Fluctuation Theorem in Driven Nonthermal Systems with Quenched Disorder
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] S e p Flu tuation theorem in driven nonthermal systems with quen hed disorderJ.A. Dro o , C.J. Olson Rei hhardt , and C. Rei hhardt Department of Physi s, Prin eton University, Prin eton, NJ 08544 Theoreti al Division, Los Alamos National Laboratory, Los Alamos, NM 87545(Dated: November 16, 2018)We demonstrate that the (cid:29)u tuation theorem of Gallavotti and Cohen an be used to hara terizethe lass of dynami s that arises in nonthermal systems of olle tively intera ting parti les drivenover random quen hed disorder. By observing the frequen y of entropy-destroying traje tories, weshow that there are spe i(cid:28) dynami al regimes near depinning in whi h this theorem holds. Hen ethe (cid:29)u tuation theorem an be used to hara terize a signi(cid:28) antly wider lass of non-equilibriumsystems than previously onsidered. We dis uss how the (cid:29)u tuation theorem ould be tested inspe i(cid:28) systems where noisy dynami s appear at the transition from a pinned to a moving phasesu h as in vorti es in type-II super ondu tors, magneti domain walls, and dislo ation dynami s.PACS numbers: 05.70.Ln,05.40.Ca,74.25.QtThe (cid:29)u tuation theorem (FT) of Gallavotti and Co-hen has been des ribed as a generalization of the se ondlaw of thermodynami s in systems outside the thermody-nami limit [1, 2, 3℄. It relates the frequen y of entropy-destroying, also sometimes alled se ond-law-violating,traje tories to entropy- reating traje tories and is su(cid:30)- iently general to apply to systems far from equilibrium.It has been demonstrated to hold analyti ally for a lassof time-reversible dynami al systems [1℄, and has beenveri(cid:28)ed numeri ally in many others [4, 5℄. Wang et al.provided the (cid:28)rst experimental veri(cid:28) ation of the rela-tion by observing the (cid:29)u tuations of a diele tri parti lepulled by an opti al trap [6℄. It has also been shown thatthe FT holds in some driven non-thermal systems su has granular materials [7℄ and a ball moving in a Sinaibilliard potential [8℄.The FT has not previously been examined in the lass of nonthermal nonequilibrium systems onsistingof olle tively intera ting parti les moving over a ran-dom ba kground, where noisy dynami s an o ur nearpinned to moving transitions. Examples of this type ofdynami s in lude the motion of magneti domain walls[9℄, depinning of ele tron rystals in solid-state materi-als [10, 11℄, plasti deformations in driven super ondu t-ing vortex matter [12, 13, 14, 15℄, and olloidal parti lesmoving over quen hed disorder [16, 17℄. Closely relatedto these systems are the dynami s of intera ting dislo- ations under a strain [18℄. Typi ally, there is a regimenear the onset of motion where the parti le traje toriesare strongly disordered and /f or ra kling noise arises.In this work, we show that the FT an be used to hara -terize the dynami al behavior of a general model of thistype, indi ating that the FT ould be applied to a mu hwider range of nonequilibrium systems than previously onsidered and may hold in the general lass of systemsexhibiting ra kling noise.We spe i(cid:28) ally examine the formulation of the FTgiven in Ref. 4. One of the main predi tions of the FTis that the probability density fun tion (PDF) of the in- je ted power p ( J τ ) obeys the following relation: p ( J τ ) p ( − J τ ) = e J τ S τ , (1)where J τ is the inje ted power, τ is the duration of thetraje tory, and S τ is some onstant. If S τ varies su hthat β τ = τ /S τ is onstant for τ ≫ Γ i , where Γ i are themi ros opi time s ales of the system, then we say that β ∞ represents an (cid:16)e(cid:27)e tive temperature.