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Extracting work from stochastic pumps

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J. Stat. Mech. (2011) P09020

(http://iopscience.iop.org/1742-5468/2011/09/P09020)

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ech.(2011)

P09020

ournal of Statistical Mechanics:J Theory and Experiment

Extracting work from stochastic pumps

Saar Rahav

Schulich Faculty of Chemistry, Israel Institute of Technology, Haifa 32000, IsraelE-mail: rahavs@tx.technion.ac.il

Received 19 July 2011Accepted 23 August 2011Published 20 September 2011

Online at stacks.iop.org/JSTAT/2011/P09020doi:10.1088/1742-5468/2011/09/P09020

Abstract. The efficiency at maximum power output of stochastic pumps,systems driven away from equilibrium by a periodic variation of parameters,is investigated by studying simple models in which the driving consists of abruptjumps of system parameters. The models are chosen to differ by qualitativeaspects of the dynamics, namely the degree of directionality and cycle control.The behaviour at maximum power output is found to depend on the choice ofparameters used to maximize the output. For a natural choice, in which theexternal force and the period of the driving are varied to increase the poweroutput, numerical and analytical results suggest that the efficiency at maximumpower output is at most 1/2.

Keywords: driven diffusive systems (theory), molecular motors (theory)

c©2011 IOP Publishing Ltd and SISSA 1742-5468/11/P09020+22$33.00

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Contents

1. Introduction 2

2. Stochastic pumps as thermodynamic machines 4

3. Employing simple models to study pumps operating at maximum poweroutput 6

4. Fully directional one-cycle stochastic pump 94.1. Maximum output conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5. Partially directional, cycle controlled pump 15

6. Fully directional pump with two competing fundamental cycles 16

7. Summary 19

References 21

1. Introduction

Biological organisms are teeming with molecular machines—molecule sized objects whichperform useful tasks. This fascinating fact has motivated an enormous research effortaiming to understand the mechanisms by which such machines operate [1, 2]. One ofthe properties characterizing any type of machine is its efficiency. In his celebrated workCarnot [3] computed the efficiency of an idealized macroscopic machine driven by thetemperature difference between a hot and a cold reservoir. He found that the efficiencyis at most ηC = 1 − T1/T2, where T1,2 are the temperatures of the cold and hot baths,respectively. The bound is realized by thermodynamically reversible driving cycles.

Molecular machines are driven by a mechanism which is quite different from the onedriving the Carnot cycle. Many machines of biological origin are driven by chemicalpotential differences, for example by hydrolysis of ATP into ADP. Thermal ratchets [4, 5]are a class of systems driven by a combination of thermal diffusion and switching betweendifferent potential landscapes. A large body of work has been dedicated to investigatingthe thermodynamical properties of such systems, and one of the properties of interest istheir efficiency [6]. Interestingly, it has been pointed out by Parrondo et al [7, 8] that theefficiency of a class of models, which they termed reversible ratchets, can be arbitrarilyclose to 1. Such machines can operate without losses.

In a realistic setting, machines are not operated in a reversible manner, lest it wouldtake an infinite amount of time to perform any useful action. It is therefore of interestto study thermodynamical machines in more relevant modes of operation. A commonlystudied alternative is that of maximal power output. This regime is obtained by optimizingparameters to increase the output. Curzon and Ahlborn [9] studied the analogue of a

Carnot cycle at maximum output and found η∗ = 1 − √T1/T2, which is smaller than

the Carnot efficiency. In recent years a renewed interest in the efficiency at maximumpower of small thermodynamics machines has led to the investigation of many interestingmodels [10]–[16]. This research effort was motivated by the possibility that there exists

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a universality in the first few powers of the expansion of η∗ as a function of ηC for smalltemperature differences—though it is clear now that this is true only under nontrivialsymmetry of the energy dissipation in the cold and hot baths [12].

For various models of microscopic machines operating near maximum output thereare several different predictions for the efficiency. Schulman et al [14] argued that foran autonomous Carnot-like system the efficiency at maximum output satisfies η∗ ≤ 1/2from thermodynamic considerations. Seifert has studied a different model and foundefficiencies which can be larger than 1/2 [16]. He further identified three broad regimescharacterizing the behaviour of microscopic machines: strong and efficient, strong andinefficient, and balanced. The apparent diversity in the literature stems from differentchoices of models and of what can be optimized in order to maximize the output. Sucha diversity is inevitable once different types of parameters can be optimized to maximizethe output. In spite of this drawback, the maximum output conditions still constitute amore representative characterization of a realistic operational regime for a machine, whencompared to a reversible driving cycle.

Recent technological advances offer us the possibility of designing and synthesizingmolecules which operate as machines [17]–[20]. These new advances also allow us todevelop new methods of driving molecular machines. One such driving mechanism employsa periodic time variation of system parameters, which is used to drive the molecularmachine out of equilibrium. This mechanism is based on the fact that the system lagsbehind its steady-state distribution, since the time variation of parameters changes thelatter. Due to the analogy with ordinary, day to day, pumps, such systems are termedstochastic pumps [21]. Several recent papers have studied stochastic pumps [22]–[28].Much of this work was motivated by no-pumping theorems, specifying conditions underwhich the driving cycle does not lead to any directed motion [29]–[34].

