On optimal protocols in stochastic thermodynamics Anomalous - - PowerPoint PPT Presentation

September 23, 2010 Erik Aurell, KTH & Aalto University 1

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On optimal protocols in stochastic thermodynamics

Anomalous transport: From Billiards to Nanosystems Sperlonga, September 20-24 2010 joint work (in progress) with Paolo Muratore-Ginanneschi, U of Helsinki Carlos Mejía-Monasterio, U of Helsinki & Madrid

September 23, 2010 Erik Aurell, KTH & Aalto University 2

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Thermodynamics of small systems

Xu Zhou, 2008 Nature blogs

• J. Liphardt et. al., Science 296, 1832, 2002

Contributions by Jarzynski, Crooks, Bustamante, Ritort, Peliti, Imparato, Seifert, Rondoni, Vulpiani, Puglisi, Kurchan, Gawedzki, Lebowitz, and many others

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“The free energy landscape between two equilibrium states is well related to the irreversible work required to drive the system from one state to the

• ther”

F W

e e

∆ − −

=

β β

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∑

=

≡ > <

S i i S S

W W

1 ) ( 1

[ ] [ ]

F W E W E

S

∆ ≥ = > <

∑

= − −

≡ > <

S i W S S W

i

e e

1 1

) (

β β

[ ]

[ ]

F S i W S S W

e e E e E

i

∆ − = − −

= = > <

∑

β β β 1 1

) (

Jarzynski estimator unbiased for exponentiated Free energy diff. But has a statistic error, scaling with sample number S

[ ] S e e E e e S E

F W S i F W i

) ( ) ( 1

2 2 1 2 ∆ − − = ∆ − −

− =       −

∑

β β β β

(protocol dep.) NB: Jarzynski estimator is biased for the Free energy difference

Why optimization & optimal protocols?

September 23, 2010 Erik Aurell, KTH & Aalto University 5

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If you admit for single small systems (the example will follow)

Other motivation(s) for optimization

Q dU W δ δ + =

and if the initial and final states are known

> < + > >=< < Q dU W δ δ ⇒

diss

W F W + ∆ = δ S T Q Wdiss ∆ + > >=< < δ

Xu Zhou, 2008 Nature blogs Molecular machines have quite high efficiency: Kinesin – 60% Ion pump on membrane – 70%

then the expected dissipated work is Minimizing expected dissipated work is maximizing efficiency

• f the small system

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[ ]

W E

What to optimize: some expectation value over the paths

[ ]

W

e E

β 2 −

Class I: given initial and final states

Which optimization tasks? What to optimize? Which constraints?

[ ]

Q E

Which constraints

For instance, equilibrium at the same T but different potentials. Connected to Schrödinger ”Über die Umkehrung des Naturgesetze”, Sitzung der Preuss. Akad. Wissen. Berlin 144 (1931); and to Guerra & Morato (1983); Dai Pra (1991)

Class II: given initial state and final control

Schmiedl & Seifert, Phys. Rev. Lett. 98 (2007); Gomez-Marin, Schiedl & Seifert, J. Chem. Phys. 129 (2008); Then & Engel, Phys Rev E77 (2008).

[ ]

W Var

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t t t t

V ω τβ λ ξ τ ξ

ξ

  2 ) , ( 1 + ∂ − =

What is stochastic thermodynamics?

(Langevin Equation)

) ( ) ; (

t i t

U t t V ξ ξ = <

(no control before initial time)

) ( ~ ) ; (

t f t

U t t V ξ ξ = >

(no control after final time)

) 2 ( ) (

t t t t

V ω τβ ξ ξ τ ξ

ξ

    − ⋅ = −∂ ⋅

(Stratonovich)

∫

− ⋅ =

f i

t t t t t

Q ) 2 ( ω τβ ξ ξ τ   

) , (

t t t t t

f i

V W λ ξ λ

λ

∫

∂ ⋅ =  U U U Q W

i f

∆ = − = − ) ( ) ( ~ ξ ξ

Sekimoto Progr. Theor. Phys.180 (1998); Seifert PRL 95 (2005)

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Class I: minimizing expected released heat with given initial and final states

[ ] ∫

> ⋅ ∂ − < =

f i

t t f i

b T b dt Q E ) | | ( , |

2 ξ

τ µ µ

) ( ]) , [ Pr(

i i i i i i

dx dx x x µ ξ = + ∈ ) ( ]) , [ Pr(

f f f f f f

dx dx x x µ ξ = + ∈

) , (

t t

V b λ ξ

ξ

−∂ =

Re-writing Q with the Ito convention gives, in expectation:

t t

b ω τβ τ ξ   2 1 + =

b (for short) is the stochastic control of the Langevin equation

m bm m

x x t 2

1 ) ( 1 ∂ = ∂ + ∂ βτ τ

which gives a forward Fokker-Planck equation for the density

) ( ) , ( dx dx t x m

i i

µ =

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Class I: minimizing expected released heat with given initial and final states

