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Cyclic Brownian Information Engine Under Error-free Measurements
Govind Paneru, Hyuk Kyu Pak
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DOI: 10.22661/AAPPSBL.2019.29.1.03

Cyclic Brownian Information Engine
Under Error-free Measurements

GOVIND PANERU,1 AND HYUK KYU PAK1,2*
1CENTER FOR SOFT AND LIVING MATTER, INSTITUTE FOR BASIC SCIENCE (IBS),
ULSAN 44919, REPUBLIC OF KOREA
2DEPARTMENT OF PHYSICS, ULSAN NATIONAL INSTITUTE OF SCIENCE AND TECHNOLOGY (UNIST)
ULSAN 44919, REPUBLIC OF KOREA
(RECEIVED 10 27 2018)

We summarize our recent experimental findings on a cyclic information engine [1, 2], consisting of a Brownian particle in a harmonic trap, with error-free measurements and ultrafast feedback control. The engine can transport the particle in one direction and extract positive work from it by utilizing information about the microscopic state of the particle. We studied the performance of the engine as a function of the cycle period τ and the distance the trap center is shifted with respect to a reference position and explored the optimal operating conditions for maximum work and power extraction. We show that the work produced by the engine increases with increasing τ and reaches a bound set by a generalized second law of thermodynamics, when τ reaches infinity. By measuring the steady state information, we could obtain the efficiency of this engine in general and the efficiency at maximum power. We also validate the generalized Jarzynski equality under error-free feedback control.

INTRODUCTION

The past few decades witnessed remarkable breakthroughs in the field of information thermodynamics [3-21]. These studies incorporate the feedback process to understand the role of Maxwell's demon [22]. A prominent example describing the relation between information and thermodynamics is the generalization of the Jarzynski equality [23] to feedback-controlled systems, i.e. information engines [7, 8, 15, 17, 24],

(1)

The exponential function averaged in Eq. (1) augments the terms from the standard Jarzynski equality - the work performed on the system W and the free energy change ΔF (in kBT = β-1 units) - with the contribution from the information circuitry: I is the information gathered by measurements, out of which a part Iu becomes unavailable due to the irreversibility of the feedback process [17]. Applying Jansen's inequality to Eq. (1) yields the generalized second law which sets a new bound on the extracted work [17],

(2)

Namely, the work extracted from the system <-βW> cannot exceed the sum of the free energy difference between final and initial states -βΔF and the available information <I>-<Iu>. In the absence of information, the inequality recaps the notion of the free energy as the maximal available work in an isothermal process,
<-W>≤-ΔF, while the additional term <I>-<Iu> sets an upper bound on extra work that can be gained from information on the system. Here, we realize an information engine that demonstrates the sharpness of the bound set by Eq. (2).

Various theoretical models for an information engine whose initial state is in thermal equilibrium have been investigated for classical [12, 17, 25] and quantum [3, 11, 26, 27] systems, and have been validated experimentally [1, 18-21, 28-31]. These studies have improved our understanding of information thermodynamics significantly; however, they are still not sufficient to explain the thermodynamics of many biological motors [32] that operate with finite cycle period and arbitrary initial state. To this end, many theoretical works on information engines operating with arbitrary initial states have been reported [9, 13, 33-37]. However, the parametric study of a Brownian information engine operating in a non-equilibrium steady state has been done rarely because of the difficulty in finding the steady state probability distribution, especially for the case of asymmetric feedback for which the probability distribution function for finite cycle period is non-Gaussian [37]. Here, we report on the experimental realization of such a cyclic information engine operating in non-equilibrium steady state where measurement and subsequent feedback control are repeated with a finite cycle time in order to find the optimal condition for maximum work, power and efficiency. Apart from maximum efficiency, we are interested in measuring the efficiency at maximum power. Finding the efficiency at maximum power has been studied extensively for stochastic heat engines with two temperature baths [38-41]; however, no experimental studies have been done so far in this direction for a single temperature bath Brownian information engine.

Our Brownian information engine consists of a colloidal particle trapped in a harmonic potential generated by optical tweezers and subjected to periodic feedback control. Each engine cycle consists of three processes: acquiring information about the position of the particle by performing a high precision measurement of the particle position, ultrafast modulation of the trap center, and relaxation of the particle for time τ. By repeating this process for a large numbers of measurement-feedback-relaxation cycles, the engine can induce one-way transport of the particle, thereby extracting work from the random thermal fluctuations of the surrounding heat bath. We found the optimal operating condition of the engine in steady state for maximum work and power. The extracted work is maximized and reaches the upper bound in Eq. (2) when τ → ∞, while, depending on the feedback control parameters, the extracted power is maximum either at finite τ or in the limit τ → 0. We also validated the generalized Jarzynski equality in Eq. (1) under error-free feedback control.

