Wiener Path Integral Research Articles

Color path-integral Monte Carlo simulations of quark–gluon plasma

Thermodynamic properties of a strongly coupled quark–gluon plasma (QGP) of constituent quasiparticles are studied by a color path-integral Monte Carlo simulations (CPIMC). For our simulations we have presented QGP partition function in the form of color path integral with new relativistic measure instead of Gaussian one used in Feynman and Wiener path integrals. For integration over color variable we have also developed procedure of sampling color variables according to the group SU(3) Haar measure. It is shown that this method is able to reproduce the available quantum lattice chromodynamics (QCD) data.

An analytical Wiener path integral technique for non-stationary response determination of nonlinear oscillators

A novel approximate analytical technique for determining the non-stationary response probability density function (PDF) of a class of randomly excited nonlinear oscillators is developed. Specifically, combining the concepts of statistical linearization and of stochastic averaging the evolution of the response amplitude of oscillators with nonlinear damping is captured by a first-order stochastic differential equation (SDE). This equation has nonlinear drift but constant diffusion coefficients. This convenient feature of the SDE along with the concept of the Wiener path integral is utilized in conjunction with a variational formulation to derive an approximate closed form solution for the response amplitude PDF. Notably, the determination of the non-stationary response PDF is accomplished without the need to advance the solution in short time steps as it is required by the existing alternative numerical path integral solution schemes. In this manner, an analytical Wiener path integral based technique is developed for treating certain stochastic dynamics problems for the first time. Further, the technique can be used as a convenient tool for assessing the accuracy of alternative, more general, approximate solution methods. The accuracy of the technique is demonstrated by pertinent Monte Carlo simulations.

Solution of the stochastic Langevin equations for clustering of particles in random flows in terms of the Wiener path integral

We propose to take advantage of using the Wiener path integrals as the formal solution for the joint probability densities of coupled Langevin equations describing particles suspended in a fluid under the effect of viscous and random forces. Our obtained formal solution, giving the expression for the Lyapunov exponent, (i) will provide the description of all the features and the behavior of such a system, e.g., the aggregation phenomenon recently studied in the literature using appropriate approximations, (ii) can be used to determine the occurrence and the nature of the aggregation-nonaggregation phase transition which we have shown for the one-dimensional case, and (iii) allows the use of a variety of approximative methods appropriate for the physical conditions of the problem such as instanton solutions in the WKB approximation in the aggregation phase for the one-dimensional case as presented in this paper. The use of instanton approximation gives the same result for the Lyapunov exponent in the aggregation phase, previously obtained by other authors using a different approximative method. The case of nonaggregation is also considered in a certain approximation using the general path integral expression for the one-dimensional case.

On the geometrical representation of the Jacobian in the path integral reduction problem

By using the formula for the scalar curvature of the manifold with the Kaluza–Klein metric we obtain the geometrical representation of the Jacobian resulted from the path integral reduction problem in Wiener path integrals for a scalar particle on a smooth compact Riemannian manifold with the given free isometric action of the compact semisimple Lie group.

Brownian simulations and unidirectional flux in diffusion

The prediction of ionic currents in protein channels of biological membranes is one of the central problems of computational molecular biophysics. Existing continuum descriptions of ionic permeation fail to capture the rich phenomenology of the permeation process, so it is therefore necessary to resort to particle simulations. Brownian dynamics (BD) simulations require the connection of a small discrete simulation volume to large baths that are maintained at fixed concentrations and voltages. The continuum baths are connected to the simulation through interfaces, located in the baths sufficiently far from the channel. Average boundary concentrations have to be maintained at their values in the baths by injecting and removing particles at the interfaces. The particles injected into the simulation volume represent a unidirectional diffusion flux, while the outgoing particles represent the unidirectional flux in the opposite direction. The classical diffusion equation defines net diffusion flux, but not unidirectional fluxes. The stochastic formulation of classical diffusion in terms of the Wiener process leads to a Wiener path integral, which can split the net flux into unidirectional fluxes. These unidirectional fluxes are infinite, though the net flux is finite and agrees with classical theory. We find that the infinite unidirectional flux is an artifact caused by replacing the Langevin dynamics with its Smoluchowski approximation, which is classical diffusion. The Smoluchowski approximation fails on time scales shorter than the relaxation time 1/gamma of the Langevin equation. We find that the probability of Brownian trajectories that cross an interface in one direction in unit time Deltat equals that of the probability of the corresponding Langevin trajectories if gammaDeltat=2 . That is, we find the unidirectional flux (source strength) needed to maintain average boundary concentrations in a manner consistent with the physics of Brownian particles. This unidirectional flux is proportional to the concentration and inversely proportional to sqrt[Deltat ] to leading order. We develop a BD simulation that maintains fixed average boundary concentrations in a manner consistent with the actual physics of the interface and without creating spurious boundary layers.