The driving mechanism of stochastic pumps differs from that of the various modelswhich were investigated under maximum output conditions. It is therefore of great interestto study similar questions for models of stochastic pumps. It should be noted thatthe results are expected to depend on the parameters used to maximize the output.Furthermore, it is technically difficult to perform the optimization over the shape ofthe driving cycle in parameter space. We circumvent this difficulty by restricting outattention to models which are driven by sudden jumps of parameters. The choice ofmodels and of the parameters varied to maximize the output, namely the external forceand period of the driving cycle, is motivated by an experimental realization of a stochasticpump [35]. It should be stressed that the simple models used in the following do notmake the problem trivial. They allow us to investigate how qualitative properties of thedynamics, such as the degree of directionality and cycle control, affect the performance ofstochastic pumps. While finding the maximum output conditions of stochastic pumps isof theoretical interest, there may also be practical consequences. A better understandingof the maximum output conditions for the models studied here will inevitably lead tobetter experimental designs for stochastic pumps.

In section 2 we use models of time-dependent jump processes to represent stochasticpumps. Thermodynamic properties such as the output power and efficiency of stochasticpumps are defined. In section 3 we describe the approach employed to study the maximumpower-output conditions. In the absence of a full solution we elect to study severalsimple models, constructed to differ by qualitative aspects of their stochastic dynamics,

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c1

c2

1

23

4

5

Figure 1. Graph depicting a stochastic jump process. The nodes correspondto the states of the system while the links represent the possible (bidirectional)transitions between states.

which we term directionality and cycle control. These dynamical properties are definedand discussed in section 3. The three different models are presented and investigatedin sections 4–6 using a combination of analytical and numerical methods. Finally, wesummarize the results in section 7 and draw conclusions on the influence of the degreedirectionality and cycle control on the performance of stochastic pumps.

2. Stochastic pumps as thermodynamic machines

Stochastic jump processes, where the system jumps between a finite number of states,with thermally activated transitions, serve as one of the representative models for smallsystems driven away from thermal equilibrium. Jump processes can be used to describemany, if not all, the mechanisms in which stochastic machines can operate. Here weemploy stochastic jump processes to study stochastic pumps, to be defined below, anddiscuss the thermodynamic quantities which characterize their operation.

The dynamics of a jump process can be helpfully described using a graph (see figure 1for an example). The nodes of the graph represent the states of the system while the linkscorrespond to the non-vanishing transitions between the states. The transitions betweenstates are assumed to be thermally activated, with rates of the form

Rij = eβ(Ej−Wij+fijθij). (1)

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In equation (1) Ej is the energy of the (isolated) system when it is in state j, Wij = Wji

denotes the energy barrier for thermally activated transitions between states i and jand fij = −fji denotes external thermodynamical (or mechanical) driving forces, due tocoupling to some external agent. The load distribution factor 0 ≤ θij ≤ 1 describes howthese forces affect the two transition rates involving states i, j [2]. By definition, it satisfies

θij + θji = 1. We note that the relation between the system parameters �λ = {E, W, f, θ}and the transition rates Rij is not unique. Different sets of parameters may result in thesame set of transition rates.

The system’s probability distribution evolves according to a master equation

dP

dt= R(�λt)P. (2)

The diagonal elements of the transition rate matrix R are set to Rii = −∑j �=i Rji, thereby

ensuring conservation of probability. The momentary rate of transitions between a pairof states i, j is Jij ≡ RijPj − RjiPi. We refer to these fluxes as probability currents. Ifthe transition rates are fixed (in time), a probability distribution evolving according toequation (2) will decay to a unique steady state as long as the transitions are bidirectionaland the system cannot be decomposed into separate, independent, subsystems [36]. Whenthe external forces vanish (fij = 0) the system relaxes to an equilibrium state with aBoltzmann probability distribution, Peq(i) ∝ e−βEj .

Models based on equations (1) and (2) are commonly employed to represent molecularmotors and machines, whose dynamics is inherently stochastic due to their small size [2].Such microscopic machines can perform useful work only when they are driven awayfrom thermal equilibrium. The two distinct driving mechanisms mentioned earlier canbe represented using jump processes. Non-vanishing time-independent external forces fij

correspond to a fixed thermodynamical force which can bring the system into a steadystate. The time-dependent driving is represented by time variation of system parameters.Here we focus on the time-dependent driving mechanism, and consider only systems whichare driven by a variation of energies and barriers.

Stochastic systems driven by a time-periodic driving protocol are often calledstochastic pumps, due to the analogy with ordinary pumps. The Floquet theorem forstochastic dynamics [4] ensures that the probability distribution will tend to an asymptoticperiodic solution, which we denote as ρ(t). Since short-time transients relax after a typicaltime, it is this periodic solution which characterizes the operation of a stochastic pump.

The probability currents Jij(t) give information about the average rate of transitionsbetween states i and j at time t. However, during the driving cycle the probabilitydistribution may slosh back and forth between various states and therefore the probabilitycurrents include a lot of spurious information which is not relevant to the thermodynamicsof the stochastic pump. Instead, many physical properties of interest can be expressed interms of the (time-)integrated currents

φij ≡∮

dt Jij(t) =

∮dt (Rij(t)ρj(t) − Rji(t)ρi(t)) (3)

expressing the accumulated transitions between states i and j during a full cycle. Suchcurrents typically exist even in absence of external forces as long as both energy levelsand barriers are varied in time [30].

