Summing Over Trajectories of Stochastic Dynamics with Multiplicative Noise Ying Tang, Ruoshi Yuan, and Ping Ao

Summing over trajectories of stochastic dynamics with multiplicative noise Ying Tang, Ruoshi Yuan, and Ping Ao

Citation: The Journal of Chemical Physics 141, 044125 (2014); doi: 10.1063/1.4890968 View online: http://dx.doi.org/10.1063/1.4890968 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/141/4?ver=pdfcov Published by the AIP Publishing

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This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.103.149.52 On: Wed, 17 Sep 2014 00:23:27 THE JOURNAL OF CHEMICAL PHYSICS 141, 044125 (2014)

Summing over trajectories of stochastic dynamics with multiplicative noise Ying Tang,1,2,a) Ruoshi Yuan,3 and Ping Ao1,2,b) 1Department of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China 2Key Laboratory of Systems Biomedicine Ministry of Education, Shanghai Center for Systems Biomedicine, Shanghai Jiao Tong University, Shanghai 200240, China 3School of Biomedical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China (Received 21 April 2014; accepted 11 July 2014; published online 30 July 2014)

We demonstrate that previous path integral formulations for the general stochastic interpretation generate incomplete results exempliﬁed by the geometric Brownian motion. We thus develop a novel path integral formulation for the overdamped Langevin equation with multiplicative noise. The present path integral leads to the corresponding Fokker-Planck equation, and nat- urally generates a normalized transition probability in examples. Our result solves the inconsis- tency of the previous path integral formulations for the general stochastic interpretation, and can have wide applications in chemical and physical stochastic processes. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4890968]

I. INTRODUCTION Generally, the choice of the stochastic interpretation depends on the system under study. Ito’s is widely used The path integral formulation is essential to applications in mathematical economics.18 Stratonovich’s is popular in in chemical and physical processes, such as sampling the rare physics19 in order to correctly describe gauge-invariant quan- events and determining the most probable path.1–7 The clas- tum dynamics.20 The fluctuation theorems21–30 can give sical Onsager-Machlup function8 and its generalization9–12 accurate estimations on free energy changes only under are applied successfully to the system with additive noise.2, 4 anti-Ito’s.31 The experiments also show different preferable However, for the system with multiplicative noise, previ- stochastic interpretations. The experiment conducted by the ous works on constructing the path integral for the general analog simulator shows that Stratonovich’s is preferable,32 stochastic interpretation13–15 (called the α-interpretation) are while force measurements33 and drift measurements34 on a still controversial. While the uniqueness of the action func- Brownian particle near a wall suggest that the system favors tion in the path integral is claimed,13 it contradicts to the fact anti-Ito’s,35–39 to ensure the Boltzmann-Gibbs distribution for that different interpretations lead to corresponding different the final steady state. Thus, the present path integral formu- processes in the Langevin formulation.16 Besides, though the lation for the general stochastic interpretation is practical for path integral in Refs. 14 and 15 depends on the stochastic in- applications to see which stochastic interpretation is prefer- terpretation, we find that the obtained transition probability is able, and can be tested experimentally. incomplete for the general stochastic interpretation when ap- This paper is organized as follows. In Sec. II, we provide plying their path integral formula to the geometric Brownian the path integral formulation and discuss its relation with the motion. previous path integral frameworks. In Sec. III, we generate This controversy on the path integral formulation for the the corresponding Fokker-Planck equation from the present system with multiplicative noise can be traced back to an path integral formulation. In Sec. IV, we obtain the transition ambiguity in choosing the integration method for the over- probabilities for the Ornstein-Uhlenbeck process and the ge- damped Langevin dynamics, which leads to different stochas- ometric Brownian motion under the general stochastic inter- tic interpretations.16 In this paper, we provide an alternative pretation. In Sec. V, we summarize our work. In Appendix A, way to construct the path integral formulation for the over- we list the previous path integral frameworks for the gen- damped Langevin equation under the α-interpretation. Our eral stochastic interpretation and apply them to the geomet- main result, Eq. (9), leads to transition probabilities directly ric Brownian motion to show the difference with our result. obeying the conservation law for general stochastic interpre- In Appendix B, we develop an equivalent form of the path tations, which is exemplified by the Ornstein-Uhlenbeck pro- integral formulation in the main text. In Appendix C, we gen- cess and the geometric Brownian motion. The present path in- eralize our path integral formula to multidimensional case. tegral formulation depends on α, and it can also generate the corresponding α-interpretation Fokker-Planck equation.17 As our formulation of path integral starts from the corresponding Langevin equation of the equivalent Stratonovich’s form, our II. PATH INTEGRAL FORMULATION result is consistent with ordinary calculus, and thus is conve- For convenience, we start from the one-dimensional over- nient for applications. damped Langevin equation with multiplicative noise

