Asymptotics of work distributions in driven Langevin systems - PowerPoint PPT Presentation

Asymptotics of work distributions in driven Langevin systems Andreas Engel, Sascha von Egan-Krieger Institute for Physics / Statistical Physics Carl von Ossietzky University Oldenburg http://www.statphys.uni-oldenburg.de/

Motivation fluctuating nano-machines large deviation properties ← → stochastic thermodynamics tails of distributions 0.18 hist P(W) P(W) 0.16 e − β W P(W) 0.14 Here: 0.12 asymptotics of P ( W ) 0.1 improving estimates for ∆ F from 0.08 0.06 e − β ∆ F = � e − βW � 0.04 0.02 0 −4 −2 0 2 4 6 8 10 12 W Aim: Combine analytical information on the tail of P ( W ) with the histogram.

Basic equations Langevin dynamics � t 1 � P ( x 0 ) = e − βV 0( x 0) 2 x = − V ′ ( x, t ) + dt ˙ ˙ β ξ ( t ) , , W [ x ( · )] = V ( x ( t ) , t ) Z 0 0 Transformation of probability x ( t 1)= x 1 dx 0 � � � e − βV 0( x 0) P ( W ) = dx 1 D x ( · ) p [ x ( · ) | x 0 , x 1 ] δ ( W − W [ x ( · )]) Z 0 x (0)= x 0 x ( t 1)= x 1 dx 0 dq � � � � D x ( · ) e − βS [ x ( · ) ,q ] = dx 1 Z 0 4 π/β x (0)= x 0 Stochastic action t 1 S [ x ( · ) , q ] = V 0 ( x 0 ) + 1 dt [1 V ] − i � x + V ′ ) 2 + iq ˙ [ ... ] = ˜ 2( ˙ 2 qW L ( ˙ x, x, t, q ) 2 0

Method of optimal fluctuation Contraction principle: The probability of an unlikely event is dominated by the probability of its most probable cause. Here: Tails of P ( W ) are dominated by ¯ x ( · ) maximizing p [ x ( · )] under the constraint W [¯ x ( · )] = W . Formally: β → ∞ and saddle-point calculation of integrals. e − βS [¯ x ( · ) , ¯ q ] P ( W ) = (1 + O (1 /β )) � Z 0 R Q ( t 1 ) /β Includes contributions from the optimal trajectory and its neighbourhood.

Specific features of the present case Euler-Lagrange equation: � � ∂ ˜ x − ∂ ˜ ∂ ˜ ∂ ˜ t =0 − V ′ d L L L L ∂x = 0 , 0 ( x 0 ) = 0 , = 0 � � dt ∂ ˙ ∂ ˙ x ∂ ˙ x � � t = t 1 One optimal trajectory ¯ x ( t ; W ) for each value of W x ( t ; W ) quantifies balance between “unlikeliness” in x 0 and ξ ( t ) ¯ Fluctuation determinant Q ( t 1 ) : V ′′ ˙ V ′′ + ( ˙ q ) ˙ Q + 2 ¯ ¨ ¯ x − ¯ V ′ ) ¯ V ′′′ ] Q = 0 Q + [(2 − i ¯ ¯ ˙ boundary conditions: Q ( t = 0) = 1 , Q ( t = 0) = 0 Constraint: Omit fluctuations violating the constraint � − 1 ˙ � t 1 � t 1 0 dt ′ ˙ δ 2 ¯ � V ′ (¯ V ′ (¯ x ( t ′ ) , t ′ ) S R = 0 dt x ( t ) , t ) δx ( t ) δx ( t ′ )

First example: The sliding parabola 40 V 0 V ( x, t ) = ( x − t ) 2 V 1 35 2 30 25 Exact solution: V 20 15 − β ( W − ( t 1 − 1+ e − t 1))2 � 10 β 4( t 1 − 1+ e − t 1) P ( W ) = 2 π 2( t 1 − 1 + e − t 1 ) e 5 0 −10 −5 0 5 10 15 20 25 30 x Asymptotic estimate: 2(2 t + e − t − e t − t 1 ) − W (2 − e − t − e t − t 1 ) x = 1 S = ( W − ( t 1 + e − t 1 − 1)) 2 ¯ ¯ 2( t 1 + e − t 1 − 1) 4( t 1 + e − t 1 − 1) 0 = ¨ Q + 2 ˙ ˙ Q, Q (0) = 1 , Q (0) = 0 Q ( t 1 ) = 1 S = − 1 2 δ ′′ ( t − t ′ ) + 1 δ 2 ¯ 2 δ ( t − t ′ ) R = 2( t 1 − 1 + e − t 1 ) S − 1 = − 2 θ ( t − t ′ ) sinh( t − t ′ ) + e t − t ′ δ 2 ¯ V ′ ≡ − 1 ˙ ,

Second example: The breathing parabola 40 25 V 0 60 20 V ( x, t ) = k ( t ) V 1 15 35 2 x 2 10 40 5 30 0 −5 20 P ( W ) = ??? 25 −10 0 20 40 60 80 100 ψ 0 V 20 10 � µ = iq − 9 / 4 → ELE is a 15 −20 8 6 Sturm-Liouville problem 10 −40 4 2 5 0 infinitly many minima of S [ x ( · )] −60 0 20 40 60 80 100 0 0 0.5 1 1.5 2 2.5 3 −20 −15 −10 −5 0 5 10 15 20 x µ 5 1 k ( t ) = 1 + t 0 � x = ± ¯ − W g ( t ) −5 ln P P ( W ) ∼ e β h ( µ 0) W −10 √− W −15 −7 −6 −5 −4 −3 −2 −1 0 W

Improving the ∆ F estimate 1.4 −0.9 1.2 −1.0 ∆ F −1.1 1.0 −1.2 0.8 −1.3 P 100 400 700 1000 n 0.6 0.4 0.2 0.0 −3.0 −2.5 −2.0 −1.5 −1.0 −0.5 0.0 W

Third example: The Blickle experiment (Phys. Rev. Lett. 96 , 070603 (2006)) V ( x, t ) = Ae − κ ( x − a ) + B ( t )( x − a ) D 0 D ( x ) = 1 + R/x � x = − D ( x ) V ′ ( x, t ) + D ′ ( x ) + ˙ 2 D ( x ) ξ ( t ) 16500 experimental values of W 0.20 1.0 0.5 1 � 2 ∆ F � x + D ( x ) V ′ ( x, t ) − D ′ ( x ) 0.15 L ( x, ˙ x, t ) = ˙ 0.0 4 D ( x ) −0.5 P 0.10 50 100 150 200 x − D ′ n x 2 + (1 − iq ) D ˙ V ′ + .... 0 = ¨ 2 D ˙ 0.05 Determine ¯ S numerically, no prefactor yet. 0.00 −5 0 5 10 15 W

Summary • Analytical expressions for the asymptotics of work distributions in driven Langevin systems. • Method of optimal fluctuation corresponding to contraction principle of large deviation theory. • May improve ∆ F estimates from the Jarzynski equation if histogram and asymptotics overlap.