Front Propagation with Rejuvenation in Flipping Processes

F ron t Propagation with Rejuv enation in Flipping Pro cesses T. An tal, 1 D. ben-Avra ham, 2 E. Ben-Naim, 3 and P . L. Krapivsk y 3, 4 1 Pr o gr am for Evolutionary Dynamics, Harvar d University, Cambridge, Massachusett s, 02138 USA 2 Physics De p artment, Cl ark son University, Potsdam, New Y ork 13699 USA 3 The or etic al Division and Center for Nonline ar Stud ies, L os Alamos National L ab or atory, L os Alamos, New Mexic o 87 545 USA 4 Dep artment of Physics, Bos ton University , Boston, Massachuset ts 02215 USA W e study a directed fl ipping process that underlies the p erformance of the random edge simplex algorithm. In this stochastic process, which tak es place on a one-dimensional lattice whose sites may b e either occupied or v acant, o ccupied sites b ecome v acan t at a constan t rate and simultaneously cause a ll sites to the right to change their state. This random pro cess exhibits rich phenomenology . First, there is a front, defi ned by the position of the left-most occupied site, that propagates at a nontrivial ve lo cit y . S econd, the fron t inv olve s a depletion zone with an excess of va cant sites. The total excess ∆ k increases logarithmically , ∆ k ≃ ln k , with the distance k from the front. Third, the fron t exhibits rejuvenation — y oung fron ts are vigorous but old fron ts are sluggish. W e inv estigate these phenomena using a quasi-static app ro ximation, direct solutions of small systems, and numerical sim ulations. P A CS n umbers: 02.50.-r, 05.40 .-a, 05.70.Ln, 89.20.Ff I. INTRO DUCTION The s implex algor ithm [1] is the fastest g eneral algo- rithm for s o lving linear problems. While efficie n t in the t ypical cas e , the deter ministic simplex a lg orithm requires an exp onen tial time in the w orst cases [2, 3]. Ra ndomized versions of the simplex a lgorithm ha ve an improv ed run- ning time that is quadratic in the num b er o f inequalities. The p erformance of the ra ndom edge simplex algor ithm on Klee-Mint y cubes [2 ] ultimately reduces to a simple asymmetric flipping pro cess in o ne dimension [4]. In this pro cess, an infinite sequence of 0 and 1 bits evolv es by flipping ra ndomly chosen 1 bits and simultaneously flip- ping all bits to the right. Figure 1 illustrates how the underlined bit flips a ll bits to the right. When flips o c- cur at a consta n t and spatially uniform ra te, the p osition of the left-most 1 bit moves to the r igh t at a constant av erage velocity . Previous formal studies were prima r- ily concer ned with establis hing the ballistic front motion rigoro usly [5], y et most of the questions co ncerning the flipping pro cess, inc luding the propag ation velo c ity , re- main largely unanswered. ...00110101... ...00001010... ...01001010... ...01001101... t FIG. 1: Illustration of the flipping pro cess. The arrow indi- cates the d irectio n of time and the line indicates the position of the adv ancing fron t. W e approach this ra ndo m pr ocess as a nonequilibr ium dynamics problem and b y utilizing a host of theoretical and computatio na l metho ds, w e find that this dir ected flipping pro cess e x hibits interesting phenomenology be- yond the ballistic fron t propa gation. W e also prop ose a mo dified proce ss wher e front pr o pagation is forbidden and show that this pr ocess, for which further theoretical analysis is p ossible, provides an excellent quantitativ e de- scription. Our starting p oint is a qua s i-static approximation. In this description, the s ha pe of the propag ating front is as- sumed to be fixed and additionally , spatia l correlations are ignored. This a ppro ximation yields a qua lita tiv e de- scription for the ov er all shap e of the fro n t and an ex- act des cription for the shap e far awa y from the front. The propagating fron t consists o f a depletion zone as the nu mber of 0 bits exceeds the n umber of 1 bits, and the cum ulative depletion grows loga r ithmically with distance from the front. Direct numerical simulations of the flipping pro cess re- veal that spatial a nd tempor al cor relations are substan- tial. In genera l, neighboring bits a r e co r related as man- ifested by the increas ed likelihoo d of finding co nsecutiv e strings of iden tical bits. Ther e ar e also aging and rejuv e- nation. The state of the sy s tem strongly depends on ag e , defined as the time elapse d s inc e the mos t r ecen t front adv ancement ev ent. In par ticular, young fronts are more rapid than old front. W e also develop a formal solution metho d that de- scrib es the evolution of a finite segment that includes the fro nt. In this approach, the time evolution of all mi- croscopic configur ations o f a finite segment is describ ed under the assumption that the system is completely ra n- dom outside the seg men t. The predictions impro ve sys- tematically as the seg men t size increases but there is a limitation since the num b er of config urations grows ex- po nen tia lly with segment size. Nevertheless, we are able to obtain accur ate estimates for quantities of in terest in- cluding the propagation v elo cit y by using