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Optimal Protocols and Optimal Transport in Stochastic Thermodynamics Erik Aurell, 1,2,3, * Carlos Mejı ´a-Monasterio, 4,5,and Paolo Muratore-Ginanneschi 5,1 ACCESS Linnaeus Centre, KTH, Stockholm, Sweden 2 Computational Biology Department, AlbaNova University Centre, 106 91 Stockholm, Sweden 3 Aalto University School of Science, Helsinki, Finland 4 University of Helsinki, Department of Mathematics and Statistics P.O. Box 68 FIN-00014, Helsinki, Finland 5 Laboratory of Physical Properties, Department of Rural Engineering, Technical University of Madrid, Avenida Complutense s/n, 28040 Madrid, Spain (Received 11 December 2010; revised manuscript received 16 May 2011; published 20 June 2011) Thermodynamics of small systems has become an important field of statistical physics. Such systems are driven out of equilibrium by a control, and the question is naturally posed how such a control can be optimized. We show that optimization problems in small system thermodynamics are solved by (deterministic) optimal transport, for which very efficient numerical methods have been developed, and of which there are applications in cosmology, fluid mechanics, logistics, and many other fields. We show, in particular, that minimizing expected heat released or work done during a nonequilibrium transition in finite time is solved by the Burgers equation and mass transport by the Burgers velocity field. Our contribution hence considerably extends the range of solvable optimization problems in small system thermodynamics. DOI: 10.1103/PhysRevLett.106.250601 PACS numbers: 05.70.Ln, 02.50.Ey, 05.60.k, 87.15.H The last two decades have seen a revolution in the understanding of thermodynamics of small systems driven out of equilibrium. Jarzynski’s equality (JE) [1] relates an exponential average of the thermodynamic work W done on a system, driven from an initial equilibrium state to another final state, to the exponentiated free energy differ- ence F between these two states: he W e F : (1) Here and in the following ¼ 1=k B T is the inverse tem- perature, k B the Boltzmann’s constant that we set to 1, and h i is an expectation over a nonequilibrium process, specified by a (time- and state-dependent) driving force or protocol. JE, and Crook’s theorem [2], from which it follows, have been used to successfully determine binding free energies of single biomolecules through repeated pull- ing experiments [3]. For stochastic thermodynamics (the setting of this Letter), such transient nonequilibrium fluc- tuation relations are comprehensively reviewed in [4]. Equally important steady-state fluctuation relations are a counterpart of the transient fluctuation relations [511]. The transient nonequilibrium fluctuation relations are identities; they hold irrespective of the protocol. Most quantities of interest, however, still depend on the protocol, and can then be varied and optimized. A first step in this direction was taken by Schmiedl & Seifert who showed that when pulling a small system by optical tweezers, (expected) heat released to the environment and (expected) work done on the small system are minimized, not by naively smoothly pulling a small system, but by protocols with discontinuities [12], a work which has generated considerable interest in the field [1315]. For technical reasons, the analysis of Schmiedl & Seifert was limited to harmonic potentials. In this Letter we show how such optimization problems in stochastic thermodynamics (minimizing heat, work, the variance of the JE estimate of free energy differences) can be mapped to problems of (deterministic) optimal trans- port. The optimal control (for any of these cases) is deter- mined by the solution of an auxiliary problem. When optimizing heat or work, this auxiliary problem is none other than the Burgers equation of fluid dynamics and cosmology, and mass transport by the Burgers field. Very efficient numerical methods have been developed to solve such problems, and these methods can be directly applied. Our contribution hence considerably extends the range of solvable optimization problems in stochastic thermodynamics. Stochastic thermodynamics and optimal protocols.—We consider dynamics in the overdamped limit described by coupled Langevin equations: _ t ¼ 1 ( @ t Vð t ;tÞþ ffiffiffiffiffiffi 2 ( s _ ! t ; (2) with initial value t 0 ¼ x 0 , drift @ t V, and _ ! t , a vector valued white noise with covariance h _ ! t _ ! t 0 ðt t 0 Þ, and mobility ( 1 . For times t<t 0 the potential is Vðx;tÞ¼ U 0 ðxÞ, and for times t>t f it is V ðx;tÞ¼ U f ðxÞ. In the control interval [t 0 , t f ] we allow the potential to be an explicit function of time Vðx;tÞ¼ Uðx;tÞ. For single sto- chastic trajectories we define W, the Jarzynski work [1], and Q, the heat released into the heat bath, as [16] PRL 106, 250601 (2011) PHYSICAL REVIEW LETTERS week ending 24 JUNE 2011 0031-9007= 11=106(25)=250601(4) 250601-1 Ó 2011 American Physical Society

