[PPT] - Stochastic thermodynamics and coarse-graining Udo Seifert II. PowerPoint Presentation

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Kyoto, Yukawa Int Seminar, August 2015

Stochastic thermodynamics and coarse-graining

Udo Seifert

II. Institut f¨

ur Theoretische Physik, Universit¨ at Stuttgart

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Stochastic thermodynamics for driven systems emb’d in a heat bath

λt λ0 W T, p

driving: mechanical shear flow (bio)chemical

Energy conservation (1st law) and entropy production

(2nd law) are defined along an individual stochastic trajectory

Review: U.S., Rep. Prog. Phys. 75 126001, 2012.

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Stochastic th’dynamics for a driven colloidal particle

D.G. Grier A revolution in optical manipulation, Nature 424, 810 (2003)

V (x, λ)

f(λ)

– Langevin dynamics ˙ x = µ[−V ′(x, λ) + f(λ)] + ζ with ζ1ζ2 = 2µkBTδ12 – external driving λ(τ)

First law [(Sekimoto, 1997)]:

dw = du + dq – applied work: dw = ∂λV (x, λ)dλ + f dx – internal energy: du = dV – dissipated heat: dq = dw − du = [−∂xV (x, λ) + f]dx = Tdsm

stochastic entropy and second law [U.S., PRL 95, 040602, 2005]

ds ≡ −d [ln p(x, t)] ⇒ exp[−∆(s + sm)] = 1 ⇒ ∆stot ≥ 0

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Exact non-eq work relations

– Jarzynski relation [ Phys. Rev. Lett. 78, 2690 (1997)] exp[−W] = exp[−∆F] – Crooks’ relation [ Phys. Rev. E 60, 2721 (1999)] p(W)/˜ p(−W) = exp[(W − ∆F)/kBT]

[Collin et al, Nature 437, 231, 2005]

– identities in stochastic (and Hamiltonian) dynamics for a thermalized initial state

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Coarse-graining

– in equilibrium ∗ spatial coarse-graining ∗ clustering states pi = exp[−β(Ei − F)] PI =

i∈I pi = exp[−β(FI(β) − F)]

– in dynamics: Markov property of dynamics is lost

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Coarse graining in stochastic thermodynamics

– fluctuating density field – NESS for two driven colloidal particles

V (x, λ)

f(λ)

V (x, λ)

f(λ)

– molecular motors with probe particles

related work:

Rahav and Jarzynski JSM 2007, Kawai et al PRL 2007, Pigolotti and Vulpiani JCP 2008, Esposito PRE 2012.....

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Coarse-graining a fluctuating density field for colloids

[T. Leonard, B. Lander, U.S., and T. Speck J. Chem. Phys. 139, 204109, 2013]

microscopic ⇒ coarse-grained – u(r) = ǫ exp[−κr]/r – free energy ∆F(T, V/N) from Crooks relation

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Microscopic density field

– overdamped Langevin ˙

rk = −

l u′(rkl)ˆ

rkl + ζk

– microscopic density ρ0(r, t) ≡

k δ(r − rk(t)) obeys

Dean’s (1996) equation ∂tρ0(r, t) = ∇

ρ0 δF

δρ0 + ξ

with ξξ ∼ 2ρ0δ....

– with free energy functional F[ρ] = FIG +

drdr′ρ(r)u(|r − r′|)ρ(r′)/2

– Work W0 ≡ −

dt ˙

V P[ρ0] from fluctuating pressure P[ρ0(r, t)] = N/V +

drdr′ρ0(r, t) f(|r − r′|)

f(r)≡−ru′(r)

ρ0(r′, t)/4V

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– coarse-grain density field on scale ℓ – ρℓ(r) ≡

k exp[−|r − rk|2/2ℓ2]/2πℓ2

– fluctuating pressure Pℓ(t) = ... = N/V +

k<l fℓ(|rk(t) − rl(t)|)/2V

with effective ”two-body” pressure fℓ(r) ≡ −ru′

ℓ(r) − 2ℓ2[u′′ ℓ(r) + u′ ℓ(r)/r]

uℓ(r) ≡

∞

0 dq q

√

1+q2 exp[−ℓ2q2]J0(qr)

2 4 6 8 10 1 2 fℓ(r) r ℓ = 0.1 ℓ = 0.2 ℓ = 0.4 ℓ = 0.0

0.4
0.2

0.0 0.2 0.4 0.6 0.8 1 2 fℓ(r) - f(r) r ℓ = 0.1 ℓ = 0.2 ℓ = 0.4 4 8 1 2 fℓ(0) ℓ

– coarse-grained fluctuating work Wℓ ≡ −

dt ˙

V Pℓ[ρℓ]

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Free energy from coarse-grained work in Crooks

