Stochastic action for tubes: Connecting path probabilities to measurement

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Stochastic action for tubes: Connecting path probabilities to measurement

Julian Kappler and Ronojoy Adhikari

Phys. Rev. Research 2, 023407 – Published 29 June 2020

Abstract

The trajectories of diffusion processes are continuous but nondifferentiable, and each occurs with vanishing probability. This introduces a gap between theory, where path probabilities are used in many contexts, and experiment, where only events with nonzero probability are measurable. Here we bridge this gap by considering the probability of diffusive trajectories to remain within a tube of small but finite radius around a smooth path. This probability can be measured in experiment, via the rate at which trajectories exit the tube for the first time, thereby establishing a link between path probabilities and physical observables. Considering $N$ -dimensional overdamped Langevin dynamics, we show that the tube probability can be obtained theoretically from the solution of the Fokker-Planck equation. Expressing the resulting exit rate as a functional of the path and ordering it as a power series in the tube radius, we identify the zeroth-order term as the Onsager-Machlup stochastic action, thereby elevating it from a mathematical construct to a physical observable. The higher-order terms reveal the form of the finite-radius contributions which account for fluctuations around the path. To demonstrate the experimental relevance of this action functional for tubes, we numerically sample trajectories of Brownian motion in a double-well potential, compute their exit rate, and show an excellent agreement with our analytical results. Our work shows that smooth tubes are surrogates for nondifferentiable diffusive trajectories and provide a direct way of comparing theoretical results on single trajectories, such as pathwise definitions of irreversibility, to measurement.

Received 27 January 2020
Revised 12 March 2020
Accepted 25 May 2020

DOI:https://doi.org/10.1103/PhysRevResearch.2.023407

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Chemical Physics & Physical Chemistry Diffusion Nonequilibrium & irreversible thermodynamics Nonequilibrium statistical mechanics

Fokker–Planck equation Langevin equation Large deviation & rare event statistics Stochastic processes & statistics

Polymers & Soft MatterStatistical Physics & ThermodynamicsPhysics of Living Systems

Authors & Affiliations

Julian Kappler ^* and Ronojoy Adhikari

DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom

^*jk762@cam.ac.uk

Article Text

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Issue

Vol. 2, Iss. 2 — June - August 2020

Subject Areas

Statistical Physics

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Images

Figure 1
For a one-dimensional system, a smooth path $φ$ is shown as a black solid line, around which a tube of radius $R$ is indicated as a gray shaded area. Initial and final positions of $φ$ are shown as horizontal dotted lines. While the blue trajectory is a realization of the Langevin Eq. (1) which stays inside the tube at all times, the orange trajectory leaves the tube before the final time $t_{f}$ , and therefore contributes to the exit rate from the tube.
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Figure 2
Potential and path considered in the numerical examples in Sec. 3. The plot on the right-hand side shows the quartic double-well potential Eq. (49) for barrier height $β U_{0} = 2$ . In the plot on the left-hand side, the potential is shown as a color map in the background, with the two minima of the potential represented by horizontal dashed lines. The reference path $φ$ , defined in Eq. (50), is shown as a solid black line. Around the reference path, a tube of radius $ε = R / L = 0.5$ is depicted by a shaded gray region. The vertical dashed lines denote the times $\tilde{t} = 0.1$ , 0.5, and 0.6, which are considered in Fig. 3.
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Figure 3
Subplots (a), (b), and (c) show the normalized probability density ${\tilde{P}}_{ε}^{n, φ} (\tilde{x}, \tilde{t})$ , defined in Eq. (53), as a function of position $\tilde{x}$ for time (a) $\tilde{t} = 0.1$ , (b) $\tilde{t} = 0.5$ , and (c) $\tilde{t} = 0.6$ , and for tube radius $ε = 0.1$ (green), $ε = 0.3$ (blue), and $ε = 0.5$ (orange). The legend given in subplot (b) is valid for subplots (a), (b), and (c). Solid colored lines denote results from numerical simulation of the FPE, Eq. (19) (see Appendix pp3-s5 for details on the numerical algorithm). Colored broken lines denote the perturbative result Eq. (53), calculated to order $ε^{5}$ . Vertical dashed lines indicate the tube center $\tilde{x} = 0$ . Subplots (d), (e), and (f) show the exit rate ${\tilde{α}}_{ε}^{φ}$ , defined in Eq. (44), as a function of time $\tilde{t}$ , for tube radii (d) $ε = 0.1$ , (e) $ε = 0.5$ , and (f) $ε = 0.7$ . From all rates the free-diffusion exit rate is subtracted and the result is divided by the free-diffusion exit rate, as defined in Eq. (57). Colored solid lines denote exit rates calculated from numerical simulation of the FPE, Eq. (19). Colored broken lines show perturbative exit rates according to Eqs. (44) and (54, 55, 56). Black solid lines denote the OM Lagrangian Eq. (55), from which the free-diffusion exit rate has already been subtracted so that in fact ${\tilde{α}}^{(0)} / {\tilde{α}}_{free}$ is plotted. Vertical dashed lines indicate the times $\tilde{t} = 0.1$ , 0.5, 0.6 used for subplots (a), (b), and (c).
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Figure 4
Exit rate as measured directly from Langevin simulations for a tube of radius $ε = R / L = 0.5$ . The black solid and blue dash-dotted lines are replots of the corresponding lines in Fig. 3. The blue solid line represents the exit rate as estimated from simulated trajectories. For this, $2.4 \times 10^{6}$ independent Langevin simulations in the quartic double-well Eq. (49) with a simulation time step $Δ \tilde{t} = 10^{- 5}$ are performed, with each trajectory starting at $x = - L$ . From these trajectories, the sojourn probability for a tube of radius $ε = R / L = 0.5$ around a reference path $φ (t)$ , as defined in Eq. (50), is evaluated directly, by counting which proportion of trajectories has never left the tube until any given time. Subsequently, the exit rate is numerically calculated via Eq. (41), and the result is smoothed using a moving average with a Hann window of width $Δ \tilde{t} = 0.003$ . From this exit rate, finally, the free-diffusion exit rate Eq. (54) is subtracted, and the result is divided by the free-diffusion exit rate [see Eq. (57)].
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Figure 5
Effect of the initial distribution ${\tilde{P}}_{i}$ inside the tube on the exit rate. The exit rate ${\tilde{α}}_{ε}^{φ}$ , defined in Eq. (41), is shown as a function of time $\tilde{t}$ , for tube radius (a) $ε = 0.1$ , (b) $ε = 0.5$ , and (c) $ε = 0.7$ . From all rates the free-diffusion exit rate is subtracted and the result is divided by the free-diffusion exit rate, as defined in Eq. (57). All data shown are obtained from numerical simulations of the FPE, Eq. (19), from which the exit rate is calculated using Eq. (41). Colored solid lines are replots of the corresponding curves in Figs. 3, and denote exit rates obtained using the instantaneous steady state as initial condition for the simulations, as explained in Appendix pp3-s5. Colored dashed lines show exit rates obtained using a delta peak at the tube center as initial condition for the simulations. Vertical dashed lines denote the initial relaxation time ${\tilde{τ}}_{rel}$ given in Eq. (C38).
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