Wiener path integrals and the fundamental solution for the Heisenberg Laplacian

In this paper, we present an explicit calculation of the heat kernel, fundamental solution and Schwartz kernel of the resolvent for the Heisenberg Laplacian using Wiener path integrals and their realizations via the Trotter product formula. This also gives another derivation of mehler’s formula.

Path integrals on a manifold with group action

A new path integral measure factorization method is proposed. By using this method the reduction procedure in Wiener path integrals for a scalar particle on a smooth compact Riemannian manifold with the given free isometric action of the compact semisimple unimodular Lie group is considered. It is shown that the path integral measure is not invariant under the reduction. The integral relation between path integrals representing the fundamental solutions of the parabolic equations on initial and reduced manifolds is derived.

Finite Dimensional Approximations to Wiener Measure and Path Integral Formulas on Manifolds

Certain natural geometric approximation schemes are developed for Wiener measure on a compact Riemannian manifold. These approximations closely mimic the informal path integral formulas used in the physics literature for representing the heat semi-group on Riemannian manifolds. The path space is approximated by finite dimensional manifolds HP(M) consisting of piecewise geodesic paths adapted to partitions P of [0, 1]. The finite dimensional manifolds HP(M) carry both an H1 and a L2 type Riemannian structures, G1P and G0P, respectively. It is proved that (1/ZiP)e−(1/2)E(σ)dVolGiP(σ)→ρi(σ)dν(σ) as mesh (P)→0, where E(σ) is the energy of the piecewise geodesic path σ∈HP(M), and for i=0 and 1, ZiP is a “normalization” constant, VolGiP is the Riemannian volume form relative to GiP, and ν is Wiener measure on paths on M. Here ρ1(σ)≡1 and ρ0(σ)=exp(−16∫10Scal(σ(s))ds) where Scal is the scalar curvature of M. These results are also shown to imply the well known integration by parts formula for the Wiener measure.

Turbulent transport of a tracer: An electromagnetic formulation

We present a formulation of the problem of advection and diffusion of a passive tracer by an arbitrary, incompressible velocity field. A Wiener path integral is employed to prove that the problem is identical to the diffusive dynamics of a charged particle in electromagnetic fields constructed from the velocity field. The case of zero diffusion has characteristics that coincide with the integral curves of the velocity field. This case is, of course, structurally unstable, and the limit of small diffusion is correctly described by the Wenzel-Kramers-Brillouin limit of our path-integral principle, wherein the tracer dynamics equals the orbits of point charges in electromagnetic fields. To lowest order, diffusive effects are accounted for within a Hamiltonian framework, and the limit of zero diffusion emerges as an unstable submanifold embedded in a six-dimensional phase space. We illustrate these ideas by considering the simple case of tracer advection-diffusion in the flow field of a time-independent, straight vortex line. We also briefly discuss generalization of the path-integral principle for the case where tracer sources and/or sinks are included. When the velocity field obeys the Navier-Stokes equation, the associated electromagnetic fields satisfy the equations of magnetohydrodynamics for a fluid with resistivity that is equal to the viscosity of the (real) fluid.

Lagrangian Solution of Dynamo Equations and Path Integrals

We discuss an explicit solution of the Cauchy problem for induction equation and suggest its generalization for equations of α2ω-dynamo. These solutions are based on concepts of multiplicative, Wiener path, and stochastic integrals. Obtained explicit solution can be useful as a tool in investigations of a dynamo with fluctuating helicity.

