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 stochastic thermodynamics ← → large deviation properties tails of distributions Here: asymptotics of P(W) improving estimates for ∆F from e−β∆F = e−βW

−4 −2 2 4 6 8 10 12 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 W hist P(W) P(W) e−β W P(W)

Aim: Combine analytical information on the tail of P(W) with the histogram.

Basic equations

Langevin dynamics

˙ x = −V ′(x, t) +

• 2

β ξ(t) , P (x0) = e−βV0(x0) Z0 , W [x(·)] = t1 dt ˙ V (x(t), t)

Transformation of probability

P (W ) =

• dx0

Z0 e−βV0(x0)

• dx1

x(t1)=x1

• x(0)=x0

Dx(·) p[x(·)|x0, x1] δ(W − W [x(·)]) =

• dx0

Z0

• dx1
• dq

4π/β

x(t1)=x1

• x(0)=x0

Dx(·) e−βS[x(·),q]

Stochastic action

S[x(·), q] = V0(x0) + 1 2

t1

• dt [1

2( ˙ x + V ′)2 + iq ˙ V ] − i 2qW [...] = ˜ L( ˙ x, x, t, q)

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. P(W) = e−βS[¯

x(·),¯ q]

Z0

• R Q(t1)/β

(1 + O(1/β)) Includes contributions from the optimal trajectory and its neighbourhood.

Specific features of the present case

Euler-Lagrange equation:

d dt ∂ ˜ L ∂ ˙ x − ∂ ˜ L ∂x = 0, ∂ ˜ L ∂ ˙ x

• t=0 − V ′

0(x0) = 0 , ∂ ˜ L ∂ ˙ x

• t=t1

= 0

One optimal trajectory ¯ x(t; W) for each value of W ¯ x(t; W) quantifies balance between “unlikeliness” in x0 and ξ(t) Fluctuation determinant Q(t1):

¨ Q + 2 ¯ V ′′ ˙ Q + [(2 − i¯ q) ˙ ¯ V ′′ + ( ˙ ¯ x − ¯ V ′) ¯ V ′′′]Q = 0

boundary conditions: Q(t = 0) = 1 ,

˙ Q(t = 0) = 0

Constraint: Omit fluctuations violating the constraint

R = t1

0 dt

t1

0 dt′ ˙

V ′(¯ x(t), t)

• δ2 ¯

S δx(t)δx(t′)

−1 ˙ V ′(¯ x(t′), t′)

First example: The sliding parabola

V (x, t) = (x − t)2 2

Exact solution:

P (W ) =

• β

2π 2(t1 − 1 + e−t1) e

−β (W −(t1−1+e−t1))2 4(t1−1+e−t1)

−10 −5 5 10 15 20 25 30 5 10 15 20 25 30 35 40 x V V0 V1

Asymptotic estimate:

¯ x = 1 2(2t + e−t − et−t1) − W (2 − e−t − et−t1) 2(t1 + e−t1 − 1) 0 = ¨ Q + 2 ˙ Q, Q(0) = 1, ˙ Q(0) = 0 δ2 ¯ S = −1 2 δ′′(t − t′) + 1 2 δ(t − t′) δ2 ¯ S−1 = −2θ(t − t′) sinh(t − t′) + et−t′ , ˙ V ′ ≡ −1 ¯ S = (W − (t1 + e−t1 − 1))2 4(t1 + e−t1 − 1) Q(t1) = 1 R = 2(t1 − 1 + e−t1)

Second example: The breathing parabola

−20 −15 −10 −5 5 10 15 20 5 10 15 20 25 30 35 40 x V V0 V1

V (x, t) = k(t) 2 x2 P (W ) = ??? µ =

• iq − 9/4 → ELE is a

Sturm-Liouville problem infinitly many minima of S[x(·)]

0.5 1 1.5 2 2.5 3 −60 −40 −20 20 40 60

µ ψ

20 40 60 80 100 2 4 6 8 10 20 40 60 80 100 −10 −5 5 10 15 20 25

k(t) = 1 1 + t ¯ x = ±

• −W g(t)

P (W ) ∼ e β h(µ0)W √−W

−7 −6 −5 −4 −3 −2 −1 −15 −10 −5 5 W ln P

Improving the ∆F estimate

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

Third example: The Blickle experiment

(Phys. Rev. Lett. 96, 070603 (2006))

V (x, t) = Ae−κ(x−a) + B(t)(x − a) D(x) = D0 1 + R/x ˙ x = −D(x)V ′(x, t) + D′(x) +

• 2D(x) ξ(t)

16500 experimental values of W

L(x, ˙ x, t) = 1 4D(x)

• ˙

x + D(x)V ′(x, t) − D′(x) 2 0 = ¨ x − D′ 2D ˙ x2 + (1 − iq)D ˙ V ′ + ....

Determine ¯ S numerically, no prefactor yet.

−5 5 10 15 0.00 0.05 0.10 0.15 0.20 W P 50 100 150 200 −0.5 0.0 0.5 1.0 n ∆ F

Summary

• Analytical expressions for the asymptotics of work distributions in driven

Langevin systems.

• Method of optimal fluctuation corresponding to contraction principle
• f large deviation theory.
• May improve ∆F estimates from the Jarzynski equation if histogram

and asymptotics overlap.