(cid:17) Wang et al. [6℄experimentally measured the quantity in Eq. 1 from thetraje tories of a olloid that was driven through a thermalsystem. In the system we onsider, there is no thermalbath; instead, the parti le traje tories are generated inthe presen e of an external drive, a random quen hedba kground, and intera tions with other parti les.We onsider olloidal spheres on(cid:28)ned to two dimen-sions and driven with an ele tri (cid:28)eld in the presen eof randomly distributed pinning sites. This parti ularmodel system has been shown to exhibit the same gen-eral dynami al features, in luding plasti (cid:29)ow and mov-ing rystalline phases [19℄, observed in other olle tivelyintera ting parti le systems driven over random disordersu h as vorti es in type-II super ondu tors [12, 14℄; thuswe believe the behavior in our system will be generi toother systems of this type. Additionally, experimental re-alizations of this system permit the dire t measurementof the parti le traje tories [17℄. We simulate a system of N c olloids with periodi boundary onditions in the x and y dire tions, and employ overdamped dynami s su hthat the equation of motion for a single olloid i is η d r i dt = f iT + f iY + f ip + f d (2)All quantities are res aled to dimensionless units, and thedamping onstant η is set to unity. The thermal for e f iT arises from random Langevin ki ks with the properties h f iT i = 0 and h f iT ( t ) f jT ( t ′ ) i = 2 ηk B T δ ( t − t ′ ) δ ij . The ol-loid intera tion for e f iY is given by the following s reenedCoulomb repulsion: f iY = P N c j = i A c ( r ij + r ij ) e − r ij ˆr ij .Here A c is an adjustable oe(cid:30) ient, r i ( j ) is the positionof vortex i ( j ) , r ij = | r i − r j | and ˆr ij = ( r i − r j ) /r ij . Thequen hed disorder introdu es a for e f ip whi h is mod-eled by N p randomly pla ed attra tive paraboli pin-ning sites of strength A p and radius r p = 0 . , f ip = P N p k =1 ( − A p r ik /r p )Θ( r p − r ik ) ˆr ik , where Θ is the Heavi-side step fun tion. The driving for e f d = f d ˆx is a on-stant unidire tional for e applied equally to all olloids.We initialize the system using simulated annealing in or-der to eliminate undesirable transient e(cid:27)e ts due to re-laxation, and then apply the driving for e. The equationsof motion are then integrated by velo ity Verlet methodfor − simulation time steps, depending on N c .The time step dt = 0 . . We ompute the longitudinaland transverse di(cid:27)usivities D α with α = x, y by (cid:28)tting h [( r i ( t + ∆ t ) − r i ( t )) · ˆ α ] i = 2 D α ∆ t .The inje ted power omputed for a single olloid i overa time period of length τ is given by: J τ = Z t + τt f d · v i ( s ) ds (3)where v i represents the instantaneous velo ity of olloid i . A parti le whi h moves opposite to the dire tion ofthe driving for e makes a negative ontribution to theentropy. We measure J τ for a series of individual par-ti les in a single run and ombine this data to obtain p ( J τ ) . We identify J τ for a variety of τ ranging froma minimum of 10 simulation time steps to roughly onetenth the duration of the entire simulation.We (cid:28)rst onsider a system at f d = 0 . with no pin-ning but with (cid:28)nite thermal (cid:29)u tuations T = 3 . at a olloidal density of ρ = 0 . . Figure 1(a) shows p ( J τ ) for τ = 0 . , . , . , . , and . . Over thisrange, p ( J τ ) is normally distributed with a slight right-ward skew due to the applied drive, and for in reasing τ the distribution sharpens. Equation 1 is ertain tobe followed sin e p ( J τ ) = C exp( − ( h J i − J τ ) / σ ) andhen e log ( p ( J τ ) /p ( − J τ )) = ( − / σ )[( h