Joint probability distribution beyond the Smoluchowski limit for brownian motion in position space

We obtain a time convolutionless partial differential equation for the joint probability distribution in position space of a non-markovian brownian particle under the influence of some potential. We discuss the corrections to the Smoluchowski limit in this context.     

The Onsager-Casimir relations revisited

We study the fate of the Onsager-Casimir reciprocity relations for a continuous system when some of its variables are eliminated adiabatically. Just as for discrete systems, deviations appear in correction terms to the reduced evolution equation that are of higher order in the time scale ratio. The deviations are not removed by including correction terms to the coarse-grained thermodynamic potential. However, via a reformulation of the theory, in which the central role of the thermodynamic potential is taken over by an associated Lagrangian-type expression, we arrive at a modified form of the Onsager-Casimir relations that survives the adiabatic elimination procedure. There is a simple relation between the time evolution of the redefined thermodynamic forces and that of the basic thermodynamic variables; this relation also survives the adiabatic elimination. The formalism is illustrated by explicit calculations for the Klein-Kramers equation, which describes the phase space distribution of Brownian particles, and for the corrected Smoluchowski equation derived from it by adiabatic elimination of the velocity variable. The symmetry relation for the latter leads to a simple proof that the reality of the eigenvalues of the simple Smoluchowski equation is not destroyed by the addition of higher order corrections, at least not within the framework of a formal perturbation expansion in the time scale ratio.     

Analytic Results in the Position-Dependent Mass Schrödinger Problem

We investigate the Schrödinger equation for a particle with a nonuniform solitonic mass density. First, we discuss in extent the (nontrivial) position-dependent mass V(x)=0 case whose solutions are hypergeometric functions in tanh²x. Then, we consider an external hyperbolic-tangent potential. We show that the effective quantum mechanical problem is given by a Heun class equation and find analytically an eigenbasis for the space of solutions. We also compute the eigenstates for a potential of the form V(x)=V₀ sinh²x.     

Half-range expansion analysis for Langewin dynamics in the high-friction limit with a singular absorbing boundary condition: Noncharacteristic case

We consider the dynamics of a Brownian particle given by the Langevin equation in a strip, under the effects of a deterministic force. The trajectories of particles originate at a source whose spatial location in the phase space coincides with the location of adsorbing boundaries. This leads to singular behavior of trajectories in the high-friction limit. We use the half-range expansion technique and systematic asymptotics to solve a boundary value problem for the Fokker-Planck operator and to calculate the steady-state transition probability density, the mean time to absorption, and the distribution of exit points. We do not make assumptions about other parameters in the problem except that they areO(1) relative to the friction coefficient. We calculate explicitly the correct location of the Milne-type extrapolation for absorbing boundary conditions for the Smoluchowski approximation to the Langevin equation.     

Exact probability distribution for soluble model with quadratic noise

A one-dimensional evolution equation transformable into a linear one coupled to a quadratic Smoluchowski (an Ornstein-Uhlenbeck) noise is considered. A one-dimensional probability distribution is obtained by way of a characteristic function which is expressed by functionals of the Smoluchowski process. It is shown that in the frame of the presented approach the probability density can be found only for a particular value of the damping constant in the linear-type relaxation equation. It is also shown that in a special case the white noise limit may be performed.Supported in part by the Polish Academy of Sciences under Contract No. MR 1-9.     

Generalized moment expansion for nonadiabatic transition reaction

We discuss the generalized moments of the nonadiabatic transition reaction by using the stochastic Liouville equation for the study of outer-sphere electron transfer in polar solvents characterized by Debye dielectric relaxation. We obtain an approximate expression for the generalized moments which incorporates the width of the transition with arbitrary initial condition far from equilibrium. For low barriers, we derive an analytical expression for the rate corresponding to harmonic potential surfaces in the overdamped regime. For Fokker-Planck operators of Smoluchowski type, we introduce a new method to solve all of the generalized moments by using the eigenfunction expansion method.     

Quantum-classical correspondence of the Dirac equation with a scalar-like potential

Quantum matrix elements of the coordinate, momentum and the velocity operator for a spin-1/2 particle moving in a scalar-like potential are calculated. In the large quantum number limit, these matrix elements give classical quantities for a relativistic system with a position-dependent mass. Meanwhile, the Klein-Gordon equation for the spin-0 particle is discussed too. Though the Heisenberg equations for both the spin-0 and spin-1/2 particles are unlike the classical equations of motion, they go to the classical equations in the classical limit.      

Semiclassical Method to Schroedinger Equation with Position-Dependent Effective Mass

In this paper, two novel semiclassical methods including the standard and supersymmetric WKB quantization conditions are suggested to discuss the Schroedinger equation with position-dependent effective mass. From a proper coordinate transformation, the formalism of the Schroedinger equation with position-dependent effective mass is mapped into isospectral one with constant mass and therefore for a given mass distribution and physical potential function the bound state energy spectrum can be determined easily by above method associated with a simple integral formula. It is shown that our method can give the analytical results for some exactly-solvable quantum systems.     

Statistical mechanics of holonomic systems as a Brownian motion on smooth manifolds

The statistical mechanics of arbitrary holonomic scleronomous systems subjected to arbitrary external forces is described by specializing the Lagrange and Hamilton equations of motion to those of the Brownian motion on a manifold. In this context, the Klein‐Kramers and Smoluchowski equations are derived in covariant form, and it is demonstrated that these equations have equilibrium solutions corresponding to the Gibbs distribution, in agreement with standard thermodynamics. At last, the Langevin dynamics corresponding to the Smoluchowski limit is found to exactly correspond to the Brownian motion on a smooth manifold. These results find significant applications in the study of several statistical properties of constrained molecular assemblies (e.g. polymers) of interest in chemistry, physics and biology.     

