two forms of nonlinearity, including power and dual-power law nonlinearity. Moreover, many new analytical exact solutions are obtained which are expressed by hyperbolic solutions, trigonometric solutions, and rational solutions. In addition, we obtained the bright soliton by HSIVM. These methods are powerful, efficient and those can be used as an alternative to establishing new solutions of different types of differential equations in mathematical physics and engineering. 相似文献
20.
We propose a parametric approach to solve self-consistency equations that naturally arise in many-body systems described by nonlinear Fokker-Planck equations in general and nonlinear Vlasov-Fokker-Planck equations of Haissinski type in particular. We demonstrate for the Hess-Doi-Edwards model and the McMillan model of nematic and smectic liquid crystals that the parametric approach can be used to compute bifurcation diagrams and critical order parameters for systems exhibiting one or more than one order parameters. In addition, we show that in the context of the parametric approach solutions of the Haissinski model can be studied from the perspective of a pseudo order parameter. 相似文献