Strong friction limit in quantum mechanics: the quantum Smoluchowski equation

For a quantum system coupled to a heat bath environment the strong friction limit is studied starting from the exact path integral formulation. Generalizing the classical Smoluchowski limit to low temperatures, a time evolution equation for the position distribution is derived and the strong role of quantum fluctuations in this limit is revealed.     

Eigenvalues of the one-dimensional Smoluchowski equation

The eigenvalues and eigenfunctions of the Smoluchowski equation are investigated for the case of potentials withN deep wells. The small parameter =kT/V, which measures the ratio of the thermal energy to a typical well depth, is used in connection with the method of matched asymptotic expansion to obtained asymptotic approximations to all the eigenvalues and eigenfunctions. It is found that the eigensolutions fall into two classes, namely (i) the top-of-the-well and (ii) the bottom-of-the-well eigensolutions. The eigenvalues for both classes of solutions are integer multiples of the squqres of the frequencies at the top or bottom of the various wells. The eigenfunctions are, in general, localized to the top or bottom of the corresponding well. The very small eigenvalues require special consideration because the asymptotic analysis is incapable of distinguishing them from the zero eigenvalue with multiplicityN. Another approximation reveals that, in addition to the true zero eigenvalue, there areN-1 eigenvalues of order exp(–). The case of other possible multiple eigenvalues is also examined.     

Multiscale derivation of an augmented Smoluchowski equation

Smoluchowski and Fokker-Planck equations for the stochastic dynamics of order parameters have been derived previously. The question of the validity of the truncated perturbation series and the initial data for which these equations exist remains unexplored. To address these questions, we take a simple example, a nanoparticle in a host medium. A perturbation parameter ε, the ratio of the mass of a typical atom to that of the nanoparticle, is introduced and the Liouville equation is solved to O(ε²). Via a general kinematic equation for the reduced probability W of the location of the center-of-mass of the nanoparticle, the O(ε²) solution of the Liouville equation yields an equation for W to O(ε³). An augmented Smoluchowski equation for W is obtained from the O(ε²) analysis of the Liouville equation for a particular class of initial data. However, for a less restricted assumption, analysis of the Liouville equation to higher order is required to obtain closure.     

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.     

Corrections to the Smoluchowski equation in the presence of hydrodynamic interactions

The systematic procedure for deriving the Smoluchowski equation and successive corrections to it from the Fokker-Planck equation is modified and extended in such a way that it now also covers the case of several interacting Brownian particles with hydrodynamic interactions. This is done by means of a suitable adaptation of the Chapman-Enskog method. The expression found for the first correction term to the Smoluchowski equation is worked out in full detail for the special case of two identical, spherically symmetric Brownian particles.     

Current characteristics of mesoscopic rings in quantum Smoluchowski regime

In normal mesoscopic metals of a ring topology persistent currents can be induced by threading the center of the ring with a magnetic flux. This phenomenon is an example of the famous Aharonov-Bohm effect. In the paper we study the current vs the external constant magnetic flux characteristics of the system driven by both the classical and the quantum thermal fluctuations. The problem is formulated in terms of Langevin equations in classical and quantum Smoluchowski regimes. We analyze the impact of the quantum thermal fluctuations on the current-flux characteristics. We demonstrate that the current response can be changed from paramagnetic to diamagnetic when the quantum nature of the thermal fluctuations increases.     

The generalized Smoluchowski equation for interacting Brownian particles with hard cores

TheT-operator method know from the kinetic theory of hard-sphere systems is applied to derive the generalized Smoluchowski equation for interacting hard spheres and for interacting hard rods. In this way the boundary conditions due to interactions of hard cores of Brownian particles are incorporated in the evolution equation. The result is a convenient starting point for the analysis of dense suspensions.     

The renewal equation for persistent diffusion

Persistent diffusion in one dimension, in which the velocity of the diffusing particle is a dichotomic Markov process, is considered. The flow is non-Markovian, but the position and the velocity together constitute a Markovian diffusion process. We solve the coupled forward Kolmogorov equations and the coupled backward Kolmogorov equations with appropriate initial conditions, to establish a generalized (matrix) form of the renewal equation connecting the probability densities and first passage time distributions for persistent diffusion.     

The Gervais-Neveu-Felder equation and the quantum Calogero-Moser systems

We quantize the spin Calogero-Moser model in theR-matrix formalism. The quantumR-matrix of the model is dynamical. ThisR-matrix has already appeared in Gervais-Neveu's quantization of Toda field theory and in Felder's quantization of the Knizhnik-Zamolodchikov-Bernard equation.     

The continuity equation and the Hamiltonian formalism in quantum mechanics

The relationship between the continuity equation and the HamiltonianH of a quantum system is investigated from a nonstandard point of view. In contrast to the usual approaches, the expression of the current densityJ is givenab initio by means of a transport-velocity operatorV _T, whose existence follows from a weak formulation of the correspondence principle. Once given a Hilbert-space metricM, it is shown that the equation of motion and the continuity equation actually represent a system in theunknown operatorsH andV _T, due to the arbitrariness on the initial condition of the quantum state. The general solution is given in some cases of special interest and a straightforward application to relativistic quantum mechanics is performed.This work was partially supported by the Ministero della Pubblica Istruzione.     

On the quantum langevin equation

The quantum Langevin equation is the Heisenberg equation of motion for the (operator) coordinate of a Brownian particle coupled to a heat bath. We give an elementary derivation of this equation for a simple coupled-oscillator model of the heat bath.Deceased.     

Turbulence in diffusion replicator equation

Dynamical behaviors in the diffusion replicator equation of three species are numerically studied. We point out the significant role of the heteroclinic cycle in the equation, and analyze the details of the turbulent solution that appeared in this system. Firstly, the bifurcation diagram for a certain parameter setting is drawn. Then it is shown that the turbulence appears with the supercritical Hopf bifurcation of a stationary uniform solution and it disappears under a subcritical-type bifurcation. Secondly, the statistical property of the turbulence near the supercritical Hopf onset point is analyzed precisely. Further, the correlation lengths and correlation times obey some characteristic scaling laws.     

The Boltzmann equation from quantum field theory

We show from first principles the emergence of classical Boltzmann equations from relativistic nonequilibrium quantum field theory as described by the Kadanoff–Baym equations. Our method applies to a generic quantum field, coupled to a collection of background fields and sources, in a homogeneous and isotropic spacetime. The analysis is based on analytical solutions to the full Kadanoff–Baym equations, using the WKB approximation. This is in contrast to previous derivations of kinetic equations that rely on similar physical assumptions, but obtain approximate equations of motion from a gradient expansion in momentum space. We show that the system follows a generalized Boltzmann equation whenever the WKB approximation holds. The generalized Boltzmann equation, which includes off-shell transport, is valid far from equilibrium and in a time dependent background, such as the expanding universe.