Work probability distribution and tossing a biased coin
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] M a r Work probability distribution and tossing a biased coin
Arnab Saha ∗ and Jayanta K Bhattacharjee † S.N. Bose National Centre for Basic Sciences, Saltlake, Kolkata 700098, India
Sagar Chakraborty ‡ NBIA, Niels Bohr Institute, Blegdamsvej 17, 2100 Copenhagen Ø , Denmark (Dated: November 5, 2018)We show that the rare events present in dissipated work that enters Jarzynski equality, whenmapped appropriately to the phenomenon of large deviations found in a biased coin toss, are enoughto yield a quantitative work probability distribution for Jarzynski equality. This allows us to proposea recipe for constructing work probability distribution independent of the details of any relevantsystem. The underlying framework, developed herein, is expected to be of use in modelling otherphysical phenomena where rare events play an important role. PACS numbers: 05.70.ln, 05.90.+m, 05.20.-y
I. INTRODUCTION
Large deviations play a significant role in non-equilibrium statistical physics[1–6]. They are difficultto handle because their effects though small, are notamenable to perturbation theory. All the conventionalperturbation theories in statistical physics are fashionedabout a Gaussian distribution, which almost by defini-tion, is the distribution with no large deviations. Thiscan be seen in static critical phenomena, critical dynam-ics, dynamics of interfacial growth, statistics of polymerchain and myriad other problems[7]. Our contention is:in the large deviation theory [8–11], the central role isplayed by the distribution associated with tossing of acoin and the simple coin toss is the “Gaussian model” ofproblems where rare events play significant role. In thispaper, we illustrate our contention by applying it to thestudy of some aspects of Jarzynski equality.Fluctuation theorems form a very important part ofnonequilibrium statistical mechanics [12–19]. There hasbeen a lot of activity in the last decade or so and var-ious forms of such theorems has been established. Oneparticular form is Jarzynski equality [14, 15]. If W is thework done during a period of duration τ , during whichan external force acts on the system and does work, thenJarzynski established that in units of K B T h e − W i = e − ∆ F (1)where angular brackets denote ensemble average and ∆ F is the free energy difference for the equilibrium free en-ergies corresponding to the initial and final states. Here K B is Boltzmann constant and T is the temperature ofthe concerned system in the initial equilibrium state or,equivalently, the temperature of the heat reservoir withwhich the system was thermalized before the process took ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] place. It is important to note that here W is a path (tra-jectory followed by the system during τ ) dependent vari-able. So, if we consider the ensemble of all possible paths(each path originating from one of the several microstatescorresponding to the initial equilibrium macrostate), dif-ferent values of W along different path can be identifiedwith a set of random variable. Now, if we define anotherrandom variable — dissipative work along a path — as W D ≡ W − ∆ F (2)Jarzynski equality (1) shows that h e − W D i = 1 (3)Clearly, to satisfy above equality, W D should take bothpositive and negative values. Again, we know that since h W i is the thermodynamic work done in going from ini-tial state to the final state, the second law of thermody-namics would assert that h W i ≥ ∆ F ( i.e. , h W D i ≥ W D are relativelyrare events, yet important enough to make the equalityof Eq.(3) to hold. II. THE STRATEGY
The strategy for demonstrating the validity of ourapproach will be as follows. As the paradigm for thedistribution of rare events we will take, as mentionedearlier, the distribution associated with tossing a coin.The random variable associated with a coin toss canrange between two finite numbers which we will taketo be 0 and 1. (It’ll be explained in details in thenext section.) The mean value p , of the variable for anunbiased coin is 1 / p = 1 /