a)Electronic mail: [email protected] b)Electronic mail: [email protected] x˙ = f (x) + g(x)ξ(t), (1)

where x denotes the position, x˙ denotes its time deriva- obtained tive, f(x) is the drift term, and g(x)ξ(t) models the stochastic | force. Here, ξ(t) is a Gaussian white noise with ξ(t)=0, P (qN tN q0t0) ξ(t)ξ(t )=δ(t − t ) and the average is taken with respect q t N N 1 1 dh to the noise distribution. The positive constant describes = Dq exp − (q˙ − h)2 + dt , (8) q t 2 2 dq the strength of the noise, corresponding to kBT in physical 0 0 systems. For this Langevin equation, an ambiguity in choos- q . N− dq N D = √ 1 1 √ n ing the integration method leads to different stochastic in- where q limN→∞ n=1 . The integral q0 2πτ 2πτ terpretations and a general notation is the α-interpretation.17 of the action function on the exponent obeys ordinary calculus The values α = 0, α = 1/2, and α = 1 correspond to Ito’s, due to the mid-point discretization and the last term comes Stratonovich’s, and anti-Ito’s, respectively. If starting from from the Jacobian. We mention that another full derivation of the second order Langevin equation with a variable friction Eq. (8) from Eq. (3) can be found in Ref. 42. coefficient, different approximation methods can also give By changing the variable reversely: x = H−1(q) with the overdamped Langevin equation with various stochastic dx/dq = g(x), we get the path integral for Eq. (1) under the interpretations.39, 40 α-interpretation, For the overdamped Langevin equation under the α- | interpretation, by modifying the drift term, we have its P (xN tN x0t0) 16 Stratonovich’s form x t 2 N N 1 1 = Dx exp − x˙ − f − α − gg 1 2 x t 2g 2 x˙ = f (x) + α − g (x)g(x) + g(x)ξ(t), (2) 0 0 2 g f 1 where the superscript prime denotes the derivative to x.The + + α − g dt , (9) 2 g 2 advantage of using this Stratonovich’s form is that ordinary 41 calculus rule can be simply applied. Then, this equation x . N− dx N D = √ 1 1 √ n where x x limN→∞ n=1 .The can be transformed to be a Langevin equation with a ad- 0 2πτg(xN ) 2πτg(xn) ditive noise by a change of variable q = H(x) with H (x) action function is dimensionless because has the same di- = 1/g(x),13 mension as energy. Though the Jacobian term comes from the measure transformation and does not belong to the conven- q˙ − h(q) = ξ(t), (3) tional action part, it is usually included in the action function for applications. where we have introduced an auxiliary function, The path integral formulation for the case with multi- −1 f (H (q)) 1 −1 plicative noise may not be absolute continuous in mathemati- h(q) = + α − g (H (q)). (4) 43 g(H −1(q)) 2 cal definition. However, we will show that the semi-classical method can still be applied with the present path integral To get the transition probability for Eq. (3),wefirstdis- to get a correct transition probability in examples, including ··· cretize the time into N segments: t0 < t1 < < tN − 1 < tN the case with multiplicative noise. For the case with additive = − = with τ tn tn − 1 small and let qn q(tn). For the sake of noise, Eq. (9) degenerates to be consistency, as we have chosen the equivalent Stratonovich’s | form, the corresponding discretized Langevin equation needs P (xN tN x0t0) the mid-point discretization x t N N 1 1 = Dx exp − (x˙ − f )2 + f dt . (10) h(q ) + h(q ) 2 n n−1 x t 2g 2 q − q − − τ = W − W − , (5) 0 0 n n 1 2 n n 1 We remark that though we use the equivalent = where W(t) is the Wiener process given by dW(t) ξ(t)dt. Stratonovich’s form above, the path integral Eq. (9) is Thus, the Jacobian for the variable transformation between for the Langevin equation (1) under the α-interpretation. Our q(t) and W(t)is result Eq. (9) can also be obtained by generalizing the result − in Ref. 42 to be applicable to the α-interpretation. We will τ N1 dh(q ) J ≈ exp − n . (6) also use other types of equivalent Langevin equations with the 2 dq n=1 n corresponding stochastic interpretation to construct the path integral formulation in Appendix B. The chosen stochastic With the conditional probability functional of the Wiener interpretation in turn assigns a corresponding discretized process11, 16 scheme for the path integral. For example, the mid-point

