Diffusion with Stochastic Resetting Martin R. Evans SUPA, School of Physics and Astronomy, University of Edinburgh, U.K. July 10, 2011 Collaborator: Satya N. Majumdar (LPTMS, Paris) M. R. Evans Diffusion with Stochastic Resetting
Introduction: Search Problems Search Problems are ubiquitous in nature and occur in a variety of contexts from foraging of animals to target location on DNA from internet searches to the mundane task of finding one’s misplaced possessions How does one search for lost keys? after a while go back to where they should be and start looking again i.e. reset the search M. R. Evans Diffusion with Stochastic Resetting
Plan: Diffusion with Stochastic Resetting Plan I Recap of diffusion equation, absorbing target, mean first passage time II Stochastic resetting III Many searchers References: M. R. Evans and S. N. Majumdar, Phys. Rev. Lett. 106, 160601 (2011) M. R. Evans Diffusion with Stochastic Resetting
I Diffusion Equation (1d) Forward Equation for p ( x , t | x 0 ) , the probability density of the position x of a diffusing particle at time t , having begun from x 0 at time 0 = D ∂ 2 p ( x , t | x 0 ) ∂ p ( x , t | x 0 ) ∂ x 2 ∂ t with initial condition p ( x , 0 | x 0 ) = δ ( x − x 0 ) . M. R. Evans Diffusion with Stochastic Resetting
I Diffusion Equation (1d) Forward Equation for p ( x , t | x 0 ) , the probability density of the position x of a diffusing particle at time t , having begun from x 0 at time 0 = D ∂ 2 p ( x , t | x 0 ) ∂ p ( x , t | x 0 ) ∂ x 2 ∂ t with initial condition p ( x , 0 | x 0 ) = δ ( x − x 0 ) . Solution (initial value Green function) exp − ( x − x 0 ) 2 1 p ( x , t | x 0 ) = √ 4 Dt 4 Dt M. R. Evans Diffusion with Stochastic Resetting
I Diffusion Equation (1d) Forward Equation for p ( x , t | x 0 ) , the probability density of the position x of a diffusing particle at time t , having begun from x 0 at time 0 = D ∂ 2 p ( x , t | x 0 ) ∂ p ( x , t | x 0 ) ∂ x 2 ∂ t with initial condition p ( x , 0 | x 0 ) = δ ( x − x 0 ) . Solution (initial value Green function) exp − ( x − x 0 ) 2 1 p ( x , t | x 0 ) = √ 4 Dt 4 Dt Also satisfies Backward Equation = D ∂ 2 p ( x , t | x 0 ) ∂ p ( x , t | x 0 ) ∂ x 2 ∂ t 0 M. R. Evans Diffusion with Stochastic Resetting
Absorbing target at the origin Boundary condition p ( x , t | x 0 ) = 0 when x or x 0 = 0 � ∞ Survival probability q ( t | x 0 ) = d x p ( x , t | x 0 ) satisfies 0 Backward equation = D ∂ 2 q ( t | x 0 ) ∂ q ( t | x 0 ) ∂ t ∂ x 2 0 with boundary/initial conditions q ( t | 0 ) = 0 and q ( 0 | x 0 ) = 1 x 0 � = 0 M. R. Evans Diffusion with Stochastic Resetting
Absorbing target at the origin Boundary condition p ( x , t | x 0 ) = 0 when x or x 0 = 0 � ∞ Survival probability q ( t | x 0 ) = d x p ( x , t | x 0 ) satisfies 0 Backward equation = D ∂ 2 q ( t | x 0 ) ∂ q ( t | x 0 ) ∂ t ∂ x 2 0 with boundary/initial conditions q ( t | 0 ) = 0 and q ( 0 | x 0 ) = 1 x 0 � = 0 � ∞ d t e − st q ( t | x 0 ) satisfies Laplace transform � q ( s | x 0 ) = 0 D d 2 � q ( s | x 0 ) − s � q ( s | x 0 ) = − 1 d x 2 0 M. R. Evans Diffusion with Stochastic Resetting
Absorbing target at the origin Boundary condition p ( x , t | x 0 ) = 0 when x or x 0 = 0 � ∞ Survival probability q ( t | x 0 ) = d x p ( x , t | x 0 ) satisfies 0 Backward equation = D ∂ 2 q ( t | x 0 ) ∂ q ( t | x 0 ) ∂ t ∂ x 2 0 with boundary/initial conditions q ( t | 0 ) = 0 and q ( 0 | x 0 ) = 1 x 0 � = 0 � ∞ d t e − st q ( t | x 0 ) satisfies Laplace transform � q ( s | x 0 ) = 0 D d 2 � q ( s | x 0 ) − s � q ( s | x 0 ) = − 1 d x 2 0 Solution which fits boundary/initial conditions q ( s | x 0 ) = 1 − e − x 0 ( s / D ) 1 / 2 � s M. R. Evans Diffusion with Stochastic Resetting
Absorbing target continued. Invert Laplace transform � � x 0 x 0 q ( t | x 0 ) = erf √ ≃ √ for t ≫ 1 D π t 1 / 2 2 Dt [ Or one can get large t behaviour from small s behaviour of � q ( s | x 0 ) (branch point at s = 0) ] The mean first passage time � ∞ d t t ∂ q ( t | x 0 ) T = − → ∞ ∂ t 0 Conclusion a purely diffusive search for a target is not efficient since mean time to absorption diverges M. R. Evans Diffusion with Stochastic Resetting
II Diffusion with resetting Now consider resetting the particle to the initial position x 0 with rate r : Forward equation for p ( x , t | x 0 ) now reads (no absorbing target) = D ∂ 2 p ( x , t | x 0 ) ∂ p ( x , t | x 0 ) − rp ( x , t | x 0 ) + r δ ( x − x 0 ) ∂ t ∂ x 2 i.e. loss rate r from all x � = x 0 provides source at x 0 M. R. Evans Diffusion with Stochastic Resetting
