Class Of Fokker-Planck Equations Research Articles

Dissipative quantum systems with a potential barrier: General theory and the parabolic barrier.

We study the real time dynamics of a quantum system with potential barrier coupled to a heat-bath environment. Employing the path integral approach an evolution equation for the time dependent density matrix is derived. The time evolution is evaluated explicitly near the barrier top in the temperature region where quantum effects become important. It is shown that there exists a quasi-stationary state with a constant flux across the potential barrier. This state generalizes the Kramers flux solution of the classical Fokker-Planck equation to the quantum regime. In the temperature range explored the quantum flux state depends only on the parabolic approximation of the anharmonic barrier potential near the top. The parameter range within which the solution is valid is investigated in detail. In particular, by matching the flux state onto the equilibrium state on one side of the barrier we gain a condition on the minimal damping strength. For very high temperatures this condition reduces to a known result from classical rate theory. Within the specified parameter range the decay rate out of a metastable state is calculated from the flux solution. The rate is shown to coincide with the result of purely thermodynamic methods. The real time approach presented can be extended to lower temperatures and smaller damping.

Quantum mechanical states as attractors for Nelson processes

In this paper we reconsider, in the light of the Nelson stochastic mechanics, the idea originally proposed by Bohm and Vigier that arbitrary solutions of the evolution equation for the probability densities always relax in time toward the quantum mechanical density ¦ψ¦2 derived from the Schrodinger equation. The analysis of a few general propositions and of some physical examples show that the choice of the L1 metrics and of the Nelson stochastic flux is correct for a particular class of quantum states, but cannot be adopted in general. This indicates that the question if the quantum mechanical densities attract other solution of the classical Fokker-Planck equations associated to the Schrodinger equation is physically meaningful, even if a classical probabilistic model good for every quantum stale is still not available. A few suggestion in this direction are finally discussed.

Analysis of Dynamic Congestion Control Protocols — A Fokker–Planck Approximation

We present an approximate analysis of a queue with a dynamically changing input rate that is based on implicit or explicit feedback information. This is motivated by proposals for adaptive congestion control algorithms, [6] and [15], where the sender's window size at the transport level is adjusted based on perceived congestion level of a bottleneck node. We develop an analysis methodology for a simplified system; however, it is powerful enough to answer important questions regarding stability, convergence (or oscillations), fairness and the significant effect that delayed feedback plays on performance.This paper quantitatively identifies the cause of the these effects, vis-a-vis the system parameters and properties of the algorithm used. Several conservation laws and balance equations governing this system are given.The model is fairly general and can be applied to evaluate the performance of a wide range of feedback control schemes. It is an extension of the classical Fokker-Planck equation. Because the Fokker-Planck equation models diffusion, it addresses traffic variability, and in a sense, extends earlier fluid approximation analyses.

Analysis of dynamic congestion control protocols

We present an approximate analysis of a queue with dynamically changing input rates that are based on implicit or explicit feedback. This is motivated by recent proposals for adaptive congestion control algorithms [RaJa 88, Jac 88], where the sender's window size at the transport level is adjusted based on perceived congestion level of a bottleneck node. We develop an analysis methodology for a simplified system; yet it is powerful enough to answer the important questions regarding stability, convergence (or oscillations), fairness and the significant effect that delayed feedback plays on performance. Specifically, we find that, in the absence of feedback delay, the linear increase/exponential decrease algorithm of Jacobson and Ramakrishnan-Jain [ Jac 88, RaJa 88] is provably stable and fair. Delayed feedback on the other hand, introduces oscillations for every individual user as well as unfairness across those competing for the same resource. While the simulation study of Zhang [Zha 89] and the fluid-approximation study of Bolot and Shanker [BoSh 90] have observed the oscillations in cumulative queue length and measurements by Jacobson [ Jac 88] have revealed some of the unfairness properties, the reasons for these have not been identified. We identify quantitatively the cause of these effects, via-a-vis the systems parameters and properties of the algorithm used. The model presented is fairly general and can be applied to evaluate the performance of a wide range of feedback control schemes. It is an extension of the classical Fokker-Planck equation. Therefore, it addresses traffic viability (to some extent) that fluid approximation techniques do not address. Comments University of Pennsylvania Department of Computer and Information Science Technical Report No. MSCIS-91-18. This technical report is available at ScholarlyCommons: http://repository.upenn.edu/cis_reports/385 Analysis Of Dynamic Congestion Control Protocols: A Fokker-Planck Approximation MS-CIS-91-18 DISTRIBUTED SYSTEMS LAB 5 Amarnath Mukherjee John C. Strikwerda Department of Computer and Information Science School of Engineering and Applied Science University of Pennsylvania Philadelphia, PA 19104-6389