W t N N 1 discretization should be used for the action function under | = D − ˙ 2 P (WN tN W0t0) W exp W dt , (7) the Stratonovich’s form, and the α-type discretization (at the W t 2 0 0 + − point αxn (1 α)xn − 1 in each interval) is needed for the W α-form. These forms with the consistent stochastic interpre- N D where the Wiener measure is deﬁned as W W 0 tation are equivalent. Thus, one can choose any speciﬁc form . − dW = 1 N 1 n limN→∞ 2πτ n=1 2πτ . By changing the measure in applications for the convenience of numerical calculations. from DW to Dq, the path integral formulation for Eq. (3) is This indicates that a particular numerical method can be

≈ adopted with adding the corresponding drift term to achieve we calculate out the first two of Ak(qn − 1)asAk(qn − 1) O(τ) the physically preferable stochastic interpretation. for k > 2, When integrating the action function, we should keep =− + A1(qn−1) h(qn−1) O(τ), using the stochastic calculus rule consistent with its dis- (13) cretized scheme. If we apply the path integral here, we need A (q − ) = + O(τ). to use Stratonovich’s calculus. When the path integral in 2 n 1 2 41 Refs. 14 and 15 is applied, the α-type integration rule is By taking the limit τ → 0, we get the Fokker-Planck equation required. Taking the Ornstein-Uhlenbeck process as an ex- for Eq. (3), ample, we will show in Appendix B that both their path in- =− + 2 tegral with the α-type integration and Eq. (10) with ordi- ∂t ρ(q,t) ∂q [h(q)ρ(q,t)] ∂q [ρ(q,t)]. (14) nary calculus can automatically give a normalized transition 2 probability. Even so, when applying their path integral to the In order to derive the Fokker-Planck equation for Eq. (1), −1 geometric Brownian motion, the transition probabilities ob- we change the variable inversely: x = H (q) with dx/dq tained by both ordinary calculus and the α-type integration are = g(x). With the aid of the corresponding transformation for 50 incomplete. the moments of the Fokker-Planck equation, we have For systems with additive noise, our result demon- =− + ∂t ρ(x,t) ∂x [(f (x) αg (x)g(x))ρ(x,t)] strates that the classical Onsager-Machlup function8 with Ito’s 16 9–12 integration and the effective action with Stratonovich’s + ∂ g2(x)ρ(x,t) , (15) calculus are equivalent.44 For the system with multiplicative 2 x noise, the consistency of the path integral and the stochas- which is the same as the conventional α-interpretation tic calculus has been noted in Ref. 13. However, only the Fokker-Planck equation.17 Langevin equation under Ito’s interpretation is considered at We emphasize that when taking the partial derivative to the beginning. Thus, their result is only for Ito’s interpreta- x in the Fokker-Planck equation, we can always use ordi- tion and not for the α-interpretation, corresponding to α = 0 nary calculus regardless of the interpretation adopted for the in Eq. (9). This explains why their result shows the unique- Langevin equation. In the Langevin dynamics, the expansion ness of the path integral. Besides, the path integral formula- for a smooth function of x is in orders of different time scales: tions on manifolds have been discussed previously,30, 42, 45–47 dW, dt, dW2 dWdt, etc. Then, dW2 should be counted up to nevertheless, their results are mainly for Stratonovich’s inter- the order of dt16, which leads to different stochastic interpre- pretation. These results in one dimension with zero curvature tations. However, on the level of the Fokker-Planck equation, are consistent with Eq. (9) when α = 1/2. The explicit differ- the expansion is in orders of the space coordinates: dx, dx2, ence between our result with α = 1/2 and the result in Ref. 42 etc. Thus, there is no ambiguity for the stochastic interpreta- comes from different ways of representing the measure, and tion for a given Fokker-Planck equation. can be fixed by normalization. Furthermore, considering the general stochastic interpretation, the path integral formulated by Wissel48 is also different from our result. IV. EXAMPLES A. The Ornstein-Uhlenbeck process III. THE FOKKER-PLANCK EQUATION This process can be described by the Langevin equation16 √ In this section, we first derive the Fokker-Planck equa- x˙ =−kx + Dξ(t), (16) tion from the path integral for the system with additive noise. Then, by the variable transformation, we obtain the Fokker- where k, D are positive constants. The temperature has been Planck equation of the α-interpretation for Eq. (1). According set to be a unit in examples for convenience. to Eq. (8), the transition probability in each interval is We apply Eq. (9), P (x t |x t ) | N N 0 0 P (qntn qn−1tn−1) xN tN 1 2 kt 2 = Dx exp − (x˙ + kx) dt + , (17) 1 τ qn 1 2D 2 = √ − + x0 t0 exp (h(qn) h(qn−1)) 2πτ 2 τ 2 = − where t tN t0. We then get the transition probability τ dh by the semi-classical method51 using ordinary calculus rule − (q − ) , (11) 2 dq n 1 in calculation of the action function P (x t |x t ) where q = q − q . The normalization condition is sat- N N 0 0 n n n − 1 | = −kt 2 isfied: P(qntn qn − 1tn − 1)dqn 1. Then, by the moment gen- k k(x − e x ) 49 = exp − N 0 . (18) erating function Dπ(1 − e−2kt) D(1 − e−2kt)