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A cyclic variation of {Ei} and {Wij} will typically generate non-vanishing integratedprobability currents. These probability currents are driven by mechanical work performedon the system, and can be used to perform work against external forces {fij}, which wetake to be time-independent. When work is taken from the cyclic driving and used to drivethe pump against the external load we will say that the system operates as a stochasticpump. It is absolutely crucial to note that stochastic pumps work in a regime which isinherently far from equilibrium when the external force is finite. This stems from the factthat for weak enough driving the external resisting force would dominate and determinethe direction of probability currents. The same would be true for very fast, or veryslow, driving cycles. Therefore, stochastic pumps are also inherently non-adiabatic. Thisconsiderably complicates their analysis.

The work done on the system during a driving cycle is given by

W ≡∑

i

∮dt

dEi

dtρi(t) = −

∑

i,j

∮dt Ei(t)Rijρj(t) = −

∑

i,ji>j

∮dt(Ei(t) − Ej(t))Jij(t). (4)

Equation (4) expresses the fact that varying the energy of a state occupied by the systemwill cost some work. In contrast, no work is required to vary any of the barriers, since thesystem is assumed to spend a negligible amount of time in their vicinity.

The driving cycle generates integrated probability currents, φij, which then performwork against the (time-independent) external forces. We define the output power of thesystem as

Φ ≡ − 1

T

∑

i,ji>j

φijfij (5)

where the sum is over all possible transitions with an arbitrary choice of direction. Tis the period of the driving cycle. When the system operates as a stochastic pumpboth the work and output power should be positive. Under those conditions the firstlaw of thermodynamics for a full period of the driving cycle is W = Q + TΦ, whereQ =

∑i,ji>j

∮dt Jij(t) ln(Rij(t)/Rji(t)) is the heat exchanged with the heat bath. Finally,

the pump efficiency is the ratio of the work that the pump performs against the externalforces and the work invested in order to generate the directed motion. For a completecycle the efficiency is

η =TΦ

W. (6)

The work, power output and efficiency characterize the thermodynamic properties of thestochastic pump. They all depend on details of the model, the driving cycle and theexternal forces.

3. Employing simple models to study pumps operating at maximum power output

The maximal power output of any model of a thermodynamical machine inherentlydepends on the choice of model parameters used in the optimization. For the modelsdescribed in section 2 these are the period of the driving cycle, the external forces and the

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shape of the driving protocol in parameter space. This is a well defined problem, but itturns out that unless one constrains the possible driving cycle the problem becomes trivial.The reason is that by increasing the driving strength (scale of energy variation during thecycle; see equation (4)) while keeping all other parameters fixed, one also increases theoutput of the pump. However, the return (in the power output) is not comparable to theincrease in work performed on the system, resulting in a vanishing efficiency at maximumoutput—a trivial result. We circumvent this difficulty by arguing that in realizations ofstochastic pumps the driving strength is bounded, due to internal properties of the system.An example for such a system is the molecular machine of [35], where the system statescan be modelled by a discrete set of chemical configurations. The results presented inthe following sections conform to this argument, showing efficiencies which decrease withincreasing driving strength.

Finding the maximum output necessitates finding the optimal path of the drivingcycle in parameter space, with the aforementioned constraint on the maximally allowedenergy variation. This is a problem whose general solution is not known. To proceedwe restrict ourselves to simple models in which this difficulty is circumvented. In thefollowing we present the class of models and argue that while they are not general, theirstudy could be quite instructive.

The models we employ have driving cycles which are composed of segments. In eachsegment of the driving cycle the master equation (2) is autonomous. The driving is per-formed by a sudden variation of energies and barriers between the different segments. Thisprescription is convenient technically, since the dynamics in the segments can be studiedemploying methods based on spectral decomposition. We stress that restricting the typeof possible driving cycle in this way does not make the problem of finding the maximumoutput conditions trivial. Even after the various assumptions specified above, the exter-nal forces and the period of the driving cycle should be varied to obtain the maximumoutput. The external force and the period are the parameters which are most likely to beexternally controlled in realistic realizations of stochastic pumps. Therefore, it is possiblethat simplified models used here may prove useful in the analysis of future experiments.

In the absence of a general theory for the maximal output conditions we choose themodels to provide a simple, qualitative understanding of stochastic pumps, with the aimof aiding the design and construction of future realizations of such models. The models weemploy are constructed to differ by qualitative aspects of the stochastic dynamics, whichwe term directionality and cycle control. We now explain what is meant by these terms.

The description of the steady state of a jump process, and, as we shall see shortly,also of the asymptotic periodic state of a stochastic pump, can be simplified considerablywith the help of network theory. Network theory, which is summarized in a seminal paperby Schnakenberg [37], allows us to describe the steady state in terms of the so-calledfundamental cycles of the graph. Closed cycles on a graph are sets of links allowing usto make transitions starting and ending in the same node without retracing our steps. Inthe network theory a smaller set of fundamental cycles is defined, so that any other cyclecan be expressed as a linear combination of the fundamental cycles. As an example, thegraph depicted in figure 1 has three simple cycles but only two fundamental cycles. Oneof the possible choices of the two fundamental cycles in this basis is depicted in the figure.The algorithm used to identify the fundamental cycles is not needed here. It can be foundin the papers by Schnakenberg [37] and Andrieux and Gaspard [38].

