On the approximation of Feynman-Kac path integrals

Transcription

1 On the approxiation of Feynan-Kac path integrals Stephen D. Bond, Brian B. Laird, and Benedict J. Leikuhler University of California, San Diego, Departents of Matheatics and Cheistry, La Jolla, CA 993, USA, University of Kansas, Departent of Cheistry, Lawrence, KS 6645, USA, University of Leicester, Departent of Matheatics and Coputer Science, Leicester, LE1 7RH, UK Eail: Version: July 3, A general fraework is proposed for the nuerical approxiation of Feynan-Kac path integrals in the context of quantu statistical echanics. Each infinite-diensional path integral is approxiated by a Rieann integral over a finite-diensional Sobolev space by restricting the integrand to a subspace of all adissible paths. Through this process, a wide class of ethods is derived, with each ethod corresponding to a different choice for the approxiating subspace. It is shown that the traditional short-tie approxiation and Fourier discretization can be recovered by using linear and spectral basis functions respectively. As an illustration of the flexibility afforded by the subspace approach, a novel ethod is forulated using cubic eleents and is shown to have iproved convergence properties when applied to odel probles. Key Words: Feynan-Kac Path Integrals, Quantu Statistical Mechanics, Functional Integration, Path Integral Methods 1. INTRODUCTION The path integral approach provides a powerful ethod for studying properties of quantu anybody systes [1]. When applied to statistical echanics [], each eleent of the quantu density atrix is expressed as an integral over all curves connecting two configurations: ρ (b, a) = D [x (τ)] exp a:b 1 h Φ [x (τ) ; β] }. (1) The sybol D [x (τ)] indicates that the integration is perfored over the set of all differentiable curves, x : [, β h] R d, with x () = a and x (β h) = b. The integer d reflects the diensionality, with d = 3N for a syste of N-particles in 3-diensional space. The functional Φ can be derived fro the classical action by introducing a relationship between teperature and iaginary tie (it = β h) [1]. In this paper, we will restrict our attention to the quantu any-body syste, for which Φ takes the following for: β h 1 d Φ [x (τ) ; β] = i ẋ i (τ) + V [x (τ)] dτ. () Calculating the path integral in (1) is a challenging task, which in general cannot be perfored analytically. It is only for siple odel probles, such as quadratic potentials, that an exact solution can be obtained. For ore coplex systes, the path integral has traditionally been estiated using either the short-tie approxiation (STA) [3] or Fourier discretization (FD) [4,5]. Many authors have proposed iproveents to the standard STA and FD, using techniques such as iproved estiators [6,7], partial averaging [8 1], higher-order exponential splittings [11], advanced reference potentials [1], seiclassical expansions [13], and extrapolation [14]. The fundaental approach is the sae in all of these ethods: the path integral is reduced to a high (but finite) diensional Rieann integral, which is then approxiated using either Monte Carlo or olecular-dynaics siulation techniques. In this paper, we investigate the discretization of path integrals by projection onto a finite diensional subspace. The idea of approxiating 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 Feynan path integrals [15], although he did not explore truncating the expansion. In a related article on Wiener integration of a different class of functionals, Caeron proposed using a finite set of orthogonal basis functions, and investigated the convergence of Fourier 1

