Spontaneous Structure Formation in Driven Systems with Two Species: Exact Solutions in a Mean-Field Theory

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1 owa State University From the SeectedWorks of Beate Schmittmann October 10, 1994 Spontaneous Structure Formation in Driven Systems with Two Species: Exact Soutions in a Mean-Fied Theory. Vifan, University of Ljubjana R. K. P. Zia, Virginia Poytechnic nstitute and State University Beate Schmittmann, Virginia Poytechnic nstitute and State University Avaiabe at:

2 VOLUME 73, NUMBER 15 PHYSCAL REVEW LETTERS 10 OcTQBER 1994 Spontaneous Structure Formation in Driven Systems with Two Species: Exact Soutions in a Mean-Fied Theory. Vif~, ' R. K. P. Zia, and B. Schmittmann J. Stefan nstitute, University of Ljubjana, P.O. Box 100, S Ljubjana, Sovenia 2Center for Stochastic Processes in Science and Engineering and Department of Physics, Virginia Poytechnic nstitute and State University, Backsburg, Virginia (Received 4 March 1994) A stochastic attice gas of partices, subject to an excuded voume constraint and to a uniform externa driving fied, is investigated. Using a mean-fied theory for a system with equa number of oppositey charged partices, exact resuts are obtained. Focusing on the current-vs-density pot, we propose an expanation for the discontinuous transition found in earier simuations. A critica vaue of the drive, beow which this transition becomes continuous, is found. These resuts are supported by a bifurcation anaysis, eading to an equation of motion for the ampitude of the soft mode. PACS numbers: Cn, Fh, Dn, Bj Compared to equiibrium phenomena, systems far from equiibrium are, in genera, extremey interesting but poory understood. t is natura to focus first on a restricted set of nonequiibrium systems, namey, ones in steady state. Within this context, driven attice gases, the simpest generaizations of the we known sing mode, were introduced a decade ago [1] and received considerabe attention since [2]. Even for this simpe mode, many unexpected phenomena were discovered, mainy through Monte Caro simuations. By contrast, anaytic resuts are much more difficut to achieve and much rarer. Recenty, a variation of this mode was introduced [3],in which there are two species of partices, driven in opposite directions by the externa fied. Further motivations for studying this and reated modes as we as possibe physica systems may be found in [4]. Since there are no interpartice interactions other than the excuded voume constraint, this mode may be arguaby even simper than the driven sing attice gas. Yet, it undergoes a discontinuous transition, from a homogeneous state to inhomogeneous ones [3,5]. n this Letter we present anaytic and numerica resuts, based on a set of continuum equations proposed in [3]. Being essentiay a mean-fied theory, it is found to dispay the equivaent of a van der Waas oop in a suitabe phase diagram, strongy indicating a first-order transition. Further, this anaysis predicts a "critica" point, beyond which the transition turns continuous. After a brief summary of the mode, simuation resuts, and the continuum description [3], we present exact steadystate soutions of the fied equations, in terms of eiptic functions. These soutions aow us to ocate the critica point exacty. A resuts are shown to satisfy scaing, in that they depend on the drive ony through its product with L, the ength of the system aong the drive. We aso present an approximation, based on the L ~ imit, which uses ony eementary functions and offers a more transparent picture. Finay, we report on the bifurcationtype anaysis from which concusions on the order of the transition are drawn. We consider a periodic square attice, partiay fied with an equa number of oppositey "charged" partices. These hop randomy to nearest neighbor empty sites (hoes), biased by an externa "eectric" fied E acting aong, say, the +y direction. Since the partice numbers are conserved, both the tota charge and tota "mass" are constants of the motion. The former is chosen to be zero for simpicity. We wi specify the atter by the mass density m, namey, the number of partices of either charge, per site. Uness the aspect ratio of the system is sizabe [5], the inear dimension transverse to the fied appears to pay itte roe. Thus, the contro parameters of this mode are m, E, and L. Eary simuations with L = 30, 60 and 0.4 ~ ~E( ~ 3.0 show that there exists a transition, as m is raised beyond a critica density, from a homogeneous steady state to an inhomogeneous one [3]. Both the current and the square of the oca charge density, the atter being a measure of inhomogeneity, jump discontinuousy at the transition, suggesting that it is first order. n order to understand the coective behavior, a continuum mode, in the form of equations of motion for the two conserved densities, was introduced [3]. n these "coarsegrained" equations, the microscopic fied ~E~ is repaced by an effective drive: 4, so that the contro parameters are now m, '4, and L. Being continuity equations, they triviay admit time-independent homogeneous soutions, which are stabe to sma perturbations provided m is ess than mh = [1 + (2n/e)2]/2, where e = XL. nhomogeneous steady-state soutions were aso obtained, but their stabiity imit has yet to be found. Moreover, acking a free energy far from equiibrium, it was uncear how to ocate a first order transition. n the foowing, we focus on JH and J, the current densities of the homogeneous and inhomogeneous states, respectivey. For fixed e, we find (i) JH(m) ~ J&(m), with /94/73(15)/2071(4)$ The American Physica Society 2071

3 VOLUME 73, NUMBER 15 PH YS CAL REVE% LETTERS 10 OcTosER 1994 equaity hoding ony at mh, (ii) J(m) is doube vaued, for mi ( m & mh, provided e exceeds a critica vaue: e and (iii) BJt/Bm ~ as m ~ mi. Thus, we conjecture that mi is the stabiity imit of the inhomogeneous soution, so that a first order transition may occur ony if a & e,. [3]: = V. (V@ + X@fy't We begin with the equations for P, the hoe density and P, the charge density and B,P = V t,pv P P)y). The mobiity, or diffusion constant, has been absorbed into the time scae. The constraints due to mass and charge conservation provide = (1 m)v and f P dv = 0, where V is the voume of the system. Of course, both P and P must be periodic in y E [O, L]. Dimensiona anaysis impies that '4 is an inverse ength, so that the parameter a is dimensioness. Thus, any steady-state soution which is inhomogeneous in y satisfies scaing, i.e., it is a function of ony dimensioness variabes: e and z = y/l. Simiary, the current densities, identified as the quantities within the [) brackets, associated with such a state aso satisfy scaing. Of course, this statement is trivia for the hoe current, which vanishes due to charge symmetry. For the charge current, it impies that Jt($, L, m) takes the form '4 j(e, m) For thė homogeneous steady state, scaing is trivia, since many quantities are independent of L and e, e.g., $0 = 1 m, Po = 0, and JH/'4 = jh = m(1 m). Focusing on steady states which are inhomogeneous ony in y, the t-independent partia differentia equations reduce to ordinary ones. Thus, each can be integrated once immediatey, the constants being the hoe and charge current densities, and P can be expressed in terms of P. Denoting d/dz by prime and etting g = /P, we arrive at an exceedingy simpe equation [3] for p: X'"/e' = JX' + X 1 mposing boundary conditions, g(0) = g(1), and the constraint f doz(1/g) = 1 m, the soution is unique. Since (1) is just the equation for a Newtonian partice in a one-dimensiona "potentia" V(g) = 3jg' 2g2 + g, it can be soved by another integration. The resut