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One of the central results of network theory is that all currents at steady state canbe described using a set of steady state currents on the fundamental cycles. The factthat the number of these independent currents is smaller than the number of all possiblecurrents is a result of conservation of probability. In a steady state the probability fluxesflowing into and out of any state must balance each other. This leads to linear relationsbetween the currents which result in a reduced number of independent currents. Thispart of network theory is simply an application of Kirchoff’s circuit laws [39] to stochasticsystems. In addition, network theory also allows us to write the entropy production atsteady state in terms of the fundamental cycle currents and of mesoscopic affinities. (Oneaffinity for each cycle.) These affinities are given in terms of the logarithm of the ratiobetween a product of transition rates traversing the cycle in one direction and a similarproduct traversing the cycle in the opposite direction. This network theory has proveduseful since it reduces the number of variables needed to describe the steady state.

The network theory applies for systems at steady state, while we consider time-dependent systems. However, we are only interested in the long-time asymptotic stateof stochastic pumps, which is periodic in time. Since this probability distribution, ρ(t),returns to its value after each driving cycle, a stroboscopic version of network theoryemerges. In this version the probability currents are replaced by the time-integratedcurrents φij, which must satisfy

∑j �=i φij = 0 for ρ(t) to be periodic. These are precisely

the same type of constraints encountered for a steady state. This means that thecurrents φij can be decomposed into fundamental cycle integrated currents in exactlythe same way that steady-state currents are decomposed. By substituting this currentdecomposition into equation (5), one can also show that the output can be written as asum of contributions from fundamental cycles.

We turn to introduce the notion of cycle control. Consider a stochastic pump whosegraph admits several possible fundamental cycles. We assume that the external force isonly coupled to one of these cycles, say c1. When the pump is driven, non-vanishingintegrated currents will typically be generated in all of the fundamental cycles. It isclear that the work invested in generating the currents in the cj , j �= 1 cycles, which arenot coupled to the external load, is wasted. Therefore, the question of cycle control—our ability to focus the probability currents to a specific cycle—is very relevant to thecharacterization of the thermodynamical properties of the pump.

The second qualitative aspect of the stochastic dynamics we use is directionality.Consider a one-cycle pump, with three states, as depicted in figure 2. Let us study asegment of the driving cycle in which the driving is designed to favour transitions fromstate 1 to state 2. For thermally activated transitions it is in general not possible todirectly control the path in which the probability flows. It can flow directly along the1 → 2 link, or indirectly through state 3. Assuming a resisting force which is equallypresent on all links of the fundamental cycle it is clear that the direct transitions (φ21)increase the output, and that the indirect transitions (φ31 and φ23) will reduce the pump’soutput. We will use the term directionality to refer to our ability to control the directionof the probability currents in each fundamental cycle of a graph. It should be evident thatdirectionality is also a relevant measure for the thermodynamical properties of the pump,since it is coupled directly to the output. For the three-state system depicted in figure 2,one can give a more precise definition of directionality by separating the integrated currentin each link into its positive and negative components φ+

ij =∮

dt Jij(t)Θ(Jij(t)), with Θ

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Figure 2. A model of a three-state stochastic pump used to illuminate the notionof directionality. The driving cycle generates probability currents, designed dowork against the external force f . Consider a segment of the cycle: the probabilityis assumed to predominantly flow from state 1 to 2. It can do so directly (φ21 > 0)but also indirectly (with φ31, φ23 > 0). Directionality measures how much thedirect route is preferred over the indirect one

the Heaviside function, and φ−ij = φij − φ+

ij. A possible measure for directionality is then

the ratio φ+ij/(|φ+

ij|+|φ−ij|). A similar notion of directionality was recently used by Efremov

and Wang to characterize track-walking molecular motors [40].

Due to the difficulty in finding the maximum output for general driving cycles we focuson qualitative aspects of this question, namely, how properties such as directionality orcycle control of a stochastic pump would affect its output and efficiency. Our approach isto construct simple models which differ either by their directionality or by the presenceof competing fundamental cycles, and compare the results obtained from the differentmodels. We expect that the qualitative insights gained from the simple models would beof relevance to other, more complicated stochastic pumps. The three different models arepresented and analysed in sections 4–6.

4. Fully directional one-cycle stochastic pump

We start from the simplest of the models, consisting of three states and one fundamentalcycle. Furthermore, we assume that the direction of probability currents is also fullydetermined, in the following sense. If a stage in the driving cycle was designed to

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predominantly drive transitions from state i to j, then during this stage the barriersto all transitions not on the wanted route from i to j are raised so that they are effectivelyprevented. (The route from i to j will typically be chosen to increase the output.) Itshould be noted that microscopic reversibility is not violated here, since the transitionrates Rij and Rji will be non-vanishing at the same time. It is the average probabilitycurrent that flows in a known, controlled, direction at each stage of the cycle. We willcall such models fully directional. We imagine a driving cycle where the probability flowtakes the path 1 → 2 → 3 → 1 between the three states.

We focus on a cycle in which one of the energies take a value of E0 < 0, while thetwo other take the value E0 + δE , where δE > 0. (δE is a measure of the strength ofthe driving.) As a result, at each stage of the driving cycle probability would flow (onaverage) to the state with energy E0. Between the stages of the driving cycle the energyof the lowest state is suddenly raised by δE while the energy of another is lowered byδE , thereby encouraging the probability distribution to flow to that state. To be morespecific, at the first stage of the cycle (0 ≤ t < τ), we have

RI23 = RI

31 = RI32 = RI

13 = 0 RI12 = exp[βE0 + βf(1 − θ)]

RI21 = exp[β(E0 + δE) − βfθ].