− k ( 1) k Note that we have directly obtained a normalized transition A (q − ) = dq (q − q − ) P (q t |q − t − ), k n 1 τk! n n n 1 n n n 1 n 1 probability. If we apply the path integral in Refs. 14 and 15 (12) with the consistent α-type integration, we can have the same

= = − FIG. 1. (a)–(i) The evolution of the probability distributions for the geometric Brownian motion Eq. (19) with k 1, σ 1 and the delta function δ(x0 1) as the initial distribution under different stochastic interpretations: α = 0, α = 1/2, and α = 1. Different colors denote results from different path integral formulations. The three graphs in the ﬁrst line correspond to Ito’s interpretation (α = 0). In this case, our result Eqs. (22) and (24) coincide. The graphs in the second line correspond to Stratonovich’s interpretation (α = 1/2). In this case, Eqs. (23) and (24) coincide. In the third line, the graphs show the result under anti-Ito’s interpretation (α = 1).

transition probability through a similar procedure. We also where the pre-factor 1/xN comes from the measure transfor- x note that their path integral with ordinary calculus cannot lead mation for N Dx. This transition probability agrees with x0 to the correct result Eq. (18). Eq. (A9) for the general stochastic interpretation and the result in Ref. 18, where two special cases Ito’s and Stratonovich’s were discussed. B. The geometric Brownian motion When applying the path integral in Refs. 14 and 15,we This process is popular in mathematical ﬁnance and re- derive two kinds of transition probabilities obtained by two cently attracts more interest in physical society.52 It can be integration rules: ordinary calculus and the α-type integration. given by the following Langevin equation:18 The detailed calculation can be found in Appendix A. First, with ordinary calculus we have x˙ = kx + σxξ(t), (19) ˆ | P (xN tN x0t0) where k, σ are positive constants. We apply Eq. (9), 1 − 1 [ln(x /x )−(k−ασ2)t]2−αkt = √ 2σ 2t N 0 | e . (23) P (xN tN x0t0) 2πtσx N x t 2 2 N N [x˙ − kx − (α − 1/2)σ x] After normalization, i.e., when αkt is eliminated, this for- = Dx exp − dt . 2σ 2x2 mula still differs from Eq. (22). x0 t0 Second, if we use the α-type integration,41 we have (20) Pˆ(x t |x t ) In the action function, to calculate the path-dependent N N 0 0 1 − 1 [ln(x /x )−(k−σ 2/2)t]2−αkt term = √ 2σ 2t N 0 e . (24) t 2 2πtσx 1 N x˙ N S = dt, (21) 0 2 2 2σ t x After normalization, Eq. (24) is still different from Eq. (22) 0 except under Ito’s interpretation. The comparison of the tran- we make a variable transformation y = ln x and then y˙ sition probabilities, Eqs. (22)–(24) is shown in Fig. 1. = x/x˙ by ordinary calculus. Thus, through the semi-classical method,51 we ﬁnally reach the result V. CONCLUSION P (x t |x t ) N N 0 0 From the overdamped Langevin equation with multi- 1 − 1 [ln(x /x )−(k+(α−1/2)σ 2)t]2 plicative noise, we have constructed the path integral formu- = √ e 2σ 2t N 0 , (22) 2πtσxN lation for the general α-interpretation. It is convenient for This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.103.149.52 On: Wed, 17 Sep 2014 00:23:27 044125-5 Tang, Yuan, and Ao J. Chem. Phys. 141, 044125 (2014)

