Using the Bethe ansatz we obtain the exact solution for the one-dimensional
asymmetric avalanche process. We evaluate the velocity of dispersive flow as a
function of driving force and the density of particles. The obtained solution
shows a dynamical transition from intermittent to continuous flow.Comment: 12 page

Ivashkevich, E. V.

Povolotsky, A. M.

Priezzhev, V. B.

English

arXiv

Using the Bethe ansatz we obtain the exact solution for the one-dimensional asymmetric avalanche process. We evaluate the velocity of dispersive flow as a function of driving force and the density of particles. The obtained solution shows a dynamical transition from dispersive to continuous flow

Ivashkevic, E. I.

DIAS Access to Institutional Repository

DIAS-STP-OO-21Exact velocity of dispersive flow in the asymmetric avalanche processE.I. Ivashkevich, A.M. Povolotsky, V.B. PriezzhevBogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna 141980, Russia(December 12, 2000)Using the Bethe ansatz we obtain the exact solution for the one-dimensional asymmetric avalancheprocess. We evaluate the velocity of dispersive flow as a function of driving force and the density ofparticles. The obtained solution shows a dynamical transition from dispersive to continuous flow.PACS numbers:64.60.Lx,05.40.+j,47.55.MhThe avalanche dynamics is a basic scenario of relaxation of unstable states in extremal systems where eachmovable element is near a border of stability. A typical long-tailed distribution of avalanche sizes leads tothe dispersive transport of particles [1]. As an illustrative example, granular systems exhibit intermittentavalanches what enables one to use granular piles (sandpiles, rice piles) for explanation of self-organized criticality in generic dissipative systems [2]. In the past decade,it has become clear that the dispersive transport can berecast in terms of interface depinning [3], [4] and various growth models [5]. Recently, a dynamical transitionfrom intermittent to continuous flow in a random sandpile model has been revealed [6]. Nevertheless, an explicittheoretical description of stochastic avalanche processesand an exact evaluation of characteristics of dispersiveflow remains an open problem.Despite drastic simplifications which were introducedto mimic real avalanches, exact results are scarce even forthe determimistic dynamics. As to stochastic dynamics,it is especially difficult as it is beyond the class of abelianmodels [7], where asymmetric processes appear to besolvable [8]. The situation is to be compared with the theory of exclusion processes where many properties, suchas steady states, average current, diffusion constant etc.have been calculated for an asymmetric one-dimensionalcase [9], [10], [11], [12]. The usual presentation of theasymmetrical exclusion process (ASEP) is given by amaster equation for the probability P (x1, ..., xp) of finding P particles at time t on sites x1, ..., Xp of a ringconsisting of N sites. During any time interval dt, eachparticle jumps with probability dt to its right if the target site is empty. This elementary restriction leads toa non-trivial problem of evaluation of the steady stateproperties, which can be solved by the Bethe ansatz.In a similar way, the simplest asymmetric avalancheprocess (ASAP) can be formulated as follows. In a stable state, each of N sites on a ring is either occupied byone particle or empty. The total number of particles P isfixed. During time interval dt, each particle jumps withprobability dt to its right. In the course of time, somesite x may get unstable with occupation number n > 1.Then it must