Fig. 1: Illustration of the Nth feedback cycle. The particle is trapped in a harmonic potential generated by an optical trap. We set a line at xm and measure the particle position at tN. (a) If the particle is on the left of xm, we do nothing. (b) If the particle is on the right of xm, we instantaneously shift the potential center to xf. After that, the particle relaxes for time τ and another cycle is repeated. By shifting the potential center, the engine extracts work equal to the change in potential energy ΔV.

We consider one dimensional motion of a Brownian particle in a heat bath of temperature T = (kBβ)-1. As shown in Fig. 1, the particle is trapped in a harmonic potential V(xB, X(t)) = (k / 2)(xB - X(t))2 ≡ 1 / 2kx2, where xB is the position of the particle, X(t) denotes the time-dependent potential center, k is the trap stiffness, and
x
xB - X(t) is the relative position of the particle with respective to the potential center. Neglecting the inertial force, the dynamics of the particle during relaxation is governed by the overdamped Langevin equation [42, 43] with the characteristic relaxation time τR = γ/k, where γ is the dissipation coefficient.

The information engine in this experiment is designed to measure the particle position and modulate the potential center depending on the measurement outcome. Each engine cycle consists of measurement, being followed by feedback control and relaxation (see Fig. 1). We set a reference line at xm and measure the particle position at t = 0 and determine which side of the reference line the particle is located at. If the particle is found on the left of xm, we do nothing. Whereas, if the particle is found on the right of xm, the potential center is instantaneously shifted (keeping the stiffness constant) to xf. By shifting the potential center, the work performed on the system W is equal to the change in potential energy ΔV. After shifting the potential, the particle evolves in time with fixed trap center until the next cycle begins. During this time period, the particle exchanges heat with the heat bath. Therefore, each engine cycle is characterized by three parameters: the setting distance xm, the potential center moving distance xf, and the time period τ for relaxation. The optimal operating condition depends on the choice of these parameters. In this work, we first optimize xm keeping xf = 2xm at infinite τ, and then vary xf at fixed (optimized) value of xm for different τ in order to find an optimal choice of the parameter values under which the average extracted work or power is maximum.

RESULTS

For an overdamped system, the kinetic energy of the particle can be neglected. According to Ref. [44], the incremental change in potential energy can be converted to work and heat within the feedback switching time. In this experiment, we use an acoustic optical deflector (AOD) to shift the potential center. The switching time for the AOD is about 20 μs. This time is very short in comparison to the characteristic relaxation time τR = γ/k~3 ms, but sufficiently larger than the momentum relaxation time τp =m /γ~0.2 μs for current setup. Hence, in this very short time scale of overdamped motion, the position of the particle cannot be far from its previous position [45], so we can neglect the heat production during the feedback process. The extracted work -W(x;tN) for the Nth cycle, when the particle is measured at the relative position x, is then given by
when xxm, and -W(x;tN) = 0 otherwise (x < xm). Here, tN = is the time for the Nth measurement, and stands for the time right after feedback.

If the process is repeated periodically for a large number of feedback cycles, the system becomes steady state. Then, the average extracted work per cycle in the steady state is given by

(3)

where pss(x) is the steady state probability distribution of the particle position at the measurement times and <…>ss denotes the steady state ensemble average. The integration in Eq. (3) begins at xm because we can extract work only when xxm. The probability of finding the particle on right of xm is given by Thus, the expression for <-W>ss takes the following form

(4)

where is the conditional mean position of the particle given that xxm.

Special case: τ → ∞

In the limit τ → ∞ the system relaxes to the equilibrium state irrespective of the measurement and feedback control. As a result, pss(x) follows the equilibrium Boltzmann distribution P(x) = (2πσ2)-1/2 exp(-x2/2σ2), with a variance σ2 = (βk)-1. Using this pss(x), we obtain an expression for <-W>ss in the limit τ → ∞,

(5)

Next, we show that our feedback protocol can achieve the upper bound on the extractable work in Eq. (2). Since the measurement is practically error-free, the net information is simply Shannon's entropy of a Gaussian variable [17, 46]:

(6)

where the limit of vanishing measurement error Δ → 0 ensures the positive-definiteness of the entropy and the correspondence between discrete and differential entropies ([47] Ch. 8). During the relaxation phase of the feedback process, part of the information in Eq. (6) becomes unavailable [17]. To calculate the unavailable information <Iu> we consider the inverse process: the particle is initially in equilibrium with the center of the trap at xf and we shift the potential center back to zero according to the same protocol.

