Large deviations in stochastic thermodynamicsAndreas Engel

University of Oldenburghttp://www.statphys.uni-oldenburg.de

„Thermodynamics is a funny subject. The first time you go through it, you don't understand it at all. The second time you go through it, you think you understand it, except for one or two small points. The third time you go through it, you know you don't understand it, but by that time you are so used to it, so it doesn't bother you any more.“ (Arnold Sommerfeld)

Traditional thermodynamics

The balances of energy and entropy

S ! 0, @S/@X ! 0 for T ! 0

dU = dW + dQ

dS = diS + deS, diS � 0

Small Systems

W ' �F ' kBT, �S ' kB

Features of small system thermodynamics

N ⇠ 1023 N = 7

−1 0 1 2 3 4 5 60

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

W

P

−1 0 1 2 3 4 5 60

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

W

P

W � �F hW i � �F

1. Fluctuating thermodynamic quantities,in particular in non-equilibrium processes.

2. Strong coupling to the reservoir(s).

3. Information acquired in measurements becomes thermodynamically relevant.

W ' �F ' kBT, �S ' kB

System with Reservoirs

SHeat

Inf

Work

Heat ! ! !

!

T1 T2

(T = 0)

(T = 1)

E

SS

S

E E

Engine

SHeat

Inf

Work

Heat

!

T1 T2

(T = 0)

(T = 1)

E

SS

S

E E�!�!

�!

Refrigerator

SHeat

Inf

Work

Heat

!

T1 T2

(T = 0)

(T = 1)

E

SS

S

E E�!

�!

�!

Maxwell’s Demon

SHeat

Inf

Work

Heat !T1 T2

(T = 0)

(T = 1)

E

SS

S

E E

�!�!

�!

Eraser

SHeat

Inf

Work

Heat !T1 T2

(T = 0)

(T = 1)

E

SS

S

E E

�!

�!

�!

Landauer limit: �S � kB ln 2

Information driven heat pump

SHeat

Inf

Work

HeatT1 T2

(T = 0)

(T = 1)

E

SS

S

E E�!

�!

�!

Quantitative analysis: Langevin dynamics

• select relevant degrees of freedom• subsume the rest into a heat bath• model the interaction with the bath by friction and noise (FDT)

works nicely if timescales separate, Here: overdamped version

x = �V

0(x,�) +p

2/� ⇠(t) h⇠(t)⇠(t0)i = �(t � t

0)

Continuous stochastic processes

PT [x(·)] = NT [x(·)] exp ��

4

Z T

0dt⇣˙

x� f(x, t)⌘2!

Stochastic differential equation:

Fokker-Planck equation:

Path measure in function space:

x = f(x, t) +

r2

�⇠(t) h⇠i(t)⇠j(t0)i = �ij�(t� t0)

@tP (x, t) = �r✓f(x, t)P (x, t)� 1

�rP (x, t)

◆

0 0.2 0.4 0.6 0.8 1t

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

x

Stochastic thermodynamics

First Law of thermodynamics for a single fluctuating trajectory.(Sekimoto, 1994)

Work and heat become stochastic variables .

What are their distributions?

W [x(·)], Q[x(·)]

Let with some protocol (driven system).�(t)f(x, t) = �rV (x,�(t))

dU = dV =@V

@�

d� +@V

@x

dx = dW + dQ

Change of energy of the system:

Transformation of probability

PT [x(·)] = NT [x(·)] exp ��

4

Z T

0dt⇣˙

x+rV (x,�)⌘2!

What to do with it?

P (W ) :=

Z (xT ,T )

(x0,0)Dx(·)PT [x(·)] �(W �W [x(·)])

P (Q) :=

Z (xT ,T )

(x0,0)Dx(·)PT [x(·)] �(Q�Q[x(·)])

W [x(·)] =Z T

0dt

@V

@�(x(t),�(t)) �(t)

Q[x(·)] =Z T

0dt rV (x(t),�(t)) · x(t)

Time inversion

−1.5 −1 −0.5 0 0.5 1 1.50

0.5

1

1.5

x

V0,V

1

Forward process

−1.5 −1 −0.5 0 0.5 1 1.50

0.5

1

1.5

x

V0,V

1

Backward process

−1.5 −1 −0.5 0 0.5 1 1.50

0.5

1

1.5

x

V0,V

1

Forward process

−1.5 −1 −0.5 0 0.5 1 1.50

0.5

1

1.5

x

V0,V

1

Backward process

Reverse process: , mirror trajectory: x(t) := x(T � t)�(t) := �(T � t)

The detailed fluctuation theorem

PT [x(·)]¯PT [¯x(·)]

=

NT [x(·)] exp✓��

4

R T0 dt

⇣˙

x+rV (x,�)⌘2◆

NT [¯x(·)] exp✓��

4

R T0 dt

⇣˙

¯

x+rV (

¯

x, ¯�)⌘2◆

=

NT [x(·)] exp✓��

4

R T0 dt

⇣˙

x+rV (x,�)⌘2◆

NT [x(·)] exp✓��

4

R T0 dt

⇣� ˙

x+rV (x,�)⌘2◆

= exp

��

Z T

0dt ˙x ·rV (x,�)

!= e���Q

Exact for arbitrarily large deviations from equilibrium!

Boundary terms

Include distribution of initial and final states.

Start in equilibrium: p[x(·)] = p0(x0)PT [x(·)] =1

Z0e��V0 PT [x(·)]

p[x(·)]p[¯x(·)] = exp[�(F0 � V0)� �(F

T

� VT

)� ��Q] = exp[�(�V ��Q��F )] = e�Wdiss[x(·)]

Start in any distribution (Seifert,2005):

�Sm[x(·)] :=�Q[x(·)]

T�s[x(·)] := ln

p0(x0)

pT (xT )�S[x(·)] := �Sm[x(·)] +�s[x(·)]

p[x(·)]p[x(·)] = e�S[x(·)]/kB

Consequences

p[x(·)]p[x(·)] = e�S[x(·)]/kB

0 X

f

X X

f(<X>)

<X>

<f(X)>

1 2

• The „emergence of irreversibility“:

• The integral fluctuation theorem:

• The Second Law as equality:

• Three faces of the Second Law:

�S[x(·)] ⇡ kB ! �S[x(·)]� kB

h�Si � 0

he��S[x(·)]/kB i = 1

S = Sa + Sna �! he��Sa[x(·)]/kB i = 1 , he��Sna[x(·)]/kB i = 1

More consequences

• Jarzynski equality (1997): equilibrium information from non-equilibrium processes.

• efficiency of molecular motors: running reliably forward.

• chemical thermodynamics at the molecular level.

• statistical mechanics very far from equilibrium.

• non-equilibrium steady states: house-keeping heat, linear response, Onsager reciprocity.

he��W i = e���F

j 6= 0

Microcanonical perspective(Cleuren et a., Phys. Rev. Lett. 96, 050601 (2006))

W

−W

E E+W

E+WE

P(W)=

P(−W)=

(Cleuren et a., Phys. Rev. Lett. 96, 050601 (2006))

P(−W)= = =P(W) e ∆ S/ kB

W

−W

E E+W

E+WE

P(W)=

P(−W)=

Microcanonical perspective

More consequences

• Jarzynski equality (1997): equilibrium information from non-equilibrium processes.

• efficiency of molecular motors: running reliably forward.

• chemical thermodynamics at the molecular level.

• statistical mechanics very far from equilibrium.

• non-equilibrium steady states: house-keeping heat, linear response, Onsager reciprocity.

he��W i = e���F

j 6= 0

Why not Gibbs? - A tale of tails

Large deviations become important in statistical mechanics.