relax immediately to the stable state bytransferring to its right either n particles with probability q(n) or n — 1 particles with the probability 1 — q(n).The quantity q(n) can be associated with a driving forceacting on the unstable group of n particles.The main difference between the ASEP and ASAPlies in the depth of reconstruction of a configurationC = {x1, ..., xp} during the time interval dt. In theASEP. the total distance 1’ covered by all particles between time 0 and t increases by 1 during dt if the configuration C differs from a new one C or remains unchanged if the motion is forbidden. In the ASAP, themotion of a particle is always possible and increase ofY is not bounded. Thus, the configuration C may becompletely different from C depending on numbers ofparticles spilled to right from each unstable site.The present formulation of the ASAP is inspired byworks [13] where a model of activated random walks isintroduced and [15] where the directed avalanche dynamics is formulated in terms of continuous variables. Underan assumption about independence of variations of theavalanche size at each time step, the probability distribution of avalanche sizes is found exactly [15]. However,the configurational space of the continuous model is toocomplicated to determine steady state features.Here, extending the Bethe anzatz approach to exclusion processes, we obtain the expression for the generating function of 1’ for the discrete ASAP in the thermodynamic limit of large N for a fixed density of particlesp = P/N. We find two phases corresponding to a dispersive flow and a continuous flow, and evaluate the exactaverage velocity in the whole range of parameters of thefirst phase. We determine the separation line betweentwo phases where avalanches are critical.Consider the ASAP consisting of P particles on a ringof N sites and denote by P(C) the probability of findingat time t the system in a configuration C. The probability P (C) satisfiesP(C) = [Mo(C,C’) + Mi(C,C’]P(C’) (1)where M1 (C, C’ )dt is the probability of going from C’to C during the time interval dt, and M0 is a diagonalmatrixMo(C,C) = — A11(C’,C) (2)C’CBefore using the Bethe ansatz, it is instructive to notethat in the region where the distances between each two1neighbouring particles exceed 1, the master equation (1)becomes “free”:=where VkPt(...,xk, ...) = Pt(...,x — 1,...) — Pt(...,xk, ...)To compensate the difference between (1) and (3) whenxk — Xk_i = 1 for some k, we introduce the boundaryconditions=—)‘xPt(...,xk—n,xk—n+l,...)where ii = q(2). This condition can be viewed as therecurrent relationwhere the probability of a unstable configurationPt(...x, x, ...) is given in terms of another unstable configuration Pt(...x — 1,x —1,...) and so on. Now, we can define the ASAP by (3) and (5) instead of (1) without evento left. This implies a recurrent relation for the probability q(n) of removing all n particles from the n-foldunstable site, q(ri) = (1 — q(n — 1))u.(3) The average velocity of the particle flow in the ASAPis determined by the average number of steps of all particles involved into an avalanche during the time intervaldt and can be written as() 18) 10VThe rest of the letter is devoted to evaluation of v in(4) the thermodynamic limit N —* oo for a fixed density ofparticles p = P/N.For a finite N, the largest eigenvalue A corresponds tothe solution {z } which converges to zj = 1, j = 1, ..., Pas 7 — 0. For small ‘y > 0, the distance Izi — 1 growsrapidly with N for all j and becomes of order of 1 in the(5) limit N —+ oc. Introducing a variable c by1 — (11)1+ie”iand assuming the solutions {c } are distributed along asmooth curve in the complex plane ci = (u + ir) withendpoints (—a + ib) and (a + ib), we obtain the Betheequation in the form1 a+ibp(c) = 2irF(i) +— J 8(a - )R(fl)dfl - i7 (12)2ir—a+ibwhere we defined as usual a function F(c) such that(6) dF/da = —R(cr)/27r and F(—a+ib) = —F(a+ib)=p/2.The functions p(o) and 8(a) are1 — eiap(c) = —i ln i + ej2J (13)andcosh(v + ia/2)8(a) = —zln( ) (14)cosh(v—z/2)the where i’ = — ln()/2. Taking the derivative in (12), weget the integral equation for R(ct)—R(u, b) + f K(u — v)R(v, b)dv = (u, b) (15)(8) 27r awithcosh ii= . . (16)sinh ii — sinh(z’ — zc)(9) andK() = sinh2u (17)cosh 2u + cos cAll that is very similar to the equations for the asymmetric 6-vertex model [16], [17](see also [18]) with anessential exception: both terms containing z and zzknowing the exact form of the matrix M(C, C’) which isvery cumbersome for the ASAP model.Specifying a configuration C by positions 1 < x1 <X2... < xp < N of the P particles, we use the Betheansatz for an eigenvector of the matrix M0 + M1 in theformA fl[ezJ)]XJwhere the sum is over all of the permutations Q of1, 2, ..., P and the factor exp(7) is the activity of a single particle step. The condition (5) fixes the two particleS-matrix asAJk — 1 (1— L)ezJ — 1te2zjzkAk — 1 — (1 — [L)e7zk — /te2zzkImposing the periodic boundary conditions givesBethe equations—N— 1 N—i1 — (1 — )ez— ezJzkZk— H i ——)ez—e2zzkThe eigenvalue AQ7) corresponding to (6) isA(7) = —P + e7 zAn important property of the ASAP to be solvable bythe Bethe ansatz, follows from the condition of ordering of particles. Due to (5), the decay of a unstable sitecan start from the rightmost argument in a block of nequal arguments in P(..., x, x, ..., x, ...) and then proceed2in (8) are negative, which is the reason for a dynamicaltransition, as we shall show below.If a = ir, equation (15) can be solved by the Fouriertransformation. To evaluate 8A, we have to find thesolution of (15) in a vicinity of the point a = ir whichcorresponds to a “conical” point, considered in [17). Following Bukman and Shore, we write the solution R(u) asan expansion in f = ir — a up to order of 0(e4)R(u) = Ro(u) + e’R1(u) +e2öR(u) +e3R(u) (18)The necessity of such a long expansion will be seen in further calculations. The Fourier transformation is definedby00X(u) = (X)e1fl= —00where X stands for R0, öRm, , K. The non-zero Fouriercoefficients of K and for b > —2v are(K)0 = (_1)ne_2= eb(1— (—1)e2),n <0Then, (16) givesIn the next order in e. we haveRo(ir)— 1) = (—1)0K)itso, that Ro(ir) = 0,(R1)= 0 and (Ro)0 = 1/(1+expb).The next terms in (18) are evaluated in [17]5R2(u) = -R(ir)in(K)0R(it)e00R3(u) = — 3it(1— (K)0)flo1 (4)= —R0 (it)Thus, in the expansion of R(u), Ô’R2 (u) 6R2 andR4(u) 5R4 are real constants and R3(u) is imaginary.Now, we are ready to start a direct evaluation of 8Ain (10) using 6yi\ = UaA/8a’7. First, we find 8a7• To thisend, we put a = a+ib in (12), and take the derivative bya at a = it. Recalling the conditions F(a + ib) =and 0(0) = 0, we obtain7t8aP + j3a7 = R(a, b) + (2ir)’9(2a)R(—a, b) +(2ii-)’ J’-a — v)8aR(V, b)dv9R(ir)8a7 = ZEit(19)e’ bRo(u) = (Ro)o + 1 — e’To evaluate r.h.s. of (27), we express the valuesR(a, b),R(—a, b) and 8(2a) by their Tailor expansions ata = it up to the order of 0(e) using (14) and (18). Theintegral in (27) is treated asf f(v)dv = f f(v)dv + B(e) (28)withB(e) = ‘{(e)mf(ml)(it) - (e)mf(ml)(it)}rri=1and, therefore, can be evaluated by the Fourier transformation. The Fourier coefficients of 0(ir— v) are(0(it— v)) = (e_2— (—1)), n 0 (29)and (0(ir— v))0 = 2-it. The only v-dependent part