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Energetics of Forced Thermal Ratchet 8 9 Hideki Kamegawa, Tsuyoshi Hondou & Fumiko Takagi 9 1 Department of Physics, Tohoku University n Sendai 980-8578, Japan a J February 1, 2008 2 1 ] t Molecular motors are known to have the to an external load: f o high efficiency of energy transformation in dx ∂V (x) ∂V (x) s the presence of thermal fluctuation[1]. Mo- =− 0 +ξ(t)+F(t)− L , (1) . dt ∂x ∂x t tivated by the surprising fact, recent stud- a m ies of thermal ratchet models[2] are show- where x represents the state of the ratchet, V0(x) ing how and when work should be extracted is a periodic potential, ξ(t) is a thermal noise sat- d- from non-equilibrium fluctuations[3, 4, 5, isfying < ξ(t)ξ(t′) >= 2kTδ(t−t′), ”< · >” is an n 6, 7, 8, 9, 10]. One of the important operator of ensemble average, F(t) is an external o finding was brought by Magnasco[6] where fluctuation,F(t+τ)=F(t), τ dtF(t)=0,andV c 0 L [ he studied the temperature dependence on is a potential due to the loaRd, ∂∂VxL = l > 0. The the fluctuation-induced current in a ratchet geometry of the potential, V(x) = V0(x)+VL(x), 1 (multistable) system and showed that the is displayed in Fig. 1. The ratchet system trans- v current can generically be maximized in a forms the external fluctuation into work (see, for 1 0 finite temperature. The interesting find- review,Ref[10]). The model[6]Magnascodiscussed 1 ing has been interpreted that thermal fluc- is a special case of the present system, where the 1 tuation is not harmful for the fluctuation- externalloadis omitted. In general,Fokker-Planck 0 induced work and even facilitates its effi- equation[13] of the system is written: 8 ciency. We show, however, this interpreta- 9 tion turns out to be incorrect as soon as ∂P(x,t) + ∂J(x,t) =0, / ∂t ∂x at we go into the realm of the energetics[11]: J =−kT∂P(x,t) +{−∂V0(x) +F(t)−l}P(x,t), ∂x ∂x m the efficiency of energy transformation is (2) - not maximized at finite temperature, even whereP(x,t)isaprobabilitydensityandJ(x,t)isa d in the same system[6] that Magnasco con- probability current. If F(t) changesslowlyenough, n sidered. The maximum efficiency is re- P(x,t)couldbetreatedasquasi-static. Insuchsit- o c alized in the absence of thermal fluctua- uation,J canbeobtainedinananalyticalform. For : tion. The result presents an open prob- slowly changing fluctuation F(t) of square wave[6] v i lem whether thermal fluctuation could facil- of amplitude A, we analytically obtain an average X itate the efficiency of energetic transforma- current over the period of the fluctuation, r tion from force-fluctuation into work. a 1 J = [J(A)+J(−A)]. (3) sqr Let us consider a forced ratchet system subject 2 It is reported[6] in the operation of the ratchet that ”there is a region of operating regime where the efficiency is optimized at finite temperature.” The result has been interpreted that the operation Figure 1: Schematic illustration of the potential, of the forced thermal ratchet is helped by thermal V(x) = V0(x)+VL(x). V0(x) is a piecewise linear fluctuation. This discovery has been followed and and periodic potential. V (x) is a potential due to confirmed by many literatures (see the references L the load, V (x) = lx. The period of the potential in Ref.