– ”joint-(Wℓ, W0) Crooks”

pexp(−Wℓ,−W0) pcom(+Wℓ,+W0) = exp[−W0 + ∆F]

– integrating out exact work W0 ln pexp(−Wℓ) pcom(+Wℓ) = ∆F + ln

dW0 pcom(W0|Wℓ) exp[−W0]

≈ ∆F + lneδWℓ − Wℓ ≡ ∆Fℓ − Wℓ

0.1 1 10 100 0.18 0.2 0.22 0.24 0.26 0.28 probability density p(w) ℓ = 0 ℓ = 1 CO EX

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4

4 8 0.18 0.2 0.22 0.24 0.26 0.28 ln[pex(-w)/pco(w)] work per particle w/N 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.0 0.5 1.0 1.5 2.0 2.5 3.0 change of free energy per particle ΔΦℓ/N coarse graining length ℓ ρ0 = 0.1, ρ1 = 0.2 ρ0 = 0.2, ρ1 = 0.3 ρ0 = 0.3, ρ1 = 0.4 ρ0 = 0.4, ρ1 = 0.5 0.20 0.24 0.28 0.0 1.0 2.0 ΔΦℓ/N ℓ

– slope 1, cg-dependent free energy ∆Fℓ

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Coarse-graining in NESSs
V (x, λ)

f(λ)

– Time-independent driving beyond linear response regime – Broken detailed-balance – Persistent “currents” with permanent dissipation

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Fluctuation theorem

p(−∆stot)/p(∆stot) = exp(−∆stot)

– long-time limit: Evans et al (1993), Gallavotti & Cohen (1995), Kurchan (1998), Lebowitz & Spohn (1999) – finite times: U.S., PRL’05 ∆stot ≡ ∆sm + ∆s = t

0 dτ ˙

x(τ)ν(x(τ)) [with ν(x) ≡ ˙ x|x = j(x)/p(x)] – experimental data [Speck, Blickle, Bechinger, U.S., EPL 79 30002 (2007)]

200 400 600 800 1000 200 400 600 800 1000

a) b)

t=2s t=20s

~

total entropy production ∆stot [kB]

t=2s t=20s t=21.5

250
200
150
100
50
250
200
150
100
50
V (x, λ)

f(λ)

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F’theorem and slow hidden degrees of freedom

[J. Mehl, B. Lander, C. Bechinger, V. Blickle and U.S., PRL 108, 220601, 2012]

– total entropy production in the NESS ∆stot ≡

t

dτ[ ˙

x1ν1(x1, x2) + ˙ x2ν2(x1, x2)] with ν1(x1, x2) ≡ ˙ x1|x1, x2

beys FT

p(∆stot)/p(−∆stot) = exp ∆stot – suppose x2 is hidden: ˜ ν1(x1) ≡

ν(x1, x2)p(x2|x1)dx2

– apparent entropy production ∆˜ stot ≡

t

0 dτ ˙

x1˜ ν1(x1)

beys FT ??

x1 x2

V (x, λ)

f(λ)

V (x, λ)

f(λ)

1

+1

x2 (R)

0,1500 0,1650 0,1800 0,1950 0,2100 0,2250 0,2400 0,2550 0,2700 0,2850 0,3000

1

+1

1

+1

x2 (R) x1 (R)

1

+1

x2 (R)

1

+1

600
300

300 600

potential (units of kBT) x1, x2 (R)

R

+

x 2

f1 f2

1

x R (a) (b) (c) (d) R

U2 U1

B R

+

min.

max.

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Experimental data

– with and without coupling [rarely:]

3
2
1

1 2 3 2 4 6 8

~

p(Dstot) ( 10

2)

Dstot

~

3
2
1

1 2 3

3
2
1

1 2 3

~ ~

ln[p(Dstot)/p(-Dstot)]

Dstot

~

(c)

3
2
1

1 2 3

3
2
1

1 2 3

~ ~ ~

ln[p(Dstot)/p(-Dstot)]

Dstot

(a)

– FT-slope

0,5 1,0 1,5 2,0

t (s)

100 200 300 0,6 0,7 0,8 0,9 1,0 1,1

slope a

G

(a) (b 14

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Molecular motor: F1-ATPase

– kinetics vs thermodynamics – first law? – efficiency(ies)?

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F1-ATPase and the fluctuation theorem

[K. Hayashi, ... H. Noji, PRL 104, 218103 (2010)]

– Γ ˙ θ = N + ζ ζ1ζ2) = 2ΓkBTδ(τ1 − τ2) ⇒ ln[p(∆θ)/p(−∆θ)] = N∆θ/kBT independent of friction coefficient – cf f’theorem ln[p(∆stot)/p(−∆stot)] = ∆stot/kB time-dependence? torque from ∆t → ∞ ?