2 (spectral) eleents [16]. An advantage of the subspace approach is that we need not require that the basis functions are orthogonal, allowing for the direct coparison of the STA and FD ethods. This is very different fro operator splitting ethods, which seek higher-order approxiations to the Boltzann operator [11, 17]. We should note that Coalson explored soe of the connections between the STA and FD ethods [18], although using a different technique. The real power of the subspace approach is that new ethods can be readily constructed using general classes of basis functions such as orthogonal polynoials or finite eleents. The structural properties of the basis functions, such as soothness and copact support, can be varied in an effort to iprove overall efficiency. In Section 3, we derive path integral ethods starting fro three different classes of basis basis functions. It is shown that while the first two choices (linear and spectral eleents) result in known ethods (STA and FD), a new ethod can be constructed using copactly supported (Herite) cubic splines (HCS). The one-diensional haronic oscillator is one of only a few systes for which the path integral can be evaluated exactly. In Section 4, we derive expressions for the average energy of the haronic oscillator, using two different energy estiators (E 1 and E ) with a general subspace ethod. Although both estiators converge to the exact average energy, we show that E (which is based on the virial equation) is far ore accurate. While the error in E 1 is first-order for all three ethods, the error in E is second, third, and sixth-order when it is calculated using the STA, FD, and HCS ethods respectively. In Section 5, we investigate the efficiency of the path integral ethods using nuerical experients. We apply each ethod to the proble of calculating the average energy of a particle in a one-diensional double well. Metropolis Monte Carlo is used to saple configurations fro each approxiating subspace. The efficiency of each ethod is easured by coparing the accuracy in the E estiator as a function of (a) subspace diension and (b) total coputation tie. It is shown that while the FD and HCS ethods provide a siilar degree of accuracy as a function of subspace diension, the HCS ethod requires far less coputation tie. This iproveent in efficiency exhibited by the HCS ethod is due to the copact support of the Herite cubic basis functions.. PATH INTEGRAL APPROXIMATION To illustrate how one can use a subspace approxiation to discretize the quantu density atrix in (1), we start by introducing a change of variables to siplify the boundary conditions and teperature dependence for each path integral: x (τ) = a + (b a) τ/β h + y (τ/β h). Since the adissible paths, x, satisfy the boundary conditions x () = a and x (β h) = b, the reduced paths given by y, will satisfy Dirichlet boundary conditions, y () = y (1) =, independent of a, b, and β. Introducing this change of variables into (1), results in the following: ρ (b, a) = D : [ y ( τ β h )] exp [ 1 h Φ a + (b a) τ ( ) ]} τ β h + y ; β. (3) β h Note that the ith coponent of each reduced path y, denoted by y i, is a real-valued function on the interval [, 1], satisfying Dirichlet boundary conditions. For the systes considered in this article, we also require that the derivative of each y i is easurable (i.e., square-integrable). Functions of this for are ebers of an infinite diensional Sobolev space [19], defined by where S 1 [, 1] = w C [, 1] w () = w (1) = and w S < }, (4) w S ẇ (ξ) + w (ξ) dξ. (5) We proceed in the following anner to discretize (3): Consider a sequence of subspaces of increasing diension V 1,, V P, S 1 [, 1], where each V P is of diension P. For convenience, let each subspace be defined as the span of a particular set of basis functions: V P = span ψ 1,, ψ P }. Now, given a coponent function y i S 1 [, 1], we can define its projection on V P uniquely by y (P ) i (ξ) P α k,i ψ k (ξ), (6) k=1

3 Using the projection y (P ) i (ξ) as an approxiation of y i (ξ) reduces the infinite-diensional path integral in (3) to a finite-diensional Rieann integral over the coefficients, α i,k : ρ (p) (b, a) = dα J exp 1 h Φ [ a + (b a) τ β h + y(p ) ( τ β h ) ]} ; β. (7) Here, we have used siplified notation for the ulti-diensional integral, with dα k,i dα k,i. The constant J reflects the particular choice of variables, and can be readily calculated (which we show later in this section). The reader should note that (7) does not depend on the basis functions chosen to represent the approxiating subspace. If both ψ 1,, ψ P } and ψ 1,, ψ P } span V P, then there is an invertible linear transforation (i.e., change of variables) U such that α = Uα. To show in detail how subspace ethods can be applied in practice, we consider the case of an N-body Hailtonian syste: Ĥ = 1 d 1 i ˆp i + V [x 1,, x d ]. (8) Here, the coordinate and oentu operators are denoted by x i and ˆp i respectively. For this syste the functional Φ is given by (), which when applied to the projected path, x (P ) (τ) a + (b a)τ/β h + y (P ) (τ/β h), results in Φ [x (P ) (τ) ; β] = β h d i [ẋ(p ) i (τ) ] + V [x (P ) (τ)] dτ. (9) After expanding the τ-integrals, introducing a change of variables ξ = τ/β h, and using the boundary conditions of each ψ k, we have Φ [x (P ) (β h ξ) ; β] = d } i (b i a i ) + α i T K α i + β h β h V [a + (b a)ξ + y (P ) (ξ)] dξ, (1) where α i [α 1,i α P,i ] T. The stiffness atrix, K R P P, has entries given by the inner-product K j,k = ψ j (ξ) ψ k (ξ) dξ. (11) Substituting (1) into (7), we obtain a siplified expression for the approxiate density atrix: } d i ρ (p) (b, a) = J exp β h (b i a i ) d } 1 i dα exp β h αt i K α i β V [x (P ) (ξ)] dξ, (1) where x (P ) (ξ) = a + (b a)ξ + y (P ) (ξ). For the Fourier case, one typically calculates J by requiring that the discretization be exact when applied to an ideal gas (i.e., V ) [4, 5, 15]. We can apply this sae technique to a general subspace ethod. Assuing that K is syetric positive definite, each integral over α i can be evaluated by diagonalizing K and applying a linear change of variables; e.g., [ ( )] 1/ 1 exp α T K α} d α = det π K. (13) Applying this forula to (1) with V allows us to solve for J in a straightforward anner: d ( ) P +1 i J = det K π β h. (14) Before discussing particular choices for basis functions, we should ention that, in general, the onediensional ξ-integral in (1) cannot be perfored analytically. This proble has been traditionally circuvented by using a discrete approxiation, such as Gaussian quadrature [5, 18]. For exaple, one can view the priitive STA as using the trapezoidal rule. If the quadrature schee is of sufficiently high order its use will not reduce the asyptotic rate of convergence of the overall ethod. An optial schee ust be efficient, since for nonlinear N-body systes evaluating V ay be coputationally expensive. 3

