Alexander N. Drozdov

Accurate calculation of quantum and diffusion propagators in arbitrary dimensions

A new approach to calculating the dynamics and equilibrium thermodynamics of an arbitrary (quantum or stochastic) system is presented. Its key points are representing the full propagator as a product of the harmonic‐oscillator propagator with the configuration function, and expanding the configuration function (its exponent) in a power series in a given function of t. Recursion relations are obtained for the expansion coefficients which can be analytically evaluated for any number of degrees of freedom. This representation is particularly attractive for two reasons. Being structurally similar to the standard Taylorlike expansions for the propagator already known in the literature, it nevertheless shows a dramatic improvement over the latter in that it converges significantly better over a much broader range of t. Another attractive feature of the present expansion is that it is amenable to subsequent approximations. With this technique a minimal computational effort is required for constructing an improve...

Improved Feynman’s path integral method with a large time step: Formalism and applications

We describe an efficient path integral scheme for calculating the propagator of an arbitrary quantum system, as well as that of a stochastic system in special cases where the Fokker–Planck equation obeys strict detailed balance. The basic idea is to split the respective Hamiltonian into two exactly solvable parts and then to employ a symmetric decomposition of the time evolution operator, which is exact up to a high order in the time step. The resulting single step propagator allows rather large time steps in a path integral and leads to convergence with fewer time slices. Because it involves no system-specific reference system, the algorithm is amenable to all known numerical schemes available for evaluating quantum path integrals. In this way one obtains a highly accurate method, which is simultaneously fast, stable, and computationally simple. Numerical applications to the real time quantum dynamics in a double well and to the stochastic dynamics of a bistable system coupled to a harmonic mode show our...

High-accuracy discrete path integral solutions for stochastic processes with noninvertible diffusion matrices. II. Numerical evaluation

We present a fast, high precision and easily implementable path integral method for numerically solving Fokker–Planck equations. It is based on a generalized Trotter formula, which permits one to attain an adequate description of dynamical and equilibrium properties even though the time increment τ=t/N is rather large. A remarkable property of the symmetric Trotter splitting is used to systematically eliminate the lower-order errors resulting from time discretization. This means a significant reduction of the number of time steps that are required to retain a given accuracy for a given net increment t=Nτ, and, therefore, significantly increasing the feasibility of path integral calculations. Yet another attractive feature of the present technique is that it allows for equations with singular diffusion matrices that are known to present a special problem within the scope of the path integral formalism. The favorable scaling of the fast Fourier transform is used to numerically evaluate the path integral on ...

Two novel approaches to the Kramers rate problem in the spatial diffusion regime

At present, there are two general theoretical approaches to calculating the rate of thermally activated escape of a Brownian particle over a barrier out of a metastable well in the spatial diffusion regime. A direct approach involves techniques entirely based on the underlying Fokker–Planck equation, such as the Kramers flux over population method, the mean first passage time formalism, and the eigenmode expansion. An alternative consists of replacing the original one-dimensional stochastic dynamics by an infinite dimensional Hamiltonian system. The rate is then calculated using reactive flux methods. Both approaches are rather efficient when treating bistable potentials with high parabolic barriers. However, complications arise if the barrier is not parabolic. In such a case, large deviations of theoretical predictions from exact numerical rates are observed in the intermediate friction region. The latter holds true even though the barrier is infinitely high, to say nothing of low barriers for which the ...

Decay of metastable states: Mean relaxation time formulation

The mean relaxation time formalism introduced by Nadler and Schulten [J. Chem. Phys. 82, 151 (1985)] in their generalized moment expansion method is extended to a general diffusion process in arbitrary dimensions. The utility of the approach is demonstrated by calculating analytically the rate of noise-induced transitions in a bistable system with an isolated transition point. The rate formula obtained summarizes in a uniform manner much of what had been done before in this field. Limitations of its validity are discussed and a perturbation procedure to systematically improve it is proposed. The validity of our theoretical predictions for the rate is confirmed by comparing with exact numerical results.

Thermally driven escape over a barrier of arbitrary shape

The Kramers theory for the thermally activated rate of escape of a Brownian particle from a potential well is extended to a barrier of arbitrary shape. The extension is based on an approximate solution of the underlying Fokker–Planck equation in the spatial diffusion regime. With the use of the Mel’nikov–Meshkov result for the underdamped Brownian motion an overall rate expression is constructed, which interpolates the correct limiting behavior for both weak and strong friction. It generalizes in a natural way various different rate expressions that are already available in the literature for parabolic, cusped, and quartic barriers. Applications to symmetric parabolic and cusped double-well potentials show good agreement between the theory and estimates of the rates from numerical calculations.

Improved power series expansion for the time evolution operator: Application to two-dimensional systems

The power series expansion formalism is used to construct analytical approximations for the propagator of the partial differential equation of a generic type. The present approach is limited to systems with polynomial coefficients. Three typical two-dimensional examples, a Henon–Heiles anharmonic resonating system, a system–bath Hamiltonian, and a Fokker–Planck chaotic model are considered. All results are in excellent agreement with those of an established numerical scheme in the field. It is found that the power series expansion method accurately describes the dynamics of very anharmonic processes in the whole time domain.

Higher-order quasilinear approximations for the propagator of the Kramers equation

Abstract We present a systematic procedure for constructing higher-order quasilinear approximations for the propagator of the Klein-Kramers equation describing the motion of a Brownian particle in a general force field. Its key points are splitting the full force field into a linear contribution and an anharmonic correction, replacing the underlying Langevin equations by difference equations and solving these equations iteratively. An accurate single step propagator is then derived in terms of known statistical properties of the noise terms. Its use in a path integral shows this approach to be advantageous over a Taylor series expansion for the propagator recently derived employing standard techniques.

Quantum Statistics of Multidimensional Nonlinear Oscillators

This paper presents two perturbation approaches to calculating the quantum statistics of multidimensional nonlinear oscillators in a simple, economic way. These are the power series expansion formalism and the system-specific split operator method. The former involves a harmonic reference system, while the latter allows for the use of physically motivated (anharmonic) zeroth-order representations. Both approaches are outlined with some improvements, and their possible limitations are discussed. The relative efficacy of the approaches is tested on two typical models that are often used by researchers as benchmarks for different numerical methods. Although no system-specific reference system is involved in the power series expansion of the Boltzmann operator, it quite accurately describes the quantum statistics in the entire temperature range, even though the coupling strength is rather large. Yet another important advantage of this approach is that it is essentially analytical, and therefore its numerical implementation does not require any computational effort. This makes the power series expansion technique particularly attractive for treating many-body problems. In contrast, the split operator method is found to be efficient only in the limit of separable motion. Otherwise, this approach results in a less than optimally efficient approximation for the density matrix, or even one which is worse than the standard Trotter approximation.