2. Dissipative work along a path W D ranges from −∞ to + ∞ and we will first carry outa transformation that maps it onto the range 0 to 1.Further, according to the second law of thermodynamics,the events corresponding to W > ∆ F (or, W D >
0) aremore likely than the ones corresponding to
W < ∆ F (or, W D <
0) and hence there will be an asymmetryor bias about the events corresponding to W = ∆ F (or, W D = 0). The amount of bias in the statistics of W due to irreversibility is clearly: h W i − ∆ F . Thecorresponding coin has to be biased as well and hence weshall take the asymmetric situation of p = 1 /
2. Needlessto say that for reversible process, since h W i = ∆ F , thisbias is zero. Having defined the mapping — which,of course, is not unique — the first check would beto verify if Jarzynski equality in the form of (3) issatisfied. This can be tested since we have an explicitdistribution — namely, the one associated with thebiased coin toss. Such a check is depicted in figure(1), details regarding which follows later in this paper.From the distribution for W D one can also have thedistribution for W . The distributions of work anddissipative work have drawn a lot of attraction. Forvarious systems these distributions are now known fromexperiments as well as numerical studies ( e.g. [20–27]).Here we obtain these distributions from very generalrequirements, which are independent of the dynamics(that usually varies from system to system) followedby the systems. Hence it is important to ask whetherthe experiments on the systems obeying widely differentdynamics really exhibit similar distributions. We findthe answer here by comparing our results with actualexperimental results and numerical simulations. Theprobability distribution for W D , P ( W D ), was obtainedexperimentally by Liphardt et al. [24]; and, P ( W ) hasbeen obtained by Blickle et al. [25] for a different system.We have also used an anharmonic oscillator driven bya linear time dependent force to simulate the dynamicsand numerically construct P ( W ) [26]. A similar systemhas also been studied in [27].In this work, we calculate P ( W ) from the biased cointoss distribution which we have taken as a starting ansatzbased on the principle of large deviations. The only con-nection between the experiments and the method we use,is the fact that, the experiment is carried out far fromequilibrium and hence must feature negative values of W D (large deviations) and we have started with a dis-tribution which has large deviations built in it. As willturn out, our method will have two parameters whichwe fix by comparing with the distribution obtained fromthe experiment. The appropriateness of P ( W ), we cal-culate employing the theory of large deviations, is borneout by comparison as well be demonstrated below. Thepoint we want to stress here is that the present theory,which explicitly takes care of large deviations, does notrequire explicit knowledge of dynamics. Consequently, ithas wide range of applicability. The parameters of thedistribution need to be fixed in each case from the scalesof measured distribution — e.g. , peak position and peakmagnitude. III. COIN TOSS
We begin by recalling the situation of coin tossing ex-periment. If we assign a value 1 to the outcome ‘heads’and 0 to the outcome ‘tails’, then the mean after N trialsis M N = 1 N N X i X i (4)This is an experimental mean which belongs to a set ofindependent and identically distributed ( i.i.d ) randomnumbers lying between 0 and 1 since each individual X i is either 0 or 1. For an unbiased coin where “heads” and“tails” are equally probable, this mean goes towards 1 / N → ∞ . However, for a biased coin where the prob-ability of obtaining “heads” is p ( = 1 / p as N → ∞ . If in N trials ‘heads’appear X times, then the probability of finding a mean M N = X/N is P (cid:18) XN (cid:19) = N C X p X (1 − p ) ( N − X ) (5)This is the binomial distribution, which is the mostcommonly used example of a theory exhibiting largedeviations i.e. even when N ≫