A comment on early-time solutions of the Smoluchowski equation

We present a simple derivation of classes of early-time solutions of the Smoluchowski equation in the presence of boundaries, simplifying and generalizing an analysis by van Kampen.     

用变分法解Fokker—Planck方程计算核裂变几率

发展了一种用变分法解Fokker-Planck方程的方法,计算了一维核裂变几率,研究了F-P方程与Smoluchowski方程的关系,为推导多维的质量系数和粘滞系数随形变参量变化并具有交叉项的Smoluchowski方程和解位能形式比较复杂的多维的F-P方程打下了基础.     

Zeno dynamics for open quantum systems

In this paper, we formulate limit Zeno dynamics of general open systems as the adiabatic elimination of fast components. We are able to exploit previous work on adiabatic elimination of quantum stochastic models to give explicitly the conditions under which open Zeno dynamics will exist. The open systems formulation is further developed as a framework for Zeno master equations, and Zeno filtering (that is, quantum trajectories based on a limit Zeno dynamical model). We discuss several models from the point of view of quantum control. For the case of linear quantum stochastic systems, we present a condition for stability of the asymptotic Zeno dynamics.     

Form-Preserving Transformations of Time-Dependent Schrödinger Equation with Time- and Position-Dependent Mass

We study space-time transformations of the time-dependent Schrödinger equation (TDSE) with time- and position-dependent (effective) mass. We obtain the most general space-time transformation that maps such a TDSE onto another one of its kind. The transformed potential is given in explicit form.     

A Remark on the Kramers Problem

We present new point of view on the old problem, the Kramers problem. The passages from the Fokker–Planck equation to the Smoluchowski equation, including corrections to the Smoluchowski current, is treated through an asymptotic expansion of the solution of the stochastic dynamical equations. The case of an extremely weak force of friction is also discussed.     

Study of a class of models for self-organization: equilibrium analysis

A new class of nonlinear stochastic models is introduced with a view to explore self-organization. The model consists of an assembly of anharmonic oscillators, interacting via a mean field of system size range, in presence of white, Gaussian noise. Its properties are explored in the overdamped regime (Smoluchowski limit). The single oscillator potential is such that for small oscillator displacements it leads to a highly nonlinear force but becomes asymptotically harmonic. The shape of the potential can be a single-or double-well and is controlled by a set of parameters. Through equilibrium statistical mechanical analysis, we study the collective behavior and the nature of phase transition. Much of the analysis is analytic and exact. The treatment is not restricted to the thermodynamic limit so that we are also able to discuss finite size effects in the model.     

On the derivation of Smoluchowski equations with corrections in the classical theory of Brownian motion

Differential equations governing the time evolution of distribution functions for Brownian motion in the full phase space were first derived independently by Klein and Kramers. From these so-called Fokker-Planck equations one may derive the reduced differential equations in coordinate space known as Smoluchowski equations. Many such derivations have previously been reported, but these either involved unnecessary assumptions or approximations, or were performed incompletely. We employ an iterative reduction scheme, free of assumptions, and calculate formally exact corrections to the Smoluchowski equations for many-particle systems with and without hydrodynamic interaction, and for a single particle in an external field. In the absence of hydrodynamic interaction, the lowest order corrections have been expressed explicitly in terms of the coordinate space distribution function. An additional application of the method is made to the reduction of the stress tensor used in evaluating the intrinsic viscosity of particles in solution. Most of the present work is based on classical Brownian motion theory, but brief consideration is given in an appendix to some recent developments regarding non-Markovian equations for Brownian motion.Supported by the National Science Foundation.     

A Position-Dependent Two-Atom Entanglement in Real-Time Cavity QED System

We study a special two-atom entanglement case in assumed cavity QED experiment in which only one atom effectively exchanges a single photon with a cavity mode. We compute two-atom entanglement under position-dependent atomic resonant dipole-dipole interaction （RDDI） for large interatomic separation limit. We show that the RDDI, even t, hat which is much smaller than the maximal atomic Rabi frequency, can induce distinct diatom entanglement. The peak entanglement reaches a maximum when RDDI strength can compare with the Rabi frequency of an atom.     

Analytic Results in the Position-Dependent Mass SchrSdinger Problem

We investigate the Schr6dinger equation for a particle with a nonuniform solitonic mass density. First, we discuss in extent the （nontrivial） position-dependent mass V（x） = 0 case whose solutions are hypergeometric functions in tanh2 x. Then, we consider an external hyperbolic-tangent potential. We show that the effective quantum mechanical problem is given by a Heun class equation and find analytically an eigenbasis for the space of solutions. We also compute the eigenstat, es for a potential of the form V （x） = Vo sinh2 z.     

Optimal reactions for the synthesis of superheavy nucleus 270Hs

The superheavy nucleus 270 Hs iS expected to be a "double-magic" deformed nucleus.We have calculated its cross sections of evaporation residue for the reactions 248Cm(26Mg,4n)270Hs,244pu(30Si,4n)270Hs,238U(36S,4n)270Hs and 226Ra(48Ca,4n)270Hs using a two-parameter Smoluchowski equation.It is found from our results that 226Ra(48Ca,4n)270Hs and 238U(36S,4n)270Hs are two optimal reactions for the synthesis of the superheavy nucleus 270Hs due to their large negative Q-values.