Quantum and classical Fokker-Planck equations for a Gaussian-Markovian noise bath.

The quantum Fokker-Planck equation for a Gaussian-Markovian bath is deduced by applying a method proposed by Tanimura and Kubo [J. Phys. Soc. Jpn. 58, 101 (1989)]. The results are expressed in the form of simultaneous differential equations in terms of density operators and can treat strong system-bath interactions where the correlated effects of the noise play an important role. The classical Fokker-Planck equation for a Gaussian-Markovian noise is obtained by performing the Wigner transformation, and its equilibrium state is shown to be the Maxwell-Boltzmann distribution. The method is convenient for numerical studies. Calculations for quantum-system harmonic oscillators and the double-well potential problems are demonstrated for cases of Gaussian-white noise and Gaussian-Markovian noise.

On a kinetic equation for the transport of particles in turbulent flows

The work described is part of a long-term study in pursuit of a kinetic equation for particles suspended in a turbulent flow. This equation represents the transport of the average particle phase space density and can be used to derive the continuum equations and constitutive relations for a two-fluid model of a dispersed particle flow and to establish the correct form of boundary conditions. A key role in this study is played by the equation of state of the dispersed particles, which is derived here for a Langevin equation of motion with a random driving force not limited to white noise. A suitable form of kinetic equation is then sought that reproduces this equation of state and predicts Gaussian spatial diffusion in the long-term limit with the correct form for the diffusion coefficient in homogeneous stationary turbulence. It is shown that both these requirements are automatically satisfied by constructing forms for the phase space diffusion coefficient that preserve invariance to a random Galilean transformation. The most general form is shown to be an expansion in successively higher-order cumulants of the aerodynamic driving force. When this force is a Gaussian process all terms apart from the first term contract to zero, i.e., the phase space diffusion current is of the form −[μ⋅(∂/∂v) +λ⋅(∂/∂x)]〈W〉 , where 〈W〉 is the average phase space density of particles with velocity v and position x at time t and μ and λ are phase space diffusion tensors whose components are memory integrals involving the correlation of the aerodynamic force along a particle trajectory. With both this form and the more general form of phase space diffusion current, the kinetic equation contracts to the classical Fokker–Planck equation in the white noise limit. Finally with the Gaussian form for the diffusion current, the form of the average phase space density is examined for the important case of a point source diffusing in an unbounded homogeneous stationary flow.

Green's function for one-dimensional Fokker-Planck equations with exponentially dependent drift and diffusion coefficients.

The class of Fokker-Planck equations considered in this paper bridges the gap between the generalized Verhulst-Landau process and the Rayleigh process, which were apparently unrelated up to now. The equations are solved analytically for the transition probability density function of the underlying stochastic processes. The solution method uses operational calculus, and simultaneously produces all components of the spectral problem: spectrum, eigenfunctions, and their normalization. The general results obtained are benchmarked with the known results for the above-mentioned processes by inserting suitable parameters.

Test-particle motion in a relativistic plasma: I. Weakly coupled medium

Starting from the relativistic Landau kinetic equation we derive a classical relativistic Fokker-Planck equation for a test particle in a plasma. A H-theorem is proven and the behaviour of the friction 4-vector and diffusion tensor is analyzed for various dynamical conditions. It is shown that no phenomenological Langevin-type theory can cover all cases.