(7)

Here f is the external force that the system is supposed to do work against, while 0 ≤ θ ≤ 1is a load distribution factor. In this stage the energy E2 = E0 is the lowest energy, leading toprobability flow from state 1 to state 2 (assuming the periodic solution ρ(t)). The systemis then left to relax for a time τ . The role of the barriers is implicit in equations (7), butit should not be discounted. The rates connecting state 3 in equation (7) vanish due toinfinite barriers between states 2 and 3 and similarly 1 and 3. There is also a finite-heightbarrier between states 1 and 2, and its height is chosen to be the origin of the energy axis.We note that during this stage of the cycle only transitions 2 ↔ 1 are possible. This isthe reason we use apply the term fully directional to this model.

At the end of the first stage of the driving cycle, but before the second stage, theenergy of state 3 is suddenly reduced to E0 while the energy of state 2 is raised to E0 + δE ,and the barriers are adjusted so that the non-vanishing rates for τ ≤ t < 2τ are

RII23 = exp[βE0 + βf(1 − θ)] RII

32 = exp[β(E0 + δE) − βfθ]. (8)

This stage is equivalent to stage I, with the roles of the states interchanged according to1 → 2 → 3 → 1, so that transitions 2 → 3 are favoured. The same is true for stage III(2τ ≤ t < T = 3τ), where

RIII31 = exp[βE0 + βf(1 − θ)] RIII

13 = exp[β(E0 + δE) − βfθ], (9)

are the non-vanishing rates. Note that we have chosen a model where all the stages of thedriving cycle are equivalent up to a permutation of states. This simplifies the followinganalysis considerably.

This simple driving cycle has an advantage over more general driving protocols. Ateach stage the system undergoes a simple relaxation process which can be readily solved.Let us define P0 as the initial probability distribution, PI as the distribution after the firststage, etc. Then, after the first stage we have

PI = TIP0, (10)

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with

TI =

⎛

⎜⎜⎜⎝

1 + yx

1 + y

1 − x

1 + y0

y(1 − x)

1 + y

y + x

1 + y0

0 0 1

⎞

⎟⎟⎟⎠

. (11)

Here y ≡ exp[βδE − βf ] and x ≡ exp[−τ exp(βE0 − βfθ){exp βδE + exp βf}]. y carriesinformation on the strength of the driving, δE , compared to the external resisting force f ,while x is a measure of the amount of relaxation during each stage of the driving cycle.

Using the equivalence of the different stages of the driving cycle, up to a cyclicpermutation of states, we immediately find

PII = TIIPI, (12)

with

TII =

⎛

⎜⎜⎜⎜⎝

1 0 0

01 + yx

1 + y

1 − x

1 + y

0y(1 − x)

1 + y

y + x

1 + y

⎞

⎟⎟⎟⎟⎠

, (13)

for the second stage. Similarly,

PIII = TIIIPII, (14)

with

TIII =

⎛

⎜⎜⎜⎝

y + x

1 + y0

y(1 − x)

1 + y0 1 0

1 − x

1 + y0

1 + yx

1 + y

⎞

⎟⎟⎟⎠

, (15)

for the third stage. The propagator of the full driving cycle is given by

T = TIIITIITI (16)

where

T =

⎛

⎜⎜⎝

(1+y)(x+y)(1+xy)+(1−x)3y3

(1+y)3(1−x)(x+y)[1+y(1+(1−x)y)]

(1+y)3y(1−x)(x+y)

(1+y)2

(1−x)y(1+xy)(1+y)2

(x+y)(1+xy)(1+y)2

1−x1+y

(1−x)(1+xy)[1+y(1+(1−x)y)](1+y)3

(1−x)[1+y+y2+x(−1+y2(x+y))](1+y)3

(x+y)(1+xy)(1+y)2

⎞

⎟⎟⎠ . (17)

We are interested in the long-time (asymptotic) periodic solution of the system, wherePIII = P0. The existence of such a solution is guaranteed by the Floquet theory fordissipative systems [4]. For this model the periodic solution, ρ, can be found analyticallyto satisfy

ρ0 = ρ(0) =1

Δ

⎛

⎝x + y1 + xy1 + xy

⎞

⎠ , (18)

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where Δ = 2 + x + y + 2xy. The stages of the cycle are identical up to an exchange ofstate indices, allowing us to write

ρI = ρ(τ) =1

Δ

⎛

⎝1 + xyx + y1 + xy

⎞

⎠ , (19)

and

ρII = ρ(2τ) =1

Δ

⎛

⎝1 + xy1 + xyx + y

⎞

⎠ . (20)

Given the solution for the probability distribution at the end of each stage, togetherwith the directionality, we can readily calculate the power output of the pump. Duringthe first stage of the driving protocol the only possible current is from state 1 to state2. Furthermore the integrated current is φ21 = ρI(2) − ρ0(2), whose contribution to theoutput of the pump is f [ρI(2)−ρ0(2)]. Collecting the contributions from all stages, whichare calculated similarly, we find

Φ =f

3τ[ρI(2) − ρ0(2) + ρII(3) − ρI(3) + ρ0(1) − ρII(1)] (21)

=f

τΔ(y − 1)(1 − x). (22)

