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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)
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Traditional thermodynamics
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The balances of energy and entropy
S ! 0, @S/@X ! 0 for T ! 0
dU = dW + dQ
dS = diS + deS, diS � 0
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Small Systems
W ' �F ' kBT, �S ' kB
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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
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System with Reservoirs
SHeat
Inf
Work
Heat ! ! !
!
T1 T2
(T = 0)
(T = 1)
E
SS
S
E E
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Engine
SHeat
Inf
Work
Heat
!
T1 T2
(T = 0)
(T = 1)
E
SS
S
E E�!�!
�!
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Refrigerator
SHeat
Inf
Work
Heat
!
T1 T2
(T = 0)
(T = 1)
E
SS
S
E E�!
�!
�!
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Maxwell’s Demon
SHeat
Inf
Work
Heat !T1 T2
(T = 0)
(T = 1)
E
SS
S
E E
�!�!
�!
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Eraser
SHeat
Inf
Work
Heat !T1 T2
(T = 0)
(T = 1)
E
SS
S
E E
�!
�!
�!
Landauer limit: �S � kB ln 2
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Information driven heat pump
SHeat
Inf
Work
HeatT1 T2
(T = 0)
(T = 1)
E
SS
S
E E�!
�!
�!
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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)
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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
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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:
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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)
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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)
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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!
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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
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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
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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
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Microcanonical perspective(Cleuren et a., Phys. Rev. Lett. 96, 050601 (2006))
W
−W
E E+W
E+WE
P(W)=
P(−W)=
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(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
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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
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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
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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
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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)
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The gist of it
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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
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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 )
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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(·)]
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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
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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
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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):
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−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?
![Page 35: Large deviations in stochastic thermodynamicspperso.th.u-psud.fr/.../files/hh12/Andrea_talk_1.pdf · 2017. 6. 12. · Continuous stochastic processes P T [x(·)] = N T [x(·)] exp](https://reader035.vdocument.in/reader035/viewer/2022071006/5fc397a81c42be417c2dd0f2/html5/thumbnails/35.jpg)
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)
![Page 36: Large deviations in stochastic thermodynamicspperso.th.u-psud.fr/.../files/hh12/Andrea_talk_1.pdf · 2017. 6. 12. · Continuous stochastic processes P T [x(·)] = N T [x(·)] exp](https://reader035.vdocument.in/reader035/viewer/2022071006/5fc397a81c42be417c2dd0f2/html5/thumbnails/36.jpg)
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)
Recent reviews: