Table Of ContentOptimal finite-time processes in stochastic thermodynamics
Tim Schmiedl and Udo Seifert
II. Institut fu¨r Theoretische Physik, Universit¨at Stuttgart, 70550 Stuttgart, Germany
For a small system like a colloidal particle or a single biomolecule embedded in a heat bath,
7
the optimal protocol of an external control parameter minimizes the mean work required to drive
0
the system from one given equilibrium state to another in a finite time. In general, this optimal
0
protocolobeysanintegro-differentialequation. Explicitesolutionsbothforamovinglasertrapand
2
a time-dependent strength of such a trap show finite jumps of the optimal protocol to be typical
n both at thebeginning and the end of the process.
a
J
PACSnumbers: 82.70.Dd,87.15.He,05.40.-a,05.70-a
3
2
Introduction.–Theconceptsofclassicalthermodynam- cycles. Inourcontext,thesystemalwaysremainsincon-
]
h ics like applied work and exchanged heat can be ap- tact with a single heat bath of constant temperature T.
c plied to soft and biomatter systems as the study of the Moreover, this heat bath provides thermal fluctuations
e drivendynamicsofsinglecolloidalparticlesandofsingle which require a stochastic formulationin contrastto the
m
biomolecules has shown [1]. Since thermal fluctuations deterministicdescriptioninmacroscopicfinite-timether-
- contributesubstantially,workandheatacquireastochas- modynamics.
t
a ticcontributionandmustbedescribedbyprobabilitydis- The model.– Paradigmatically, a Langevin equation
t
s tributions. Exact results constraining such distributions describes the driven overdamped motion of a single de-
.
t liketheJarzynskirelation[2]andvariousgeneralizations gree of freedom with co-ordinate x in a time-dependent
a
thereof have been derived theoretically [3, 4, 5, 6, 7] and one-dimensional potential V(x,λ(τ)) as
m
tested experimentally [8, 9, 10, 11, 12]. Typically, these
- ∂V(x,λ)
d exact relations hold for any time-dependent driving de- x˙ =−µ +ζ. (1)
∂x
n scribed by an external control parameter λ(τ).
o Here, µ is the mobility and time-derivatives are denoted
c In this paper, we ask for the optimal protocol λ∗(τ) by a dot throughout the paper. The thermal fluctua-
[ that minimizes the mean work required to drive such a
tionsaremodelledasGaussianwhitenoisehζ(τ)ζ(τ′)i=
system from one equilibrium state to another in a finite
1 2µk Tδ(τ −τ′), with k as Boltzmann’s constant. The
B B
v time t. The emphasis on a finite time is crucial since for time evolution of the probability distribution p(x,τ) to
4 infinitetimetheworkspentinanyquasi-staticprocessis
observe the particle at position x at time τ is then gov-
5 equal to the free energy difference of the two states. For
erned by the Fokker-Planck equation
5
finitetimethemeanworkislargerandwilldependonthe
1
protocolλ(τ). Knowingtheoptimalprotocolλ∗(τ)could ∂V
0 ∂ p(x,τ)=∂ µ +µk T∂ p(x,τ). (2)
7 interaliaimprovetheextractionoffreeenergydifferences τ x ∂x B x
(cid:20) (cid:21)
0 fromfinite-timepathsamplingbothinvariousnumerical
Initially, the system is in thermal equilibrium in the
/ schemes[13, 14,15,16,17,18, 19,20]andinexperimen-
t potential V(x,λ ). During the time-interval 0 ≤ τ ≤ t,
a talstudies[21,22]. Quitegenerally,thesmallerthemean i
m the control parameter λ(τ) is varied from λ to the final
workis,thebetterthestatisticsforfreeenergyestimates i
value λ . The mean work spent in this process
- becomes[17, 23]. Apriori,one mightexpectthe optimal f
d
n protocol connecting the given initial and final values to t ∂V
o besmoothasitwasfoundrecentlyinacasestudywithin W[λ(τ)]= dτλ˙ ∂λ(x(τ),λ(τ)) (3)
c the linear response regime [24]. In contrast, as our main Z0 (cid:28) (cid:29)
v: result, we find here for genuine finite-time driving that becomes a functional of the protocol [λ(τ)] where the
i the optimal protocol involves discontinuities both at the average h....i is over the initial thermal distribution and
X
beginning and the end of the process. over the noise history. For notational simplicity, we set
r k T = µ = 1 in the following by choosing natural units
a B
For macroscopicsystems, optimalprocesseshave been
for energies and times. We will first investigate two case
investigatedunderthelabeloffinite-timethermodynam-
studies motivated by previously set up experiments on
ics for quite some time [25, 26, 27]. Indeed, jumps were
colloidal particles and then analyze the general case.
found there as well [26] despite the significant differ-
CasestudyI:Movinglasertrap.–Asanalmosttrivial,
ences both in the role of heat baths and the equations
but still instructive introductory example, we consider a
of motion between macroscopic and stochastic thermo-
colloidal particle dragged through a viscous fluid by an
dynamics. For the former, heat reservoirs of different
optical tweezer with harmonic potential
temperature are typically involved in a search for opti-
maladaptionsof,e.g.,Carnot-likemachinestofinitetime V(x,τ)=(x−λ(τ))2/2. (4)
2
The focus of the optical tweezer is moved according to Case study II: Time-dependent strength of the trap.–
a protocolλ(τ). In previous experiments, such protocols Inthisexample,fluctuationsarecrucial. Weconsiderthe
havebeenusedtotestthefluctuationtheorem[8]andthe motionofacolloidalparticleinatrapwhosestrengthbe-
Hatano-Sasa relation [10]. The optimal protocol λ∗(τ) comes time-dependent whereas its position remains con-
connectinggivenboundaryvaluesλ =0andλ inatime stant. The corresponding potential reads
i f
t minimizes the meantotalwork(3) whichweexpressas
V(x,τ)=λ(τ)x2/2 (11)
a functional of the mean position of the particle u(τ) ≡
hx(τ)i as with λ(0) = λ and λ(t) = λ > λ as boundary condi-
i f i
tions. Such a potential with a sudden jump protocolhas
t t
W[λ(τ)] = dτλ˙(λ−u)= dτ(u˙ +u¨)u˙ been investigated experimentally as test of the fluctua-
Z0 Z0 tion theorem [9]. We first derive the equation of motion
= tdτu˙2+ u˙2 t /2. (5) for the variance w(τ)≡ x2(τ)
0
Z0 w˙ =(cid:10)−2λw(cid:11)+2 (12)
(cid:2) (cid:3)
Here, we have used
by multiplying eq. (2) with x2 and integrating over x.
The mean work (3) can then again be cast in a local
u˙ =(λ−u) (6)
functional of the new variable w and its first derivative
which follows from averaging the Langevin equation. by solving (12) for λ˙
The Euler-Lagrange equation corresponding to (5), t w 1 1 t w˙2
u¨=0,issolvedbyu(τ)=mτ,whereu(0)=0isenforced W[λ(τ)]= dτλ˙ = [λw−lnw]t + dτ (.13)
2 2 0 4 w
bytheinitialcondition. Eq. (6)thenrequiresthebound- Z0 Z0
ary conditions u˙(0)= λ −u(0)=0 and u˙(t)=λ −mt The minimization of the work functional then requires
i f
which can only be met by discontinuities in u˙ at the solving the Euler-Lagrangeequation
boundaries which correspond to jumps in λ. Note that
w˙2−2ww¨ =0. (14)
these “kinks” do not contribute to the integral in the
second line of eq. (5). The yet unknown parameter m Its general solution
follows from minimizing the mean total work
w(τ)=c (1+c τ)2 (15)
1 2
W =m2t+(λ −mt)2/2 (7)
f
contains two constants. The thermal initial distribution
which yields m∗ = λf/(t+2). The minimal mean work w(0) = 1/λi fixes c1 = 1/λi. The second constant c2
W∗ =λ2/(t+2)vanishesinthequasi-staticlimitt→∞. follows from minimizing the total mean work
f
The optimal protocol then follows from eq. (6) as (c t)2 1 1
W = 2 −ln(1+c t)+ (λ /λ )(1+c t)2− (16)
2 f i 2
λ∗(τ)=λ (τ +1)/(t+2), (8) λit 2 2
f
which leads to
for 0<τ <t. As a surprising result, this optimal proto-
−1−λ t+ 1+2λ t+λ λ t2
col implies two distinct symmetrical jumps of size c∗t= f i f i . (17)
2 2+λ t
p f
∆λ≡λ(0+)−λ =λ −λ(t−)=λ /(t+2) (9)
i f f
The optimal protocol derived from eq. (12)
at the beginning and the end of the process. λ −c∗(1+c∗τ)
A priori, one might have expected a continuous linear λ∗(τ)= i 2 2 (18)
(1+c∗τ)2
protocolλlin(τ)=λ τ/ttoyieldthelowestworkbutthe 2
f
explicite calculation shows that for 0 < τ < t again implies jumps at the beginning and
endoftheprocessasshowninFig. 1a. Boththeminimal
Wlin =(λf/t)2(t+e−t−1)>W∗ (10) work W∗, see Fig. 1b, and the scaled optimal protocol
λ∗(τ/t)/λ depend only on two parameters (λ /λ ) and
i f i
for any t >0, with a maximal value Wlin/W∗ ≃1.14 at
λ t.
i
t≃2.69.
For the two limiting cases of an immediate jump, t→
Inmacroscopicfinite-timethermodynamics,theoccur-
0, and a quasi-static process, t → ∞, respectively, the
rence of such jumps has previously been rationalized by values of W∗ also follow from general principles. For an
pointingoutthespecialnatureofthis typeofvariational
immediate jump, the minimal work
problem where the highest derivative (here u¨ in eq. (5))
occurs linearly [26]. In the present model where fluctua- Wjp ≡ limW∗ = (λf −λi)x2 = 1((λ /λ )−1)
f i
tions are irrelevant to the mean work, these jumps have t→0 2 2
(cid:28) (cid:29)λi
the same formal origin. (19)
3
(a) (b)
4.5
45 λλλλλλffffff//////λλλλλλiiiiii======222555,,,,,,λλλλλλiiiiiitttttt======1010110..011 3 .345 λλλλλλiiiiiitttttt==→=→=00110∞..0022
2.5
∗
λi 3 W 2
/
∗
λ
1.5
2
1
1 0.5
0
-0.2 0 0.2 0.4 0.6 0.8 1 1.2 2 3 4 5 6 7 8 9 10
τ/t λ /λ
f i
(c) (d)
10 100
ln(W∗/Wq s ) 4 6 0.9 1
1 21 45 10 00..76 0.9
3 0.55 0.8
0.7
2
λti 0.1 01 λti 1 00..56
0.01 0.1 ∆λ/λf
(W∗−Wqs)/Wjp
0.001 0.01
1 10 100 1000 10000 100000 10 100 1000 10000
λ /λ λ /λ
f i f i
FIG.1: (coloronline)Optimizationresultsforatime-dependentstrengthλ(τ)ofalasertrapfordifferentvaluesof(λf/λi)and
∗ ∗
λit (case study II): (a) Optimal protocols λ /λi as a function of the scaled time τ/t. (b) Minimal work W . (c) Logarithmic
fraction ln(W∗/Wqs) of the optimal work and the quasistatic work. (d) Relative height ∆λ/λ of the jump of the optimal
f
protocol at t=0 in unitsof (W∗−Wqs)/Wjp.
is equaltothe difference inenergyevaluatedinthe ther- The originof this surprising features lies in the fact that
mal initial ensemble. In this limit, the optimal protocol the relaxation time scales like 1/λ. For large λ, the par-
λ∗(τ)/λ ≈((λ /λ )+1)/2 is constant for 0<τ <t but ticle can follow a larger change of the control parameter
i f i
hasdiscontinuitiesatτ =0andτ =t. Inthequasi-static almost quasi-statically. Therefore, the optimal protocol
limit, the minimal work can become quite steep towards the end of the process
for large λ .
f
1
Wqs ≡ lim W∗ = ln(λ /λ )=∆F (20) General case. – For a general non-harmonic potential,
f i
t→∞ 2 it is not possible to express the mean work as a local
isequaltothefreeenergydifference∆F betweenthefinal functional of just one variable as we have done for the
and the initial state. In this limit, the optimal protocol two harmonic cases. Rather, our optimization problem
is continous at τ =0 and τ =t and takes the form becomesnon-localintime sincechangingthe protocolat
a time τ affects the mean work increments for all later
λ∗(τ/t)≈ λi . (21) times τ′ > τ. This fact becomes obvious by expressing
2
the mean work as a path integral average
1−(τ/t)+ (λ /λ )(τ/t)
i f
(cid:16) p (cid:17) W[λ(τ)]= d[x(τ)]p[x(τ)] tdτλ˙∂V (22)
For (λf/λi) ≫ 1, the minimal work is of the order ∂λ
of the quasi-static work for any λ t∗ ≫ 2/ln(λ /λ ) as Z Z0
i f i over all possible trajectories x(τ) with weight
a simple analysis of eqs. (16) and (17) shows. Thus,
the larger the change of the control parameter λ, the p[x(τ)]=Np(x,0)exp − tdτ (x˙ +∂xV)2 − ∂x2V ,
smaller is the time-interval required to essentially reach 4 2
(cid:20) Z0 (cid:18) (cid:19)(cid:21)
thequasi-staticwork,asquantitativelyshowninFig. 1c. (23)
4
where N is a normalization constant. Minimizing the cyclic processes combining mechanical steps with chemi-
meanwork(3) thenrequiressolvingthe non-localEuler- calreactionsgivenafinitecycletime. Theseperspectives
Lagrange equation to be investigated in future work demonstrate that the
optimizationproblemintroducedhereforstochasticther-
d ∂V δW[λ(τ)]
= (24) modynamics has not only a broad fundamental signifi-
dτ ∂λ δλ(σ)
(cid:28) (cid:29)|τ=σ cance. Its ramificationscould ultimately also lead to the
wheretherighthandsidecanbeexpressedbycorrelation construction of “optimal” nano-machines. Finally, it is
functions as tempting to speculate which, if any, biological processes
onthe cellularandsubcellular levelhavebeen optimized
d ∂V ∂2V
= λ˙ + (25) during evolution for their finite-time performance in the
dτ ∂λ ∂λ2
(cid:28) (cid:29)|τ=σ (cid:28) (cid:29)|τ=σ noisy cellular environment.
t λ˙ ∂3V ∂2V ∂V
dτ −(x˙ +∂ V) · .
Zσ 2 *(cid:18)∂λ∂2x x ∂λ∂x(cid:19)|τ=σ ∂λ|x(τ)+
Ingeneral,thisintegro-differentialequationsolvedbythe
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