Prob(�S/kB ��) e

��

−2 0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Stot /N

Prob

N ≅ 1N ≅ 100

dominant trajectories in are atypical for .P [x(·)]he��S/kB i

Large deviations

standard statistical physics: small fluctuations O(1/pN) �! hf(W )i ' f(hW i)

he��W i ' e��hW i �! hW i ' �F WRONG!

Why? Rare events contribute substantially to averages!

Mathematical framework: Large deviation theory

0 0.5 1 1.5 2w

0

0.5

1

1.5

2

2.5

3

3.5

4

I

P (W ) = e�NI(w)+o(N)

I(hwi) = 0, I(w) � 0

I(w) ⇡ I 00

2(w � hwi)2

Transformation of rare probabilities

P

x

(x) ⇠ e

�NI

x

(x)

x, P

x

(x); y = f(x) �! P

y

(y) =?P

y

(y) =

ZdxP

x

(x) �(y � f(x))

�! P

y

(y) =

Zdx e

�NI

x

(x)�(y � f(x))

saddle-point approximation:

Py(y) ⇠ e�NIy(y)

contraction principle

I

y

(y) = minx:y=f(x)

I

x

(x)

The gist of it

Asymptotics of work distributions

−4 −2 0 2 4 6 8 10 120

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)

D. Nickelsen, A.E., Eur. Phys. J. B, 82,207 (2011)

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

• is not known exactly.• Asymptotics for small is crucial.• This region is badly sampled.

P (W )

W

The idea

−15 −10 −5 0

10−6

10−4

10−2

100

W

P(W

)

atmpar: t0 = 0 ; t1 = 100 ; r = 0 ; β = 1 ; M = 5e7 ; RelTol = 1e−07 % ; bins = 100 ; jakob = 3 ; N = 1 ; n = 5e7 ; S = 100 ; W* = −10..−2

SimulationAsymptotik

1

−7 −6 −5 −4 −3 −2 −1 00

0.1

0.2

0.3

0.4

0.5

0.6

W

P(W

)

atmpar: t0 = 0 ; t1 = 100 ; r = 0 ; β = 1 ; M = 5e7 ; RelTol = 1e−07 % ; bins = 100 ; jakob = 3 ; N = 1 ; n = 5e7 ; S = 100 ; W* = −10..−2

SimulationAsymptotik

1

Crucial: existence of an overlap

Get analytical information about the tail of .Combine it with the histogram in integrals.

P (W )

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 are dominated by maximizing under the constraint .

Formally: and saddle-point approximation for functional integral.

P (W ) x(·)W [x(·)] = W

� !1

Includes contributions from the optimal trajectory and its neighbourhood.

P (W ) =Z

dx0 p0(x0)Z

dx

f

(xf ,tf )Z

(x0,t0)

Dx(·) P [x(·)] �(W �W [x(·)])

P (W ) =e��S[x(·)]

Z0

pdetM/�

(1 +O(1/�))

P [x(·)]

The optimal trajectory

�S

�x

= 0 �!

@S

@q= 0 �!

P (W ) = NZ

dx0

Z0

Zdx

T

Zdq

4⇡/�

x(T )=xTZ

x(0)=x0

Dx(·) e

��S[x(·),q]

0

x

ttf

S[x(·), q] = V0(x0) +TZ

0

dt

h14(x + V

0)2 +iq

2V

i� iq

2W

One solution of ELE for each value of the work combining unlikely initial conditions with strange realization of the noise.

¨x + (1� iq) ˙

V

0 � V

0V

00 = 0˙x0 � V

00 = 0,

˙xT + V

0T = 0

W =Z T

0dt ˙V

The pre-exponential factor

• Contributions from quadratic neighbourhood of the optimal trajectory.• Includes neighbourhood of initial and final points.• Constraint suppresses fluctuations orthogonal to it.

