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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 Articles you may be interested in Study of non-Markovian dynamics of a charged particle in presence of a magnetic field in a simple way J. Appl. Phys. 113, 124905 (2013); 10.1063/1.4798356 From Entropic Dynamics to Quantum Theory AIP Conf. Proc. 1193, 48 (2009); 10.1063/1.3275646 Stochastic dynamics of the scattering amplitude generating K -distributed noise J. Math. Phys. 44, 5212 (2003); 10.1063/1.1611264 Stochastic Resonance, Brownian Ratchets and the FokkerPlanck Equation AIP Conf. 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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 interpre- tation generate incomplete results exemplified by the geometric Brownian motion. We thus develop a novel path integral formulation for the overdamped Langevin equation with multiplica- tive 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) 0021-9606/2014/141(4)/044125/8/$30.00 141, 044125-1 © 2014 AIP Publishing LLC 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 044125-2 Tang, Yuan, and Ao J. Chem. Phys. 141, 044125 (2014) 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.
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