is g' = eg2(u V), where U is the anaog of the "energy" of the "partice. " From here, a further integration gives the soution. Note that the constants j and U are unknowns at this point, to be fixed by the constraints above. Potting V against g, it is cear that a periodic "motion" (in z) is possibe ony if j ( 4 with suitabe vaues of U. Unfortunatey, U is an unknown here, making it an inconvenient parameter. nstead, we wi use p = 5/(g+ gi), where 5 = g+ g and g& ~ y ( g+ are the three roots of U = V(g). Thus, ~(z) ies between and g+, whie g ' and g+' are the imits of the hoe density. Writing 2(U V) as (2j/3)(g+ g)(yg )(g gi), a three roots are expressed in terms of j and p. The soution is now written expicity using Jacobian eiptic functions [6]: for the haf period z E [0, 2]. For the second haf, g simpy retraces from g to g+. At z = 2, u must take the vaue of the compete eiptic integra EC(p). So, K/e = Qj b /24p Afte.r some tedious agebra, 5 is eiminated in favor of j and p, so that (2) 1 4j=R (1 p+ p), (3) where R = [4K(p)/e]2. Before continuing, note that (3) can be expoited to find other important quantities, such as /g+ = (1 R [1 p + p ])/2[1 + R + pr], the minimum hoe density, and n = 5/g+ = 3pR/(1 + R + pr). Now, (3) is an expicit expression for j(e, p). But our goa is j(e, m), for which we need a formua for m(e, p). Casting the constraint 1 m = f dz/y = f dg/g'g in terms of a compete eiptic integra of the third kind [6], m. (nip), we obtain 1 m = (1 R'[1 p + p'])n. (nip) 2[1+ R + pr]sc(p) nstead of inverting (4) to express the unknown p in terms of m, we generate the function j (e, m) parametricay. For each fixed e, (3) and (4) can be used to pot a ine in the m-j pane by etting p vary from 0 to po(e), a vaue to be discussed. By anayzing (3), we find that j(p) is a monotonicay decreasing function. Since K(1) diverges, j wi vanish before p reaches 1, for any a ~ ~. Thus, the equation j(e, po) = 0 defines a po(e), which is a maximum aowed vaue of p. Since po ( 1, a factors in (3) are finite. But the factor 1 R [1 p + p ] is just 4j, so that 1 m(po) aso vanishes. n the imit p 0, the partice executes simpe harmonic "motion" around yo, the oca minimum of V. Since the average of g is triviay go, it is constrained to be 1/(1 m). At this point, j = jh. From (1), the frequency of this "osciation" is eg2jgo 1, which must fit the "period" [0,1]. This fixes m to be precisey mh. Thus, for a given a, a ine is traced out in the m- j pane, from (mh, jh) to (1,0). Exampes of these, for various a from 10 to 200, are shown in Fig. 1. %e see that jh(m) ) ji(e, m), with the equaity hoding at mh(e) Whie j(p) is monotonic for any e, Bm/Bp may vanish. n particuar, for sma p, we find m = mh + 3p co[1 + 2' 23cu2]/(8[1 + cu])2 +, where cu = (2m. /e)2. On the other hand, j = j(0) 15cu2p2/64 + Thus, ines j(m) begin with sopes with different signs, diverging at e, = 2' ( 424 1) 'i = (5) 2072

4 ~ g VOLUME 73, NUMBER 15 PHYSCAL REVEW LETTERS 10 OcTQBER ~ ~ i t r y t + ~ r t g ~ ~ ~ ' ' \ \ ' ~ri r' ~ ~ FG. 1. Steady-state current characteristics for various s. For the homogeneous states, jh is e independent (bod ine). For the inhomogeneous states, j(m) is singe vaued (thin soid ines) provided e ( e,. Otherwise, j is doube vaued (dashed and thin soid ines). The dash-dotted ine is the conjectured imit of stabiity of these states. Given an e, the homogeneous states become unstabe at the junction: j = jh FG. 2. An a-m phase diagram, showing the homogeneous (H) and inhomogeneous () phases. The transition from one to the other is continuous across the bod soid ine. Beyond the soid circe, the transition is "first order" with the "spinodas" shown as dot-dashed and thin soid ines. n between, H +, one phase is stabe and the other one is metastabe. The coexistence ine, obtained numericay for d = 2, is represented