II Diffusion with resetting Now consider resetting the particle to the initial position x 0 with rate r : Forward equation for p ( x , t | x 0 ) now reads (no absorbing target) = D ∂ 2 p ( x , t | x 0 ) ∂ p ( x , t | x 0 ) − rp ( x , t | x 0 ) + r δ ( x − x 0 ) ∂ t ∂ x 2 i.e. loss rate r from all x � = x 0 provides source at x 0 For t → ∞ the stationary state probability density is � p st ( x | x 0 ) = α 0 2 exp ( − α 0 | x − x 0 | ) where α 0 = r / D Nonequilibrium stationary state M. R. Evans Diffusion with Stochastic Resetting
Survival probability with resetting The Backward equation for the survival probability q ( t | z ) when there is an absorbing target at the origin reads = D ∂ 2 q ( t | z ) ∂ q ( t | z ) − rq ( t | z ) + rq ( t | x 0 ) ∂ z 2 ∂ t with boundary/initial conditions q ( t | 0 ) = 0 and q ( 0 | z ) = 1 z � = 0 z is the starting position (variable) and x 0 is the fixed resetting position M. R. Evans Diffusion with Stochastic Resetting
Survival probability with resetting The Backward equation for the survival probability q ( t | z ) when there is an absorbing target at the origin reads = D ∂ 2 q ( t | z ) ∂ q ( t | z ) − rq ( t | z ) + rq ( t | x 0 ) ∂ z 2 ∂ t with boundary/initial conditions q ( t | 0 ) = 0 and q ( 0 | z ) = 1 z � = 0 z is the starting position (variable) and x 0 is the fixed resetting position Laplace transform satisfies D d 2 � q ( s | z ) − ( s + r ) � q ( s | z ) = − 1 − r � q ( s | x 0 ) d z 2 Solution which fits boundary/initial conditions � � 1 − e − α z � 1 + r � q ( s | z ) = q ( s | x 0 ) s + r M. R. Evans Diffusion with Stochastic Resetting
Survival probability with resetting The Backward equation for the survival probability q ( t | z ) when there is an absorbing target at the origin reads = D ∂ 2 q ( t | z ) ∂ q ( t | z ) − rq ( t | z ) + rq ( t | x 0 ) ∂ z 2 ∂ t with boundary/initial conditions q ( t | 0 ) = 0 and q ( 0 | z ) = 1 z � = 0 z is the starting position (variable) and x 0 is the fixed resetting position Laplace transform satisfies D d 2 � q ( s | z ) − ( s + r ) � q ( s | z ) = − 1 − r � q ( s | x 0 ) d z 2 Solution which fits boundary/initial conditions � � 1 − e − α z � 1 + r � q ( s | z ) = q ( s | x 0 ) s + r Then solve self-consistently for � s + r � 1 / 2 q ( s | x 0 ) = 1 − e − α x 0 � where α = s + r e − α x 0 D M. R. Evans Diffusion with Stochastic Resetting
Mean first passage time (MFPT) � ∞ d t t ∂ q ( t | x 0 ) = � MFPT T = − q ( 0 | x 0 ) is now finite for 0 < r < ∞ ∂ t 0 T = e x 0 ( r / D ) 1 / 2 − 1 r M. R. Evans Diffusion with Stochastic Resetting
Mean first passage time (MFPT) � ∞ d t t ∂ q ( t | x 0 ) = � MFPT T = − q ( 0 | x 0 ) is now finite for 0 < r < ∞ ∂ t 0 T = e x 0 ( r / D ) 1 / 2 − 1 r T ( r ) has a minimum at r ∗ d T d r = 0 ⇒ y 2 = 1 − e − y y = x 0 ( r / D ) 1 / 2 where y = distance from target : typical distance diffused between resets Optimal y ∗ = 1 . 5936 ... M. R. Evans Diffusion with Stochastic Resetting
Survival probability The long-time behaviour of the q ( t | x 0 ) is now controlled by q ( s | x 0 ) = 1 − e − α x 0 simple pole of � s + r e − α x 0 at s 0 = − r exp − x 0 [( r + s 0 ) / D ] 1 / 2 M. R. Evans Diffusion with Stochastic Resetting
Survival probability The long-time behaviour of the q ( t | x 0 ) is now controlled by q ( s | x 0 ) = 1 − e − α x 0 simple pole of � s + r e − α x 0 at s 0 = − r exp − x 0 [( r + s 0 ) / D ] 1 / 2 For y = x 0 ( r / D ) 1 / 2 ≫ 1 s 0 ≃ − r exp − y and q ( t | x 0 ) ≃ exp ( − rt e − y ) which has the form of a Gumbel distribution which gives the cumulative distribution for the maximum of independent r.v.s M. R. Evans Diffusion with Stochastic Resetting
Survival probability The long-time behaviour of the q ( t | x 0 ) is now controlled by q ( s | x 0 ) = 1 − e − α x 0 simple pole of � s + r e − α x 0 at s 0 = − r exp − x 0 [( r + s 0 ) / D ] 1 / 2 For y = x 0 ( r / D ) 1 / 2 ≫ 1 s 0 ≃ − r exp − y and q ( t | x 0 ) ≃ exp ( − rt e − y ) which has the form of a Gumbel distribution which gives the cumulative distribution for the maximum of independent r.v.s Explanation On average there are rt resets. For each reset the process is “renewed” and the particle trajectory is independent. The particle must not reach the origin in any reset to survive. So survival is probability that max excursion to left, out of ≃ rt resets, is less than x 0 M. R. Evans Diffusion with Stochastic Resetting
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