Physica A 391 (2012) 930–936

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Physica A

journal homepage: www.elsevier.com/locate/physa

Monte Carlo calculation of the mean work required to drive abistable system

Wen Bao a, Fang Lin b,∗

a College of Natural Sciences, University of Texas at Austin, Austin, TX 78741, USAb College of Physical Science and Technology, Sichuan University, Chengdu 610064, China

a r t i c l e i n f o

Article history:Received 5 August 2011Received in revised form 15 September2011Available online 10 October 2011

Keywords:Stochastic thermodynamicsMean workBistable systemBrownian computer

a b s t r a c t

The mean work required to drive a bistable system from one equilibrium state to anotheris calculated by using a Langevin simulation combined with Monte Carlo sampling. Theresultingwork depends not only on the proposal formbut also on the temperature, becausethe particle subjected to thermal fluctuation passes over the barrier during a finite time.This shows that the mean work of a periodic signal done on a particle in a double-wellpotential is a non-monotonic function of the temperature when the energetic barrieris encountered. By applying this to information erasure in a Brownian computer, it isdiscovered that the work dissipated into the environment for 1-bit information erasurefrom two states to a single state can be minimized at a finite temperature.

© 2011 Elsevier B.V. All rights reserved.

1. Introduction

Recently, the nonequilibrium stochastic thermodynamics of small systems has become a topic of great interest [1].One can obtain equilibrium properties from nonequilibrium work measurements based on the Jarzynski relation [2]or the fluctuation theorem [3]. Various theoretical implications have been proposed, and experimental studies of thethermodynamics of a colloidal particle using laser traps generated a harmonic potential with a time-dependent center orspring constant have been reported [4–7]. The theory has been confirmed by experiments, and the dependence of the workdistribution on the transition protocols has been discussed [8]. Nevertheless, for nonlinear systems, it is not possible toacquire a simple protocol for minimizing the dissipated work [9]: this work should depend on the temperature. Duringthe actualizing process of a protocol, the particle exchanges its position between different wells through passing over theenergetic barriers, so the variation of the center of distribution is influenced by not only the external driving force but also thethermal fluctuation. Studying the dependence of the mean and dissipated work in a nonlinear system on the environmenttemperature is indeed interesting.

Bistable systems have various applications. A well-known phenomenon for such a system is stochastic resonance(SR), characterized by a large noise-induced response to a weak periodic signal [10], namely, the signal-to-noise ratio ismaximized at finite temperature. Only recently, the work fluctuation of a Brownian particle driven by an ac force in apotential of this kind was considered [11]; the probability function of the injected and dissipated energy in a colloidalparticle trapped in a double-well potential periodically modulated by an external perturbation has also been studiedexperimentally [12]. Nevertheless, the negative effect of SR in this regard has not yet been investigated, but the heatdissipated into the medium should be deleterious to the apparatus when resonance occurs. It is important to determine

∗ Corresponding author.E-mail address: linfang@scu.edu.cn (F. Lin).

0378-4371/$ – see front matter© 2011 Elsevier B.V. All rights reserved.doi:10.1016/j.physa.2011.09.021

W. Bao, F. Lin / Physica A 391 (2012) 930–936 931

Fig. 1. (Color online) The initial distribution density function calculated by Langevin simulation and Monte Carlo sampling (dashed line). The (red) solidline is the theoretical result and the dotted line is the potential. The parameters used are c0 = 0.0, c2 = 1.0, c3 = 0.1, c4 = 1.0, and T = 0.05.

whether the energy consumption can reach a minimum at a finite temperature. Landauer [13] discussed the limitation of theefficiency of computers (i.e., information erasure of ‘‘restore to one’’ or as ‘‘restore to zero’’) imposed by physical laws. Basedon the second law of thermodynamics, he argued that the erasure of 1 bit of information, i.e., changing two states into onesingle state, requires a minimal heat generation of kBT ln 2; however, no particular operation was presented.

This paper focuses on the temperature dependence of the mean work done on a bistable system driven by a periodicsignal from one equilibrium state to another when an energetic or entropic barrier is present. In particular, the dissipatedwork which characterizes the negative effect of the stochastic resonance is considered. Last, but by no means least, theMonte Carlo method is used to generate the initial equilibrium distribution of the system in a one-dimensional (1D) ortwo-dimensional (2D) double-well potential. Application to information erasure in a Brownian computer is attempted.

2. The model and comparison of various methods

The system ismodeled as an overdamped particle in contactwith a heat bath,whichmoves in a time-dependent potentialdescribed by the following Langevin equation:

x = −∂U(x, λ(t))

∂x+ η(t), (1)

where η(t) is a Gaussian white noise with ⟨η(t)⟩ = 0 and ⟨η(t)η(t ′)⟩ = 2β−1δ(t − t ′), and β = (kBT )−1 is the inversetemperature. The mean work spent in a finite-time process is defined by [2]

⟨W ⟩ =

∫ τ

0λ(t)

∂U(x(t), λ(t))

∂λ

dt, (2)

and the dissipated work is determined by

Wdiss = ⟨W ⟩ − ∆F , (3)

where ∆F denotes the free energy difference of the system between the final and initial states.Within stochastic thermodynamics, the work done on the system is usually calculated between two equilibrium states.

A key point is to generate the initial coordinate of a Langevin trajectory sampling from the equilibrium distribution atvarious temperatures. It is easy to generate the initial equilibrium probability density function (pdf) by running a Langevindynamics (for instance, Eq. (1) in the presence of λ(0)) for some time before switching on the driving protocol. However,the time consumed should be very long when the temperature is much less than the potential barrier. Here, we suggestusing the Monte Carlo (MC) sampling method [14], given in the Appendix A, to get the coordinates of the initial equilibriumdistribution of the system.

In Fig. 1, we plot the initial equilibrium pdf of a particle in a one-dimensional asymmetrical double-well potential

U0(x) = c0 −12c2x2 +

13c3x3 +

14c4x4, (4)

with c2 > 0 and c4 > 0. The above two methods are applied to generate the initial coordinates of the test particles. In theLangevin simulation (LS), bymeans of the second-order stochastic Runge–Kutta algorithm,we used the time step∆t = 0.01and the running time t = 200. In theMonte Carlo sampling scheme, for simplicity,we chose the trial pdf at low temperaturesΓ (x) = (2L)−1 for x ∈ [−2, 2]; thus L = 2.0. The common parameters are N = 2× 105 and ∆x = 0.1. It is obvious that the

932 W. Bao, F. Lin / Physica A 391 (2012) 930–936

Fig. 2. (Color online) Comparison of the molecular-field theory (dashed lines) and Monte Carlo simulation (solid lines). From top to bottom, the meanwork required for paralleled-moving the double-well potential with three types of protocol: square increasing with time, linearly increasing with time,and the optimal protocol used in [9], are shown respectively. The parameters used are a = b = 4.0, T = 0.2, and λτ = 2.0.

Monte Carlo data are in good agreementwith the theoretical distribution, furthermore, the simulating process is fast and theresult is accurate. Unfortunately, the results of the LS disagree with the equilibrium distribution even when a long runningtime (t = 200.0) is used, and the pdf of the LS depends on the initial prediction of test particles when the two potentialwells are asymmetric. Moreover, it is difficult for test particles to cross over the potential barrier at low temperature.

We consider a situation in which a bistable potential is paralleled-moved U(x, λ(t)) = Φ(x − λ(t)), where Φ(x) is thesymmetrical double-well potential, given as

Φ(x) =b4

x2 −

ab

2. (5)

Here, three types of protocol are tested: (i) λ(t) = λτ t/τ ; (ii) λ(t) = λτ (t/τ)2; and (iii) λ(t) = 0 for t < 0, λ(t) =

λτ (t + 1)/(τ + 2) for 0 < t < τ , and λ(t) = λτ for t ≥ τ . The third scheme was used in the optimal protocol of a colloidalparticle manipulated through a viscous fluid by an optical tweezer with a harmonic potential [9].

In Fig. 2, the results of themeanwork done on the system calculated byMC simulation are comparedwith those from themolecular-field approximation (see Appendix A). It can be found that the molecular-field theory is in good agreement withMonte Carlo data if variation of the protocol is slow. It is assumed that the better proposal in the bistable system for gettingless dissipated work is still a step function of time such as a harmonic oscillator. For τ → ∞, the mean work approaches aminimum, resulting in the quasi-static limit, as expected.