The work done on the pump is due to the change in the well energies between the stages(see equation (4)). Between the first and second stage the system has the probabilitydistribution ρI, and the energy of state 2 (3) is raised (lowered) by δE , giving a contributionof δE [ρI(2) − ρI(3)]. Collecting the contributions from all the stages leads to

W =3δEΔ

(y − 1)(1 − x). (23)

The pump efficiency is calculated from equation (6). We find

η ≡ 3τφ

W=

f

δE . (24)

The efficiency is simply the ratio of the resisting force and the driving field δE . This simpleresult stems from the full directionality of the model, which restricts the probability flow,leading to a strong coupling between the work and the output. It should be noted thatthis model works as a stochastic pump only in the parameter regime where 0 < f < δE .When f < 0 the force and driving cycle work in tandem and it is not possible to arguethat one of them drives the system against the other. When f > δE the force overcomesthe driving cycle and drives the system in the wrong direction. It is also interesting tonote that an efficiency arbitrarily close to unity is obtained for forces close to the stallforce f = δE , independently of the period of the driving cycle.

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4.1. Maximum output conditions

We wish to identify the conditions of maximum output of this model, with the aim ofstudying thermodynamical properties of this stochastic pump working in, or close to,maximum output conditions. Our model, by its construction, does not allow us to optimizethe shape of the driving cycle, or even the size of the barriers. It does, however, allow usto vary the period of the driving protocol, T = 3τ , and the strength of the resisting force,f . We therefore maximize the output with respect to these parameters.

We recast the output (22) as a function of the force, and a rescaled time:

t ≡ τ exp(βE0 − βfθ)[exp βδE + exp βf ]. (25)

Written as a function of those variables, the output takes the form

Φ(t, f) =f

teβE0−βfθ

[eβδE + eβf

](eβδE−βf − 1

)(1 − e−t)

2 + e−t + eβδE−βf + 2eβδE−βfe−t. (26)

The maximum output conditions f ∗, t∗ are found by demanding that the partial derivativesof Φ vanish. This can be reduced to a single transcendental equation for the scaled timet∗:

24t∗ sinh2

(t∗

2

)[sinh t∗ − t∗] + ln

[eβδE (−1 + 3t∗ + cosh t∗ − 3 sinh t∗)

−1 − 3t∗ + cosh t∗ + 3 sinh t∗

]

× {3t∗[9 − 2t∗(−1 + 3t∗ + 2θ)] − 27t∗ cosh 2t∗

+ 6t∗(2θ − 1) cosh t∗(t∗ − sinh t∗) − [16 + 6t∗(1 − 9t∗ − 2θ)]

× sinh t∗ + 2 sinh 2t∗ + 4 sinh 3t∗} = 0. (27)

The force at maximum output is obtained from

f ∗ = δE + β−1 ln

[−2 + et∗ + e2t∗ − 3t∗et∗

1 + et∗ − 2e2t∗ + 3t∗et∗

]. (28)

The efficiency at maximum power can be calculated by substituting f ∗ and t∗ inequations (28) and (24), though it is clear that equation (27) can only be solvednumerically.

We have solved equation (27) numerically for various values of inverse temperature,β, and driving strength, δE . The results are presented in figure 3(a), while the resultingresisting force at maximum output is depicted in figure 3(b). Both t∗ and f ∗ dependlinearly on δE at high temperatures, or for small driving strength, with more pronounceddeviations from linearity as the temperature is lowered. This linear dependence can becalculated from a Taylor expansion of equations (27) and (28). We find t∗ � βδE/4 atweak driving and t∗ → 1.035 66 . . . in the limit δE → ∞. The optimal force is also linearat weak driving, with f ∗ � δE/2. For strong driving the optimal force approached thevalue βf ∗ → 1.428 57 . . .. It should be noted that while the scaled times in figure 3(a)are all of the same order of magnitude, the actual optimal periods of the driving cyclecan differ by orders of magnitude, depending on the temperature. This can be seen byinverting equation (25).

The maximal power output is depicted as a function of δE in figure 3(c). This poweroutput increases with the temperature, at least for the temperature range investigatedhere. At first this may seem odd, since at low temperatures driving via energy differences

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(a) (b)

(c) (d)

Figure 3. Numerical results for the fully directional model for several values of thetemperature: (a) scaled period and (b) force at maximal output; (c) the maximaloutput and (d) the efficiency at maximal output. Other model parameters areE0 = −4, θ = 0.7.

is ‘more efficient’. However, it should be noted that in the current model 1/β is alwayssmall compared to the infinite barriers. Namely, the dynamics of this model does notlose its full directionality as the temperature is raised. In addition, the power output isalso dependent on the period of the cycle, which is considerably shorter for the highertemperature. As figure 3(c) shows, the output also increases with increasing drivingstrength δE .

The efficiency at maximum output is depicted in figure 3(d). The efficiency is highestat the limit of weak driving and decreases monotonically with δE . The efficiency alsodecreases with decreasing temperature. The highest attainable efficiency at maximumpower is η∗ = 1/2 at the limit of vanishing driving force.

The data presented in figures 3(c) and (d) exhibit increasing output and decreasingefficiency as a function of δE . This is the reason we have not optimized the output powerwith respect to the driving strength. Optimization with respect to δE would have ledto infinite driving strength and vanishing efficiency. In practice, the driving strength δEshould be set as a compromise between the need to obtain high output and the desireto achieve reasonable efficiency. We expect that this compromise will be different fordifferent applications.