Shanks extrap- olation. In the directed flipping pr ocess, the system do es not 2 reach a steady sta te b ecause o f the perp etual motion o f the front, yet when the front is pinned, the system do es settle into a steady state. W e therefore also examined a mo dified pro cess in which the flipping of the leftmost bit is forbidden. Remark a bly , this pinned front pro cess provides an excellent quantitativ e approximation of the original propaga ting fro n t pro cess. In this case, w e are able to obtain sev eral exact r esults. F or example, w e can show that a pair of neighbo ring sites is cor related. More- ov er, the sma ll system solution is now exact and com- bined with the Shanks transformation, yields excellent results for the velocity . The rest of this pa per is org a nized as follows. In s ection II, titled “ propagating fronts”, we inv estigate the o riginal flipping pro cess. W e b egin with a quasi-static approxi- mation for the shape of the front, co n tinue with n umer- ical sim ulations tha t eluc ida te spatial and temp oral cor- relations, and finis h with analysis of sma ll segments. In section I II, titled “pinned fronts”, w e exa mine the corre- sp onding b eha viors in a mo dified flipping pr ocess w he r e the front is pinned and hence further theoretical analysis is p ossible. Conclusions ar e presented in section I V. II. PR OP A GA TING FRONTS The flipping pro cess takes place on an infinite one - dimensional la ttice whose sites may be in one of t wo states. If σ i denotes the state of i th site then σ i = 1 corres p onds to a n o ccupied site, a 1-bit, and σ i = 0 cor - resp onds to a v acant site, a 0-bit. In the flipping pro cess, each occupied site may “flip” from the o ccupied state to the v aca n t state a nd consequently ca us e a ll sites to the right to sim ultaneously change their state. F o r example, when the j th site flips, σ i → 1 − σ i , for all i ≥ j. (1) The flipping proces s is uniform: all o ccupied sites flip at a uniform r ate, set to one without los s o f generality . Note that the interaction rang e is infinite: every flip even t affects a n infinite n umber of sites! This is in con trast, for example, with constrained spin dynamics such as the east mo del [6, 7, 8] wher e the flipping is ca used o nly by the neighboring spin on the left. V acan t sites with no occupied sites to their left remain v acant forever. Mo reo ver, the left-mos t o ccupied site de- fines a front that adv ances to the r igh t, as s ho wn in fig - ure 1. W e consider the natura l initial condition where all sites left of the origin are v acant, σ i ( t = 0) = 0 for all i < 0, the orig in is occ upied σ 0 ( t = 0) = 1, and a ll sites right of the origin are randomly o ccupied: with equal probabilities σ i ( t = 0) = 1 or σ i ( t = 0) = 0 for all i > 0. A. F ron t Profile and De pletion W e index the system using a refere nc e frame that is moving with the front. Sp ecifically , we c haracteriz e lat- 0 10 20 30 40 k 0 0.2 0.4 0.6 0.8 1 ρ k FIG. 2: The densit y p rofile ρ k , obtained f rom t he quasi-static approximatio n. tice sites by their distance k fro m the fro n t, and by defi- nition, σ 0 = 1. The profile o f the adv ancing fro n t is b est describ ed by the de ns it y ρ k ( t ), th e av er age o ccupation at distance k from the front at time t , ρ k ( t ) ≡ h σ k ( t ) i , where the brack ets indicate an av erag e ov er a ll realiza tions of the random pr ocess. Our theoretical descr iption in volv es tw o simplifying as- sumptions. If w e ov erlo ok the motion o f the front, the densities satisfy d h σ k i dt = * k − 1 X j =0 σ j ! (1 − σ k ) + − * 1 + k − 1 X j =0 σ j ! σ k + (2) for k > 0. The gain term on the right-hand size ac- counts for v a can t sites changing in to o ccupied sites and conv ersely , the loss term represents oc cupied sites c hang- ing into v acant sites. Since every o ccupied site to the left can cause a v ac a n t site to c hange, the gain ra te at the k th site equa ls the total num b er of o ccupied s ites to the left. The loss rate, how e v er, is larger b y one b ecause a flip at the site itse lf can also cause an o ccupied site to change. The evolution equations (2) are hier arc hical: the equa- tion for one-s ite av erag es inv olves tw o-sites av erages , the equation for tw o -site averages inv olves thr ee-site aver- ages, etc. If we ig nore p ossible co rrelations betw een dif- ferent sites and appr o ximate tw o-site a verages b y the pro duct of the resp ective single site averages h σ j σ k i → h σ j ih σ k i , the densities satisfy the closed equation dρ k dt =   k − 1 X j =0 ρ j   (1 − ρ k ) −   1 + k − 1 X j =0 ρ j   ρ k . (3) The flipping rates in equations (2)-(3) reflect the fact that o ccupied sites change at a higher rate than v aca nt sites. Our final assumption is that in the reference frame moving with the fron t, the sys tem is quasi-s ta tic. In- deed, b y definition dρ 0 /dt = 0 , and we further ass ume 3 10 0 10 1 10 2 10 3 k 0 2 4 6 ∆ k ln k simulation FIG. 3: The t otal excess of empty sites ∆ k versus distance k . The sim ulations are in a system of size L = 1000. dρ k /dt = 0 for