by the dashed ine. Corresponding to j, = and m, = , this point is shown as a soid circe in Fig. 1. For a ( a both Bj/Bp and Bm/Bp have definite sign, so that j(m) is singe vaued. Thus, we arrive at a simpe picture, for a = 10, say: The system remains in the homogeneous state (heavy soid ine in Fig. 1) unti m reaches mh(e) = 0.7, beyond which ony the inhomogenous state (ight soid ine) is stabe. Thus, mh(e) marks a supercritica bifurcation. However, for each e ) e Bm/Bp = 0 at one point, which we abe by mi(e) and dispay as the dotdashed ine. As Bj/Bm changes sign, j(m) is a doube vaued function of mt & m & mh. Thus, the picture for, e.g., a = 30 is far more compex: As m is raised from 0, the homogeneous state (heavy soid ine) is the ony avaiabe branch for the system. Stabe against sma perturbations, this branch may be foowed unti mh = 0.52, where a subcritica bifurcation takes pace. At that point, we may retrace aong the upper branch of j(m), shown as a dashed ine in Fig. 1, foowed by the ower branch (ight soid ine). Such a oop is reminiscent of a van der Waas oop. We have numerica evidence that the upper (ower) branch is unstabe (stabe), with mi being the imit of stabiity of the inhomogeneous states. Thus, mi and mh provide the "spinodas" in a phase diagram (dot-dashed and ight soid ines in Fig. 2), and we expect "first order" type transitions, if a ) e,. Unike in equiibrium, however, there is no free energy to determine the ocation of the associated coexistence curve. Therefore, we numericay investigated the evoution, under the equations of motion, of an initia configuration with a domain wa, separating the homogeneous and inhomogenous phases. Since such a wa is expected to move toward the metastabe phase, vanishing of its veocity can be used to determine the coexistence curve. The resut, for d = 2, is shown as the dashed ine in Fig. 2. These resuts for inhomogeneous states, though exact, are not easiy visuaized. Fortunatey, there is a simpe and exceent approximation for a )) 1, so that p ~ 1. Then, sinhu ~ tanhu, 6 ~ 3/1 4j/2j, g ~ 2/(1 + Q 4j ), and u (ez/2)[1 4j]'j4, so that g(z) is a soiton in the Korteweg de Vries equation [7], i.e., + A sech u. Of course, we shoud use the interva z C [ 2, 2] instead of [0,1]. Even so, periodic boundary conditions wi not be satisfied for g, though the discrepancy is exponentiay sma. Since 1 /g(z) is the partice density, the majority of partices is accumuated in a finite "wa" around z = 0. Thus, any m vaue can be achieved by cutting this function off at z = ~2 and adjusting j. n the imit of arge me, this procedure gives us j = 6e 'i, which aso provides a sense of where in the m-a pane can we expect sma j. From g, we obtain the mass density 1 1/g and the charge density, = di g'/eg. n Fig. 3, we give an exampe of the various densities with e = 30 and m = 0.4. Note that a the partices are essentiay ocked in the centra region, giving rise to a minute current. To provide additiona carification regarding the order of the transitions, and to define the order parameter, we have performed a bifurcation-type anaysis in the neighborhood of mh, the stabiity imit of the homogeneous phase. Beginning with the fu equations of motion for hoe and charge density, we first identify the sow modes and then derive effective equations of motion for the atter, by adiabatic eimination [8] of the fast modes. Writing (@,P) = ($0, 0) + (g, g), we empoy the Fourier components (gk, gk), where k = (q, 2m n/l) 2073

