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Erik Aurell, KTH & Aalto University 1March 15, 2011

Optimal protocols and optimal transport in stochastic

termodynamics

KITPC/ITP-CAS ProgramInterdisciplinary Applications of Statistical Physics and

Complex NetworksWorkshop A – March 14-15 2011

E.A., Carlos Mejia-Monasterio, Paolo Muratore-Ginanneschi [arXiv:1012.2037]

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Erik Aurell, KTH & Aalto University 2September 28, 2010

Nonequilbrium physics of small systems

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

Contributions by Jarzynski, Bustamante, Cohen, Crooks, Evans, Gawedzki, Kurchan, Lebowitz, Moriss, Peliti, Ritort, Rondoni, Seifert, Spohn, and many others

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Erik Aurell, KTH & Aalto University 3September 28, 2010

“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 other”

FW ee

Fluctuation relations

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Erik Aurell, KTH & Aalto University 4September 28, 2010

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

QdUW

then you can optimize expected dissipated work or released heat

QdUW

Xu Zhou, 2008 Nature blogs

Related to efficiency of the smallsystem e.g. molecular machinessuch as kinesin or ion pumps

Another motivation is thevariance of JEas an estimator SeeEee

SE FW

S

i

FW i

)()(1 22

1

2

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Erik Aurell, KTH & Aalto University 5March 15, 2011

tttt V 2),(1

The stochastic thermodynamics model

(Langevin Equation)

)();( tit UttV (no control before initial time)

)(~);( tft UttV (no control after final time)

)2()( tttt V (Stratonovich sense)

f

i

t

t tttQ )2( ),( t

t

t ttf

i

VW

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

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Erik Aurell, KTH & Aalto University 6March 15, 2011

Released heat with initial & final states

f

i

t

t xfi bbdtQtStS 121 ||)()(

)(]),[Pr( iiiiii dxdxxx )(]),[Pr( ffffff dxdxxx

re-writing δQ with the Itô convention gives in expectation:

Density evolution, forward Fokker-Planck

Optimal control, Bellman equation

)( ii dx )( ff dx

SRb xx 21*

),(log),( 1 txmtxR

SSbbbSt211121 )()|(|

),( ttVb

),( ff txS),( ii txS

mbmm xxt211 )(

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Erik Aurell, KTH & Aalto University 7March 15, 2011

)()(])([ 22

124121

21 SRSRSSR xxxxxt

mmSRm xxxt21

21 )(

)(21 RS

Optimal control b* depends both on forward and backward processes

An ”instantaneous equilibrium” ansatz for the control

xxRb * 0)( vmmt

xv 02

21 xt

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Erik Aurell, KTH & Aalto University 8March 15, 2011

0221 xt

Burgers equation

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Erik Aurell, KTH & Aalto University 9March 15, 2011

02

2

dt

xdBurgers is free motion if no shocks

)()()det(

xmam

ax

f

i

]2

)()(max[arg)(2

taxxax f

solved by Hopf-Cole transformation if there are

and by Monge-Ampere equation if only initial and final mass distributions are known

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Erik Aurell, KTH & Aalto University 10March 15, 2011

Burgers’ equation with initial and finaldensities is well-known in Cosmology

Frisch et al Nature (2002), 417 260; Brenier et al MNRAS (2003), 346 501

ita, ftx,

(with average over initial or final state) is minimal released heat by a small system

21 )( axt

...but here we see that it comes up also in mesoscopics.Monge-Ampere equation and Hopf-Cole transformationcan be combined into a minimization of quadratic cost

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Erik Aurell, KTH & Aalto University 11March 15, 2011

21* )( axSSQ tiffi

Expected generated heat between initial and final states has one entropy change term, and one ”Burgers term” (released heat):

ift mmUUaxW loglog~)( 12*

The quadratic penalty term means Monge-Ampere-Kantorovich optimal transport

2 RS

This quadratic penalty term can be minimized by discretization, and looking for minimal transport cost.

Similarly for minimal expected work done on the small system.

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Erik Aurell, KTH & Aalto University 12March 15, 2011

T. Schmiedl & U. Seifert ”Optimal Finite-time processes in stochastic thermodynamics”, Phys Rev Lett 98 (2007): 108301Initial state in equilibrium. Final state is not fixed: final control is.

The examples of Schmiedl & Seifert

221)( ii xxU

2

2)(~ fc

f xxU

.),( 221 ConstxtxR iii

.),(2

2 ConstqxtxR fr

ff

Optimizing over r and q in ”Burgers formula” for the work gives

2

tctcq (Seifert’s ”protocol jump formula”)

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Erik Aurell, KTH & Aalto University 13March 15, 2011

FW ee

More complicated optimal transport tooptimize protocols in stochastics

We 2

0log222261 mxxxxt

0])[(1 mm xxt

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

Estimating free energydifferences using Jarzynski’s equation has statistical fluctuations – which can be minimized in the same wayas for heat and work above

…with some auxiliary field

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Erik Aurell, KTH & Aalto University 14September 28, 2010

Conclusions and open problems

We can solve the problems of optimal protocols in the nonequilibrium physics of small systems

The solutions are in terms of optimal (deterministic) transport.

For released heat or dissipated work, the optimal transport problem is Burgers equation and mass transport by the BurgersField. Very efficient methods have been worked out in Cosmology.

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?

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Erik Aurell, KTH & Aalto University 15March 15, 2011

Thanks to Carlos Meija-MonasteiroPaolo Muratore-Ginanneschi

Ralf EichhornStefano Bo