of(20) ãaR(v,b) is öR3(v). Using (24)-(26),(20) and (29), weobtain the explicit expression for r.h.s. of (27)(21)eR(it) - 9R(it) + (30)Due to (22), the first and third terms in r.h.s. of (30) are(22) real, the second one is imaginary. Therefore, we have(31)(23) The expression for 8aA can be found in a similar way.In this case, we take the derivative by a in= R(u,b)(z(u,b) - 1)du (32)(24)where z(rz, b), according to (11), is1 — eiu_b(25) z(u, b)= 1 + eiu_21(33)The obtained derivative is similar to (27) but containsz(v, b) instead of 0(a— v). So, we need the Fourier coef(26) ficients of z(v, b) which are(z)0 = (1 <0 (34)and (z)o = 1.Continuing as in (27), we get1OA =9R(it)z’(it,b)-39(R (35)We can see that the term R3(u) in (18) is relevant. Asto 6R4 (u), it is sufficiently that it is a constant and does(27) not lead to a divergency by integration.3Substituting the explicit expressions for (z)._ and(oR3) gives the second term in r.h.s. of (35) in theformlfl:_2flVDue to (31), R(7r) is cancelling in UA/0y. Then, using(34) and the identity p = (R0) = 1/(1+expb) we obtainthe final result(l—p)(1+t) 1+iiv= 9+ F(i,p)(l—p(l+t))- tpF(i,p)=1:Ci—nThe velocity of flow v diverges at Pc = 1/(1 + t) whichimplies a transition to the phase of continuous flow. Thevalue of critical density Pc can be easily understood fromthe condition of a balance between gaining (Pc) and losing(1 — q(oo)) one particle at each step of a large avalanche.The considered model is a directed version of the modelof activated random walks introduced in [13] to see howa conservative dynamical system with the sandpile toppling rules approaches criticality. It has been shown in[14] that the relaxation time r and correlation lengthdiverge as T Loc — pj’ and Pc — ph’2. The exponents v and 112 have been determined numerically forseveral kinds of toppling rules. In the directed case,coinsides with T and is proportional to the average sizeof avalanches (s). On the other hand, (s) = v so we havefrom (37) (s) (Pc—p)2 and v = 1)2 = 2. An extension of this result to the symmetrical case [19] is a veryinteresting and difficult problem.One of us (VBP) is grateful to D.Dhar for helpfulcomments. VBP acknowledges T.Dorlas for reading themanuscript and DIAS, Dublin for hospitality.[1] H.Scher,M.Shlesinger, and J.Bendler, Physics Today 441,26 (1991).[2] P.Bak, C.Tang, and K.Wiesenfeld, Phys. Rev. Lett. 59,381 (1987).[3] J.Krug and H.Spohn, Solids Far From Equilibrium.C.Godreche,ed. (Cambridge University Press, 1991).(37) [4] L.-H. Tang, M.Kardar and D.Dhar, Phys.Rev.Let. 74,920 (1995).[5] M.Paczuski and S.Boettcher, Phys.Rev.Let. 77, 111(1996).[6] A.Corral and M.Paczuski, Phys. Rev. Lett. 83,57238) (1999).[7] D.Dhar, Phys. Rev. Lett. 64, 1613 (1990).[8] D,Dhar and R.Ramaswamy, Phys.Rev.Lett. 63, 1659(1989).[9] L.-H. Gwa, H.Spohn, Phys.Rev.Lett. 68, 725 (1992).[10] B.Derrida and J.L.Lebowitz, Phys.Rev.Let. 80, 209(1998).[11] B.Derrida, Phys.Rep. 301, 65 (1998).[12] M.Khorrami and V.Karimipour, J.Stat.Phys. 100,999(2000).[13] R.Dickman, M.A.Munos, A.Vespignani and S.Zapperi,Brazilian Journal of Physics 30,27 (2000);[14] A.Vespignani, R.Dickman, M.A.Munoz and S.Zapperi,Phys.Rev.E 62,4564 (2000).[15] S.Maslov and Y.C.Zhang, Phys. Rev. Lett. 75, 1550(1995).[16] I.M.Nolden, J. Stat. Phys. 67,155 (1992).[17] D.J.Bukman and J.D.Shore, J.Stat.Phys. 78, 1277(1995).[18] T.Sasamoto and M.Wadati, Phys. Rev. E 58,4181 (1998).[19] S.Manna, J.Phys.A 24,L363 (1991).(36)with4

Exact velocity of dispersive flow in the asymmetric avalanche process

Abstract

Similar works

Full text

Available Versions

DIAS Access to Institutional Repository