a ll k . The stationar y densit y profile is ρ k = P k − 1 j =0 ρ j 2 P k − 1 j =0 ρ j + 1 . (4) This recur siv e equatio n is so lv ed s ub ject to the b oundary condition ρ 0 = 1. F or s mall k w e hav e ρ k = 1 , 1 3 , 4 11 , 56 145 , k = 0 , 1 , 2 , 3 , · · · . (5) Despite the crude simplifying ass umptions, this quasi- static approximation provides the fo llo wing v aluable in- sights (see figure 2): 1. Depletion. With the exception of the o ccupied front, all sites a re mor e likely to be v acant, ρ k < 1 / 2 for all k > 0. In other words, the propa g ating front includes a depletion zone. This depletio n is a direct consequence of the fact tha t o ccupied sites change at a higher rate than v a can t sites. In other words, v acant sites hav e a larg er lifetime. 2. Monotonici ty . The densit y pro file is monotonic, ρ i > ρ j for i > j ≥ 1. The tail of the dens ity pr o file can b e obta ined by not- ing that ρ k → 1 / 2 as follows from (4). Cons equen tly , the av erage total “mass” to the left of a given site, m k = P k − 1 j =0 ρ j , grows linear ly with dis ta nce, m k ≃ k / 2. A t large distances, the rec ursion equatio n for the densit y ρ k = m k / (2 m k + 1 ) ca n b e re-written as ρ k ≃ 1 2 − 1 4 m k and ther efore, ρ k ≃ 1 2 − 1 2 k . (6) F ar awa y fro m the front, sites are occupied at random as ρ k → 1 / 2 for k → ∞ . Indeed, sites at the tail c hange their state extremely rapidly at ra tes that g ro w linearly with distance. These rapid changes effectiv ely destr o y 20 40 60 80 t 1.7622 1.7624 1.7626 v 0 20 40 60 80 100 t 0 50 100 150 200 FIG. 4: The a verage position of the leftmost b it h x i versus time t . The results are from a Mon te Carlo sim u lati on in a system of size L = 10 2 , evolv ed up to time t = 10 11 . T he inset show s the vel o cit y v = d h x i /dt versus time. spatial correla tions. Moreover, the a dv ancemen t of the front b ecomes irrelev ant at large distances. Hence, the t wo assumptions underlying our theory a re inconsequen- tial in the tail regio n and (6) is in fact exact. W e com- men t that the algebraic tail (6) is unusual beca use travel- ing wav es are t y pic a lly characterized by exp onential tails [9, 10]. The cumulativ e e x pected excess of v acant sites ov er o c- cupied sites, ∆ k = P k − 1 j =0 (1 − 2 ρ j ), measures the extent of the depletion zone. This quan tit y f ollows fro m the tail behavior (6), ∆ k = k − 2 m k , and since m k ≃ ( k − ln k ) / 2, the excess of v acant sites grows logarithmically with dis- tance, ∆ k ≃ ln k . (7) Thu s, the total excess of v aca n t sites is divergent! W e confirmed the theor e tical predictio ns for the alge- braic tail (6) and the lo garithmic g ro wth of the exc ess (7) using massive Monte Ca rlo simulations (figure 2). The nu merical simulations a re str a igh tforward. In ea ch sim- ulation step one site is chosen at random. If this site is o ccupied, the state of the site and all sites to the right change acco rding to (1), but o ther wise, nothing ha ppens. After each step, time is augmented by the inverse of the system size t → t + L − 1 where L is the n um b er o f sites in the lattice. In our implementation, the front is al- wa y s lo c a ted at the zer oth site, σ 0 = 1. Whenever the front adv ance s by n sites, all lattice sites a re appr opri- ately shifted to the left, σ i → σ i − n (the n r igh tmost sites are reo ccupied at random). Subsequently , the front p o- sition is a ugmen ted by n . This efficient implementation allows us to simulate the evolution of the system up to extremely la rge times. W e ca n evolv e a system of size L = 1 0 2 up to time t = 10 11 , and w e obtain statistical av erages from sna pshots of the system tak en at unit time int erv a ls. 4 B. F ron t Propagation Whenever the leftmost site flips, the front po s ition x adv ances by n lattice sites, · · · 0000 11111 | {z } n 0100 · · · → · · · 0 000 00 0 00 | {z } n 1011 · · · . (8) Hence, the leftmost str ing o f occupied sites governs the front propag ation. Like all other sites, the fro nt flips at a unit rate, and conseq ue ntly , the av erage front po s ition grows ballistically , h x i ≃ v t, (9) and the propag ation velo cit y v equals the av erage size of the leftmost o ccupied str ing v = h n i . Let S n be the probability that the leftmost n lattice sites including the front are all o ccupied, S n ≡ Prob (1111 1 | {z } n ) . (10) The pr obabilit y of finding a string of exact length n as in (8) is equal S n − S n +1 , and therefor e the velocity is given by v = h n i = P ∞ n =1 n ( S n − S n +1 ). Consequently , the velocity equals the sum of string probabilities v = ∞ X n =1 S n . (11) In the Quasi-Static Approximation (QSA), corr ela- tions betw een differen t sites are neglected, a nd hence, the str ing pr obabilit y (10) is a pro duct ov er the corre- sp onding densities, S uncorr n = ρ 1 ρ 2 · · · ρ n − 1 , (12) for n > 1 while S 1 = 1 . With this appr o ximate expres- sion, the propagation velocity is v = 1 + ρ 1 + ρ 1 ρ 2 + ρ 1 ρ 2 ρ 3 + · · · . W e o bta in the approximate velocity v QSA = 1 . 