Quantum Fokker-Planck equation

The dynamics of quantum systems subject to dissipation is a subject of fundamental importance which has recently been of great interest due to its relation to macroscopic quantum phenomena, In particular the behaviour of the magnetic flux trapped in a SQUID is an example of this. The authors derive the quantum mechanical analogue of the Fokker-Planck equation which describes such systems. The major difference between the quantum and classical Fokker-Planck equations arises from the presence of a quantum mechanical memory term.

Path integral approach to quantum Brownian motion

We apply the influence-functional method of Feynman and Vernon to the study of Brownian motion at arbitrary temperature. By choosing a specific model for the dissipative interaction of the system of interest with its environment, we are able to evaluate the influence functional in closed form and express it in terms of a few parameters such as the phenomenological viscosity coefficient. We show that in the limit h→0 the results obtained from the influence functional formalism reduce to the classical Fokker-Planck equation. In the case of a simple harmonic oscillator with arbitrarily strong damping and at arbitrary temperature, we obtain an explicit expression for the time evolution of the complete density matrix ϱ( x, x′, t) when the system starts in a particular kind of pure state. We compare our results with those of other approaches to the problem of dissipation in quantum mechanics.

A semiquantal evaluation ofd 2 ?/d? dE in DIC dynamically based on classical transport equations

The quantal double differential cross-section averaged over micro channels is parametrized in a way that accounts for the transfer of radial momentum and orbital angular momentum of relative motion of two colliding heavy ions into intrinsic degrees of freedom. The formfactors introduced are related to the solutions of classical Fokker-Planck equations and to semiclassical approximations of the phase shifts. An application of the model to40Ar→232Th atE lab=388 MeV which is based on familiar classical friction models and contains no further parameters yields a cross sectiond 2 σ/dθ dE in good overall agreement with the experimental data.

Stochastic modeling of generalized Fokker–Planck equations. I.

A relatively simple method is developed whereby the many-body features of a typical generalized Fokker–Planck equation (GFPE) for a diffusing molecule are first replaced by stochastic bath variables that are assumed to be Markovian. Then the combined molecular and bath variables are characterized as a multidimensional Markov process obeying a stochastic–Liouville equation, which is, in general, incomplete, because it ignores the back reaction of the molecule on the bath variables. In the final step, the equation is completed by subjecting it to the appropriate constraints required for detailed balance. In this form the augmented Fokker–Planck equation (AFPE) properly describes relaxation to thermal equilibrium, and, for the appropriate limiting conditions, it reduces to the classical Fokker–Planck equation. This procedure for stochastic modeling of GFPE is both an improvement on and a generalization of a method previously outlined by Hwang, Mason, Hwang, and Freed (HMHF). Detailed illustrations of AFPE’s are presented for the simple case of a planar rotator subjected to fluctuating torques, and these models are extended to the case of three-dimensional rotational diffusion. Examples include fluctuating torque models related to that used by HMHF. It is shown that only if the fluctuating torque is independent of the orientation of the molecule (more precisely of any fluctuating equilibrium orientation of the molecule), does the model become equivalent to the usual generalized Langevin equation. Otherwise, more general nonlinear models are obtained, which, however, are easily handled by the present methods. Models related to the slowly relaxing local structure (SRLS) model of Polnaszek and Freed are also developed. They are shown to be a consequence of requiring relaxation to the instantaneous value of the fluctuating potential associated with the torque, whereas the fluctuating torque models are a consequence of requiring relaxation to a uniform orientational distribution. They differ further in that the SRLS models are ’’nonfrictional’’. For these reasons we characterize the fluctuating torques as being ’’collision-induced’’ and the SRLS as being ’’structure-induced’’.