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Page 1: Optimal Protocols and Optimal Transport in Stochastic Thermodynamics

Optimal Protocols and Optimal Transport in Stochastic Thermodynamics

Erik Aurell,1,2,3,* Carlos Mejıa-Monasterio,4,5,† and Paolo Muratore-Ginanneschi5,‡

1ACCESS Linnaeus Centre, KTH, Stockholm, Sweden2 Computational Biology Department, AlbaNova University Centre, 106 91 Stockholm, Sweden

3Aalto University School of Science, Helsinki, Finland4University of Helsinki, Department of Mathematics and Statistics P.O. Box 68 FIN-00014, Helsinki, Finland

5Laboratory of Physical Properties, Department of Rural Engineering, Technical University of Madrid,Avenida Complutense s/n, 28040 Madrid, Spain

(Received 11 December 2010; revised manuscript received 16 May 2011; published 20 June 2011)

Thermodynamics of small systems has become an important field of statistical physics. Such systems

are driven out of equilibrium by a control, and the question is naturally posed how such a control can be

optimized. We show that optimization problems in small system thermodynamics are solved by

(deterministic) optimal transport, for which very efficient numerical methods have been developed, and

of which there are applications in cosmology, fluid mechanics, logistics, and many other fields. We show,

in particular, that minimizing expected heat released or work done during a nonequilibrium transition in

finite time is solved by the Burgers equation and mass transport by the Burgers velocity field. Our

contribution hence considerably extends the range of solvable optimization problems in small system

thermodynamics.

DOI: 10.1103/PhysRevLett.106.250601 PACS numbers: 05.70.Ln, 02.50.Ey, 05.60.�k, 87.15.H�

The last two decades have seen a revolution in theunderstanding of thermodynamics of small systems drivenout of equilibrium. Jarzynski’s equality (JE) [1] relates anexponential average of the thermodynamic work W doneon a system, driven from an initial equilibrium state toanother final state, to the exponentiated free energy differ-ence �F between these two states:

he��Wi ¼ e���F: (1)

Here and in the following � ¼ 1=kBT is the inverse tem-perature, kB the Boltzmann’s constant that we set to 1, andh� � �i is an expectation over a nonequilibrium process,specified by a (time- and state-dependent) driving forceor protocol. JE, and Crook’s theorem [2], from which itfollows, have been used to successfully determine bindingfree energies of single biomolecules through repeated pull-ing experiments [3]. For stochastic thermodynamics (thesetting of this Letter), such transient nonequilibrium fluc-tuation relations are comprehensively reviewed in [4].Equally important steady-state fluctuation relations are acounterpart of the transient fluctuation relations [5–11].

The transient nonequilibrium fluctuation relations areidentities; they hold irrespective of the protocol. Mostquantities of interest, however, still depend on the protocol,and can then be varied and optimized. A first step in thisdirection was taken by Schmiedl & Seifert who showedthat when pulling a small system by optical tweezers,(expected) heat released to the environment and (expected)work done on the small system are minimized, not bynaively smoothly pulling a small system, but by protocolswith discontinuities [12], a work which has generatedconsiderable interest in the field [13–15]. For technical

reasons, the analysis of Schmiedl & Seifert was limitedto harmonic potentials.In this Letter we show how such optimization problems

in stochastic thermodynamics (minimizing heat, work, thevariance of the JE estimate of free energy differences) canbe mapped to problems of (deterministic) optimal trans-port. The optimal control (for any of these cases) is deter-mined by the solution of an auxiliary problem. Whenoptimizing heat or work, this auxiliary problem is noneother than the Burgers equation of fluid dynamics andcosmology, and mass transport by the Burgers field. Veryefficient numerical methods have been developed tosolve such problems, and these methods can be directlyapplied. Our contribution hence considerably extends therange of solvable optimization problems in stochasticthermodynamics.Stochastic thermodynamics and optimal protocols.—We

consider dynamics in the overdamped limit described bycoupled Langevin equations:

_� t ¼ � 1

�@�t

Vð�t; tÞ þffiffiffiffiffiffiffi2

��

s_!t; (2)

with initial value �t0¼ x0, drift �@�t

V, and _!t, a vector

valued white noise with covariance h _!t _!t0 i ¼ �ðt� t0Þ,and mobility ��1. For times t < t0 the potential is Vðx; tÞ ¼U0ðxÞ, and for times t > tf it is Vðx; tÞ ¼ UfðxÞ. In the

control interval [t0, tf] we allow the potential to be an

explicit function of time Vðx; tÞ ¼ Uðx; tÞ. For single sto-chastic trajectories we define �W, the Jarzynski work [1],and �Q, the heat released into the heat bath, as [16]