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Hybrid model

[E. Zimmermann and U.S., New J. Phys. 14, 103023, 2012]

– probe particle ∗ ˙ x = µ(−∂yV (y) + fex) + ζ with y(τ) ≡ n(τ) − x(τ) – motor ∗ w+/w− = exp[∆µ − V (n + d, x) − V (n, x)] ∗ local detailed balance condition

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FT-slope from simulations vs experiment

1 2 3 4 5 6

0.5

0.5 1 1.5 p(∆x) ∆x [d] ∆t = 1 ms ∆t = 2 ms ∆t = 5 ms ∆t = 10 ms ∆t = 20 ms

2 4 6 8 10 12 14 16 0.01 0.02 0.03 0.04 0.05 0.06 0.07 α ∆t cATP = 1.4747 × 10-7 cATP = 8 × 10-7 cATP = 2.962 × 10-6 cATP = 5.9494 × 10-5

∆t → 0 limit yields average force/torque

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Fine-structured large deviations

[P. Pietzonka, E. Zimmermann and U.S., EPL 107 20002, 2014]

dynamics

∂tp(n, y, t) = (L1 + L2)p(n, y, t)

generating function

g(λ, y, t) ≡ ∞

n=−∞ eλnp(n, y, t) ≈ eα0(λ)tQ(λ, y, y0)

large deviation form with amplitude

p(n, y, t|y0) ≈ e−th(n/t)Q(λ(n/t), y, y0)

rate function

h(u) ≡ uλ(u) − α0(λ(u))

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Fine-structured fluctuation theorem
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6
4
2

2 4 6 8

2
1.5
1
0.5

0.5 1 1.5 2 Simulation

for t → ∞: discrete symmetry: P(∆x + m) = e−λ0mP(∆x)
ln P(∆x)

P(−∆x) = −2λ0∆x + ψ(∆x)

with λ0 = −(∆µ − fexd)/2. and periodic antisymmetric ψ(∆x)

slope at 0 not given by entropy production
”finite-difference slope” determines ent’ production

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Fine structure at any ”base point” nc = ut
30
20
10

10 20 30

3
2
1

1 2 3

30
20
10

10 20 30

3
2
1

1 2 3

30
20
10

10 20 30

3
2
1

1 2 3

30
20
10

10 20 30

3
2
1

1 2

50
100
150
1

1 2 3

u/v

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Generalizations

– fine structure holds for any model with spatial periodicity and hid- den degrees of freedom

8
6
4
2

2 4 6 8

2
1.5
1
0.5

0.5 1 1.5 2 Simulation

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Dynamically and thermod’y consistent coarse-graining of

molecular motor models

[E. Zimmermann and U.S., Phys Rev E 91, 022709, 2015]

– one-state motor ⇒ – conditions: v = d(Ω+ − Ω−)

Ω+ Ω− = exp[∆µ − fexd]

– coarse-grained rates Ω+ = v exp[∆µ−fexd]/d

exp[∆µ−fexd]−1

Ω− =

v/d exp[∆µ−fexd]−1

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Coarse-graining

versus traditional model ⇓

– probe particle omitted – external force assumed to act directly on the motor – exponential dependence of the rates on the external force

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Example: F1-ATPase

10−10 100 1010 20 40 10−10 10−5 100 20 40

– Ω± approach ˆ w± with decreasing probe size – non-exponential dependence of Ω± on external force

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Coarse-graining multi-state models (Example: Kinesin)

10−6 10−4 10−2 100 102 104 −10 10 20 30 100 105 1010 −10 10 20 30

[S. Liepelt et al, PRL 98 (2007)]

20 40 60 80 −10 10 20 30 −0.5 −1 0.5 1 13.6 13.8 14 14.2 14.4

– stall force depends on probe size

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Invariance of entropy production under coarse-graining

– detailed model with explicit dynamics of the probe particle: ˙ Stot =

i

γjx

i 2

pi(y)dy

probe

+

i,j
pi(y)wij(y) ln

pi(y)wij(y) pj(y + dij)wji(y + dij)dy

motor

=

i<j

∆µijjij − fexv – coarse-grained model: ˙ Stot =

i,j

PiΩij ln PiΩij PjΩji =

i<j

∆µijjij − fexv – entropy production is conserved

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Conclusions

– Coarse-graining, in general, compromises the exact ST-relations – colloidal suspension: ∗ pseudo Crooks relation for coarse-grained work ∗ extension/relation to DFT? – colloids and molecular motors in a NESS ∗ FT-slope time-dependent for small t ∗ long-time asymptotics: fine structure with modulated slope ∗ th’dynamically and dyn’y consistent coarse-graining possible