applications as ordinary calculus can be applied. The corre- further find that they just develop the path integral formula sponding α-interpretation Fokker-Planck equation has been for the Langevin equation under Ito’s interpretation. On the generated, and thus the three widely used descriptions in contrary, we start from the Langevin equation under the α- stochastic process are connected. For the system with addi- interpretation and then the action function is not unique but tive noise, our result demonstrates the equivalence of the ef- α-dependent. fective action in Ref. 4 and the classical Onsager-Machlup The second previous path integral for the general stochas- function with their corresponding stochastic integration. For tic interpretation is14, 15 the system with multiplicative noise, the present path integral ˆ | provides a complete formulation for general stochastic inter- P (xN tN x0t0) pretations directly obeying the conservation law. x t N N 1 For the high dimensional case with a non-singular diffu- = Dx exp − (x˙ − f + αgg)2dt α 2 x t 2g sion matrix, our method can be used similarly to develop the 0 0 path integral formulation. The fluctuation theorem based on t N the path integral here can be obtained. The forward and the − α f dt , (A2) t reverse dynamical processes should be defined in accordance 0 with the α-interpretation Fokker-Planck equation. Whether or x − dx N . √ 1 N 1 √ n where Dx = lim →∞ .The not the fluctuation theorem is related to the stochastic inter- x N 2πτg(x ) n=1 2πτg(x ) 0 N n pretation is an interesting topic to be explored. The influence tN symbol t α means the integrand obeys the α-type of our result on the numerical side need to be illustrated. The 0 integration:41 for a smooth function F(x(t)), connection between our path integral formulation and the op- erator orderings53 in quantum mechanics also remains to be − 1 2α 2 discovered. dF(x) ≈ F (x)dx + F (x)g (x)dt. (A3) 2

The symbol of integral without subscript is the ordinary in- t t ACKNOWLEDGMENTS N = N tegral, i.e., t 1 t . Thus, we omit the subscript 1/2 of 0 2 0 tN Stimulating discussions with Hong Qian, Alberto Im- 1 in this paper. The integral for the Jacobian term does not t0 2 parato, David Cai, Andy Lau, Xiangjun Xing, and Bo Yuan specify the stochastic interpretation and always obeys ordi- are gratefully acknowledged. We thank Jianhong Chen for nary calculus. the critical comments. This work is supported in part by We notice that it is necessary to use the α-type integration the National 973 Project No. 2010CB529200 and by the for the action function, as their corresponding discretization is Natural Science Foundation of China (NSFC) Project Nos. + − at the point αxn (1 α)xn − 1 in each interval. For the case NSFC61073087 and NSFC91029738. with additive noise, Eq. (A2) becomes