 



Fig. 2: Schematics of the optical tweezers set up with feedback control system.

The unavailable information associated with a single measurement is Iu = -ln[P(x)Δ] for -xxm, and
Iu = -ln[P(x-xf)Δ] for xmx. The average unavailable information is therefore,

(7)

The upper bound of extractable work is found from Eqs. (2), (5), (6) and (7) keeping in mind that ΔF = 0,

(8)

Thus, the present feedback protocol achieves the equality in the generalized second law in Eq. (2), <-βW>=<I> - <Iu>. This is because the feedback is instantaneous and W(x) has no time to fluctuate. In the following, we experimentally confirm the equality in Eq. (8) by measuring the average work <-βW> and comparing it to the available information <I> - <Iu> (r.h.s. in Eq. (8)). Finally, we verify that the feedback protocol satisfies the generalized Jarzynski equality in Eq. (1) by substituting the work and information terms,

(9)

Experimental setup.- The details of the experimental setup are described in Fig. 2. A laser with 1064 nm wavelength is used for trapping the particle. The laser is fed to an acoustic optical deflector (AOD) (Isomet, LS110A-XY) via an isolator and a beam expander. The AOD is controlled via an analog voltage controlled radio-frequency (RF) synthesizer driver (Isomet, D331-BS). The AOD is properly mounted at the back focal plane of the objective lens so that k is essentially constant while the potential center is shifted. A second laser with 980 nm wavelength is used for tracking the particle position. A quadrant photo diode (QPD) (Hamamatsu, S5980) is used to detect the particle position. The electrical signal from QPD is preamplified by a signal amplifier (On-Trak Photonics Inc., OT-301) and sampled at every τ with a field-programmable gate array (FPGA) data acquisition card. Our system is capable of measuring the particle position with high spatial accuracy of 1 nm. The sample cell consists of highly dilute solution of 2.0 μm diameter polystyrene particles suspended in deionized water. All experiments are carried out at 293 ± 0.1 K.

 

Fig. 3: (a) Probability distribution P(x) of the particle position in thermal equilibrium (red solid circles), and measured every 25 ms during each engine cycle (blue solid squares) for xm = 0.5σ. The shifted probability distribution P(x-2xm) is obtained by performing the backward protocol (green open circles). The black solid curve is a fit to the Boltzmann distribution, P(x) ∝ exp[-x2/(2σ2)]. (b) Histogram of the extracted work -βW for xm = 0.5σ. (c) Extracted work as a function of xm/σ. The solid curve is a fit to Eq. (5). (d) Distribution of (-βW -I+Iu) in order to verify the generalized Jarzynski equality in Eq. (1).

Experimental testing of the information engine bounds. We first calibrate the parameters of the trap (Fig. 2). By fitting the probability distribution of the particle position in thermal equilibrium without feedback to the Boltzmann distribution P(x) = (2πσ2)-1/2 exp[-x2/(2σ2)], we obtain the standard deviation σ = 28 nm. Prior to actual operation of the information engine, we set the region R from xm > 0 to infinity. The QPD measures the particle position periodically at intervals of 25 ms. The FPGA board generates a bias voltage that corresponds to the initial position of the potential center. This bias voltage is applied to the AOD via the RF synthesizer driver. If the particle is found in R, the FPGA board generates an updated bias voltage that corresponds to the shift of the potential center to xf = 2xm. The decision whether to update the bias voltage and thereby shift of the potential center is taken within 20 μs. After shifting the potential center, we wait for 25 ms, about eight times the relaxation time τ~3 ms. Finally, the potential center is instantaneously shifted back (within ~20 μs) to the initial position, we wait for 25 ms for full relaxation of the particle and then the cycle is repeated.