3. Dependence of mean work on temperature

Themeanwork done on the system is studiedwhen the SR occurs, where the rocking double-well potential is considered:

U(x, λ(t)) = U0(x) − λ(t)x, λ(t) = A sin(2πT−1Ω t). (6)

The condition of SR is Tk =12TΩ , where the Kramers mean barrier passage time is given by Tk = β

x+x−

dy exp[βU(y, λ(0))] y−∞

dz exp[−βU(z, λ(0))] [15], where x+ and x− are the coordinates of the right and left well bottoms.Fig. 3 shows the calculated mean work for various signal frequencies. Here, the final potential shape is the same as that

at the initial time if the protocol time is chosen as an integer multiple of TΩ ; thus, ∆F = 0, and all the work is dissipated.Indeed, the Kramers mean barrier passage times corresponding to the maximal mean work for TΩ = 50.0, 20.0, and 5.0 areequal to Tk = 25.1962, 10.2614, and 4.0792, respectively. These values obey the condition of the SR, especially in the large-TΩ cases; this means that the work measurements show an SR-like phenomenon. Unfortunately, a large energy dissipationarises.

Does the above effect occur in the presence of an entropic barrier [16–18]? The equation of motion of a particle in arocking 2D double-well potential is as follows:

x(t) = −∂V (x, y)

∂x+ A sin(2πT−1

Ω t) + ξx(t),

y(t) = −∂V (x, y)

∂y+ ξy(t), (7)

W. Bao, F. Lin / Physica A 391 (2012) 930–936 933

Fig. 3. Temperature-dependent mean work done on a signal-driven particle in a double-well potential. The parameters used are τ = 5TΩ , c0 = 1.0,c2 = c4 = 4.0c3 = 0.0, and A = 0.2. The open circles are numerical data and lines are curves smoothly linking the points; subsequent figures are similarlypresented.

Fig. 4. (Color online) Mean work done on a rocking bistable system with an entropic barrier. The solid and dashed lines respectively represent the resultsof full 2D potential and 1D effective potential. The inset depicts the effective potential at various temperatures. The parameters used are A = 0.3 andτ = NTΩ = 20.0.

where ξx and ξy are two independent Gaussian white noises. The potential is chosen as

V (x, y) = U1(x) +12C(x)y2. (8)

From exp(−βVeff(x)) =

∞

−∞exp(−βV (x, y))dy, we obtain the effective potential with an entropic barrier:

Veff(x) = U1(x) − kBT ln

2πβC(x)

1/2

. (9)

Fig. 4 shows the mean work done on a signal-driven particle in a 2D double-well potential of U1(x) = 0 and C(x) =

U0(x)+1with c0 = 0.0, c2 = c4 = 4.0, and c3 = 0.0. The result of the 1D LS for Eq. (1) with the effective potential (Eq. (10))is in agreement with the 2D result for a slow driving force at low temperature; however, the accuracy of the descriptionof the effective entropic barrier deteriorates in the cases of high temperature and non-adiabatic driving force. The meanwork is a monotonically decreasing function of the temperature. At zero temperature, the energetic barrier vanishes, andthen the particle possesses a deterministic motion: ⟨x(t)⟩ = x0 − ATΩ(2π)−1 cos(2π t/TΩ). Thus, the mean work reaches amaximum, given by ⟨W ⟩ =

12A

2τ , while τ = NTΩ . With increasing temperature, the entropic barrier increases, which leadsto the decrease of the mean displacement of the center of the pdf and the mean work done on the system.

4. Application to Brownian computer

The final illustration considers the Landauer restore to one or zero process in a Brownian computer [19–21], in whichthe system is equally distributed with the equilibrium form in the two states at t = 0 [19,20]. An alternative protocol for

934 W. Bao, F. Lin / Physica A 391 (2012) 930–936

a b

Fig. 5. (Color online) (a) Plot of a proposal for erasing the left potential well. The parameters used are Ub = 1.0, ωb = 1.0, and ωb =√2. (b) Bit factor,

mean work, free energy difference, and dissipated work as functions of temperature in the protocol of raising left well. The parameters used are Ub = 1.0,ωb = 1.0, ωb =

√2, and τ = 300.0.

erasing one state is proposed for which the time-dependent potential is

U(x, λ(t)) =

U− +

12ω2

0(x − x−)2, −∞ < x ≤ x−

s ;