1, we find a P which falls off slower than Gaussian. P ( X/N ) ∼ exp[ − N ( X/N − p ) /σ ]. Taking log of both side of eq.(5), we arrive at:ln P (cid:18) XN (cid:19) = X ln p + ( N − X ) ln(1 − p ) +ln N ! − ln X ! − ln( N − X )! . (6)We know from Stirling’s formula that for large N , N ! ≃ N N e − N √ πN . Applying Stirling’s approximation forlarge N , X , ( N − X ), the above relation becomesln P ( x ) = − N J ( x ) , (7)where, J ( x ) ≡ x ln xp + (1 − x ) ln (cid:18) − x − p (cid:19) +12 N ln x (1 − x ) + 12 N ln N (8)and x ≡ X/N . This leads to P ( x ) ∼ p N x (1 − x ) exp( − N I ( x )) (9)where, I ( x ) ≡ x ln xp + (1 − x ) ln 1 − x − p (10)For large enough N the pre-factor changes slowly com-pared to the exponential term and we get Chernoff’s for-mula where I ( x ) is the rate function. IV. THE MAPPING
In order to appreciate the analogy between the largedeviation theory and the Jarzynski equality, we need toconsider evolution of a relevant system to have a stochas-tic component, i.e. there is a regular time dependentforce acting on the system from t = 0 to t = τ as itproceeds from an initial equilibrium state to a final stateand in addition there is a random component. Langevindynamics of a particle of mass m can be considered as asimple example of such a stochastic dynamics: m ¨ x = − ∂V∂x − λ ˙ x + f ( t ) + η ( t ) (11)where V is a potential function, f ( t ) is a regular time de-pendent force, λ is damping coefficient and η ( t ) is randomnoise. Here dots represent time derivative. Fluctuationtheorems were proven for such a system by Kurchan [28]and subsequently by several authors [29–32]. The timedependent force is switched on at t = 0 and switchedoff at t = τ . At t = 0 the system resides in a macro-scopic equilibrium state, corresponding to which thereexist a large number of microstates. Since we are con-sidering stochastic evolution, we can start from the samemicrostate and do the experiment N times, each timegetting different value of the dissipative work. If w iD isthe dissipative work for i th realisation then we can define W D [the analogue of M N is Eq.(4)] as W D = 1 N N X i =1 w iD . (12)The distribution of W D is sought from the large deviationprinciple.We now note that h W i ≥ ∆ F (or, h W D i ≥
0) accord-ing to the second law of thermodynamics — the equalityholds for reversible processes. To implement our scheme,we need to define a transformation which maps W D toanother variable Z such that, 0 ≤ Z ≤
1, in accordancewith the experimental mean in coin toss scenario, givenin Eq.(4). We consider the variable W D + c , where ‘ c ’ isa quantity that we shall fix later. The class of transfor-mation we consider here is Z ( W D ) = 12 [1 − tanh α ( W D + c )] , (13)where ‘ α ’ is a parameter which eventually will have tobe fixed using experimental results. Actually, only thepositive constant ‘ α ’ defines this class of transformationsbecause a constraining relation for c will be established.Our ansatz is that, Z , like W D , satisfies large deviationprinciple and the rate function for the coin toss problemis the rate function for Z . So, the rate function for Z is I ( Z ) = Z ln Zp + (1 − Z ) ln 1 − Z − p . (14)The probability distribution for Z is simply P ( Z ) ∼ p N Z (1 − Z ) e − NI ( Z ) (15) < e xp (- W D ) > FIG. 1: In this figure we show the convergence of h exp( − W D ) i with respect to N for different values of α . The thick-dotted, dot-dashed, and dashed lines are respectively for α = 0 . , . , .
96. The curves in the figure are obtainedby numerically integrating equation (16) by employing Simp-son’s one-third rule. where N is the number of trajectories used in construct-ing the experimental mean of the i.i.d variables. Inour case, X i in coin tossing experiment and dissipativework w iD in Eq.(12) are i.i.d variables. We note that I ( Z = 1) = ln(1 /p ), while I ( Z = 0) = ln[1 / (1 − p )]. For p < / I ( Z = 0) < I ( Z = 1). The function I ( Z ) hasa minimum at Z = p . Thus the probability P ( Z ) hasa peak at Z = p and is exponentially small at Z = 0and Z = 1, but with P ( Z = 0) > P ( Z = 1), becauseof the inequality in I ( Z ). From second law of thermo-dynamics, we need h W D i > i.e. realizations with the outcome W D > W D <