ˆ | P (xN tN x0t0) APPENDIX A: THE PREVIOUS PATH INTEGRAL FOR x t THE GENERAL STOCHASTIC INTERPRETATION N N 1 = Dx exp − (x˙ − f )2dt α 2g2 In this appendix, we list two kinds of previous path in- x0 t0 tegral formulations for Eq. (1) under the general stochastic tN interpretation. The ﬁrst one is13 − α f dt . (A4) t | 0 P (xN tN x0t0) x t 2 We ﬁnd that for the Ornstein-Uhlenbeck process the transi- N N 1 1 = Dx exp − x˙ − f + gg tion probability calculated by Eq. (A4) directly with ordinary 2g2 2 x0 t0 calculus is not normalized automatically. On the contrary, the corresponding α-type integration directly leads to a normal- g f 1 + − g dt , (A1) ized transition probability. Therefore, the consistent stochas- 2 g 2 tic integration is necessary to calculate the action function.

x . N− dx However, for the geometric Brownian motion, we will N D = √ 1 1 √ n where x x limN→∞ n=1 and show in the following that the transition probabilities by ap- 0 2πτg(xN ) 2πτg(xn) the superscript prime denotes the derivative to x. According plying Eq. (A2) with both ordinary calculus and the α-type to their generation on the path integral, the action function integration are not consistent with the known result.18 Before obeys ordinary calculus. that, we ﬁrst provide the transition probability for the geomet- Note that their path integral is independent of α and ric Brownian motion under the general stochastic interpreta- thus the authors claim the uniqueness of their action function by generalizing the derivation in Ref. 18. tion. If applying their path integral with ordinary calculus,13 For the one-dimensional geometric Brownian motion un- we notice that the transition probability is always the same der the α-interpretation, the equivalent equation under Ito’s for any stochastic interpretation. However, it is known that interpretation is for the geometric Brownian motion different stochastic interpretations lead to the corresponding different results.18 We x˙ = (k + ασ2)x + σxξ(t). (A5)

The Ito’s formula,16 i.e., α = 0inEq.(A3), tells that for F(x) be transformed to be a Langevin equation with additive noise = ln x, q˙ − h(q) = ξ(t), (B2) 1 dF(x) ≈ k + ασ2 − σ 2 dt + σdW(t). (A6) 2 with −1 As a result, the solution to Eq. (A5) given initial condition x f (H (q)) 1 − 0 h(q) = + α − g (H 1(q)). (B3) −1 at time t0 is g(H (q)) 2 1 For the sake of consistency, as we have chosen the x(t) = x exp k + α − σ 2 t + σW(t) . (A7) 0 2 α-interpretation, the corresponding discretized Langevin equation should be By the distribution function of x(t) − + − 2 q − q − − [αh(q ) + (1 − α)h(q − )]τ = W − W − . W(t) [ln(xN /x0) (k (α 1/2)σ )t] n n 1 n n 1 n n 1 P √ ≤ √ , (B4) t σ t (A8) Thus, the Jacobian for the variable transformation between we get the transition probability q(t) and W(t) becomes | N−1 P (xN tN x0t0) dh(q ) J ≈ exp − ατ n . (B5) 1 − 1 [ln(x /x )−(k+(α−1/2)σ 2)t]2 = dqn = √ e 2σ 2t N 0 . (A9) n 1 2πtσx N Then, with the property of Wiener process and the It is consistent with Eq. (22) for the general stochastic inter- Chapman-Kolmogorov equation,13 the transition probability pretation and the previous result in Ref. 18, where Ito’s and is obtained Stratonovich’s cases were discussed. q t N N 1 Now, we apply Eq. (A2) and have the action function | = D − − 2 P (qN tN q0t0) q exp α [q˙ h(q)] dt q t 2 tN 2 0 0 1 x˙ 2 x˙ 2 2 Sˆ = −2(k−ασ ) +(k − ασ ) dt+αkt. t 2 α 2 N 2σ t x x dh(q) 0 − α dt . (B6) t dq (A10) 0 | We then use two different integration rules separately. First, To get the transition probability P(xNtN x0t0)forEq.(1),we we use ordinary calculus. For the first term of the action change the variable reversely, function, we make a variable transformation y = ln x and x t N N 1 = P (x t |x t ) = Dx exp − (x˙ − f )2dt thus y˙ x/x˙ . Then, with the semi-classical method on a free N N 0 0 α 2 51 x t 2g particle, we have Eq. (23). 0 0 Second, if we use the α-type integration for the action t N f 1 function, then − α g + α − g dt . (B7) t g 2 t 0 1 N x˙ 1 x 1 dt = ln N + − α σ 2t . 2 α 2 The different forms of the present path integral come from 2σ t x 2σ x 2 0 0 the equivalent Langevin equation under different stochastic (A11) interpretations. With the consistent calculus rule, all the forms Besides, for the variable transformation y = ln x, we should are equivalent. The corresponding integration rule should be have y˙ = x/x˙ − (1/2 − α)σ 2 and finally obtain Eq. (24) after applied with the specific form of the present path integral. similar procedure. Therefore, one can conveniently choose the path integral form with the integration rule for the problem considered. For additive noise cases, Eq. (B7) becomes APPENDIX B: PATH INTEGRAL FORMULATION α x t OF THE EQUIVALENT -FORM N N 1 P (x t |x t ) = Dx exp − (x˙ − f )2dt In this appendix, we provide another way to develop the N N 0 0 α 2g2 x0 t0 path integral formulation starting from the one-dimensional tN Langevin equation under the α-interpretation. Instead of mod- − α f dt , (B8)

