Dynamics Reading Group Optimal paths: Revisited Paul Ritchie - PowerPoint PPT Presentation

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 ) + ˙ 2 Dη ( t ) x T δ Gate x The optimal path is the most probable path for the transition between a given starting point x 0 Path at time t 0 to a given end position x T at time T end . x 0 Δ t 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: = D∂ 2 P ( x, t ) ∂P ( x, t ) − ∂ ∂x ( f ( x, t ) P ( x, t )) ∂x 2 ∂t where a potential U ( x, t ) satisfies: ∂U ( x, t ) = − f ( x, t ) ∂x Paul Ritchie , Supervisor: Jan Sieber Optimal paths: Revisited 19th November 2015

Introduction 1 P(x,0) P(x,3) 0.8 0.6 Density 0.4 0.2 0 −10 −8 −6 −4 −2 0 2 4 x Fokker-Planck run for ˙ x = − 1 + η , x 0 = 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 ( x, t ) e − ikx d x P ( k, t ) = −∞ Dirac delta identity � ∞ f ( x ) δ ( x − x 0 )d x = f ( x 0 ) −∞ Inverse Fourier Transform � ∞ P ( x, t ) = 1 ˆ P ( k, t ) e ikx d k 2 π −∞ Gaussian integral � ∞ � π e − αx 2 d x = α −∞ Paul Ritchie , Supervisor: Jan Sieber Optimal paths: Revisited 19th November 2015

Notation t k = t k − 1 + ∆ t , for k = 1 , .., N + 1 x k = x ( t k ) : Realisation of random variable x at time t k conditioned on having passed through gates 1 , ..., k − 1 x k : Location of path and represents centre of gate k at time t k ˜ P k ( x k ) : Probability density function for being at x k assuming passed through gates 1 , ..., k − 1 at time t k P k : Probability of passing through gate k conditioned on having passed through gates 1 , ..., k − 1 ˜ P k ( x k ) : Probability density function for being at x k assuming passed through gates 1 , ..., k at time t k 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 � T end � N +1 � 2 � � − 1 � d˜ � δ x P (˜ x, δ, ∆ t ) = exp d τ √ 4 D d τ 4 πD ∆ t t 0 Paul Ritchie , Supervisor: Jan Sieber Optimal paths: Revisited 19th November 2015

Key book results Case 1: Pure diffusion � T end � N +1 � 2 � � − 1 � d˜ � δ x P (˜ x, δ, ∆ t ) = exp d τ √ 4 D d τ 4 πD ∆ t t 0 Case 2: Absorbing medium � T end � N +1 � 2 � � 1 � d˜ � δ x P (˜ x, δ, ∆ t ) = exp + A (˜ x ( τ ))d τ √ − 4 D d τ 4 πD ∆ t t 0 Paul Ritchie , Supervisor: Jan Sieber Optimal paths: Revisited 19th November 2015

Key book results Case 3: Fokker-Planck equation � T end � N +1 � � U ( x 0 ) − U ( x T ) � δ P = exp L (˜ x ( τ ))d τ √ − 2 D 4 πD ∆ t t 0 where � 2 1 � d x L ( x ) = + V s ( x ) 4 D d τ and � 2 d 2 U 1 � d U − 1 V s ( x ) = d x 2 4 D d x 2 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 ∂ L x = 0 d τ ∂ ˙ A 2 nd order BVP is derived that the most likely trajectory will satisfy: � x = 2 D d V s x ( t 0 ) = x 0 ¨ d x , x ( T end ) = x T where � 2 d 2 U 1 � d U − 1 V s = d x 2 4 D d x 2 Paul Ritchie , Supervisor: Jan Sieber Optimal paths: Revisited 19th November 2015

Ornstein-Uhlenbeck example Consider the Ornstein-Uhlenbeck process: √ x = − ax ( t ) + ˙ 2 Dη ( t ) Optimal path satisfies: � x ( t 0 ) = x 0 x = a 2 x, ¨ x ( T end ) = x T Solution can be obtained analytically: x ( t ) = x 0 sinh( a ( T end − t )) + x T sinh( a ( t − t 0 )) sinh( a ( T end − t 0 )) Paul Ritchie , Supervisor: Jan Sieber Optimal paths: Revisited 19th November 2015

Ornstein-Uhlenbeck example a = 0 . 5 a = 0 . 2 D = 0 . 05 D = 0 . 1 Paul Ritchie , Supervisor: Jan Sieber Optimal paths: Revisited 19th November 2015

Time dependent potentials U ( x, t ) New PDE is: ˙ ∂t = D∂ 2 P s 2 − U ′ 2 ∂P s � U ′′ U � ∂x 2 + 4 D + P s 2 D 2 nd order BVP remains the same: x = 2 D d V s ¨ d x where ˙ V s = U ′ 2 4 D − U ′′ U 2 − 2 D Paul Ritchie , Supervisor: Jan Sieber Optimal paths: Revisited 19th November 2015

Ornstein-Uhlenbeck example Consider the Ornstein-Uhlenbeck process: √ x = − a ( t ) x ( t ) + ˙ 2 Dη ( t ) where a is not constant, instead a ( t ) = a 0 − ǫt The optimal path satisfies: x = a ( t ) 2 x + ǫx ¨ To be solved numerically Paul Ritchie , Supervisor: Jan Sieber Optimal paths: Revisited 19th November 2015

Ornstein-Uhlenbeck example a 0 = − 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