Distribution Function In Configuration Space Research Articles

Quantum effects in the Brownian motion of a particle in a double well potential in the overdamped limit

Quantum effects in the noninertial Brownian motion of a particle in a double well potential are treated via a semiclassical Smoluchowski equation for the time evolution of the reduced Wigner distribution function in configuration space allowing one to evaluate the position correlation function, its characteristic relaxation times, and dynamic susceptibility using matrix continued fractions and finite integral representations in the manner of the classical Smoluchowski equation treatment. Reliable approximate analytic solutions based on the exponential separation of the time scales of the fast intrawell and slow overbarrier relaxation processes are given. Moreover, the effective and the longest relaxation times of the position correlation function yield accurate predictions of both the low and high frequency relaxation behavior. The low frequency part of the dynamic susceptibility associated with the Kramers escape rate behaves as a single Lorentzian with characteristic frequency given by the quantum-mechanical reaction rate solution of the Kramers problem. As a particular example, quantum effects in the stochastic resonance are estimated.

Smoluchowski equation approach for quantum Brownian motion in a tilted periodic potential

Quantum corrections to the noninertial Brownian motion of a particle in a one-dimensional tilted cosine periodic potential are treated in the high-temperature and weak bath-particle coupling limit by solving a quantum Smoluchowski equation for the time evolution of the distribution function in configuration space. The theoretical predictions from two different forms of the quantum Smoluchowski equation already proposed-viz., J. Ankerhold [Phys. Rev. Lett. 87, 086802 (2001)] and W. T. Coffey [J. Phys. A 40, F91 (2007)]-are compared in detail in a particular application to the dynamics of a point Josephson junction. Various characteristics (stationary distribution, current-voltage characteristics, mean first passage time, linear ac response) are evaluated via continued fractions and finite integral representations in the manner customarily used for the classical Smoluchowski equation. The deviations from the classical behavior, discernible in the dc current-voltage characteristics as enhanced current for a given voltage and in the resonant peak in the impedance curve as an enhancement of the Q factor, are, respectively, a manifestation of relatively high-temperature nondissipative tunneling (reducing the barrier height) and dissipative tunneling (reducing the damping of the Josephson oscillations) near the top of a barrier.

Dielectric relaxation: an overview

The classical theory of dielectric relaxation of an assembly of polar molecules is briefly reviewed as a prelude to the derivation of the fractional diffusion equation, which is used to extend the Debye–Fröhlich model of relaxation over a potential barrier to anomalous diffusion governed by the Cole–Cole equation. A detailed derivation of the fractional diffusion analogue of the classical Smoluchowski equation for the evolution of the distribution function in configuration space excluding inertial effects is given by considering the continuum limit of the discrete orientation or site model of dielectric relaxation with the waiting times at the sites given by a Lévy distribution.

Microscopic and mesoscopic results from non-equilibrium molecular dynamics modeling of fene dumbbell liquids

The microscopic and the mesoscopic results are presented for several finitely extensible non-linear elastic (FENE) dumbbell fluids investigated under imposed flow according to SLLOD dynamics. The contracted distribution functions are calculated in both position and velocity spaces, and the fluid structure is probed by two conformation tensors. It is observed that dumbbells form a variety of short and long range structures depending on the imposed shear rate and the size of a single dumbbell. The assumption of Maxwellian distributed bead velocities, which is often used in the elastic dumbbell theories, is shown not to be satisfied except at low shear rates. Under shear, the distribution of the end-to-end distances is similar to the Gibbs equilibrium distribution function in configuration space if intra-molecular interactions are much stronger than inter-molecular forces. On the average, the longest dumbbells are found at between 30° and 50° to the direction of flow, and the shortest at between −50° and −30°.