ifying the drift term and writing down its equivalent Langevin t0 equation in Stratonovich’s form, we directly use Eq. (1).The which is the same as Eq. (A4). This demonstrates that the α-type chain rule, Eq. (A3), should be applied to do the vari- present path integral formulation and that in Refs. 14 and able transformation, q = H(x), 15 show no difference for the additive noise cases. It should 1 − 2α dH(x) ≈ H (x)dx + H (x)g2(x)dt, (B1) be emphasized that though Eq. (B8) explicitly contains an 2 α-term, it is independent of α after integration. Because the tN where the superscript prime denotes the derivative to x. Then, integral α should be done through the α-type integration, t0 with H (x) = 1/g(x) and H (x) =−g(x)/g2(x)Eq.(1) can which will eliminate the α-term on the exponent.

APPENDIX C: PATH INTEGRAL FORMULATION is obtained IN HIGH DIMENSION | P (qN tN q0t0) In this appendix, we follow the method in the main q t m text to implement the path integral formalization for the m- N N 1 = Dq exp − (q˙k − hk(q))2 dimensional Langevin equation with multiplicative noise q t 2 0 0 k=1 i = i + ij j x˙ f (x) g (x)ξ (t), (C1) dhk(q) + dt , (C7) where xi denotes the ith component of the m-dimensional po- dqk sition vector, x˙i denotes its time derivative, fi(x)istheith q m N− dqk j N D ≡ √ 1 1 √ n component of the drift term. The noise ξ (t)isthejth com- where q limN→∞ m k=1 n=1 .The q0 ( 2πτ) 2πτ ponent of the l-dimensional Gaussian white noise with ξ j(t) integral on the exponent obeys ordinary calculus, and the last = 0, ξ s (t)ξ j (t )=δsj δ(t − t ) with the average taken with term comes from the Jacobian. respect to the noise distribution, and gij(x)ξ j(t) models the | To get the transition probability P(xNtN x0t0)forEq.(C1) stochastic force. Here, gij(x)isanm × l tensor and Dik(x) under the α-interpretation, we change the variable by xi = gij(x)gkj(x)/2 (k = 1, ..., m) denotes the m × m diffu- = (H−1)i(q) with ∂(H−1)i/∂qk = gik, sion matrix. The Einstein’s summation rule has been used. | We mention that l > m means the dimension of the noise may P (xN tN x0t0) be larger than the dimension of the system. However, here we x t N N 1 i i = Dx − 1 − discuss the case where the tensor gij is invertible for conve- exp dt x˙ f x t 2 2 −1 ki ij = ij −1 jk = 0 0 nience of calculation, and thus (g ) g g (g ) δik. Following the derivation for the one-dimensional case, 1 τ − α − f i (g−1)ki(g−1)kj we next implement the path integral formalization. The 2 Eq. (C1) under the α-interpretation has the equivalent Stratonovich’s form j j 1 j × x˙ − f − α − f 1 2 x˙i = f i (x) + α − f i (x) + gij (x)ξ j (t), (C2) 2 1 d 1 + gik [(g−1)kj f j + α− f j , i where the drift f i(x)is17 2 dx 2 i = kj ij (C8) f (x) g ∂xk g . (C3) − j k x m N− [g 1(x )]kidxi N D ≡ √ 1 1 √ n n where x x limN→∞ m k=1 n=1 . 