0. All the above constraintsare met since Z → W D → ∞ and Z → W D → −∞ . We now return to Eq.(13); noting that e − W D = e c [ Z/ (1 − Z )] / α , we have h e − W D i = e c R (cid:16) Z − Z (cid:17) α P ( Z ) dZ R P ( Z ) dZ (16)The right hand side of equation (16) is plotted as a func-tion of N in figure (1) for different values of α . As N → ∞ , we find h e − W D i converges to unity for all α ,as it should according to Jarzynski equality. From fig-ure (1) one can see that as α decreases, lesser number oftrajectories (or, realizations) are required for the conver-gence. The convergence is verified for various values of p . The result shown in figure (1) is for p = 0 . V. COMPARISION WITH EXPERIMENTS
We will now obtain P ( W ) from P ( W D ) following thetheoretical technique discussed above and compare withthe work distribution function obtained experimentallyand numerically. From eq. (13) we write W = ∆ F − c + 12 α ln 1 − ZZ (17)We fix c = ∆ F to get the following simple form W = 12 α ln 1 − ZZ (18)We can now find P ( W ) by noting the normalisation con-dition: Z P ( Z ) dZ = Z ∞−∞ P ( Z = f ( W )) (cid:12)(cid:12)(cid:12)(cid:12) dZdW (cid:12)(cid:12)(cid:12)(cid:12) dW = 1 (19)where, P ( Z ) is given in Eq.(15). The work distributionis then found as P ( W ) = P ( Z = f ( W )) (cid:12)(cid:12)(cid:12)(cid:12) dZdW (cid:12)(cid:12)(cid:12)(cid:12) (20)with f ( W ) = (1 − tanh αW ) / P ( Z ) as given byEq.(15). The parameter N in Eq.(15) can be linked to thewidth σ of the distribution by appealing to the Gaussianlimit which shows that σ = 2 p (1 − p ) /N . In principlewe need to fix three unknown parameters in Eq.(20), viz. , σ , α and p . To reduce the task of parameter-adjustments,we pre-assign a value of p . As a result, only α and σ willbe used as fitting parameters.We now show the comparison between our assertion ofthe form of P ( W D ) [and hence P ( W )] and different ex-perimental and numerical results. A. Experiment by Liphardt et al.
This experiment tests Jarzynski equality by strechinga single RNA molecule between two conformations —both reversibly and irreversibly. The experiment hasbeen done for three different molecular end-to-end ex-tensions and for each extension, three different strechingrates are considered. For all combinations of extensionsand pulling rates, the experiment provides P ( W D ). Forthe present purpose, we consider three distributions cor-responding to 15 nm extension, which are shown in figure(2a). Other distributions can also be taken care of simi-larly. In this work, P ( W D ) we compute, depends on twoparameters viz. , α and σ . We determine these two param-eters by comparing with P (0) and P max ( W D ) [maximumvalue of P ( W D )] of the corresponding distribution givenby Liphardt et al. . After fixing these two parametersas ( α, σ ) ≃ (0 . , . , (0 . , . , (0 . , .
20) for threedifferent pulling rates, we arrive at the full distributionsfor every pulling rate. This is shown in figure (2b). Wefix p = 0 .
48 here.
B. Experiment by Blickle et al.
This experiment deals with the thermodynamics of anoverdamped colloidal particle in a time dependent non-harmonic potential. Blickle et al. have not only measured -2.5 0 2.5 5 7.5 10 W D P(W D ) FIG. 2: (
Colour online ) The figure of the inset is taken from[24] where dissipated work probability distributions are ex-perimentally obtained for a particular end-to-end extension(= 15 nm ) of P5abc RNA molecule but for three differentpulling rates indicated by three different curves — dashed,solid and dotted. In main figure we obtain P ( W D ) by fixing α and σ , as required by the theory presented here. P ( W ), they have also computed P ( W ) from the relevantFokker-Planck dynamics. Our explanation of their P ( W )is dependent on the choice of the two parameters α and σ .We determine these parameters by comparing P (0) and P max ( W ). This fixes α ≃ . σ ≃ .