We next focus on the energetics of the information engine. We set the region from xm = 0.5σ = 14 nm to infinity and perform the measurement with feedback control as described above. The distribution (blue squares in Fig. 3(a)), which is obtained from 100,000 feedback cycles, is indistinguishable from the equilibrium distribution (red) with the same σ = 28 nm. Fig. 3(b) shows the distribution of the measured extracted work, -βW(x) = β[V(x)-V(x-2xm)] for xxm and -βW(x) = 0 for xxm, whose average is <-βWexp> = 0.197±0.001. We also calculate the average extractable work from the model in Eq. (5), <-βWmodel> = 0.198±0.002, which agrees well with the experimental value. This shows that the feedback protocol is capable of extracting positive work from the information of the system immersed in a single heat bath, thus exceeding the standard bound of the second law of thermodynamics (<-W> ≤ -ΔF = 0).

Using the equilibrium probability distribution P(x) and the shifted one P(x-2xm) from Fig. 3(a), we measure the available information <I> - <Iu> from definitions in Eqs. (6) and (7) where integration is approximated by discrete summation. The experimental value of the available information <I> - <Iu> = 0.200±0.002, is close to the measured extracted work. This demonstrates the sharpness of the bound set by the generalized second law. To find the optimal feedback protocol, we evaluate the extracted work as a function of xm /σ (red solid circles in Fig. 3(c)). The fit to the theoretical curve in Eq. (8) agrees well with the measurement, implying that the engine indeed achieves the upper bound of the generalized second law in Eq. (2) for any xm > 0. The maximum of Eq. (5) is obtained at xm ≈ 0.612σ. We also demonstrate that the generalized Jarzynski equality in Eq. (1) is satisfied by the feedback protocol. To this end, we evaluate the integration in Eq. (9) for xm = 0.5σ, xf = 2xm and find it to be equal to 1.02 ± 0.06 (see Fig. 2(d)) as in Eq. (1).

Optimal tuning of the engine. In the first part of our study, the optimal value of xm for maximum work extraction is found to be xm ~ 0.6σ. Hence, in the following study we fix xm ~ 0.6σ and vary xf and τ in order to find the optimal condition for maximum work and power extraction. The experimentally measured average extracted work per engine cycle in the steady state <-βW> as a function of xf for seven different τ with xm fixed at 16 nm is shown in Fig. 4(a). For xf > 0, <-βW> increases with increasing τ and saturates when system relaxes fully, corresponding to τ ≥ 5τR. The global maximum of <-βW> is obtained when xf = 2xm and τ ≥ 5τR. From Ref. [37] we note that given the particle position x > xm, the extracted work is maximum if the particle is shifted to the center of the potential (in relative frame of reference). Thus, xf should be taken as the conditional mean position of the particle <x>ss which is ~1.2σ when xm ~ 0.6σ. This is the reason why the global maximum of <-βW> is at xf = 2xm. For ττR, the optimal value of xf for the maximum <-βW> decreases and goes to zero as τ → 0. The solid curves obtained by plotting Eq. (A9) in ref. [2], fit well with the experimentally measured <-βW> and also fit to the theoretical curve (yellow solid curve) in Eq. (5). It is worth mentioning that in comparison to work extraction by always moving the trap center in one direction (without feedback), where the extracted work -βW is not always positive [48], our feedback controlled information engine is capable of extracting positive work for all cycles (see inset of Fig. 4(a)). Hence, the feedback control enhances the magnitude of the average work extracted significantly.

 

Fig. 4: (a) Average extracted work per engine cycle as a function of xf for seven different τ with xm fixed at 16 nm. The black solid squares, red open squares, green solid circles, blue open circles, cyan solid triangles, magenta open triangles and dark yellow solid diamonds correspond to the average work values for τ = 0.2, 0.5, 1.0, 2.0, 5.0, 10.0, and 20.0 ms, respectively. The solid curves are a plot of Eq. (A9) in ref. [2]. The topmost yellow solid curve is a plot of Eq. (3). Inset: Histogram of the extracted work,
-βW = β[V(x)-V(x-xf)] for xm = 16 nm τ = 0.2 ms and xf = 2xm. (b) Average extracted power as a function of xf for different τ with xm fixed at 16 nm. The black solid squares, red open squares, green solid circles, blue open circles, cyan solid triangles, magenta open triangles and dark yellow solid diamonds correspond to the average power for τ = 0.2, 0.5, 1.0, 2.0, 5.0, 10.0, and 20.0 ms, respectively. The solid curves are the plots of <-βW>ss /τ from Eq. (A9) in ref. [2].