4 (a) Linear (b) Spectral (c) Cubic1 Cubic FIG. 1: Saple basis functions are shown above for the (a) linear, (b) spectral, and (c) cubic eleent ethods. 3. SUBSPACE METHODS As we entioned 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 pseudospectral or finite-eleent basis functions, a diverse group of discretizations can be constructed. Direct coparisons can be ade between basis functions of varying soothness and support. However, for brevity, we restrict our attention in this paper to three different types of basis functions: linear, spectral, and cubic eleents. Representative basis functions fro each of these discretizations are shown in Figure 1. The traditional STA ethod can be constructed by considering polygonal paths, which can be represented by piecewise linear basis functions [19]. For a given nuber of linear segents, P +1, we can define an approxiating subspace V P as the span of basis functions ψ 1,, ψ P }, where each ψ k is defined by the following forula ψ k (ξ) := φ lin (ξ (P + 1) k), with φ lin 1 u u [ 1, 1] (u) := otherwise. For this discretization, one can show that the reduced path y (P ) (ξ) = α k ψ k (ξ) satisfies Dirichlet boundary conditions, and fors a polygonal path with corners at the interior grid points (j/(p + 1), α j ) for integers 1 j P. The stiffness atrix, K R P P, can be calculated in a routine anner, using its definition given in (11): K sta := (P + 1) (15) We should note that in the traditional forulation of the STA ethod,the coefficients represent points on the true path, rather than the reduced path. The two forulations are actually equivalent, with the coefficients related through a trivial linear change of variables. In a siilar anner, the FD ethod can be derived using the subspace approach by considering spectral basis functions of the for ψ k (ξ) = 1 sin (k πξ). (16) k 4