J i− J τ ) − ( h J i + J τ ) ] ∝ J τ . The validity of Eq. 1 for this system is illus-trated by the linear (cid:28)ts in Fig. 1(b).We next repeat the pro edure used to obtain Fig. 1 in asystem with f d = 0 . , no thermal (cid:29)u tuations ( T = 0 ),and whi h ontains pinning sites with A p = 0 . . For f d < . the parti les are pinned and there is no non-transient motion, as illustrated in Fig. 2(a). Just abovethe depinning transition at f d = 0 . , the parti le motionpersists with time and the traje tories are highly disor-dered as shown in Fig. 2(b). Approximately one thirdof the olloids are pinned at any given time; however,all of the parti les take part in the motion. In Fig. 3(a)we plot the strongly non-Gaussian p ( J τ ) that appear inthe absen e of thermalization for τ = 0 . , 4.02, 8.02,12.02, and 16.02 at f d = 0 . . The τ = 0 . urve,most losely representative of the instantaneous distri-bution, peaks at J τ = 0 and is skewed in the positivedire tion by the applied drive. Despite the strong non-Gaussianity of the PDF's, the ratio of entropy-produ ingto entropy- onsuming traje tories is in agreement withthe (cid:29)u tuation theorem of Eq. 1, as shown in Fig. 3(b). −3 l n [ p ( J τ ) / p ( − J τ ) ] J τ −0.04 −0.02 0 0.02 0.04 0.0610 −5 −4 −3 −2 −1 J τ p ( J τ ) τ = 0.42 τ = 1.02 τ = 1.62 τ = 2.22 τ = 2.82 ba Figure 1: Demonstration of the FT for driven thermal par-ti les in the absen e of pinning at ρ = 0 . , T = 3 . , and f d = 0 . . (a) Probability density fun tion p ( J τ ) of inje tedpower for all observed traje tories with τ = 0 . , 1.02, 1.62,2.22, and 2.82 (from enter bottom to enter top). (b) Fit toEq. 1 for τ = 0 . , 1.02, 1.62, 2.22, and 2.82, from bottom totop. Linearity at various τ indi ates agreement with the FT. x(a)y x(b)y Figure 2: Colloid positions (dots) and traje tories (lines) dur-ing 40000 simulation time steps at ρ = 0 . , A p = 0 . , and (a) f d = 0 . ; (b) f d = 0 . .The quality of the (cid:28)t to Eq. 1 for (cid:28)xed ρ and A p de-pends on f d . It is known from earlier studies that systemswith depinning transitions an exhibit a number of dif-ferent dynami al regimes as a fun tion of external drive,in luding a ompletely pinned phase where there is nomotion, a stable (cid:28)lamentary hannel phase just at depin-ning where a small number of parti les move in periodi orbits [12℄, haoti (cid:29)ow at higher drives when the (cid:28)la-ments hange rapidly with time [12, 14℄, and a dynam-i ally re rystallized phase at even higher drives where −0.05 0 0.05 0.1 0.15 0.210 −4 −3 −2 −1 J τ p ( J τ ) τ = 0.02 τ = 4.02 τ = 8.02 τ = 12.02 τ = 16.020 1 2 3 4 5 6x 10 −3 l n [ p ( J τ ) / p ( − J τ ) ] J τ ba Figure 3: Demonstration of FT in a nonthermal system withquen hed disorder. (a) p ( J τ ) for all observed traje tories at ρ = 0 . with A p = 0 . and f d = 0 . at τ = 0 . , 4.02,8.02, 12.02, and 16.02 (from upper right to lower right). (b)Fit to Eq. 1 showing agreement with the FT. Bottom to top: τ = 0 . , 4.02, 8.02, 12.02, and 16.02.the parti le paths are mostly ordered [14℄. To quantifythe quality of the (cid:28)ts to Eq. 1 we al ulate the Pearsonprodu t-moment orrelation oe(cid:30) ient r [20℄, whi h isa measure of the linearity of the relation between twovariables. In Fig. 4(a) we plot the mean dissipation η h v i /f d versus f d for the system in Fig. 2(b), along withthe orresponding longitudinal and transverse di(cid:27)usiv-ities D x and D y . In Fig. 4(b) we show the value of r for varied f d and for all τ < τ c ( f d ) , where τ c ( f d ) =sup { t | r ( f d , τ ) ≥ . ∀ τ < t } . Agreement with the FT,indi ated by r ≈ , holds over the largest range of τ at f d ≈ . , near the depinning threshold