Fig. 4(b) shows the plot of experimentally measured average extracted power <βP> = <-βW>/τ as a function of τ and xf, respectively, when xm is fixed at 16 nm. The global maximum of <βP> is obtained when ττR and xfxm. Our finding is in agreement with the recent theoretically realized Brownian information engine [37], which shows that the extracted power is maximum when τ → 0 and xf → 0. However, for xfxm, <βP> is found to be maximum at finite τ. The solid curves are the plots of <-βW>ss / τ from Eq. (A9) in ref. [2] and fit well with the experimentally measured powers.

 

Fig. 5: Plot of engine efficiency as a function of xf for different τ with xm fixed at 16 nm. The black solid squares, red open squares, green solid circles, blue open circles, cyan solid triangles, magenta open triangles and dark yellow solid diamonds correspond to the efficiency for τ = 0.2, 0.5, 1.0, 2.0, 5.0, 10.0, and 20.0 ms, respectively. The solid curves are the plots of Eq. (A11) in ref. [2].

Another quantity of interest is the efficiency of the Brownian information engine in the steady state which is defined as η = <-βW>ss / <I>, where <I> is the mean information acquired through the measurement [34]. Considering there is negligible error in estimation of the particle position in this experiment, <I> is given by the Shannon information [1, 46]. In the current feedback scheme, each engine cycle has two discrete measurement outcomes; with probability pR the particle is found on the right of xm and with probability 1 - pR the particle is found on the left of xm. Thus, the acquired information is simply given by <I> = -pR ln pR -(1-pR)ln(1-pR). Fig. 5 shows a plot of the engine's efficiency measured as a function of xf for different τ. We found that, for a given xf, efficiency increases with increasing τ. The maximum efficiency of ~35% was obtained when xf = 2xm and τ ≥ 5τR. Our experimentally measured efficiencies fit well with the solid curves obtained by using Eq. (A11) in ref. [2]. The nearly error-free position measurement and instantaneous shift of the potential center allow us to extract the maximum amount of the information. However, some part of the information gain is not fully utilized during the relaxation part of the engine cycle. Also, the acquired information is useless for work extraction when the particle is found on the left of xm. These are the two reasons why the measured efficiency is less than the unity. Considering the true available information, we expect that this engine may achieve the upper bound of the generalized second law [1, 17]. We also measured the efficiency of this engine at maximum power and found that for xf < xm, the engine has vanishing efficiency at maximum power (which is for xf ~ 0 and τ ~ 0). However; for xfxm, the efficiency at maximum power is found to be highest (equal to ~19%) for xf = 2.5xm and near τR(τ = 2 ms).

CONCLUSIONS

In conclusion, we studied the cyclic Brownian information engine consisting of a colloidal particle trapped in a harmonic potential. Each engine cycle consists of three steps: particle position measurement, shift of the potential center based on measurement outcome, and relaxation. Each process is characterized by three parameters: xm for measurement, xf for the feedback, and τ for the relaxation. By performing the nearly error-free position detection with ultrafast feedback, we were able to convert all available information to work extraction thereby achieving equality in the generalized second law. We found that the global maximum of extracted work per engine cycle is obtained when the system is fully relaxed and xf = 2xm. On the other hand, the global maximum of extracted power is obtained when both xf and τ approaches to zero. For xfxm, the extracted power is maximum at finite τ. Although, the global maximum of power is obtained when τ → 0; however, it is somewhat useless because in this case both the extracted work and the efficiency are vanishing. On the other hand, the extracted work and the efficiency are maximum when the system is fully relaxed for which the extracted power is vanishing. On the basis of these observations, the optimized parameters for maximum power extraction with non-zero work and efficiency are xm ~ 0.6σ, xf ~ 2xm and τ ~ τR. Our study shall assist in the design and understanding of efficient synthetic, as well as biological, motors.

Acknowledgements: This work was supported by the Korean government under the Grant No. IBS-R020-D1.

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Govind Paneru is a Research fellow at the Institute for Basic Science, Center for Soft Living Matter, Ulsan, South Korea. He received his PhD in Physics from Kansas State University, Manhattan, Kansas, USA in 2014. He is an experimental physicist with research interest in soft matter, specifically in the field of non-equilibrium statistical mechanics and information thermodynamics.

Hyuk Kyu Pak is a professor of Physics, UNIST Korea and a research fellow at the Center for Soft and Living Matter in the Korea Institute for Basic Science. After receiving a Ph.D. from the University of Pittsburgh in 1992, he worked at Duke University, Kansas State University, and Pusan National University before joining UNIST in 2014. His research field is experimental soft matter and statistical physics.

 
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