2. This compari-son is shown in figure (3a) and (3b). We fix p = 0 .
24 here.The moments found by Blickle et al. and us compare asfollows (Table I)Moments Values from experiment Values from theby Blickle et al. theory presented here h W i h W i h W i α and σ ) to getthe distribution, it implies that only h W i and h W i havebeen used. This leaves the h W i as a prediction whichcan be compared with the experimental data. A moresensitive quantity to measure asymmetry of a distribu-tion is h ∆ W i , where ∆ W = W − h W i . Our distributionshows that this moment is nonzero. If p = 1 /
2, then |h ∆ W i| / ( σ | − p |h ∆ W i / ) is a constant, the value ofwhich in our case is 10. Nonzero h ∆ W i corresponds toasymmetry of P ( W ) that has been observed wheneverthe dynamics has been nonlinear[26, 27]. In those casesthe cause of the asymmetry is the strength of the non-linear term. Here the role is played by (1 − p ) (though,no particular dynamics is explicitly involved here) and itcan be considered as a measure of asymmetry. -4 0 4 8 12 16 W P ( W ) FIG. 3: (
Colour online ) The figure of the inset is takenfrom [25], which shows experimentally as well as numericallyobtained work probability distribution function for an over-damped colloidal particle in a time-dependent nonharmonicpotential. In the main figure we obtain P ( W ) from the theorypresented here. C. Driven anharmonic oscillator
We consider here a Brownian particle, trapped by thepotential V ( x ) = kx + γx (where k and γ are con-stants) and driven by a linearly time-dependent force f ( t ). The evolution is taken to be governed by follow-ing overdamped Langevin dynamics, λ ˙ x + ∂V∂x = f ( t ) + η ( t ) . (21)Here η ( t ) is the random noise coming from heat bath. Weassume h η ( t ) i = 0 and h η ( t ) η ( t ′ ) i = 2 T λδ ( t − t ′ ), where T is temperature of the bath. The force f ( t ) acts from t = 0 to t = τ and P ( W ) (where W = − R τ ˙ f ( t ) x ( t ) dt ) isnumerically obtained. The comparison between numer-ically obtained P ( W ) and that obtained from Eq.(20)with α ≃ .
15 and σ ≃ .
062 is shown in figure (4). Wefix p = 0 .
28 here.
VI. CONCLUSIONS
In conclusion, we re-stress that the discussed frame-work is very general and simple because we require onlyfew parameters from experiments and elementary resultsfrom large deviation theory to construct a full work prob-ability distribution, bypassing the ‘nitty-gritty’ of dy-namics. We believe that it is possible to make better con-tact with experiments by constructing more appropriateform for the function Z . Readers would also appreciatethat we could derive results concerning Jarzynski equalitymerely by focusing on the rare events — rare negative dis-sipation — that enter into Jarzynski equality and map-ping them onto the biased coin-toss-experiments. Here -5 -4 -3 -2 -1 0 1 2 3 4 5 W P ( W ) FIG. 4: (
Colour online ) In this figure we show how P ( W ),calculated by simulating the dynamics of a driven Brownianparticle obeying Eq.(21) (shown in red dots) collapses to the P ( W ) calculated from the distribution for tossing a biasedcoin (shown in black solid line), as it is prescribed here. Forsimulating the dynamics we take k = 1 , T = 1 , λ = 1 and γ = 0 . we have discussed situations where the evolution had astochastic component in addition to the regular time de-pendent force. We hope to extend it to the deterministicsituations. In the deterministic case, we envisage thefollowing picture. The evolution of a nonlinear systemunder time dependent drive is intrinsically chaotic andwe can exploit that to define an “experimental mean”for w iD . In this case, we need to consider the differentinitial conditions around an ǫ -neighbourhood ( ǫ →
0) ofa given microstate and since the evolution of each initialmicrostate (from same initial macrostate) will be differ-ent from each other due to the chaotic flow, we can define W D as in Eq.(12). Therefore, in accordance with our con-tention that the simple coin toss is the ‘Gaussian model’for the problems where rare events play significant role,one might speculate that the phenomenon of intermit-tency (and hence multifractality) in fluid turbulence canbe obtained by treating rare events in the energy dissipa-tion rate in the similar fashion outlined in this paper. Aswe have reported elsewhere[33] that, for fluid turbulence,the rare events present in the distribution of energy inthe real space, when mapped appropriately on the phe-nomenon of large deviations found in simple coin toss areenough to yield anomalous exponents which are knownto be the signatures of multifractality in fluid turbulence.Within this very framework, we hope to model variousother physical phenomena where rare events play a sig-nificant role; after all, now we have a working approachto arrive at quantitative results for such processes thatcannot be usually solved otherwise. Acknowledgments
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