5 VOLUME 73, NUMBER 15 PH YS CAL REVE% LETTERS 10 OcToaER 1994 ~ W Q p P z FG. 3. Density profies in the inhomogeneous steady state, with e = 30 and m = p+ and p are the densities of + and partices, respectivey. Their sum and difference, denoted by p and P, are the tota mass and charge densities, respectivey. consists of a transverse wave vector q and an integer n for k~. The eigenvaues of the inearized equation are 1 r-(q, n) =, k' ( + Pp) ~ mna L k '. 6 Since the conservation aws prohibit q and n from both vanishing, ony r (0, 1) 0 when Pp 2(1 co), i.e., m mh. Thus, we find a singe sow mode: M = $(0, 1), with s" = 2miM/aPp, which can serve as an order parameter. Of course, in the thermodynamic imit with e fixed, there woud be bands of sow modes. Focusing on ongitudina modes ony, we report a preiminary, but intriguing, resut when ony the q = 0 fast modes are eiminated: Beyond the inear eve, we find an equation of the time-dependent Ginzburg-Landau type [9], dm/dt = (rm + gm~m ~~ + ), where r is r (0, 1) and g(a) ) 0 for a ( e,. This strongy suggests that these transitions are continuous and provides a possibe framework for first order transitions for a ) e,. At s g vanishes and it is tempting to abe this as a tricritica point. %e refrain from doing so, since M is compex. Ceary, this approach has potentia and shoud be pursued. n particuar, to study the possibe existence of infrared singuarities and anomaous behavior, we shoud incorporate the noise, as we as the fu11 spatia dependence, into a Lang evin equation for the order parameter. To concude, we found that the exact soutions of the mean-fied theory, proposed for the biased diffusion of two species, provide cear insight into the spontaneous formation of inhomogeneous structures observed in simuations [3]. Simpe scaing prevais, i.e., the drive appears ony through the parameter e. Given an e, inhomogeneous states cannot exist if m ( mt(e). On the other hand, if m ) mh(s), the inhomogeneous state is unstabe. At a critica a given exacty in (5), m, = mh. Thus, the transition is continuous for e ~ e, and is beieved to be first order otherwise. Near the continuous transitions, we found a sow mode, M. Preiminary investigations show that it obeys an equation typica for an order parameter in a second order phase transition. Ceary, this study raised many interesting questions concerning the nature of phase transitions in nonequiibrium states. %e thank Z. Racz and K. Basser for many enightening discussions. This research is supported in part by grants from the Nationa Science Foundation through the Division of Materias Research and the Jeffress Memoria Trust. One of us (.V. ) acknowedges the hospitaity of the Physics Department at Virginia Poytechnic nstitute and State University. Note added. After competition of this work, we became aware of a manuscript by Foster and Godreche [10], consisting of anaytic and simuations resuts. Whie the atter showed good data coapse confirming the scaing of EL, the former reied ony on the soiton approximation, so that the continuous transitions for a & a, are missed. [1] S. Katz, J. L. Lebowitz, and H. Spohn, Phys. Rev. B 28, 1655 (1983); J. Stat. Phys. 34, 497 (1984). [2] B. Schmittmann and R. K. P. Zia, in "Phase Transitions and Critica " Phenomena, edited by C. Domb and J. L. Lebowitz (Academic, New York, to be pubished); B. Schmittmann, nt. J. Mod. Phys. B 4, 2269 (1990). [3] B. Schmittmann, K. Hwang, and R. K. P. Zia, Europhys. Lett. 19, 19 (1992). [4] B. Schmittmann, K. Basser, K. Hwang, and R. K. P. Zia, Phyisica (Amsterdam) A (to be pubished). [5] K. Basser, B. Schmittmann, and R. K. P. Zia, Europhys. Lett. 24, 115 (1993). [6] We use the notation of M. Abramowitz and.a. Stegun, Handbook of Mathematica Functions (Dover, New York, 1970), Chap. 16 and 17. [7] See, e.g., P. G. Drazin and R. S. Johnson, Soitons: An ntroduction (Cambridge University Press, Cambridge, 1989). [8] Z. Racz (private communication). See, e.g., H. Thomas, in Noninear Dynamics in Soids, edited by H. Thomas, (Springer, Berin, 1992). [9] P. C. Hohenburg and B.. Haperin, 435 (1977). Rev. Mod. Phys. 49, [10] D. P. Foster and C. Godreche, Sacay report, 1993 (to be pubished). 2074