5 One attractive feature of this basis is that the stiffness atrix, K R P P, is diagonal: K fd := π 1... (17). 1 A new ethod can be constructed by approxiating the space of paths using piecewise (Herite) cubic splines (HCS) [19]. Each spline is defined on an interval of width /P, with its shape uniquely deterined by its function value and derivative at the ends of the interval. It is assued here that P is an even integer. Each piecewise cubic path has a continuous derivative, and is described by linear cobinations of the basis functions φ ψ k = hcs 1 (ξp/ k) 1 k < P/ φ hcs (ξp/ + P/ k) P/ k P, (18) where and φ hcs 1 (u) := φ hcs (u) := (1 u ) ( u + 1) u [ 1, 1] otherwise u (1 u ) u [ 1, 1]. otherwise One can verify that the reduced path y (P ) (ξ) = α k ψ k (ξ) satisfies Dirichlet boundary conditions, and interpolates the interior grid points (j/p, α j ) for integers 1 j < P/. The derivative of the path at all the grid points is deterined by the reaining P/ + 1 coefficients, α k. Due to the copact support of the basis functions, the stiffness atrix is banded, with block structure: K hcs = P K 1 K 3, (19) 6 K T 3 K where the blocks are given by K 1 = and K 3 =, K = , (). (1) Note that the blocks are not all the sae size, with K 3 of diension ( P 1) ( P + 1). The deterinant of K hcs ay be calculated exactly, but for ost purposes it is enough to know that it is a constant, which will cancel out when (1) is used to calculate averages. 4. HARMONIC OSCILLATOR In this section we present a siple procedure for exactly evaluating the path integrals which arise when the haronic oscillator density atrix is discretized using an arbitrary subspace ethod. This is one of the few cases where both the approxiate and exact density atrices can be evaluated analytically. We will restrict our attention to the proble of selecting a suitable energy estiator for calculating the average energy. The partition function is defined as the trace of the density operator: Q Tr [ˆρ] = ρ (a, a) da. () 5

6 [ ] Since the density operator is proportional to exp βĥ, we can calculate the average energy by differentiating the log of the partition function with respect to β: ] E = Tr [Ĥ [ ˆρ Ĥ ρ (x, a)] β ln Q = = da x=a. (3) Tr [ˆρ] ρ (a, a) da To calculate the partition function for the approxiate density atrix, ρ (p) (b, a), we start by inserting (1) into ρ (p) (a, a) da: Q (p) = J exp β h αt K α β } V [x (P ) (ξ)] dξ d αda. (4) Inserting the potential for the haronic oscillator, V (x) = ω x /, into the forula above results in Q (p) = J exp β h αt K α β [ 1 P ω a + α i ψ i (ξ)] dξ d α da. (5) Expanding the quadratic ter, integrating over a, and siplifying yields π Q (p) = J ω exp β β h αt K α β ω α T ( A c c T ) } α d α, (6) where A i,j = ψ i (ξ) ψ j (ξ) dξ and c i = Applying (13) to (6), we can integrate over α, which results in Q (p) = det K β h ω where we have inserted the forula for J given in (14), ψ i (ξ) dξ. (7) ( [ det K + (β h ω) ( A c c T )]) 1/, (8) J = det K ( ) P +1 π β h. (9) Distributing the deterinant of K, we can reduce (8) into the following for: Q (p) = 1 β h ω ( [ det I + K 1 (β h ω) ( A c c T )]) 1/. (3) Using the eigenvalues of K 1 ( A c c T ), we can reduce Q (p) into its final for λ 1,, λ P } = spec [ K 1 ( A c c T )], (31) Q (p) = 1 β h ω P [ 1 (β h ω) λ i ] 1/. (3) To obtain the first energy estiator, we differentiate the logarith of the approxiate partition function with respect to β: [ ] E 1 (p) = 1 P (β h ω) λ i 1 β 1 (β h ω). (33) λ i In Figure, a plot of E 1 /( hω) is shown as a function of T/( hω), using linear eleents with P = 8, 16, 3, and 64. For non-zero teperatures, we find that this energy estiator converges to the true energy 6

7 E 1 / (h ω) T / (h ω) FIG. Average energy as a function of teperature for the one-diensional haronic oscillator. The average energy is calculated using the energy estiator E 1 and linear eleents, with P = 8, 16, 3, and 64. The dotted line represents the exact value, P =. (represented by the dotted curve) as P is increased. This estiator provides a reasonable approxiation to the exact energy curve for teperatures sufficiently far fro the zero-point. A popular ethod for calculating the average energy can be derived using the virial theore [6], E = 1 x V (x) + V (x). (34) For the haronic oscillator, this eans that the total average energy is siply twice the average of the potential energy. To obtain a second energy estiator, we find the average of the virial equation using the approxiate density function: E (p) = 1 Q (p) Inserting the potential for the haronic oscillator, we find E (p) = J ω a exp Q (p) β h αt K α β ω [ V (a) + 1 ] a V (a) ρ (p) (a, a) da. (35) [ a + P α i ψ i (ξ)] dξ d α da. (36) Expanding the quadratic, integrating over a, and siplifying results in J π [1 E (p) = + β ω Q (p) β β ω α T c c T α ] exp [ β h αt K + (β h ω) ( A c c T )] } α d α. (37) To rewrite (37) in the for of a Gaussian integral, we insert (6) for Q (p) and introduce a duy variable, γ: E (p) = 1 β + β γ ln exp β h αt K α β h (β h ω) α T [ A γ c c T ] } α d α. (38) γ=1 Applying (13), we can integrate over α which results in E (p) = 1 β 1 β [ det γ ln K + (β h ω) ( A γ c c T )] γ=1. (39) 7