A := � d2

dt2+ (V 00)2 + V 0V 000 � (1� iq) ˙V 00

V 000 'n(0)� 'n(0) = 0, V 00

T 'n(T ) + 'n(T ) = 0

dn :=Z T

0dt 'n(t) ˙V 0(t)

P (W ) =Np2Z0

e��S

qdetA h ˙V 0|A�1| ˙V 0i

�1 +O(1/�)

�

X

n

d2n

�n= h ˙V 0|A�1| ˙V 0i

Determine eigenvalues and eigenfunctions of the Hessian, as well as the projections of on the gradient of the constraint:

�n, 'n'ndn

The breathing parabola

−20 −15 −10 −5 0 5 10 15 200

5

10

15

20

25

30

35

40

x

V

V0

V1•

• Experimentally accessible.

• is not Gaussian.

• is not known analytically.

P (W )

V (x, t) =k(t)2

x

2

P (W )

P (W ) =NZ0

px0 xT

|W | e

��iq2 |W | �1 +O(1/�)

�⇠ C1

s�

|W | e

�� C2 |W |

Exact results for the asymptotics (for all protocols):

−7 −6 −5 −4 −3 −2 −1 00

0.1

0.2

0.3

0.4

0.5

0.6

W

P(W

)atmpar: t0 = 0 ; t1 = 100 ; r = 0 ; β = 1 ; M = 5e7 ; RelTol = 1e−07 % ; bins = 100 ; jakob = 3 ; N = 1 ; n = 5e7 ; S = 100 ; W* = −10..−2

SimulationAsymptotik

1

−15 −10 −5 0

10−6

10−4

10−2

100

W

P(W

)

atmpar: t0 = 0 ; t1 = 100 ; r = 0 ; β = 1 ; M = 5e7 ; RelTol = 1e−07 % ; bins = 100 ; jakob = 3 ; N = 1 ; n = 5e7 ; S = 100 ; W* = −10..−2

SimulationAsymptotik

1

−20 −15 −10 −5 00

0.5

1

1.5

2

2.5

3

W

P(W

) ⋅e−

W

atmpar: t0 = 0 ; t1 = 100 ; r = 0 ; β = 1 ; M = 5e7 ; RelTol = 1e−07 % ; bins = 100 ; jakob = 3 ; N = 1 ; n = 5e7 ; S = 100 ; W* = −10..−2

SimulationAsymptotik

1

−25 −20 −15 −10 −5 010−3

10−2

10−1

100

101

102

103

W

P(W

) ⋅e−

W

atmpar: t0 = 0 ; t1 = 100 ; r = 0 ; β = 1 ; M = 5e7 ; RelTol = 1e−07 % ; bins = 100 ; jakob = 3 ; N = 1 ; n = 5e7 ; S = 100 ; W* = −10..−2

SimulationAsymptotik

1

Comparison with simulations

Work distribution in quasi-static processes always Gaussian?

Entropic saddle-points

1.0

U(x)/(f2 c/4)

x

f∗ = 1/2

(a)

0f∗ = 1

0.2

0.4

f∗ = 2

0.6

0.8h(σ)

1

-1

ARW

0 1

h∗(σ)

2 3

h(σ)

σ

(b)

-4

-2

0

2

0.0 0.2 0.4 0.6 0.8

f∗ = 0

10time τ/τ0

0

1

2

0 1 2 3potential

0

1

2

-20 -10 0

positionx

Driving a colloidal particle around a periodic potential. Large deviation function for the entropy production .�

For a single optimal trajectory dominates.For an ensemble of nearly optimal trajectories dominates.

|�| � 1|�| ⌧ 1

T. Speck, A.E., U. Seifert, JSTAT, P12001 (2012)

Thank you!

C. Jarzinsky, Ann. Rev. Condens. Matter Physics 2, 329 (2011)U. Seifert, Rep. Prog. Phys. 75, 126001 (2012)J. M. Parrondo et al., Nature Physics, 11, 131 (2015)

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