5 3 4070 (13) by substituting the densities from (4) in to (12) and then summing numerically . T he velocity (13) obeys the obvi- ous b ounds 1 ≤ v ≤ 2. The lo wer b ound reflec ts that the fro n t m ust adv ance by at lea st one lattice site, and the upp er b ound cor responds to the completely random configuratio n, ρ k = 1 / 2 and S n = 2 − ( n − 1) . The numerical simulations c onfirm that the fro n t ad- v ances ba llistically (figure 4) but the propagation veloc- it y is la rger than the v alue predicted by the q ua si-static approximation v MC = 1 . 7 6 24 ± 0 . 0001 . (14) Strong s patial correla tions ar e primarily resp onsible for the discrepancy b et ween (13) a nd (14). Indeed, if w e substitute the densities ρ k obtained from the Monte 0 10 20 30 40 50 60 70 n 10 -12 10 -10 10 -8 10 -6 10 -4 10 -2 10 0 S n simulation n -1 0.745 n FIG. 5: The string probabilit y S n versus th e string length n . The results are from a Monte C arlo si mulation i n a system of size L = 200, evolv ed up to time t = 10 11 . Carlo simulations into the pro duct expression (12) and per form the summation in (11), we obtain the v alue v = 1 . 5329 ± 0 . 00 0 1 that is surpris ingly close to the quasi- static approximation (1 3). W e therefore co nclude that spatial correlatio ns b e t ween neighboring sites hav e a sig - nificant effect on the velocity . C. Correlations, Aging, and Rejuv enation Spatial structures and s pa tial c o rrelations c a n b e quan- tified in multiple w ays and we fo cus on the likelihoo d of o ccupied strings S n . Numerically , we find that this qua n- tit y decays exp onen tially (figure 5), S n ∼ n − ν λ n , (15) as n → ∞ with λ = 0 . 7 4 5 ± 0 . 001 and ν ≈ 1. The quasi-static approximation yields muc h more rapid de- cay , λ = 1 / 2 and ν = 1 as follows from the alg ebraic tail (6) and the pro duct expression (12). Of course, when sites a r e completely uncorrelated, one also has λ = 1 / 2 . The fact that λ is larg e r than 1 / 2 r eflects that the sys- tem is strongly co rrelated. T he r e is a significant enhance- men t of str ings of consecutively o ccupied sites and this enhancement is largely responsible for the larger veloc it y (14). Even though spatial correla tio ns are sig nifican t a nd af- fect quantit ies of in terest such as the v elo cit y , they a re limited in extent as indicated b y the exp onential decay of the string likelihoo d. F o r this r eason, n umerical sim- ulations may b e perfor med in rela tiv ely small systems. Given the s pa tial extent of strings shown in figure 5, we per formed the simulations using a relatively small sys- tem, L = 200. This system size is used thro ughout this inv estig ation, unless no ted otherwis e. W e a lso prob ed the correlation b et ween t wo successive front “ jumps” as a measure of temp oral c o rrelations. Let n and n ′ be the siz es o f t wo co nsecutiv e jumps, resp ec- tively . If the front adv ances via a r enew al pro cess then 5 0 1 2 3 4 5 6 τ 0 1 2 3 4 u FIG. 6: The velocity u v ersus age τ . h nn ′ i = h n i 2 = v 2 . How ever, the num erica l simulations yield h n n ′ i = 2 . 95 9 ± 0 . 001 while v 2 = 3 . 10 60 ± 0 . 00 01. Thu s, fron t adv ancement even ts are co rrelated, so the state of the system just after a jump is correla ted with the state o f the system just b efore a jump. This temp oral c orrelation affects, in par ticular, the dif- fusion co efficient D that quan tifies the uncertaint y in the front po sition, h x 2 i − h x i 2 ≃ 2 Dt. (16) Numerically , w e find D = 2 . 8 56 ± 0 . 001. In contrast with the velocity (11) that follows from average quantities such as the average segmen t density , the diffusion co efficien t requires mor e detailed informatio n abo ut tempo ral cor- relations [11]. T o further characterize the dyna mics , we define the age τ as the time elapsed since the most recent fro n t jump. Moreov er, we define the age-dep enden t velocity u ( τ ) a s the a verage size of t he leftmost string n as in (8 ) at age τ bec ause this quantit y gov erns the fro n t pro pagation. The simulations show that the v elo cit y r apidly decays with age (figure 6). O f cours e , since long-living fro n ts outliv e any of their o ccupied neigh b ors, u → 1 as τ → ∞ . Ag- ing fron ts are t herefor e sluggish. In con trast, newly-bor n fronts a re muc h more vigoro us beca use u (0) > v . Since the flipping pro cess is co mpletely random, the surviv al probability of a config ur ation decays exp onen tially with age. T he av er age velo cit y in (11) is the weigh ted integral of the age-dep endent v e locity v = Z ∞ 0 dτ u ( τ ) e − τ , (17) and the weight equals the exp onen tial surviv a l pro ba bil- it y . The age- de p endence of the v elo cit y implies that the shap e of the fro nt must a lso b e age - dependent. W e there- fore measur ed the density profile c k ( τ ) = h σ k ( τ ) i , defined as the average o ccupation at distance k from the front at age τ . W e find interesting evolution with a ge. The profile 0 10 20 30 40 k 0 0.2 0.4 0.6 0.8 1 c k τ=0 τ=1/4 τ=1/2 τ=1 τ=2 τ=4 FIG. 7: The density profile c k versus d is tance k at d ifferent ages. of long -living fr on ts has a depletion zo ne and is qualita- tively