8 1 (a) Linear Fourier Cubic 1 (b) 1 Linear Fourier Cubic E 1 E / (h ω) 1 1 E E / (h ω) P P FIG. 3 Results for the haronic oscillator at a teperature of T =.1 hω/k B. The error in the average energy is shown as a function of P, using (a) E 1 and (b) E for the energy estiator. The pluses, circles, and triangles represent results using linear, Fourier, and cubic eleents respectively. [ ] Factoring out the constant det K + (β h ω) A E (p) = 1 β 1 β γ ln [I det γ fro the inside of the deterinant, we find ( ) ] 1 K + (β h ω) A c c T. (4) γ=1 To reduce E (p) into its final for, we evaluate the deterinant, and differentiate with respect to ( 1 γ. Note that the atrix K + (β h ω) A) c c T is rank-one, with a single non-zero eigenvalue of λ = c T (K + (β h ω) A) 1 c. Applying this to (4), we are left with E (p) = β 1 ( [1 ( h ω β) c T K + ( h ω β) A ) 1 c ]. (41) In Figure 3, we show the error in the average energy of the haronic oscillator as a function of P, using linear, Fourier, and cubic eleents. The average energy is calculated using (a) E 1 and (b) E, at a fixed teperature of T =.1 hω/k B. We find that the E 1 estiator is accurate to first-order in 1/P for all three ethods. On the other hand, the E estiator is far ore accurate, with a convergence rate of second, third, and sixth-order for the linear, Fourier, and cubic eleents respectively. Although we can t expect sixth-order for a general syste, this result suggests that we ay be able to do better than first-order for specific systes, depending on the estiator used to calculate therodynaic averages. One of the ore interesting features of purely haronic systes is that they can be used to define a reference potential for ore coplex systes. If the actual syste is strongly haronic, one can expect iproveents in efficiency over a ore traditional ethod which uses the free-particle as its reference syste [4]. To illustrate how this is accoplished for a one diensional syste, with a general subspace ethod, we start by adding and subtracting the potential for the haronic oscillator in (4): Q (p) = J exp } β h αt K α βω x (P ) (ξ) dξ β V [x (P ) (ξ)] dξ d αda, (4) where the perturbed potential is defined as V (x) = V (x) ω x /. After expanding the quadratic ter, and soe siplification, we are left with: Q (p) = J exp [ β h α T B α + (β h ω a) ] } β V [x (P ) (ξ)] dξ d αda. (43) We have cobined the stiffness and ass atrices into a single positive-definite atrix, B := K + (β h ω) [ A c c T ]. One could further reduce (43) by replacing B by its Cholesky factorization, which in turn induces a linear change of variables. Discussion of how, in the Fourier case, these new variables relate to distorted waves can be found in an article by Miller [4]. 8