Ub −12ω2

bx2, x−

s ≤ x ≤ x+

s ;

12ω2

0(x − x+)2, x+

s ≤ x < ∞,

(10)

where x+s =

2Ub/[ω

2b(1 + ω2

b/ω20)], x

−s = −

2(Ub − U−)/[ω2

b(1 + ω2b/ω

20)]; x± = (1 + ω2

b/ω20)x

±s . Here, the height of

the bottom of the left well (U− = U−(λ(t))) is regarded as a control parameter, and it increases with time until U− = Ub.Namely, the left well in the ‘‘ZERO’’ state will vanish, and the right well in the ‘‘ONE’’ state remains fixed. Three potentialconfigurations corresponding to the proposal process

U− = Ubtτ

(11)

are shown in Fig. 5(a).The quality factor for 1-bit information erasure from two states to a single state is defined as

B =

∞

0 exp[−βU(x, λ(τ ))]dx∞

−∞exp[−βU(x, λ(0))]dx

. (12)

For the present model, the above factor is evaluated by

B = 1 − ierf

i

βω2

b

2x+

s

+ωb

ω0eβVb

1 − erf

βω2

0

2x+

s

, (13)

where erf(x) is the error function.Fig. 5(b) shows a number of quantities as functions of the temperature. Thework is calculated by Eq. (2) with ∂λU(x, λ) =

1− ω20ω

−1b (x− x−)

(1 + ω2

b/ω20)/[2(Ub − U−)] under the proposal (11). At low temperature, the pdf of the particle in the

left well undergoes a large displacement into the right well when the left well vanishes; the change in the free energy ofthe system from the double states to the single state is large at high temperatures, meaning that the mean work done onthe system is large in both cases. Indeed, ∆F ≃ kBT ln 2 for B → 1. An important discovery shown in Fig. 5(b) is that thedissipatedwork reaches aminimumat a finite temperature. This result implies that the Landauer restore to zero processwillexpend less dissipated work and heat if the system in two states exhibits correlation, i.e., both the two initial equilibriumdistributions in the left and right wells have finite width.

5. Conclusion

The temperature-dependent mean and dissipated work of an external driving force done on a particle in a double-wellpotential are considered. This is in terms of Langevin simulation for the protocol process andMonte Carlo method sampling

W. Bao, F. Lin / Physica A 391 (2012) 930–936 935

for the initial equilibrium coordinate. The mean work is found to be a non-monotonic function of the temperature of theenvironment when an energy barrier exists, whereas in the case of an entropic barrier it shows a monotonic decrease withtemperature. Under the condition of stochastic resonance, a particle in a double-well potential shows a large-amplitudeoscillation between the two wells. This leads to the surprising result that the mean work done on the system and theenergy dissipated into the environment reach maxima simultaneously at finite temperature. In particular, a protocol hasbeen presented to investigate information erasure in a Brownian computer, revealing that this process should expend lessdissipated work and heat if the system in two states exists correlation. The technique and analysis presented here could beapplied to a wide variety of situations. Further experiments are required into the work distribution of bistable systems.

Appendix A. The molecular-field approximation and Monte Carlo sampling

A.1. The molecular-field approximation

In the case of bothweak noise and slowly driving force, we follow Suzuki [22] andmake amolecular-field approximation.Letting y(t) = x(t) − λ(t), and replacing y3 by y⟨y2⟩ [22,23], we obtain

d⟨y(t)⟩dt

=a − b⟨y2(t)⟩

⟨y(t)⟩ − λ(t), (A.1)

d⟨y2(t)⟩dt

= 2a − b⟨y2(t)⟩

⟨y2(t)⟩ − 2λ(t)⟨y(t)⟩ + β−1. (A.2)

This set of coupled equations has to be solved numerically and self-consistently with the initial equilibrium condition

[⟨x(0)⟩, ⟨x2(0)⟩] = Z−10

∫∞

−∞

dx [x, x2] exp(−βU(x, λ(0))), (A.3)

where Z0 is the partition function of the system in the initial equilibrium state, given by

Z0 =π

2exp

βa2

4b

[I−

14

βa2

8b

+ I 1

4

βa2

8b

], (A.4)

where Iν(x) is the modified Bessel function of the first kind of order ν.