Density evolution, forward Fokker-Planck Optimal control, Bellman equation

) (

i i dx

µ i

t

) (

f f dx

µ f

t

) (

f f dx

µ

x t, y dt t , −

[ ]

) ( , |

i f i

t S Q E = µ µ

∫

= dx t x S t x m t S

f )

; , ( ) , ( ) ( µ S S b b T b S

t 2 2

1 ) ( ) | (| ∂ + ∂ ⋅ + ⋅ ∂ + = ∂ − τβ τ τ

S R b ∂ − ∂ =

∗

m T R log =

Problem: b* depends both on the forward and the backward proceses

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Class I: minimizing expected released heat with given initial and final states

) ( 2 1 4 | ) ( | ] 1 2 ) ( [

2 2 2

S R S R S S R

x x x x x t

− ∂ − − ∂ − = ∂ + ∂ − ∂ + ∂ βτ τ τβ τ

m m S R m

x x x t 2

1 ] 2 ) ( [ ∂ = − ∂ ∂ + ∂ βτ τ

m T R log =

) ( = ⋅ ∂ + ∂ vm m

t

) ( 2 1 R S + − = τ ψ ψ

x

v ∂ =

) ( = ∂ ⋅ + ∂ v v v

t

Burgers’ equation and mass transport by Burgers’ field

September 23, 2010 Erik Aurell, KTH & Aalto University 11

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Burgers’ equation with initial and final densities is a problem we know well

Monge-Ampère-Kantorovich Reconstruction of the early Universe

Frisch et al Nature (2002), 417 260; Brenier et al MNRAS (2003), 346

501

) ( ) ( ) det( a m x m a x

i f

= ∂ ∂

i

t a,

f

t x,

) ( ) , ( a t a v

a i

ψ ∂ = ] 2 ) ( ) ( max[ arg ) (

2

t a x a x a

• ∆

− − = ψ t x a x t a v t x v

i f

∆ − = = ) ( ) , ( ) , (

Valid as long as there are no shocks in the field

September 23, 2010 Courtesy Sobolevskii & Mohayaee 12

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« True » equations

• f early Universe

« Zeldovich » approximation

« Burgers » equation

and mass transport

v

x

v

shock

caustic caustic

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MAK-reconstruction applied to optimal protocols in stochastic thermodynamics

f i i f i

TS TS v t Q − + > < ∆ = > <

∗ 2

τ

rel f rel f

U U Q > < − > =< > < ~ ~

In relaxation, no work is done: One can now express released heat from initial to final state as

> < + > >=< < Q U W

So using the equivalence We have also

i f i rel i

F F v t W − + > < ∆ = > <

∗ 2

τ

Since

i a i f

v t T a U t v a U F F > ∂ ∆ + − − ∆ + =< − ) 1 log( ) ( ) ( ~

It should also be possible to optimize over final state f (Class II)

September 23, 2010 Erik Aurell, KTH & Aalto University 14

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Comparison with Seifert’s examples

We have not worked this out yet..... (that’s why this is work in progress...)

• T. Schield & U. Seifert ”Optimal Finite-time processes in stochastic

thermodynamics”, Phys Rev Lett 98 (2007): 108301

September 23, 2010 Erik Aurell, KTH & Aalto University 15

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Some other problems can be solved this way also (but not easily)

[ ]

W

e E

β 2 −

Mimimizing the statistical error of the Jarzynski estimator Can be turned into a non-linear transport problem

] ) log [( 2 = ∂ ⋅ ∂ − ∂ φ τ φ m

x x t

] ) log [( 2 = ∂ ⋅ ∂ + ∂ m m

x x t

φ τ

Which is a bit more difficult than mass transport by Burgers’ eq.

September 23, 2010 Erik Aurell, KTH & Aalto University 16

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Conclusions and open problems

Work out the mixed backward-forward equations for other problems in stochastic thermodynamics. Are there other examples that are as solvable as Burgers’ eq.? Compare to Seifert’s state-independent protocols for the harmonic trap (shame on us, we have not done so as yet). What do shocks and caustics in the optimal control problem mean for stochastic thermodynamics? Does any of this generalize to other systems e.g. jump processes?

Thanks to

Paolo Muratore-Ginanneschi Carlos Mejía-Monasteiro

For posing and framing the problem Collaborators on the project

Udo Seifert

For contributing slides

Xu Zhou (through Nature blogs) Andrei Sobolevskii & Roya Mohayaee