similar to the av erage pr ofile discussed ab o ve, but the pr ofile of newly bor n fronts has an enha ncemen t of o ccupied sites ov er v aca n t sites (fig ure 7). This r ejuv e- nation is intuitiv e: the state o f the system just after a jump is a mirro r image of the state of the system just be- fore the jump. Long living fronts are follow e d b y a lar ge string of v acant sites, and these fronts are necessar ily slow. Y et, up on flipping, such slug gish fronts rejuvenate as the string of v a can t sites b ecome a string of o ccupied sites. In terestingly , the density pro file may even b e non- monotonic at in termediate ages. In co nclusion, the flipping pr ocess in volves all the hall- marks of no nequilibrium dynamics including spatial co r- relations, temp oral co rrelations, ag ing, and rejuvenation [12]. D. Small Se gmen ts W e complete the analysis with a direct solution for the state of small segments containing the front. The k left- most sites can be in an y o ne of 2 k − 1 po ssible configura - tions. The equations describing the c onfiguration proba- bilities are hier arc hical: due to the fro n t mo tion, the state of small segments containing the fr on t is coupled with the state of larg er segments. T o ov ercome this closure issue , we pr opose an appr o ximation where the state of the sys - tem outside the se g men t of interest is completely r andom, as in our simulation method. Clearly , this a ppro ximation bec omes exact as k → ∞ . A segment of length tw o can b e in one o f tw o config- urations: 1 0 o r 11 . The resp e ctiv e probabilities P 10 and P 11 evolv e acco r ding to dP 10 dt = − P 10 + P 11 + 1 2 P 11 + 1 2 P 10 (18a) dP 11 dt = − 2 P 11 + 1 2 P 11 + 1 2 P 10 . (18b) 6 k v (0) k v (1) k v (2) k v (3) k v (4) k 2 1.50000 0 3 1.53571 4 1.41894 7 4 1.58716 5 1.82620 5 1.7 79225 5 1.62950 3 1.77309 9 1.7 65862 1.764458 6 1.66220 1 1.76673 0 1.7 64592 1.758245 1.762322 7 1.68710 8 1.76512 9 1.7 63533 1.770104 1.765175 8 1.70598 7 1.76433 0 1.7 62272 1.761669 9 1.72025 1 1.76375 4 1.7 61864 10 1.730 993 1.76 3313 11 1.739 055 T ABLE I: The velocity v , obtained b y successive iterations of the Shanks transformation (propagating fron ts). W e e x plain the la tter equation in detail. The loss rate in (18b) equals t wo b ecause a n y of the tw o o ccupied s ites may flip. If the front flips, there is adv ancement, and since the second site is o ccupied with probabilit y 1 / 2, the gain terms a re 1 2 P 11 and 1 2 P 10 . The stea dy s tate so lution is ( P 10 , P 11 ) = 1 4 (1 , 3); therefore ρ 1 = 1 / 4. W e denote b y v k the velocity obtained from a segment o f leng th k . F or k = 2 w e have v 2 = P 10 + 3 P 11 since the front adv ances by o ne site when the fron t flips in the state 10, but it adv ances three sites in the state 1 1 (t wo sites plus an av erage of one, given the random o ccupation outside the segment). F or k = 3, the governing equations are dP 100 dt = − P 100 + 3 2 P 101 + 1 4 P 110 + 5 4 P 111 (19a) dP 101 dt = − 3 2 P 101 + 5 4 P 110 + 1 4 P 111 (19b) dP 110 dt = 1 2 P 100 − 7 4 P 110 + 5 4 P 111 (19c) dP 111 dt = 1 2 P 100 + 1 4 P 110 − 11 4 P 111 . (19d) The stea dy state solution is ( P 100 , P 101 , P 110 , P 111 ) = 1 56 (27 , 1 1 , 12 , 6 ); th us, the densities a re ρ 1 = 9 / 28, and ρ 2 = 17 / 5 6 a nd the velo cit y is v 3 = 43 / 28. F urther- more, v 4 = 1090 7 / 6872 and the a ppr o ximation steadily improv es as k increases. W e can compute the configur ation of s egmen ts with k ≤ 12 as detailed in Appendix A. T o ex trapola te the velocity , we use the Shanks transfor mation [13] v ( m +1) k = v ( m ) k − 1 v ( m ) k +1 − v ( m ) k v ( m ) k v ( m ) k − 1 + v ( m ) k +1 − 2 v ( m ) k (20) where v ( m ) n is the velocity estimate after m iterations. Repe a ted Shanks tra nsformations g iv e a useful estimate for the propaga tio n v elo cit y (see T able I I), v shanks = 1 . 76 ± 0 . 01 . (21) The Shanks transforma tion c a n b e used to e stimate other quantities as well. F or example, we obtain quantit y propagating fron ts pinned fron ts v 1 . 7624 1 . 7753 D 2 . 856 3 . 178 λ 0 . 74 0 . 75 ρ 1 0 . 3492 1 / 3 ρ 2 0 . 3400 1 / 3 ρ 3 0 . 3479 41 / 120 T ABLE I I: The velocity v , the d iffusio n coefficient D , the deca y constan t gov erning th e string probability λ , and the first few den sities ρ k for propagating and pinned fronts. The results are from Mon te Carlo simulations in a system of size L = 200 evolved up to time t = 10 11 . The exact solution for the density profile is detailed below. an excellen t estimate for the densit y of the first site, ρ 1 = 0 . 3492 ± 0 . 