On the quantum-mechanical Fokker-Planck and Kramers-Chandrasekhar equation

In the classical theory of Brownian motion we can consider the Langevin equation as an infinitesimal transformation between the coordinates and momenta of a Brownian particle, given probabilistically, since the impulse appearing is characterized by a Gaussian random process. This probabilistic infinitesimal transformation generates a streaming on the distribution function, expressed by the classical Fokker-Planck and Kramers-Chandrasekhar equations. If the laws obeyed by the Brownian particle are quantum mechanical, we can reinterpret the Langevin equation as an operator relation expressing an infinitesimal transformation of these operators. Since the impulses are independent of the coordinates and momenta we can think of them as c numbers described by a Gaussian random process. The so resulting infinitesimal operator transformation induces a streaming on the density matrix. We may associate, according to Weyl functions with operators. The function associated with the density matrix is the Wigner function. Expressing, then, these operator relations in terms of these functions we can express the streaming as a continuity equation of the Wigner function. We find that in this parametrization the extra terms which appear are the same as in the classical theory, augmenting the usual Wigner equation.

Transient analysis of the first-order phase-locked loop (Corresp.)

The problem of obtaining the transient solution to a class of Fokker-Planck equation derived from the first-order nondetuning phase-locked loop is considered. Two methods are proposed to obtain the eigenvalues and the eigenfunctions of the spatial operator for the Fokker-planck equation. One is a perturbation method in which the signal-to-noise ratio is adopted as a parameter. The other is a method similar to the Ince-Goldstein method for solving the Mathieu equation. Numerical calculations show that the latter is superior to the former in accuracy and in covering a wider range of signal-to-noise ratios.

Some properties of Fokker-Planck equations with memory under detailed balance

Some properties of the Fokker-Planck equations with memory under the condition that detailed balance holds are discussed. The appropriate conditions which the drift and diffusion coefficients for a system in detailed balance should satisfy are obtained. The analysis is then restricted to a special class of Fokker-Planck equations namely the one studied previously by Zwanzig. Analog of Onsager's reciprocity relation is obtained and the eigenfunction expansion of the conditional probability is given. Finally the linear response theory of such systems is also briefly discussed.

Green’s function for simple brownian kinetics and its application to plasma

An analysis is outlined concerning the derivation and application of the Green’s function for a class of Fokker-Planck equations with strictly time-dependent external fields and constant viscosity-diffusion coefficients, the kinetic equation commonly used in the theory of Brownian motion. The form of Green’s function for this general class of partial differential equations is given explicit expression in terms of the collisionless time-development operator on the set of initial velocities under the influence of external fields and viscous force. Application of such a singular solution is made to simplify the formulation for the near-equilibrium kinetics of excitation of neutral atoms by electrons and to electromagnetic propagation problems where a linearized self-consistent field treatment of the electronic constituent of a three-fluid plasma is necessary.

Useful identities for half-space problems in linear transport theory: R. E. Aamodt and K. M. Case, Department of Physics, The University of Michigan, Ann Arbor, Michigan

Starting from the relativistic Landau kinetic equation we derive a classical relativistic Fokker-Planck equation for a test particle in a plasma. A H-theorem is proven and the behaviour of the friction 4-vector and diffusion tensor is analyzed for various dynamical conditions. It is shown that no phenomenological Langevin-type theory can cover all cases.

Kinetic Equation for a Plasma with Unsteady Correlations

As a generalization of the Boltzmann equation, the kinetic equation for a plasma is derived in the form of a generalized Fokker-Planck equation, by considering unsteady correlations, including non-Markovian and nonlinear behavior. Both the binary and ternary correlations are used for many kinds of particles with different temperatures. The coefficients of the kinetic equation depend on the law of interaction for a pair of particles and are influenced by relaxation. The effective potential of friction consists of two parts: the static part corresponds to the Debye potential and is isotropic, the dynamical part is axially symmetrical about the direction of motion, and causes a dynamical friction. The results show that the friction is proportional to velocity for slow particles, and inversely proportional to the square of velocity for fast particles. This tendency of the fast particles to overcome repulsion is a property connected with the "run-away" of electrons. A criterion for maximum friction is derived. The triplet interaction, which mainly affects the shielding phenomena, assures the convergence of the coefficients in case of distant interaction. Since the length scales of interaction are well determined in this way, the kinetic equation can be expected to be valid over a longer range than does the Boltzmann equation. The large scale agrees with the Debye radius, when the shielding term is linearized, as should be expected. When time relaxation is left aside and linearization is made, the kinetic equation degenerates to the classical Fokker-Planck equation with convergent coefficients.