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Dynamics Reading Group Optimal paths: Revisited

Paul Ritchie Supervisor: Jan Sieber 19th November 2015

Paul Ritchie , Supervisor: Jan Sieber Optimal paths: Revisited 19th November 2015

Overview

Stochastic differential equation: ˙ x = f(x(t), t) + √ 2Dη(t) The optimal path is the most probable path for the transition between a given starting point x0 at time t0 to a given end position xT at time Tend.

Δ δ

Path Gate x t xT x0 Limit: δ ≪ ∆t ≪ 1 Optimisation problem: Optimal path derived from optimising a functional of the probability for passing through gates along a path.

Paul Ritchie , Supervisor: Jan Sieber Optimal paths: Revisited 19th November 2015

Introduction

Probability density function P(x, t) of the random variable x(t) is governed by the Fokker-Planck equation: ∂P(x, t) ∂t = D∂2P(x, t) ∂x2 − ∂ ∂x(f(x, t)P(x, t)) where a potential U(x, t) satisfies: ∂U(x, t) ∂x = −f(x, t)

Paul Ritchie , Supervisor: Jan Sieber Optimal paths: Revisited 19th November 2015

Introduction

−10 −8 −6 −4 −2 2 4 0.2 0.4 0.6 0.8 1 x Density P(x,0) P(x,3)

Fokker-Planck run for ˙ x = −1 + η, x0 = 0, T = 3

Paul Ritchie , Supervisor: Jan Sieber Optimal paths: Revisited 19th November 2015

Identities

Fourier Transform ˆ P(k, t) of P(x, t) ˆ P(k, t) = ∞

−∞

P(x, t)e−ikxdx Dirac delta identity ∞

−∞

f(x)δ(x − x0)dx = f(x0) Inverse Fourier Transform P(x, t) = 1 2π ∞

−∞

ˆ P(k, t)eikxdk Gaussian integral ∞

−∞

e−αx2dx = π α

Paul Ritchie , Supervisor: Jan Sieber Optimal paths: Revisited 19th November 2015

Notation

tk = tk−1 + ∆t, for k = 1, .., N + 1 xk = x(tk): Realisation of random variable x at time tk conditioned

• n having passed through gates 1, ..., k − 1

˜ xk: Location of path and represents centre of gate k at time tk Pk(xk): Probability density function for being at xk assuming passed through gates 1, ..., k − 1 at time tk Pk: Probability of passing through gate k conditioned on having passed through gates 1, ..., k − 1 ˜ Pk(xk): Probability density function for being at xk assuming passed through gates 1, ..., k at time tk P(T)

k : Total probability of passing through first k gates

P = P(T)

N+1: Probability of passing through all N + 1 gates

Paul Ritchie , Supervisor: Jan Sieber Optimal paths: Revisited 19th November 2015

Key book results

Case 1: Pure diffusion P(˜ x, δ, ∆t) =

• δ

√ 4πD∆t N+1 exp

• − 1

4D Tend

t0

d˜ x dτ 2 dτ

• Paul Ritchie , Supervisor: Jan Sieber

Optimal paths: Revisited 19th November 2015

Key book results

Case 1: Pure diffusion P(˜ x, δ, ∆t) =

• δ

√ 4πD∆t N+1 exp

• − 1

4D Tend

t0

d˜ x dτ 2 dτ

• Case 2: Absorbing medium

P(˜ x, δ, ∆t) =

• δ

√ 4πD∆t N+1 exp

• −

Tend

t0

1 4D d˜ x dτ 2 +A(˜ x(τ))dτ

• Paul Ritchie , Supervisor: Jan Sieber

Optimal paths: Revisited 19th November 2015

Key book results

Case 3: Fokker-Planck equation P =

• δ

√ 4πD∆t N+1 exp U(x0) − U(xT ) 2D − Tend

t0

L(˜ x(τ))dτ

• where

L(x) = 1 4D dx dτ 2 + Vs(x) and Vs(x) = 1 4D dU dx 2 − 1 2 d2U dx2

Paul Ritchie , Supervisor: Jan Sieber Optimal paths: Revisited 19th November 2015

Key book results

To minimise L, solve the Euler-Lagrange equation: ∂L ∂x − d dτ ∂L ∂ ˙ x = 0 A 2nd order BVP is derived that the most likely trajectory will satisfy: ¨ x = 2DdVs dx ,

• x(t0)

= x0 x(Tend) = xT where Vs = 1 4D dU dx 2 − 1 2 d2U dx2

Paul Ritchie , Supervisor: Jan Sieber Optimal paths: Revisited 19th November 2015

Ornstein-Uhlenbeck example

Consider the Ornstein-Uhlenbeck process: ˙ x = −ax(t) + √ 2Dη(t) Optimal path satisfies: ¨ x = a2x,

• x(t0)

= x0 x(Tend) = xT Solution can be obtained analytically: x(t) = x0 sinh(a(Tend − t)) + xT sinh(a(t − t0)) sinh(a(Tend − t0))

Paul Ritchie , Supervisor: Jan Sieber Optimal paths: Revisited 19th November 2015

Ornstein-Uhlenbeck example

D = 0.05 D = 0.1 a = 0.5 a = 0.2

Paul Ritchie , Supervisor: Jan Sieber Optimal paths: Revisited 19th November 2015

Time dependent potentials U(x, t)

New PDE is: ∂Ps ∂t = D∂2Ps ∂x2 + U ′′ 2 − U ′2 4D + ˙ U 2D

• Ps

2nd order BVP remains the same: ¨ x = 2DdVs dx where Vs = U ′2 4D − U ′′ 2 − ˙ U 2D

Paul Ritchie , Supervisor: Jan Sieber Optimal paths: Revisited 19th November 2015

Ornstein-Uhlenbeck example

Consider the Ornstein-Uhlenbeck process: ˙ x = −a(t)x(t) + √ 2Dη(t) where a is not constant, instead a(t) = a0 − ǫt The optimal path satisfies: ¨ x = a(t)2x + ǫx To be solved numerically

Paul Ritchie , Supervisor: Jan Sieber Optimal paths: Revisited 19th November 2015

Ornstein-Uhlenbeck example

a0 = −0.2, ǫ = −0.05 D = 0.05 D = 0.1

Paul Ritchie , Supervisor: Jan Sieber Optimal paths: Revisited 19th November 2015

References

• M. Chaichian and A. Demichev. Path Integrals in Physics: Volume I

Stochastic Processes and Quantum Mechanics. Institute of Physics, 2001.

• S. Bayin Mathematical methods in science and engineering. John

Wiley&Sons, New York, 2006. B.W. Zhang. Theory and Simulation of Rare Events in Stochastic Systems. ProQuest, 2008. C.-L. Ho and Y.-M. Dai. A perturbative approach to a class of Fokker-Planck equations Modern Physics Letters B, 22(07): 475-481, 2008. W.-T. Lin and C.-L. Ho. Similarity solutions of a class of perturbative Fokker-Planck equation. Journal of Mathematical Physics, 52(7): 073701, 2011.

• A. J. McKane and M. B. Tarlie. Physical Review E, 69(4): 041106, 2004.
• K. Morita. IOS Press, 1995.
• H. Aratyn and C. Rasinariu. World Scientific, 2006.

Paul Ritchie , Supervisor: Jan Sieber Optimal paths: Revisited 19th November 2015