On the approximation of Feynman–Kac path integrals
Introduction
The path integral approach provides a powerful method for studying properties of quantum many-body systems [1]. When applied to statistical mechanics [2], each element of the quantum density matrix is expressed as an integral over all curves connecting two configurationsThe symbol indicates that the integration is performed over the set of all differentiable curves, , with and . The integer d reflects the dimensionality, with d=3N for a system of N-particles in 3-dimensional space. The functional Φ can be derived from the classical action by introducing a relationship between temperature and imaginary time (it=βℏ) [1]. In this paper, we will restrict our attention to the quantum many-body system, for which Φ takes the following form:
Calculating the path integral in (1) is a challenging task, which in general cannot be performed analytically. It is only for simple model problems, such as quadratic potentials, that an exact solution can be obtained. For more complex systems, the path integral has traditionally been estimated using either the “short-time” approximation (STA) [3] or “Fourier discretization” (FD) [4], [5]. Many authors have proposed improvements to the standard STA and FD, using techniques such as improved estimators [6], [7], partial averaging [8], [9], [10], higher-order exponential splittings [11], advanced reference potentials [12], semi-classical expansions [13], and extrapolation [14]. The fundamental approach is the same in all of these methods: the path integral is reduced to a high (but finite) dimensional Riemann integral, which is then approximated using either Monte Carlo or molecular-dynamics simulation techniques.
In this paper, we investigate the discretization of path integrals by projection onto a finite-dimensional subspace. The idea of approximating path integrals using a finite subset of basis functions has been suggested before in the literature. Davison was one of the first to consider the use of orthogonal function expansions in the representation of Feynman path integrals [15], although he did not explore truncating the expansion. In a related article on Wiener integration of a different class of functionals, Cameron proposed using a finite set of orthogonal basis functions, and investigated the convergence of Fourier (spectral) elements [16]. An advantage of the subspace approach is that we need not require that the basis functions are orthogonal, allowing for the direct comparison of the STA and FD methods. This is very different from operator splitting methods, which seek higher-order approximations to the Boltzmann operator [11], [17]. We should note that Coalson explored some of the connections between the STA and FD methods [18], although using a different technique.
The real power of the subspace approach is that new methods can be readily constructed using general classes of basis functions such as orthogonal polynomials or finite elements. The structural properties of the basis functions, such as smoothness and compact support, can be varied in an effort to improve overall efficiency. In Section 3, we derive path integral methods starting from three different classes of basis basis functions. It is shown that while the first two choices (linear and spectral elements) result in known methods (STA and FD), a new method can be constructed using compactly supported (Hermite) cubic splines (HCS).
The one-dimensional harmonic oscillator is one of only a few systems for which the path integral can be evaluated exactly. In Section 4, we derive expressions for the average energy of the harmonic oscillator, using two different energy estimators (E1 and E2) with a general subspace method. Although both estimators converge to the exact average energy, we show that E2 (which is based on the virial equation) is far more accurate. While the error in E1 is first-order for all three methods, the error in E2 is second-, third-, and sixth-order when it is calculated using the STA, FD, and HCS methods, respectively.
In Section 5, we investigate the efficiency of the path integral methods using numerical experiments. We apply each method to the problem of calculating the average energy of a particle in a one-dimensional double well. Metropolis Monte Carlo is used to sample configurations from each approximating subspace. The efficiency of each method is measured by comparing the accuracy in the E2 estimator as a function of (a) subspace dimension and (b) total computation time. It is shown that while the FD and HCS methods provide a similar degree of accuracy as a function of subspace dimension, the HCS method requires far less computation time. This improvement in efficiency exhibited by the HCS method is due to the compact support of the Hermite cubic basis functions.
Section snippets
Path integral approximation
To illustrate how one can use a subspace approximation to discretize the quantum density matrix in (1), we start by introducing a change of variables to simplify the boundary conditions and temperature dependence for each path integral: . Since the admissible paths, x, satisfy the boundary conditions and , the reduced paths given by y, will satisfy Dirichlet boundary conditions, , independent of a, b, and β. Introducing this change of variables
Subspace methods
As we mentioned in the previous section, the real benefit of using a general subspace approach is the flexibility afforded through the choice of basis functions. By considering a general class of pseudo-spectral or finite-element basis functions, a diverse group of discretizations can be constructed. Direct comparisons can be made between basis functions of varying smoothness and support. However, for brevity, we restrict our attention in this paper to three different types of basis functions:
Harmonic oscillator
In this section we present a simple procedure for exactly evaluating the path integrals which arise when the harmonic oscillator density matrix is discretized using an arbitrary subspace method. This is one of the few cases where both the approximate and exact density matrices can be evaluated analytically. We will restrict our attention to the problem of selecting a suitable energy estimator for calculating the average energy.
The partition function is defined as the trace of the density
Double well potential: a numerical experiment
As a numerical experiment, we apply each path integral discretization to the problem of calculating the average energy of a particle in a one-dimensional double-well. We have chosen the same double-well potential considered in [5], which is as follows:The parameter values are all in atomic units, with ω=0.006, A=0.009, a=0.09, and m=1836. At low temperatures, the energy is just above 0.006, which is below the barrier height of 0.009. The potential energy, and the first
Conclusion
We show that the problem of approximating Feynman–Kac path integrals can be addressed using the finite-dimensional subspace approach. This general framework allows for the ready construction of broad classes of new methods through the choice of a suitable set of basis functions. In addition, traditional approaches, such as the short-time approximation and Fourier discretization methods, can be formulated and compared using this formalism. As an illustration, we demonstrate that, by considering
Acknowledgements
The authors gratefully acknowledge support for this work from the National Science Foundation under Grant No. DMS-9627330. S.D.B. is supported in part by the Howard Hughes Medical Institute, and in part by NSF and NIH grants to J.A. McCammon. B.B.L. acknowledges the support of the National Science Foundation under Grant CHE-9970903. B.J.L. acknowledges the UK Engineering and Physical Sciences Research Council Grant GR/R03259/01.
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