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5. Partially directional, cycle controlled pump

We turn to study a model where the stochastic motion is no longer fully directional.This is achieved by placing finite barriers of height δW instead of the infinite barriersthat were responsible for vanishing rates in section 4. In particular, the model studiedbelow will have an identical dependence of energies on time (or driving mechanism) asthe model presented in section 4. Therefore, comparing the two models would give usinformation regarding the influence of the degree of directionality on the behaviour ofstochastic pumps.

In the first stage of the driving cycle the transition rates are given by

RI12 = eβE0+βf(1−θ) RI

21 = eβ(E0+δE)−βfθ

RI23 = RI

31 = eβ(E0+δE−δW )+βf(1−θ) RI32 = eβ(E0−δW )−βfθ

RI13 = eβ(E0+δE−δW )−βfθ.

(29)

In equation (29) the rates RI12 and RI

21 are identical to those of equation (7), while theremaining rates take a similar form, but with a finite barriers of height δW . As a resulttransport is roughly suppressed by a factor of e−βδW in the links 13 and 23 when comparedto transitions between states 1 and 2. The rates during the other stages of the drivingcycle can be obtained from equation (29) by a cyclic permutation of the states.

The dynamics of this model can be solved explicitly via a spectral decompositionof the transition rate matrix. However, this turn out to be too complicated to give thesimple insights we are looking for. Instead, we have solved this model numerically by (i)maximizing the power output (over a full driving cycle) with respect to the period, T , andexternal force, f , and (ii) calculating the output and efficiency and plotting the results asa function of the driving strength. The results are given in figure 4. The scaled times infigure 4(a) are based on equation (25) which, while not exact for the current model, stillcaptures the most dominant dependence of the period on temperature.

Interestingly, the behaviour of this model as a function of temperature is qualitativelydifferent from that of the fully directional model of section 4. The force at maximumoutput is considerably smaller in figure 4(b) compared to that of figure 3(b). In additionthe force decreases with temperature, showing an opposite trend compared to figure 3(b).The reason for this is that the effect of finite barriers on the dynamics depends ontemperature, and the temperature range studied here includes results with βδW > 1as well as βδW < 1. This dependence also leads to the nonmonotonic dependence of theoutput on the temperature in figure 4(c).

The efficiencies obtained for this model are considerably smaller than those of thefully directional model. In particular, they no longer approach 1/2 for weak driving.Comparison of figures 3(d) and 4(d) demonstrates that the efficiency at maximum outputis strongly dependent on the degree of directionality. (The model with βδW > 0 haspartial directionality.)

In order gain a better understanding of the role of directionality we have fixed thetemperature and varied the barrier height δW . The results are compared to the fullydirectional model in figure 5. The finite-barrier model converges to the fully directionalpump in the limit of large barriers—as expected. Figures 5(b) and (c) show that the forceand maximum output of the model depend monotonically on the barrier height. Theyare both maximal for the fully directional model. A similar behaviour can be observed

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Figure 4. Numerical results for the finite-barrier model for several values of thetemperature: (a) scaled period of driving cycle (based on equation (25)) and(b) force at maximal output; (c) the maximal output and (d) the efficiency atmaximal output. Other model parameters are E0 = −4, θ = 0.7, and δW = 1.

for the efficiency at maximum power. Both the efficiency and power are reduced by aconsiderable amount (to less than half their value for the fully directional model) evenfor barriers with βδW = 2. The results clearly show that large degree of directionalityis required to obtain an efficient stochastic pump with respectable output power. Thenumerical results presented in figures 5(b) and (c) suggest that barriers of size ∼3/β oreven ∼4/β are needed to obtain efficiencies and output comparable to the fully directionalmodel of section 4.

6. Fully directional pump with two competing fundamental cycles

The last model considered in this paper exhibits fully directional stochastic dynamics. Itdiffers from the model of section 4 by being composed of two different fundamental cycles.The graph representing this model is depicted in figure 6. The model is constructed so thatone of the fundamental cycles, consisting of states 1, 2 and 3, is essentially identical to thefully directional model studied in section 4, including the resisting force. The existence

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Figure 5. Comparison between the finite-barrier model and the fully directionalmodel of section 4 (solid line) for several values of the barrier δW . Other modelparameters are E0 = −4, θ = 0.7, and β = 1.

of states 4 and 5 allows for an additional fundamental cycle, with transitions that areassumed to be decoupled from the external force. Therefore, probability currents drivenin this cycle cannot contribute to the output, and the work done to drive these currentsis wasted. Thus the model shows a competition between useful and wasteful cycles, andwe expect that control over the division of work invested in the cycles can improve theoutput of the pump. We again consider a driving cycle with three stage as in section 4,but in the current model the energies of states 2 and 4, as well as those of states 3 and 5,are coupled so they are varied together. The transition rates during the first stage of thedriving cycle are given by

RI12 = eβE0+βf(1−θ) RI

21 = eβ(E0+δE)−βfθ

RI14 = MeβE0 RI

41 = Meβ(E0+δE),(30)

while other transition rates vanish. Here, M is a parameter controlling how easy is it tomake transitions through the right cycle. It is equivalent to barriers of magnitude − 1