9 V(x) x FIG. 4: The double well potential energy function in atoic units. The horizontal bars illustrate the first four energy levels, with the lowest energy state below the energy barrier. 5. DOUBLE WELL POTENTIAL: A NUMERICAL EXPERIMENT As a nuerical experient, we apply each path integral discretization to the proble of calculating the average energy of a particle in a one-diensional double-well. We have chosen the sae double-well potential considered in [5], which is as follows: V (x) = 1 ω x + A (x/a) + 1. (44) The paraeter values are all in atoic units, with ω =.6, A =.9, a =.9, and = At low teperatures, the energy is just above.6, which is below the barrier height of.9. The potential energy, and the first four energy levels, are shown in Figure 4. To easure the accuracy of each ethod, we copute the energy at a fixed teperature of T =.1 hω/k, using Metropolis Monte Carlo to generate the canonically distributed configurations. The onediensional line-integrals of the potential are approxiated using Sipson s rule for the FD and HCS ethods, and the traditional trapezoidal rule for the STA ethod. The nuber of integration nodes is set equal to the nuber of basis functions, P, resulting in the sae nuber of potential evaluations for each ethod. For the STA ethod this results the potential is evaluated at the end points of each polygonal segent (consistent with its traditional ipleentation). Since the ai of our nuerical experients is to easure the accuracy of the discretization, no attept is ade to optiize the efficiency of the Monte Carlo algorith. While the application of ore advanced sapling techniques such as staging [], and haronic reference potentials [4] would certainly iprove the sapling efficiency for this proble, this would not iprove the accuracy of the underlying discretization. It has been previously observed that averaged quantities (such as energy) converge at different rates, depending on the syste, reference potential, and the for of the estiator [5, 6, 1]. We use a virial estiator of the energy [6], E = V (x) + 1 x V (x), (45) which was shown to exhibit iproved convergence properties in the previous section. The accuracy of each average is deterined by coparing with the exact solution, coputed by suing over the 15 lowest energy levels as calculated with Nuerov s ethod []. In Fig. 5, the error in the coputed energy is shown as a function of (a) the nuber of basis functions and (b) noralized CPU tie. When the nuber of basis functions (or potential evaluations) is used as a easure of the work, we find that the FD and HCS ethods are coparable, and both are ore efficient than the STA ethod. However, when copared on the basis of CPU tie, the HCS ethod is draatically ore efficient than both other ethods. The inefficiency of the FD ethod for lowdiensional probles can be explained by considering the work required to copute P points on the path. This work scales like O(P ) for the FD ethod, since the spectral basis functions are not copactly 9

10 1 (a) Linear Fourier Cubic 1 (b) Linear Fourier Cubic E E /E E E /E P Work FIG. 5: Results for the double well at a teperature of T =.1 hω/k B. The difference between E and the exact average energy is shown as a function of (a) subspace diension, P and (b) CPU tie (work). The pluses, circles, and triangles represent results using linear, Fourier, and cubic eleents respectively. supported. On the other hand, for the STA and HCS ethods this cost scales linearly with P. Although for very high-diensional probles, the cost of evaluating the potential should doinate, and we expect that the differences in coputational cost would not be as pronounced. 6. CONCLUSION We show that the proble of approxiating Feynan-Kac path integrals can be addressed using the finite-diensional subspace approach. This general fraework allows for the ready construction of broad classes of new ethods through the choice of a suitable set of basis functions. In addition, traditional approaches, such as the short-tie approxiation and Fourier discretization ethods, can be forulated and copared using this foralis. As an illustration, we deonstrate that, by considering Herite cubic splines (HCS), a new ethod can be constructed that exhibits draatically iproved efficiency over standard ethods when applied to two one-diensional odel probles: the haronic oscillator and an anharonic double-well. Of course, whether this iproveent over standard ethods can be sustained in applications to the ore coplex ulti-diensional probles of interest in cheistry and physics is as yet an open question; however, the enhanceent seen in the one-diensional probles is significant enough to warrant further study. In addition, it ust be pointed out that the HCS discretization is only one of infinitely any possible ethods that can be constructed within the generalized subspace foralis approach presented here. 7. ACKNOWLEDGEMENTS The authors gratefully acknowledge support for this work fro the National Science Foundation under Grant No. DMS SDB is supported in part by the Howard Hughes Medical Institute, and in part by NSF and NIH grants to J. A. McCaon. BBL acknowledges the support of the National Science Foundation under grant CHE BJL acknowledges the UK Engineering and Physical Sciences Research Council grant GR/R359/1. REFERENCES [1] R. P. Feynan and A. R. Hibbs, Quantu Mechanics and Path Integrals (McGraw-Hill, New York, 1965). [] R. P. Feynan, Statistical Mechanics (Benjain, Reading, Mass., 197). [3] K. S. Schweizer, R. M. Stratt, D. Chandler, and P. G. Wolynes, J. Che. Phys. 75, 1347 (1981). 1

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