Introduction to Path-integral Methods in Physics and Polymer Science

In this paper, we consider the Langevin equation from an unusual point of view, that is as an archetype for a dissipative system driven out of equilibrium by an external excitation. Using path integral method, we compute exactly the probability density function of the power (averaged over a time interval of length τ ) injected (and dissipated) by the random force into a Brownian particle driven by a Langevin equation. The resulting distribution, as well as the associated large deviation function, display strong asymmetry, whose origin is explained. Connections with the so-called "Fluctuation Theorem" are thereafter discussed. Finally, considering Langevin equations with a pinning potential, we show that the large deviation function associated with the injected power is completely insensitive to the presence of a potential.

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“…Let us consider the stochastic equation (3). The propagator P (v 1 , τ |v 0 , 0) can be easily expressed in terms of a path integral [11,12,13]:…”

Section: Characteristic Functionsmentioning

confidence: 99%

“…v0 designates an average over the realizations of v such that v(0) = v 0 . The path integral in (11) is well-known and its value can be exactly computed [11]:…”

Section: Characteristic Functionsmentioning

confidence: 99%

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Farago¹

2002

Journal of Statistical Physics

138

219

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“…is called the Onsager-Machlup (OM) action "functional" [27,28] (for its derivation and pedagogic discussions, see, for example, [29]) because it contains all the information of the whole history of a path x(t). Here, ∆U = U(x fin ) − U(x ini ) is the potential energy difference between an initial state with x ini = x(0) and a final state with x fin = x(t), D is the diffusion constant, which is related to a friction constant ζ and the absolute temperature T by Einstein's relation D = k B T /ζ (k B is the Boltzmann constant).…”

Section: Introductionmentioning

confidence: 99%

“…The statistical weight is P (x 1 , x 2 , · · · , x N ) ∝ e −S/2D , and the path search problem is therefore mapped onto the statistical mechanics of a polymer under the effective potential function [28]. (This isomorphology is similar to that between a quantum particle and a ring polymer [33].)…”

Section: Introductionmentioning

confidence: 99%

Onsager–Machlup action-based path sampling and its combination with replica exchange for diffusive and multiple pathways

Fujisaki

Shiga

Kidera

2010

The Journal of Chemical Physics

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For sampling multiple pathways in a rugged energy landscape, we propose a novel action-based path sampling method using the Onsager-Machlup action functional. Inspired by the Fourier-path integral simulation of a quantum mechanical system, a path in Cartesian space is transformed into that in Fourier space, and an overdamped Langevin equation is derived for the Fourier components to achieve a canonical ensemble of the path at a finite temperature. To avoid "path trapping" around an initially guessed path, the path sampling method is further combined with a powerful sampling technique, the replica exchange method. The principle and algorithm of our method is numerically demonstrated for a model two-dimensional system with a bifurcated potential landscape. The results are compared with those of conventional transition path sampling and the equilibrium theory, and the error due to path discretization is also discussed.

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“…The series is in powers of the reflection coefficients D n = σ n −σ n+1 σ n +σ n+1 and, unlike the Born series, converges for all contrasts. The series expansion of the kernel R is obtained through a calculation of the Green function of (1.1) via path integration [16,20,28,32]. In section 2 we explain the derivation of the multiscattering series via path integration.…”

Section: Introductionmentioning

confidence: 99%

A multiscattering series for impedance tomography in layered media

Borcea

Ortiz

1999

Inverse Problems

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Abstract. We introduce an inversion algorithm for tomographic images of layered media. The algorithm is based on a multiscattering series expansion of the Green function that, unlike the Born series, converges unconditionally. Our inversion algorithm obtains images of the medium that improves iteratively as we use more and more terms in the multiscattering series. We present the derivation of the multiscattering series, formulate the inversion algorithm and demonstrate its performance through numerical experiments.

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Introduction to Path-integral Methods in Physics and Polymer Science

Cited by 184 publications

References 0 publications

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Onsager–Machlup action-based path sampling and its combination with replica exchange for diffusive and multiple pathways

A multiscattering series for impedance tomography in layered media

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