Mesoscopic kinetic theory of polymer melts

A novel approach to phase space kinetic theory of dense polymer systems and to the study of their equilibrium and nonequilibrium properties is proposed. Phase space kinetic equations are constructed as continuity equations of the time evolution equation of a representative dumbbell subjected to an averaged field force. A thermodynamic equation of state, an expression for the pressure tensor involving the configuration space distribution function and an equation governing the time evolution of the configuration space distribution function are obtained as three consequences of a single phase space kinetic equation.

Macromolecules in nonhomogeneous velocity gradient fields

Some dilute solution kinetic theory results for bead-spring-type macromolecular models (linearly elastic dumbbell and Rouse) in nonuniform velocity gradient fields are obtained. It is shown that contrary to the homogeneous flow result, in general, a macromolecular solute does not move with the local center-of-mass solvent velocity in a nonhomogeneous flow. This results essentially from a unique coupling between segmental Brownian motion of the macromolecule in the flowfield and the center-of-mass translational diffusion possible only in nonhomogeneous flow. Poiseuille flow between parallel plates and circular Couette flow problems are solved for the dumbbell and some results are obtained for the Rouse model. Migration along streamlines occurs in the Poiseuille flow and migration across streamlines occurs in the Couette flow. Additional energy dissipation results for the Rouse model in Poiseuille flow relative to the simple shear flow case. Nonuniform concentration profiles are calculated in Couette flow by taking the appropriate average of the exact equation for the dumbbell configuration space distribution function.

Inertial effects and dipole-dipole coupling

It is indicated how Kramers' equation for the distribution function (in configuration-angular velocity space) underlying the motion of a pair of dipoles compelled to rotate about an axis through their common centre under the influence of their mutual dipole-dipole interaction may be derived. A solution procedure (analogous to that originally given by Brinkman in a discussion of translational brownian movement) for the equation is then set up, whereby the velocity-dependent portion of the distribution function is expressed in terms of two dimensional Hermite polynomials, whence a hierarchy of equations for the Laplace transform of the distribution function in configuration space is obtained. The lowest order approximation to the solution of the hierarchy is taken. This equation is then used to calculate the ensemble averages appropriate to the decay of electric polarization of an assembly of pairs of dipoles (rotating under the influence of their dipole-dipole interaction) following on the removal of a uni...

An application of the generalized Fokker–Planck equation to the dynamics of dilute polymer solutions

The nonlinear formalism developed by Zwanzig and Mori is utilized to derive a kinetic equation for the distribution of monomer phase space coordinates and a coarse-grained momentum density. Several simplifying approximations are then introduced into the exact kinetic equation. The resulting approximate description is shown to be closely related to the starting equations of the Freed–Edwards theory. The former differs, however, due to the presence of a non-Stoke’s frictional term which accounts for dissipation of monomer momentum fluctuations relative to the local velocity field of the solvent. Two applications of the approximate description are considered. A derivation of an equation for the two-time configuration space distribution function ψ (y, y′, t) is presented, where y denotes the collection of monomer position vectors. It is demonstrated that ψ (y,y′,t ) satisfies an equation similar to the Kirkwood–Riseman equation. Nonlinear couplings of the polymer distribution function to monomer momenta and the momentum density of the solvent lead to a diffusion tensor in which hydrodynamic interactions are characterized by a coarse-grained Oseen tensor. The correlation function formulation of the intrinsic viscosity proposed by Stockmayer et al. is extended to finite wavevectors and polymer concentrations. The specific viscosity is identified as the sum of two terms involving the mechanical contribution to the polymer momentum flux tensor and the diffusion current of chain segments.

Dielectric Relaxation and Hindered Internal Rotation

The effect of hindered internal rotation on the dielectric relaxation in a liquid whose molecules contain two polar groups of equal size with a common rotation axis is calculated for twofold symmetric cosine and parabolic potential barriers. The dielectric properties are calculated from the configuration space distribution function which is obtained by a perturbation expansion of a differential equation resulting from a rotational Brownian motion argument. The results are combined with a similar calculation involving an asymmetric potential to interpret the effects of symmetry, barrier height, and barrier shape on the relaxation process.