000 1 . II I. PINNED FRO NTS As discussed abov e, the qua si-static a ppr o ximation ne- glects the movemen t of the front and p ossible corr elations betw een sites. Of these t wo assumptions, the latter is more s ignifican t. W e ther efore mo dify the origina l flip- ping pro cess and forbid the front fro m changing state. This minor modification pins the front and allows us to fo cus on the role of corr e lations. In the pinned pro cess, a flip ev ent at every site other than the orig in changes the state o f the system exactly as in (1), but a flip even t a t the origin yields σ i → 1 − σ i , for all i > 0 . (22) Hence, the site at the o r igin is a lw ays o ccupied, σ 0 ( t ) = 1. Remark a bly , pinning the fron t res ults in only minor quantitativ e c hanges. All quantities of interest including the velocity v , the diffusion co efficien t D , the decay con- stant underlying the de c a y o f the segment dens it y λ , and the density pr ofile ρ k are all within a few p ercent of the corres p onding v a lues for propag ating fron ts (T able I I). In particular, the discrepancy in the pro pa gation velo cit y is smaller tha n 1 %, v pinned = 1 . 7753 ± 0 . 000 1 . (2 3 ) W e note that the v elo city and the diff usion coefficient are obtained b y using the running total of seg ment lengths at the time when the origin causes a flip as a surr ogate for the fro n t p osition x . Finally , we ca n not exclude the po ssibilit y that the str ing pro babilit y S n is c haracteriz ed by the same parameter λ in b oth pro cesses (T able I I). In addition, pinned fronts and propag ating fronts have very similar density profiles (figure 8). The quasi- static approximation, which is b etter suited for pinned fronts, bec omes slightly more accurate. Of course, the exa ct tail behavior (6) and the log arithmic excess (7) extend to pinned fronts. 7 0 10 20 30 40 k 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ρ k static front moving front FIG. 8: The density ρ k versus distance k for pinned and prop- agating fronts. Both profiles are obtained using Mon te Carlo sim ulations. A. Correlations The hierarchical evolution equation (3) for the aver- age o ccupation o nly assumes that the front is pinned and hence, this equation provides a n exact description. Therefore, the single- site a verages and the t wo-site av er- ages a re rela ted, h σ k i − k − 1 X j =0 h σ j i = − 2 k − 1 X j =0 h σ j σ k i , (24) at the s tea dy state. Of cours e , h σ 0 i = 1 . W e can obtain the nea r est-neigh b or correlatio n h σ k σ k +1 i , a q uan tit y that evolv es acco rding to d h σ k σ k +1 i dt = − *   2 + k − 1 X j =0 σ j   σ k σ k +1 + (25) + h (1 − σ k ) σ k +1 i + *   k − 1 X j =0 σ j   (1 − σ k )(1 − σ k +1 ) + . This equation is very similar to the equation gov erning the one-site co rrelation. The rate of change for tw o o c- cupied sites is 2 + P k − 1 j =0 σ j bec ause either one of the t wo sites can flip. In genera l, the equation for tw o-site cor re- lations inv olves three-site cor relations, but in the par tic- ular case of neighboring sites, the three-s ite cor r elation cancels in (2 5)! W e therefore obtain a r elation b et ween av erage densities and tw o-site co rrelations k − 2 X j =0 h σ j i = h σ k − 1 σ k i + k − 2 X j =0 h σ j σ k − 1 i + k − 1 X j =0 h σ j σ k i . (26 ) There are tw o different r elations b et w een the av erag e density a nd the tw o -site co rrelation: equatio ns (24) and k ρ k S k +1 v k +1 0 1 1 1 1 1 3 1 3 4 3 2 1 3 1 6 3 2 3 41 120 23 240 383 240 4 76121 21600 0 25577 43200 0 71497 7 43200 0 T ABLE I II: The density ρ k , the string densit y S k , and the velocit y v k = P k n =1 P n , obtained b y direct solution of t he microscopic evolution equations. (26). By manipulating the tw o, we obtain the nea rest- neighbor c o rrelation in terms of the average densit y , h σ k σ k +1 i = 1 2 h σ k +1 i (27) for k > 0. This relation demonstra tes that neighbor ing sites are po sitiv ely co rrelated, h σ k σ k +1 i − h σ k ih σ k +1 i =  1 2 − h σ k i  h σ k +1 i . (28) W e also note that correla tio ns deca y slowly at large distances a s equations (6) and (2 8) imply h σ k σ k +1 i − h σ k ih σ k +1 i ≃ (4 k ) − 1 . F or completeness, we mention that the corr elation be- t ween three consecutive sites can also b e w r itten as a function of low er-order cor relations h σ k σ k +1 σ k +2 i = 1 2 h σ k +2 (1 − σ k ) i . (29) B. Small Sy stems When the fr on t is pinned, the system reaches a sta- tionary s tate. This steady state ca n be obtained exa c tly for small system b y co ns idering the evolution of a ll p ossi- ble configurations. F or pinned fronts, finite s egmen ts are not affected by flipping o utside the s egmen t, a nd co nse- quently , the evolution equations are now closed. Consider for example a system with tw o sites. The r e are tw o poss ible configura tions: 10 and 11 with the r e- sp ectiv e proba bilities P 11 and P 10 . Thes e probabilities evolv e acco r ding to dP 10 dt = − P 10 + 2 P 11 (30a) dP 11 dt = − 2 P 11 + P 10 . (30b) Hence, at the steady state, ( P 10 , P 11 ) = 1 3 (2 , 1) and con- sequently , ρ 1 = S 2 = 1 / 3. Next we consider the first three sites with the four co n- figurations 1 00, 101, 110, 1 11. The evolution equations 8 k v (0) k v (1) k v (2) k v (3) k v (4) k 1 1. 