[10]) with various situations. We first con- L is λ=λ +λ , and ∆=λ −λ (>0) is a symme- firm the previous report and then analyze it ener- 1 2 1 2 try breaking amplitude. For a finite work against getically. We can distinguish the behavior of the load, we assume the condition l ≤ A throughout currentJ onthetemperatureintothreeregimes, sqr this paper. (a) A < Q +l < Q −l, (b) Q +l < A < Q −l, λ1 λ2 λ1 λ2 1 and (c) Q +l < Q −l < A; whereas the distinc- Because J(−A) <0, Eq. (8) is rewritten, λ1 λ2 J(A) tion is not explicitly described in the paper[6]. In rfiengiitmeetsem(ap)eraantdur(eb)(,FJisgq.r2ias,cbe)r.taIinnlryegmimaxeim(ci)z,edJsaqrt η = Al 1− 12+|J|J(J−((A−A))A|)|. (9) is a monotonically decreasing function of the tem- J(A) perature (Fig. 2c). In the limit |J(−A)| → 0, the maximum efficiency We have to notice at this stage that the J(A) of the energy transformation for given load l and fluctuation-induced current J is not an energetic force amplitude A is realized: η = l . quantity and therefore J is only the mimic of the max A In Eq. (9), we can discuss the effect of thermal energetic efficiency. The lack of discussion of the fluctuation on the energetic efficiency of the forced forced ratchet system by this real efficiency is at- thermal ratchet. As demonstrated in Fig. 3, it is tributed to the lack of construction of energetics proved that the efficiency is a decreasing function of the systems described by Langevin or equiva- of temperature in all the three regimes: Because, lently by Fokker-Planck equations. Recently, an energetics of these systems was constructed by |JJ(−(AA))| is monotonically increasing function of the Sekimoto[11]. Therefore, we will go into the realm temperature as found in Fig. 4, the efficiency η of the energetics of the forced thermalratchet, and is a decreasing function of the temperature. This analyze the real efficiency. certainlyshowsthatthepresenceofthermalfluctu- According to the energetics[11], the input en- ation does not help efficient energy transformation ergy R per unit time from external force to the by the ratchet, which is in contrast to the previous ratchet and the work W per unit time that the interpretation that thermal fluctuation could facil- ratchet system extracts from the fluctuation into itates the efficiency. the work are written respectively: We can learn here that the efficiency should be discussedenergetically: Theconditionofmaximum 1 x=x(tf) current does not correspond to that of the maxi- R[F(t)]= F(t)dx(t), (4) t −t Z mum efficiency. The difference is attributed to the f i x=x(ti) observation that the efficiency is a ratio of the ex- 1 x=x(tf) tracted work W to the consumed energy R. The W = dV(x(t)). (5) extractedworkW issurelyproportionaltothecur- t −t Z f i x=x(ti) rentJ = 1[J(A)+J(−A)](Eq. 3). However,the sqr 2 For the square wave, they yield: consumed energy is not a constant but varies sen- sitively according to the condition. Therefore the 1 <R >= [<R(A)>+<R(−A)>] efficiency η is not simply proportional to the in- sqr 2 duced current J. The important problem was left 1 for future study whether the existence of thermal = A[J(A)−J(−A)], (6) fluctuation could facilitate the efficiency of energy 2 transformation in more general forced ratchets. 