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�W ¼Z tf

t0

@tVð�t; tÞdt; (3)

�Q ¼ �Z tf

t0

_�t � @�tVð�t; tÞdt: (4)

The definition of the work (3) follows after requiring that�W � �Q is the integral of an exact differential, namely

�W � �Q ¼ Ufð�tf; tfÞ �U0ð�t0

; t0Þ; (5)

which expresses the 1st law over [t0, tf]. It holds if and

only if the stochastic integral in �Q is defined in the senseof Stratonovich.

The stochastic differential Eqs. (2) lead to a (control-dependent) probability density �ðx; tÞ defining the expec-tation value of local quantities as

hGð�t; tÞi ¼ZRd

dx�ðx; tÞGðx; tÞ:

Straightforward application of the Ito lemma (see, e.g.,[17]) yields the average heat release

h�Qi ¼Z tf

t0

dt

�hk @�t

U k2 � 1

�@2�t

Ui; (6)

where we switch from V to U to emphasize that thepotential now has the meaning of a control. Given initialand final states, the minimal variance of the heat (or work)can be written as a Kullback-Leibler distance between acontrolled and uncontrolled process, and this connectionhas been thoroughly explored in the literature [18,19].

The Burgers equation in optimal stochastic control.—We first focus on heat minimization. Following [20] welook for a function Aðx; tÞ such that

��@t � 1

�@xUðx; tÞ � @x þ 1

��@2x

�Aðx; tÞ

¼ � 1

�k @xUðx; tÞ k2 � 1

�@2xUðx; tÞ

�; (7)

with � evolving according to the Fokker-Planck equation

@t�� 1

�@x � ð�@xUÞ ¼ 1

��@2x�: (8)

If such function A can be found, the left-hand side of (7)becomes the average of a stochastic differential so that

h�Qi ¼ hAð�t0; t0Þ � Að�tf

; tfÞi: (9)

Introducing the (fictitious) potential Rðx; tÞ as if it wouldhave been in equilibrium, i.e., R :¼ ��1 ln�, the stationarycondition is obtained by taking the functional variation of(7) with respect to U, yielding ð@xR � @x þ ��1@2xÞðA�2U� RÞ ¼ 0. In the absence of constant flux solutions for�, the stationarity condition is

U� ¼ A� R

2þ�; (10)

where � is an arbitrary function of time alone [21].From (10) we obtain for A and R the following coupled

backwards-forwards equations

@tA� k @xA k2 � k @xR k24�

¼ �@2xðAþ RÞ2��

; (11)

@t�� @x ��@xðA� RÞ

2��

�¼ 1

��@2x�: (12)

We note that these equations have the property that if wesplit the drift into an equilibrium piece @xR and a remain-der specified by the gradient of

c ¼ �Aþ R

2; (13)

the Eqs. (11) and (12) reduce to the deterministic transportequations

@tc þ k @xc k22�

¼ 0; (14)

@t�þ 1

�@x � ½ð@xc Þ�� ¼ 0: (15)

Equation (14) is the Burgers equation (for the velocitypotential), and (15) is the equation of mass transport bythe corresponding velocity field. These two equations arethe first main result of this Letter: we have reduced acomplicated stochastic optimization problem to a classicalproblem of optimal deterministic transport.Optimal heat between given initial and final states.—In

the theory of deterministic optimal control it is well justi-fied to construct unique solutions in viscosity sense [22].The mixed forwards-backwards nature of the problem hereconsidered renders the application to the Burgers equation(14) of viscosity solution theory a nontrivial extension.On the other hand, without shocks the solutions of theBurgers equation are free-streaming motion, which wecan specify by an inverse Lagrangian map x0 ¼ xf � ðtf �t0Þvðxf; tfÞwhere the velocity (constant along streamlines)

is vðxf; tfÞ ¼ 1� @xfc ðxf; tfÞ. By mass conservation the

inverse Lagrangian map must satisfy the Monge-Ampereequation

det@x0@xf

¼ �fðxfÞ�0ðx0Þ ; (16)

where �0ðxÞ � �ðx; t0Þ is the initial state and �fðxÞ ��ðx; tfÞ is the final state. In 1D this equation is immediately

solved in terms of the cumulative mass functionsdMf

dx ¼ �f

and dM0

dx ¼ �0. The inverse Lagrangian map is then deter-

mined by M0ðx0Þ ¼ MfðxfÞ. We note that in dimensions

higher than one the free-streaming motion has the follow-ing potential representation:

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x 0 ¼ @xf

�k xf k22

� tf � t0�

c ðxf; tfÞ�:¼ @xf�; (17)

and (16) becomes a partial differential equation in thegradient of the scalar field � � �ðxf; tf; t0Þ

det@2�

@x�f @x�f

¼ �fðxfÞ�0ð@xf�Þ : (18)

Combining (9) with (13) the optimal released heat can bewritten as

h�Qi ¼ ���1�Sþ 2hc ð�tf; tfÞ � c ð�t0

; t0Þi; (19)

where �S ¼ �hln�fð�tfÞ � ln�0ð�t0

Þi is the entropy

change. Similarly, the minimal expected work is

h�Wi ¼ hUfð�tf; tfÞ �U0ð�t0

; t0Þi þ h�Qi: (20)

The last term in Eq. (19) represents the dissipated workWdiss. Noticing that in the shockless Burgers equation2½c ðxf; tfÞ � c ðx0; t0Þ� ¼ � k xf � x0 k2 =ðtf � t0Þ, and

using x0 given by (17) and mass conservation, the dissi-pated work can be written as

Wdiss ¼ �

tf � t0hk �tf

� @�tf�ð�tf

; tf; t0Þ k2i: (21)

Equation (21) means that the initial and final states

can be specified by mass points fxð1Þ0 ; xð2Þ0 ; . . . ; xðNÞ0 g and

fxð1Þf ; xð2Þf ; . . . ; xðNÞf g, and a possible inverse Lagrangian

map by a one-to-one assignment xðiÞf ! xðjÞ0 . The inverse

Lagrangian map solving (18) is then given by the assign-ment which minimizes the quadratic cost function (21), anapproach which has been used with great success to recon-struct velocity fields in the early universe [23,24]. Theinterpretation of this quadratic cost function as dissipatedwork in stochastic thermodynamics is, up to now, to thebest of our knowledge, new.

Optimal work with given final control.—Optimal workwith given initial state and given final control can be seenas a variational problem over the (unknown) final stateminimizing the functional (20). This is the second resultof our Letter, and, as we will show below, for the examplespreviously introduced by Schmiedl & Seifert [12] theminimization can be done exactly. For general initial statesand final controls the minimization can be carried outnumerically using the Monge-Ampere-Kantorovich ap-proach or the variational equation; this is, however, some-what involved and will be left to a future contribution.

Optimal work in transitions in an optical trap.—Wecombine here the two examples discussed [12] movingthe center and changing the stiffness of a harmonic poten-tial. The initial potential isU0ðxÞ ¼k x k2 =2 and the initialstate, in equilibrium with this potential, is �0ðxÞ �expf��U0ðxÞg. The final control potential is UfðxÞ ¼ c kx� h k2 =2 and thus characterized by the position h and

stiffness c, which in the examples in [12] is � 1. We nowevoke that the transition is effectuated by changing theharmonic potential in some (time-dependent) manner. Thestochastic process inside the control interval is then ageneralized Ornstein-Uhlenbeck process, and it followseasily that the final state must also be Gaussian. We there-fore make the ansatz

Rðx; tÞ ¼ �rk x��t k2

2þ d

2�ln

r

2�: (22)

All the terms in (20) are now determined by (22), and weare left with an optimization problem in the two parameters� and r which is

Qð�; rÞ ¼ 1

2�

�c

r� 1

�þ c

2jq��j2 þ logr

2�

þ 1

tf � ti

�jqj2 þ ð1� ffiffiffi

rp Þ2

r�

�: (23)

The last term above follows from similarity when mappingan initial Gaussian state to a final Gaussian state by aLagrangian map which is a linear transformation.Variation of Q with respect to � and r gives