βlnM

placed on the links connected to states 4 and 5. When M is small the probability currentflows only through the left cycle. When M � 1 the current would flow more or less equallythrough both cycles and some of the work done on the pump would be wasted. (Note

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Figure 6. A directional stochastic pump with two cycles. The external force isapplied on one of the cycles. Driving currents in the other cycle wastes the workdone on the system.

that the existence of a resisting force in one cycle breaks the symmetry between the cycleseven for M = 1.) The non-vanishing rates for the second and third stages of the drivingcycle are given by

RII23 = eβE0+βf(1−θ) RII

32 = eβ(E0+δE)−βfθ

RII45 = MeβE0 RII

54 = Meβ(E0+δE),(31)

and

RIII31 = eβE0+βf(1−θ) RIII

13 = eβ(E0+δE)−βfθ

RIII51 = MeβE0 RIII

15 = Meβ(E0+δE).(32)

Numerical results for the period and force at maximum power output, the maximaloutput and the efficiency at maximum output are given in figure 7. The results arequalitatively similar to those obtained for the model of section 4. However, both theoutput and the efficiency of the pump are reduced due to the existence of the wastefulcycle.

It is instructive to have a quantitative comparison of this model with the one-cyclemodel of section 4. This could be done by varying the parameter M . Such a comparisonis depicted in figure 8 for β = 1. Interestingly, the force at maximum output is essentiallyunaffected by the presence of a competing cycle, while the period is only mildly affected.The output is found to be weakly dependent on M , or equivalently the heights of barriers inthe unproductive cycle. In contrast, the efficiency shows dependence on M , and convergesto that of the model of section 4 in the limit M → 0.

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Figure 7. Numerical result for the two-cycle fully directional model for severalvalues of the inverse temperature β. Other model parameters are E0 = −4,θ = 0.7 and M = 1.

These results can be qualitatively explained by realizing that in the periodic solution,ρ(t), there is some probability of finding the system in states 4 or 5 which belong to theunproductive fundamental cycle. The parameter M determines whether this probability(and some of the probability of finding the system in state 1) will flow in the unproductivecycle, thereby generating a current. Thus some of the probability distribution is effectivelyexcluded from the productive cycle, and therefore does not contribute to the output.The numerical results suggest that the amount of probability involved in the integratedcurrent through the productive fundamental cycle is approximately independent of M .When M � 1 work is not invested in the unproductive cycle, and the system efficiencyis essentially that of the one-cycle fully directional model. Finally, from figure 8(d) onecan deduce that even for a value of M = 0.1 the efficiency is close to that of the one-cyclemodel. Based on figures 5(d) and 8(d), loss of directionality seems to have a strongereffect on the efficiency than loss of cycle control.

7. Summary

The behaviour of stochastic pumps operating close to their maximum power output wasstudied with the help of simple models. This approach was taken since finding the

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Figure 8. A comparison between the two-cycle model and the fully directionalone-cycle model of section 4 (solid line), as a function of M . Other modelparameters are E0 = −4, θ = 0.7 and β = 1.

maximum output for models based on the general parametrization presented in section 2turns out to be very difficult. The models were constructed to differ by qualitativeaspects of the dynamics, namely the degree of directionality and cycle control. It wasfound that both directionality and cycle control are coupled to the maximal power outputand efficiency, though the results suggest that the efficiency is more sensitive to loss ofdirectionality.

Some general aspects of the behaviour of systems at maximum power-outputconditions can be deduced from the results presented. The results presented in figure 3demonstrate that if one were to also maximize the output with respect to the drivingstrength the maximal output would be found for δE → ∞, where the efficiency willvanish. In a similar vein, if one decides not use the external resisting force to maximizethe power output then it is clear from equation (24) that any efficiency at maximal poweris attainable. The behaviour of systems near their maximum power output inherentlydepends on the choice of parameters which are used to maximize the output. Here wechose the external resisting force and the period of the driving cycle as the parametersvaried to increase the output. This choice was motivated by the fact that both parametersare connected to external, possibly macroscopic, objects. It is therefore likely that theseparameters will be controllable in realizations of stochastic pumps.

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With the specific choice of f and T as the external parameters used to increase thepower output, all the models studied show efficiencies bounded by 1/2. This bound wassuggested for autonomous systems not far from equilibrium [10, 14]—a different type ofsystem. However, it was shown by Seifert that autonomous systems can exhibit efficienciesat maximum power which are larger than 1/2 [16]. For the periodically driven systemsstudied here, with T and f as the optimization parameters, the value η∗ = 1/2 was foundto be the maximal efficiency for a ‘best case’ model, with fully directional dynamics. Thissuggests that η∗ = 1/2 is an upper bound for the class of stochastic pumps defined insection 2. It will be of interest to give a more solid theoretical foundation for this.

Finally, the results for the three models presented here suggest simple guidelines whichmay help the design and synthesis of realizations of stochastic pumps. For instance, fromthe results it is clear that directionality and cycle control are desired properties in suchsystems. It is obvious that barriers much larger then β−1 would ensure directionality.However, knowledge concerning the transition region where βδW ∼ 1, namely on theamount of directionality attained by barriers which are neither large nor small, canbe useful. The results presented in section 5 show that barriers of βδW � 3 or 4 areneeded to achieve a reasonable amount of directionality. It is our belief that such simpleguidelines can be applied to more complicated systems, and can therefore prove helpfulto experimentalists attempting to design efficient realizations of stochastic pumps.

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