Nonstationary generalized Duru-Kleinert transformation of path integrals for systems of differential equations in one-dimensional space

We obtain new formulas for the transformations of Wiener path integrals corresponding to the parabolic systems of two differential equations with time-dependent coefficients in one-dimensional space. These formulas determine the transformation of the path integrals under a rheonomous-homogeneous-pointwise transformation of integration variables and the path reparameterization transformation. These formulas allow us to obtain an integral relation between the Green's functions of related systems of differential equations. We show how to obtain the generalized Shepp formula from this relation for the path integral under consideration. We derive these new formulas using the properties of random processes under phase transitions and a random change in time.

From Feynman-Kac formula to Feynman integrals via analytic continuation

By using a calculus based on Brownian bridge measures, it is shown that under mild assumptions on V (e.g. V is in the Kato class) the fundamental solution (FS) q ( t,x,y) for the heat equation ∂ tu = 1 2 Δ − V)u can be represented by the Feynman-Kac formula. Furthermore, it has an analytic continuation in t over C +, where C +={z ∈ C, Re z > 0} , and q(ε + i t,x,y) can be expressed via Wiener path integrals. For small ε > 0 it can be considered as an approximation of the FS for the Schrödinger equation ∂ tψ=i( 1 2 Δ − V)ψ . We also give an estimate of q( t,x,y) for t ∈ C +.

A new method for the asymptotic evaluation of a class of path integrals

A general method of calculating the asymptotic behavior of a class of Wiener path integrals is given. These integrals are averages of functionals of the ’’local time.’’ The technique is essentially a variation on the well-known ’’replica’’ method now widely used in condensed matter physics, combined with the Laplace method for evaluating integrals containing a large parameter. The leading term is given, and from the construction one sees that the error is typically of order 1/t.

Accurate Evaluation of Stochastic Wiener Integrals with Applications to Scattering in Random Media and to Nonlinear Filtering

In 1973 A. J. Chorin [Math. Comp., 27 (1973), pp. 1–15] reported quadrature formulas for the approximation of a class of Wiener path integrals by n-fold ordinary integrals with an error Similar formulas are presented here for certain stochastic Wiener function space integrals in which n-fold integral approximations result in errors $O( n^{ - 2} )$ or $O( n^{ - 3/ 2} )$. Stochastic Wiener integrals arise in the physics of wave propagation in random media and in signal estimation problems in communication theory. The present formulas provide simple, accurate computational tools for these applications, among others. This point is illustrated by several examples in the paper.

A quantum-statistical Monte Carlo method; path integrals with boundary conditions

A new Monte Carlo method for problems in quantum-statistical mechanics is described. The method is based on the use of iterated short-time Green’s functions, for which ’’image’’ approximations are used. It is similar to the use of Feynman or Wiener path integrals but with a modification to take account of hard-core boundary conditions. It is applied to two one-dimensional test problems: that of a single particle in a hard-walled box and that of two hard particles in a hard-walled box. For these test problems, the results are in excellent agreement with exact quantum-mechanical results both at high temperatures (near the classical limit) and at very low temperatures such that essentially only the ground state is occupied. Generalizations to three-dimensional systems, to many-body systems, and to more realistic potentials are discussed briefly.

Experiments on a superposition approximation for Wiener path integrals

The pair correlation function for He 4 gas is calculated in the following way: Two- and three-particle Slater sums are expressed in terms of Wiener path integrals. The three-particle path integral is approximated by a product of two-particle path integrals. From tabular values of these path integrals obtained previously, the pair correlation function is obtained by quadrature. Comparison is made with existing calculations employing the three-particle term.

Quantum-Mechanical Pair-Correlation Function of Hard Spheres

The density-independent part of the quantum-mechanical direct and exchange pair-correlation functions for hard spheres has been calculated for λT / σ = 1.4, 2, 2.93761, 4, 6, 8, and 10. In addition, points were obtained very near the surface of the sphere for λT / σ = 0.025 and 0.05. The results delineate the approach of the direct (Boltzmann) correlation function to the classical limit and emphasize the rapid disappearance of the effects of statistics with increasing temperature. These features are explained using Wiener (path) integrals. Analytical expressions valid near the surface of the sphere are derived and compare well with numerical results. Virial coefficients obtained from the correlation functions are in satisfactory agreement with very precise value calculated by Boyd, Larsen, and Kilpatrick.