A.2. The Monte Carlo sampling

The equilibrium distribution pdf of the system reads

Peq(x) = Z−10 exp[−βU0(x)], (A.5)

where U0(x) = U(x, λ(0)) and Z0 is the partition function of the system at the initial time.The one-dimensional (1D) sampling process for x0 is as follows. (i) Generate a random variable X1 from a trial pdf Γ (x).

(ii) Use the MC selection approach

x0 = X1, ifMξ ≤ H(X1), (A.6)

where H(x) = Peq(x)/Γ (x), M = max[H(x)], and ξ is an uniform random number in [0, 1]. For the two-dimensional (2D)case, the equilibrium pdf of the system is rewritten as Peq(x, y) = f2(y|x)f1(x), where f2(y|x) is the conditional pdf and f1(x)is the marginal pdf.

∞

−∞f2(y|x)dy = 1 for any x and

∞

−∞f1(x)dx = 1 are assured because

∞

−∞

∞

−∞Peq(x, y)dxdy = 1.

We first sample x0 from f1(x) and then inject it into f2(y|x0); the random variable y0 is sampled from f2(y|x0). Note that theMonte Carlo selection sampling method itself is an exact algorithm [14].

The pdf is calculated numerically by

p(x) =∆Nx→x+∆x

N∆x, (A.7)

where N is the total number of test particles and ∆Nx→x+∆x is the number of test particles in the region [x, x + ∆x].

References

[1] For a review, see F. Ritort, C. R. Phys. 8 (2007) 528.[2] C. Jarzynski, Phys. Rev. Lett 78 (1997) 2690; Phys. Rev. E 56 (1997) 5018.[3] G.E. Crooks, Phys. Rev. E 60 (1999) 2721; Phys. Rev. E 61 (2000) 2361.[4] E.H. Trepagnier, C. Jarzynski, F. Ritort, G.E. Crooks, C.J. Bustamante, J. Liphardt, Proc. Natl. Acad. Sci. USA 101 (2004) 15038.

936 W. Bao, F. Lin / Physica A 391 (2012) 930–936

[5] G.M. Wang, J.C. Reid, D.M. Carberry, D.R.M. Williams, E.M. Sevick, D.J. Evans, Phys. Rev. E 71 (2005) 046142.[6] F. Douarche, S. Joubaud, N.B. Garnier, A. Petrosyan, S. Ciliberto, Phys. Rev. Lett. 97 (2006) 140603.[7] F. Blickle, S. Speck, A. Helden, I. Seifert, V. Bechinger, Phys. Rev. Lett. 96 (2006) 070603.[8] R. Kawai, J.M.R. Parrondo, C Van den. Broeck, Phys. Rev. Lett. 98 (2007) 080602.[9] T. Schmiedl, U. Seifert, Phys. Rev. Lett. 98 (2007) 108301.

[10] L. Gammaitoni, P. Hänggi, P. Jung, F. Marchesoni, Rev. Modern Phys. 70 (1998) 223.[11] S. Saikia, R. Roy, A.M. Jayannavar, Phys. Lett. A 369 (2007) 367.[12] A. Imparato, P. Jop, A. Petrosyan, S. Ciliberto, J. Stat. Mech. (2008) P10017.[13] R. Landauer, IBM J. Res. Dev. 15 (1961) 183.[14] M.H. Kalos, P.A. Whitlock, Monte Carlo Methods, Vol. I: Basics, (A Wiley-Interscience Publication), 1986.[15] P. Hänggi, P. Talkner, M. Borkovec, Rev. Mod. Phys. 62 (1990) 251.[16] G.A. Cecchi, M.O. Magnasco, Phys. Rev. Lett. 76 (1996) 1968.[17] D. Reguera, G. Schmid, P.S. Burada, J.M. Rubí, P. Reimann, P. HFanggi, Phys. Rev. Lett. 96 (2006) 130603.[18] J.D. Bao, Phys. Rev. E 62 (2000,) 4606; 63 (2001) 061112.[19] K. Shizume, Phys. Rev. E 52 (1995) 3495.[20] B. Piechocinska, Phys. Rev. A 61 (2000) 062314.[21] J.M.R. Parrondo, Chaos 11 (2001) 725.[22] M. Suzuki, J. Stat. Phys. 16 (1977) 11.[23] P. Grange, H.C. Pauli, H.A. Weidenmüller, Z. Phys. A 296 (1980) 107.