and oin idingwith peaks in both D x and D y . Here, the olloids (cid:29)owin plasti (cid:29)u tuating hannels, as shown in Fig. 2(b). Atlower drives f d < . , D x and D y are mu h smaller,the motion in the system is very unstable, and the par-ti les (cid:29)ow only through short-lived (cid:28)laments, as shownin Fig. 2(a). We are unable to determine whether theFT fails to hold for f d < . sin e our measurement inthis regime is dominated by rare events and our statisti sremain poor over omputationally a essible time s ales.For higher drives f d & . , the olloids begin to forman ordered rystal stru ture similar to that found in vor-tex systems at su(cid:30) iently high driving [13, 14℄, and both D x and D y drop. The FT ontinues to hold for small τ at in reasing f d , with the maximum value of τ at whi h r > . de reasing with in reasing f d . On short time −2 −4 f d D η < v > /f d f d τ x D y ab Figure 4: Limits of regime in whi h FT is veri(cid:28)ed in a sys-tem with ρ = 0 . and f p = 0 . . (a) Solid urve: meandissipation η h v i /f d vs f d , relating olloid displa ements toapplied drive. Upper rosses: longitudinal di(cid:27)usivity D x vs f d . Lower rosses: transverse di(cid:27)usivity D y vs f d . (b)Pearson produ t-moment orrelation oe(cid:30) ient r of the (cid:28)t log( p ( J τ ) /p ( − J τ )) = mJ τ + b as a fun tion of τ and f d . Val-ues loser to indi ate better agreement with the FT. TheFT holds over the largest range of τ in the (cid:29)u tuating plasti (cid:29)ow regime near f d ≈ . illustrated in Fig. 2(b).s ales, the parti les experien e a (cid:16)shaking temperature(cid:17) T s whi h de reases as T s ∝ /f d [13℄. Due to parti le-parti le intera tions, on longer time s ales the parti lesare e(cid:27)e tively aged in a o-moving referen e frame andno longer undergo long time di(cid:27)usion. As a result, theFT fails to hold on longer time s ales.As des ribed previously, the FT allows the de(cid:28)nition ofan (cid:16)e(cid:27)e tive temperature(cid:17) β τ →∞ when su(cid:30) ient entropy-destroying traje tories of duration ex eeding the mi ro-s opi time s ales of the system an be sampled. Thisne essarily involves a balan e of time s ales sin e these ond law of thermodynami s guarantees that p ( J τ <
0) = 0 as τ → ∞ . In Fig. 5, we plot β τ versus τ showingthe existen e of an asymptoti e(cid:27)e tive temperature β ∞ in a nonthermal system with quen hed disorder. Whenwe vary the initial on(cid:28)gurations of the parti le positionsby hanging the random simulation seed, we onsistently(cid:28)nd an asymptoti value of β ∞ ≈ . for τ & .Equivalently, this indi ates that the slope of our (cid:28)ts ob-tained as in Fig. 3(b) s ales su h that τ /S τ rea hes a onstant value at large τ .We do not observe signi(cid:28) ant variation in β ∞ with f d ;however, as noted previously, we an only de(cid:28)ne an ef-fe tive temperature for those values of f d where the FT τ β τ β ρ Figure 5: E(cid:27)e tive temperature β τ in a nonthermal systemwith quen hed disorder at ρ = 0 . , A p = 0 . , and f d = 0 . .We overlay ten urves, ea h representing one realization witha unique random seed. The asymptoti value β ∞ ≈ . .Inset: β = h β <τ< i al ulated over a range of ρ .holds over a wide range of τ , whi h limits us to drivesnear the depinning threshold where plasti (cid:29)ow o urs.To ompare β ∞ a ross di(cid:27)erent ρ , we perform our mea-surement at f d = 1 . f c for ea h ρ , where f c is the de-pinning for e at that value of ρ . This pla es us withinthe plasti (cid:29)ow regime for every ρ onsidered here. Theinset of Fig. 5 indi ates that there is an apparently linearin rease in β ∞ with in reasing parti le density saturatingat ρ ≈ .6