2 1.33333 3 1.66666 6 3 1.5 1.72549 1.76973 7 4 1.59583 3 1.75074 2 1.7 73156 1.775020 5 1.65503 9 1.76261 6 1.7 74362 1.775178 1.775278 6 1.69322 8 1.76852 1 1.7 74849 1.775239 1.775289 7 1.71856 5 1.77157 6 1.7 75065 1.775267 1.775293 8 1.73570 9 1.77320 5 1.7 75170 1.775280 1.775293 9 1.74747 3 1.77409 5 1.7 75223 1.775287 10 1.755 632 1.77 4593 1.775252 11 1.761 337 1.77 4876 12 1.765 350 T ABLE IV: It era ted Shanks transfo rmations for the vel o cit y . The zeroth column is from the small system solution (pinned fron ts). for the resp ectiv e pr obabilities ar e dP 100 dt = − P 100 + P 110 + P 111 + P 101 (31a) dP 101 dt = − 2 P 101 + 2 P 110 (31b) dP 110 dt = − 2 P 110 + P 101 + P 111 (31c) dP 111 dt = − 3 P 111 + P 100 . (31d) The stea dy state solution is ( P 100 , P 101 , P 110 , P 111 ) = 1 6 (3 , 1 , 1 , 1). Therefore ρ 1 = ρ 2 = 1 / 3 and S 3 = 1 / 6. Results for k ≤ 4 ar e summarized in table I II. In gener a l, there are 2 k − 1 microscopic configur ations in a system of size k . W e can compute the s tationary probabilities fo r systems o f size k ≤ 1 2 as detailed in Appendix B. Knowledge of these steady state pr obabili- ties yields the density ρ k , the string proba bilit y S k , and hence, an estimate for the velocity v k = P k n =1 S n . The v elo cit y , as well as other quantities of interest, can be obtained very accura tely using the Shanks tra ns- formation. W e find v shanks = 1 . 7753 ± 0 . 0 001, in per - fect agree ment with the Mont e Carlo simulations (23) as shown in T able IV. C. Aging a nd Rejuvenation W e also examined the evolution with age and found that pinned and propaga ting front s display very simi- lar behaviors, as evident from the a ge-dependent densit y c k ( τ ) (figure 9). F or pinned fron ts, the zero age co nfiguration is the exact mirro r imag e of the configuration just b efore the flip and s ince the front flips at rando m, c k ( τ = 0) = 1 − ρ k (32) 0 10 20 30 40 k 0 0.2 0.4 0.6 0.8 1 c k τ=0 τ=1/4 τ=1/2 τ=1 τ=2 τ=4 FIG. 9: The density c k at d ifferen t ages for pinned fronts (broken lines) and propagating fron ts (solid lines). for all sites except the or igin, k > 0. This expres- sion demonstr ates the enhancement of o ccupied s ites for newly b orn config ur ations. Aging can b e con venien tly studied using small sys- tems. F or the first site, w e have dc 1 /dτ = − c 1 and therefore, c 1 ( τ ) = c 1 (0) e − τ . The initial condition, c 1 (0) = 1 − ρ 1 = 2 / 3 follows from (32). Therefore, c 1 ( τ ) = 2 3 e − τ . (33) F or the first tw o sites, there a re four configura tions: 100, 101 , 110 , 111 , and the resp ective probabilities evolve according to dP 100 dτ = P 111 + P 101 (34a) dP 101 dτ = − P 101 + P 110 (34b) dP 110 dτ = − P 110 + P 111 (34c) dP 111 dτ = − 2 P 111 . (34d) These equations differ from (31 ) in that flipping even ts caus e d by the front are excluded. The initial condition a gain mirr o rs the stationary state ( P 100 , P 101 , P 110 , P 111 )   τ =0 = 1 6 (1 , 1 , 1 , 3). By solving the evolution equations, the age-dep enden t density of the sec- ond s ite, c 2 = P 101 + P 111 , is c 2 ( τ ) = 1 3 (2 τ − 1) e − τ + e − 2 τ . (35) Already , w e can justify the non-monotonic behavior s een in figures (7 ) and (9): c 1 > c 2 for τ < τ ∗ with τ ∗ = 0 . 8742 while c 1 < c 2 for otherwise . In gener al, all densities exhibit a simple exponential decay with age, c k ( τ ) ∝ e − τ as τ → ∞ . W e conclude that pinned fronts faithfully capture aging a nd rejuvenation. 9 IV. CONCLUSIONS In conclusio n, we reformu lated the bit flipping pro- cess underlying the s implex a lgorithm as a nonequilib- rium dynamics pro ble m and studied spatia l and tempo- ral proper ties using theoretical and computational meth- o ds. Ov erall, we find that the infinite interaction range leads to rich phenomenology . There is a front that prop- agates ballistica lly with a nontrivial v elo cit y that is gov- erned by the leng th of the occupied strings containing the front. The pr opagating fro n t includes a deep deple- tion zo ne: v acant sites outnum be r o ccupied sites with the total excess of uno ccupied sites growing log arithmi- cally with depth. The flipping pr ocess is characterized by significant spatia l correlations. F or example, the like- liho od of finding strings of consecutively o ccupied sites is s trongly enhance d. The flipping process also exhibits non trivial dynamics. Successive front jumps are correla ted and additionally , there are aging and rejuvenation a s y o ung fro n ts are fast but old fronts are slow. Underlying this b ehavior is the fact that the state of the system just a fter a jump mir rors the state o f the system just b efore a jump. W e slightly modified the or iginal flipping pro cess b y pinning the front. Qualitatively and quantitativ ely , pinned fronts and propag a ting fronts are very close. W e demonstrated analytically muc h of the interesting phe- nomenology including spatial correlations, ag ing, a nd re- juv enation for pinned fronts. Aging is usually characterized by using t wo different times [14]. Here, in co n tr ast, the time elapsed since the latest front yields a natural definition of age and a charac- terization of the dynamics that complements time itself. W e comment that there is an alternative wa y of study- ing the density profile throug h an av erage at a given lat- tic e site over all r ealizations [15]. The c o rresp o nding a v- erage density ˜ ρ k ( t ) reaches a sta tionary fo r m once the av erage and the v ar