1 <W >= l[J(A)+J(−A)]. (7) Finally, we mention that the complementarity sqr 2 relation[12] of the ratchet system. We found that Therefore, we obtain the efficiency of the energy the maximum efficiency, η = 1, can be realized max transformation η: if all of the the following conditions are satisfied, A→Q/λ +l+0 and Q→+0 and T →0. In this 1 <W > l[J(A)+J(−A)] η = sqr = . (8) limit, the speed of energy transformation goes to <Rsqr > A[J(A)−J(−A)] zero. Thatistosay,thismaximumefficiencyofthe forced ratchet is realized in quasistatic process. As weincreasethevelocityofthisengine,theefficiency Figure 2: Plot of the current J as a function of isdecreased. Theresultalsoemphasizestheimpor- sqr kT/Q. The first regime (a), A < Q +l < Q −l tance of time scales of the operation of the ratchet (λ = 1.0, ∆ = 1.0, l = 0.01, A = 1λ.10); the sλe2cond as Ju¨licher et al. pointed out[10]. Detailed analy- regime (b), Q +l < A < Q −l (λ = 1.0, ∆ = sis of the loss of the efficiency may be analyzed by λ1 λ2 1.0, l = 0.01, A = 1.2); and the third regime (c), Q +l < Q −l < A (λ = 1.0, ∆ = 0.6, l = 0.01, λ1 λ2 A = 6.0). Regimes (a), (b) and (c) correspond to Figure 3: Plot of the efficiency η as a function of thelow,moderateandhighamplitudeforcinginthe kT/Q. In each regime (a), (b) and (c), the con- description of Magnasco[6]respectively. In regimes dition is the same as in Fig.2. In all the regimes, (a) and (b), increasing temperature results first in increasingtemperature resultsin decreasingthe ef- a rise and then a fall in the current. ficiency. 2 the formal theory of complementarity relation[12] [11] Sekimoto, K. Kinetic characterization of heat between the time lapse of thermodynamic process bath and the energetics of thermal ratchet and the irreversible heat. models. J. Phys. Soc. Jpn. 66, 1234-1237 (1997). [12] Sekimoto,K.& Sasa,S.ComplementarityRe- References lation for Irreversible Process Derived from Stochastic Energetics. J. Phys. Soc. Jpn. 66, 3326-3328(1997). [1] Forexample,Woledge,R.C.,Curtin,N.A.,& Homsher,E.,Energetic Aspect of Muscle Con- [13] Gardiner,C.W.,HandbookofStochasticMeth- traction (Academic Press, New York, 1985). ods 2nd ed. (Springer-Verlag,Berlin, 1990). [2] Feynman,R.P.,Leighton,R.B.,&Sands,M., Acknowledgements. We would like to thank K. The Feynman Lectures in Physics (Addison- Sekimoto, S. Sasa, T. Fujieda and T. Tsuzuki for Wesley, Massachussets, 1966). helpful comments and discussions. This work is supportedbytheJapaneseGrant-in-AidforScience [3] Bu¨ttiker, M. Transport as a consequence of ResearchFundfromtheMinistryofEducation,Sci- state-dependentdiffusion.Z. Phys.B 68,161- ence and Culture. 167 (1987). Correspondence and requests for materials should [4] Landauer, R. Motion out of noisy States. J. be addressed to T.H. Stat. Phys. 53, 233-248(1988). (e-mail: [email protected]). [5] Vale, R. D. & Oosawa, F. Protein motors and Maxwell’s demons: Does mechanochem- ical transduction involve a thermal ratchet? Adv. Biophys. 26, 97-134 (1990). [6] Magnasco, M. O. Forced thermal ratchet. Phys. Rev. Lett. 71, 1477-1481(1993). [7] Ajdari, A., Mukamel, D., Peliti, D., & Prost, J.Rectifiedmotioninducedbyacforcesinpe- riodic structures. J. Phys. I (France) 4, 1551- 1561 (1994). [8] Doering, A. R., Horsthemke, W., & Riordan, J. Nonequilibrium fluctuation-induced transport. Phys. Rev. Lett. 72, 2984-2987(1994). [9] Hondou, T., & Sawada, Y. Dynamical be- haviour of a dissipative particle in a periodic potentialsubjectto chaoticnoise: Retrievalof chaotic determinism with brokenparity.Phys. Rev. Lett. 75, 3269-3272(1995). [10] Ju¨licher, F., Ajdari, A., & Prost, J. Modeling molecular motors. Rev. Mod. Phys. 69, 1269- 1281 (1997). Figure 4: Plot of currents J(A), J(−A) and J(−A) . The condition is the same as in the sec- (cid:12) J(A) (cid:12) o(cid:12)nd reg(cid:12)ime (b) in Fig. 2. |J(−A)| increases slower (cid:12) (cid:12) than |J(A)| when kT increases from zero temperature. This difference is attributed to the symmetry breaking of the potential as illustrated in Fig. 1. 3