� tf ¼c�th

�tcþ 2�and r ¼ K2

ð�tÞ2 (24)

with K ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�tðc�tþ 2�Þ þ �2

p � �. For these examples,as found in [12], there are discontinuities in the controlboth at the initial and the final time.Optimizing the variance of the Jarzynski estimator.—We

now turn our attention to a different expectation value. TheJarzynski equality (1) is an equality in expectation, butdoes not hold for a finite number of samples [25]. Let therebe N independent measurements of the work; then the freeenergy difference is estimated as �F ¼ ���1 lnð1N �PN

i¼1 e��WiÞ, with a statistical error determined by

Var½e��W�=N. Moreover, expectation and variance of afinite sampling will depend upon the details of the drift. Ittherefore makes sense to study the expectation valueg�ðx; tÞ ¼ he���Wix;t, with respect to the initial state that

we assume in equilibrium at inverse temperature �. Usingthe approach of [9] g� can be shown to satisfy for any givenU a controlled diffusion equation which we can write forA� :¼ �ð��Þ�1 lng� as �@tA� þð2�� 1Þ@xU � @xA� þ��1@2xðA� �UÞ ¼ ð�� 1Þ k @xU k2 þ� k @xA� k2 . Nowthe extremum condition for the drift gives @xU� ¼ @x½ð1�2�ÞA� � R�=½2ð1� �Þ�, with R ¼ ��1 ln�. If we againsplit the drift into an equilibrium piece @xR and a remain-der @xc � ¼ �ð1� 2�Þ@xðA� þ RÞ=ð1� �Þ, we obtainthe generalized optimal transport equations

@t�þ 1

�@x � ð�@xc �Þ ¼ 0; (25)

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Page 4: Optimal Protocols and Optimal Transport in Stochastic Thermodynamics

@tc þ k @xc k22�ð1� 2�Þ þ

�ð@2xc Þ��ð1� �Þ ¼

�ð@xc Þ � ð@x�Þ���ð�� 1Þ : (26)

These equations are not immediately solved, and deservefurther study.

A toy model of nucleation.—As an illustration of newphysics which can be investigated by the methods intro-duced in this Letter, we consider the minimization of heatrelease in a toy model of nucleation in one dimension. Wemodel nucleation as a transition between states specifiedby assigning �Rðx; tÞ at times t0, tf such that it has local

minima at x �x, and such that at the initial time t0 thewell at x � �x has the lowest energy, while at the finaltime tf the global minimum is at x þ �x. An example is

�RiðxÞ ¼ x2ð2� x2= �x2Þ þ ð�1Þi‘xþ ��1 lnZið�Þ, i ¼ 0denoting the initial state and i ¼ 1 the final one while‘ > 0 governs the asymmetry of the wells. We expect atrade-off between dissipated work, which by (21) can bewritten as �th½@xc ðx; t0Þ�2i=�, for �t ¼ tf � t0, and not

moving the probability mass from the left to the rightpotential well. The latter action (or lack of action) entailsa cost �R, namely, the energy difference in the final statebetween the right and left potential wells. In the exampleabove �R 2�x‘. If the probability mass is moved,the ‘‘Burgers’’ velocity v ¼ @xc =� in the left wellcan be chosen to be uniform, i.e., @xc ðx; t0Þ 2� �x=�t.Balance is then obtained for 4� �x2=�t 2�x‘ or �t 2� �x=‘. If the duration is much shorter, it ‘‘pays’’ to leavethe mass where it is, while if it is much longer, it pays tomove. Finding the exact optimal protocol at around thecritical time requires solving the variational equations inthis Letter, which in one dimension reduce to the equation0 ¼ lnj@2x�j þ �R0ð@x�Þ ¼ R1ðxÞ for the Lagrangianmap � entering (17). Solving this equation is certainlynot trivial, even in one dimension, and depends on thedetails of the boundary conditions. Details of this calcu-lation will be reported elsewhere.

In summary, we have shown how stochastic optimiza-tion problems are solved by the methods of optimal con-trol. The solution is built on an auxiliary problem ofoptimal transport. When minimizing heat or work of asmall system this optimal transport is a classic of fluidmechanics and cosmology, namely, Burgers equation.Between any prescribed initial and final states, these prob-lems can be solved numerically with the Monge-Ampere-Kantorovich method, introduced to reconstruct velocityfields in the early Universe. The direct connection betweenoptimal transport and optimal protocols in small systemthermodynamics was wholly unexpected, and is promising,as it applies to a whole wide class of related optimizationproblems.

This work was supported by the Swedish ResearchCouncil through Linnaeus Center ACCESS and theFIDIPRO program Grant No. 129024 of the Academy ofFinland (E.A.), by the European Research Council

(C.M.M.), and the center of excellence ‘‘Analysis andDynamics Research’’ of the Academy of Finland(C.M.M. and P.M.G.). We thank two anonymous refereesfor insightful and constructive comments.

*[email protected][email protected][email protected]

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