0 ( 2πτ) 2πτ Now, the ordinary calculus can be applied to the variable tN The symbol t 1 means that the integral on the exponent transformation due to Stratonovich’s calculus. Thus, by mul- 0 2 tiplying (g−1)ki, and using the variable transformation qk obeys Stratonovich’s calculus. Note that Eq. (C8) when α = = k k = k = −1 ki 1/2 is consistent with the result in Ref. 42. H (x) with ∂xi q ∂xi H (g ) , we get the Langevin equation with the additive noise 1H. K. Janssen, Lect. Notes Phys. 104, 25 (1979). q˙k = hk(q) + ξ k(t), (C4) 2D. M. Zuckerman and T. B. Woolf, Phys.Rev.E63, 016702 (2000). 3K. R. Glaesemann and L. E. Fried, J. Chem. Phys. 118, 1596 (2003). where hk(q) = (g−1)ki[H−1(q)] · {fi[H−1(q)] + (α − 1/2)fi 4P. Faccioli, M. Sega, F. Pederiva, and H. Orland, Phys. Rev. Lett. 97, −1 −1 108101 (2006). [H (q)]}, and H is the inverse of the variable transforma- 5 −1 i k = ik J. Wang, K. Zhang, H. Lu, and E. Wang, Phys. Rev. Lett. 96, 168101 tion with ∂(H ) /∂q g . (2006). To get the transition probability for Eq. (C4),wedis- 6G. Gobbo, A. Laio, A. Maleki, and S. Baroni, Phys. Rev. Lett. 109, 150601 cretize the time into N segments: t < t < ··· < t − (2012). 0 1 N 1 7 < t with τ = t − t small, and let qk = qk(t ). As we J. Wang, K. Zhang, and E. Wang, J. Chem. Phys. 133, 125103 (2010). N n n − 1 n n 8S. Machlup and L. Onsager, Phys. Rev. 91, 1512 (1953). have chosen the equivalent Stratonovich’s form, the corre- 9R. Graham, Z. Phys. B: Condens. Matter 26, 281 (1977). sponding discretized Langevin equation should be 10P. Arnold, Phys.Rev.E61, 6099 (2000). 11M. Chaichian and A. Demichev, Stochastic Processes and Quantum Me- k + k k k h qn h qn−1 k k chanics, Path Integrals in Physics (IOP, Bristol, 2001), Vol. I. q − q − − τ = W − W − , (C5) 12 n n 1 2 n n 1 D. Chatterjee and B. J. Cherayil, Phys. Rev. E 82, 051104 (2010). 13K. L. Hunt and J. Ross, J. Chem. Phys. 75, 976 (1981). k k 14A. W. C. Lau and T. C. Lubensky, Phys.Rev.E76, 011123 (2007). where W (t) is the Wiener process given by dW (t) 15 = k Z. G. Arenas and D. G. Barci, Phys. Rev. E 81, 051113 (2010). ξ (t)dt. Thus, the Jacobian for the variable transformation 16C. W. Gardiner, Handbook of Stochastic Methods (Springer, Berlin, between q(t) and W(t) is a multiplication of the Jacobian for 2004). each component of the vector 17J. Shi, T. Chen, R. Yuan, B. Yuan, and P. Ao, J. Stat. Phys. 148, 579 (2012). 18B. Øksendal, Stochastic Differential Equations: An Introduction with Ap- − m τ N1 dhk(q ) plications, 3rd ed. (Springer, Berlin, 1992). ≈ − n 19 J exp k . (C6) N. Van Kampen, J. Stat. Phys. 24, 175 (1981). 2 dq 20 k=1 n=1 n L. Schulman, Techniques and Applications of Path Integration (Wiley, New York, 1981), Vol. 140. Then, with the property of Wiener process and the 21D. J. Evans, E. G. D. Cohen, and G. P. Morriss, Phys. Rev. Lett. 71, 2401 Chapman-Kolmogorov equation,16 the transition probability (1993).

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