iance are taken into account, ˜ ρ k ( t ) → Φ  k − v t √ D t  (36) with Φ( −∞ ) = 0 and Φ( ∞ ) = 1 / 2. This approach has a disa dv antage: the sca ling function Φ( x ) is dominated by fluctuatio ns in the p osition of the front. In other words, the dens it y profile ρ k is smeared beca use of dif- fusion. These less interesting diffusiv e fluctuations are suppressed whe n the fr on t profile is prob ed in a reference frame moving with the front. W e als o pr esen ted a sy stematic s olution metho d of small systems a nd success fully demonstrated how to ex- trap olate relev ant parameters fo r infinite systems. Y et, since the complexity grows exp onen tially with system size, such co mput ations q uic k ly become pr ohibitiv e. W e hav e also seen how most quant ities of interest require an infinite hierar c hy of equations. Finding an a ppropriate theoretical framework with closed evolution equations re- mains a formidable challenge. Nevertheless, the pinned front pro cess provides a p ow er ful theoretica l framew ork. Ackno wl edgmen ts W e are grateful for fina nc ia l suppo rt from NIH grant R01GM078 986, DOE grant DE-AC52-06NA253 96, NSF grants CHE-0532 9 69 and PHY-055 5312; w e a lso thank Jeffrey E ps tein for support of the Prog r am fo r Evolu- tionary Dyna mics at Harv ard Univ ersity . [1] F or a review of earlier w ork, see G. B. Dan tzig, Lin- e ar Pr o gr amming and Extensions (Princeton Universit y Press, Princeton, NJ, 1963). [2] V. Klee and G. Mint y , “How goo d is the simplex algo- rithm?” I n : Ine qualities , ed. O. Sisha (Academic Press, New Y ork, 1972). [3] V. Klee and P . Kleinschmidt, Math. Op er. Researc h 12 , 718 (1987). [4] B. G¨ artnert, M. Henk, and G. M. Ziegler, Combinatorica 18 , 349 (1998). [5] J. Balogh and R. Peman tle, Rand. S truct. Alg. 30 , 464 (2007). [6] F. Ritort and P . S ol lich, Adv. Ph ys. 52 , 219 (2003); S. L´ eonard, P . Ma yer, P . Sollich, L. Berthier, and J P . Garrahan, J. Stat. Mec h. P07017 (2007). [7] P . Sollic h and M. R. Evans, Phys. Rev. Lett. 83 , 3238 (1999). [8] D. A ldous and P . Diaconis, J. Stat. Phys. 107 , 945 (2002). [9] W. v an Saarloos, Physics Rep orts 385 , 2 (2003). [10] E. Brunet and B. Derrida, Ph y s. Rev. E 56 , 259 7 (1997 ). [11] It is simple to show for a renewal pro cess, D renew = 1 2 h n 2 i , but this v alue underestimates the diffusion co efficien t , D renew = 2 . 750 ± 0 . 001. [12] V . Privman, None qui librium Statistic al Physics in One Dimension (Cam bridge Universit y Press, Cambridge, 2005). [13] C. M. Bender and S. A. Orzag, A dvanc e d Mathe mati- c al Metho ds f or Scientist s and Engine ers (Sp ringer, New Y ork 1999). [14] L. F. Cugli andolo, J. Ku rchan, and G. P arisi, J. de Physique 4 , 164 1 (1994). [15] J. R iordan, D . b en-Avraham, and C. R. D oering, Phys. Rev. Lett. 75 , 565 (1995). APPENDIX A: TRANSITION M A TRIX FOR PR OP A GA TING FR ONTS The e v o lution eq ua tions for the config ur ation proba- bilities in a finite segment o f s ize k can b e represented in the matrix form d P dt = M P . (A1) 10 Here, P is the vector P = { P 1 j | 0 ≤ j ≤ U − 1 } where j , written a s a bina ry , is in incr easing order and U = 2 k − 1 . F or example, when k = 5 the state v ector is ( P 10000 , P 10001 , · · · , P 11111 ), with U = 16 en tries. The elements of this vector e qual the pr obabilities that the system is in the res p ective configura tio n. Also, M is the U × U transition matr ix whose elements equal the tr an- sition rates betw een the cor responding co nfigurations. The transition matrix M is a sum o f three matrices M = M 1 + M 2 + M 3 . (A2) W e quote the first tw o for k = 5, M 1 =                                 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0                                 and M 2 = 1 16 ×                                 8 4 2 1 1 8 4 2 1 1 8 4 2 1 1 8 4 2 1 1 8 4 2 1 1 8 4 2 1 1 8 4 2 1 1 8 4 2 1 1 8 4 2 1 1 8 4 2 1 1 8 4 2 1 1 8 4 2 1 1 8 4 2 1 1 8 4 2 1 1 8 4 2 1 1 8 4 2 1 1                                 The third matrix M 3 is diag onal and it g uaran tees that each column of M sums to zero. W e note that the transition matrix is spa rse. The steady state pro ba - bilit y equals the zer o th eigenv ector, M P = 0 . Fi- nally , the velocity follows from the av er age adv ance- men t exp ected in each configur ation. This a dv ance- men t is represented by the vector J and for example, J = (1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 2 , 2 , 2 , 2 , 3 , 3 , 4 , 6) for k = 5. The velocity is simply the scalar pro duct, v = J · P . APPENDIX B: TRANSITION MA TRIX F OR PINNED F RONTS Using the matrix no tation in (A 1 ), the ev olution eq ua- tions for k = 4 inv olve the following transitio n matrix M =                               − 1 1 1 2 − 2 1 2 − 2 1 2 − 3 2 1 − 2 1 1 1 − 3 1 1 − 3 1 1 − 4                               . (B1) In this ca se, the steady state proba bilities are P =               P 1000 P 1001 P 1010 P 1011 P 1100 P 1101 P 1110 P 1111               = 1 240 ×               92 28 22 18 27 13 17 23               (B2) Thu s, the density is ρ 3 = 41 / 120 and the string proba- bilit y is P 4 = 23 / 240.