Reentrant and forward phase diagrams of the anisotropic three-dimensional Ising spin glass

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1 HYSI REVIEW E 77, Reentrant and forward hase diagrams of the anisotroic three-dimensional Ising sin glass an Güven,. Nihat Berker,,, Michael Hinczewski, and Hidetoshi Nishimori Deartment of hysics, Koç University, Sarıyer 5, Istanbul, Turkey Deartment of hysics, Massachusetts Institute of Technology, ambridge, Massachusetts 9, US eza Gürsey Research Institute, TÜBITK Boshorus University, Çengelköy, Istanbul, Turkey Deartment of hysics, Tokyo Institute of Technology, Oh-okayama, Meguro-ku, Tokyo 5-55, Jaan Received ebruary ; ublished 9 June The satially uniaxially anisotroic d= Ising sin glass is solved exactly on a hierarchical lattice. ive different ordered hases, namely, ferromagnetic, columnar, layered, antiferromagnetic, and sin-glass hases, are found in the global hase diagram. The sin-glass hase is more extensive when randomness is introduced within the lanes than when it is introduced in lines along one direction. hase diagram cross sections, with no Nishimori symmetry, with Nishimori symmetry lines, or entirely imbedded into Nishimori symmetry, are studied. The boundary between the ferromagnetic and sin-glass hases can be either reentrant or forward, that is either receding from or enetrating into the sin-glass hase, as temerature is lowered. However, this boundary is always reentrant when the multicritical oint terminating it is on the Nishimori symmetry line. DOI:./hysRevE.77. S number s : 75..Nr,..aq, 5.7.h, 5..c I. INTRODUTION The Ising sin glass yields a hase diagram with a distinctively comlex ordered hase, in d=. wide accumulation of methods and results has occurred for this system. Most remarkably, in site of its high satial dimension and comlex ordering behavior, exact or recise information is being obtained for this system. Thus, in the hase diagram in terms of temerature and concentration of antiferromagnetic bonds, the occurrence of the Nishimori symmetry line has been deduced, and the accurate location of the multicritical oint has been redicted,. urthermore, in systems with the Nishimori symmetry, it has been shown that the ferromagnetic hase cannot extend to antiferromagnetic bond concentrations beyond that of the multicritical oint,. The two remaining otions being a straight line or a reentrance situation, subsequent works 9, on hierarchical lattices have shown that for these systems, the sin-glass hase diagram is reentrant, namely, that below the multicritical oint, the ferromagnetic hase recedes from the sin-glass hase as temerature is lowered. Exact results recently have also been extended to otts sin glasses. These results comlement recent recise calculations, using Monte arlo simulations, on cubic lattices. satially uniaxially anisotroic d = system is studied in this work, to our knowledge the first study of quenched randomness and frustration in a satially anisotroic higherdimensional system. In fact, both anisotroy and quenched randomness have acquired increased relevance from hightemerature suerconductivity results,. Our calculation is exact for a hierarchical lattice and aroximate for a cubic lattice. We find a rich hase diagram e.g., ig. with five different ordered hases, namely with ferromagnetic, an- = =5. =. =.5 =. = IG.. olor online onstant-temerature cross sections of the global hase diagram for / =.5, as a function of and, which are the concentrations of antiferromagnetic xy and z bonds, resectively. t low temeratures high, the central sin-glass hase searates the corner ferromagnetic, columnar, antiferromagnetic, and layered hases. The diagrams are twofold symmetric along each axis, but not fourfold symmetric, due to the difference between longitudinal = and transverse = sin glasses. s temerature increases, the aramagnetic hase aears at the central oint, first reaches the transverse sin-glass system and eliminates the sin-glass hase, then reaches the longitudinal sin-glass system and eliminates the sin-glass hase. In the latter system, the sin-glass and aramagnetic hases simultaneously occur for a very narrow range of temeratures, as also seen in the inset in the lower left anel of ig //77 / 7 - The merican hysical Society

2 GÜVEN et al. IG.. onstruction of the uniaxially anisotroic d= hierarchical model. Two grahs are mutually and reeatedly selfimbedded. Note that for =, =, and =, the system reduces, resectively, to the d=, isotroic d=, and isotroic d = systems. tiferromagnetic, layered, columnar, and sin-glass order. The sin-glass hase is more extensive when randomness is introduced within the lanes than when it is introduced in lines along one direction. The global hase diagram includes cross sections with no Nishimori symmetry, cross sections with Nishimori symmetry lines, and a cross section entirely imbedded within Nishimori symmetry. Thus, the multicritical oint between the sin-glass, ferromagnetic, and aramagnetic hases, reviously found to occur on the Nishimori symmetry line, is also found here at oints with no Nishimori symmetry, but renormalizes to a fixed distribution of interaction robabilities that obeys Nishimori symmetry. Nevertheless, we find that the boundary between the ferromagnetic and sin-glass hases can be either reentrant or forward, that is either receding from or enetrating into the sin-glass hase, as temerature is lowered. When the multicritical oint is not on the Nishimori symmetry line, the ferromagnetic-sin glass boundary can be reentrant or forward. However, when the multicritical oint is on the Nishimori symmetry line, this boundary is always reentrant 9,, consistently with the rigorous result,. II. UNIXIY NISOTROI SIN GSS The uniaxially anisotroic Ising sin-glass system has the Hamiltonian H = u K u ij s i s j, ij u where s i = at each site i, ij u denotes a sum over nearestneighbor airs of sites along the z direction u=z or in the xy lane u=xy, and the bond strengths K u ij are equal to K u with robability u and K u with robability u, resectively corresonding to ferromagnetic and antiferromagnetic interaction. When imbedded in a cubic lattice, the Hamiltonian yields a uniaxially anisotroic d= system. Hierarchical lattices are d-dimensional lattices yielding exact renormalization-grou solutions to comlex statistical mechanics roblems. These lattices are constructed by the reeated self-imbedding of a grah into a bond. The shortest ath between the external vertices of the grah gives the length rescaling factor b and the number of bonds in the grah gives the volume rescaling factor b d, from which the dimension d is determined. Hierarchical lattices have been used to study a wide variety of roblems, including chaotic rescaling 7,, sin-glass 9, random-field 9, Schrödinger equation, lattice-vibration, dynamic scaling, random-resistor network, aeriodic magnet, comlex hase diagram 5, directed-ath,7, heteroolymer, directed-olymer 9, and, most recently, scale-free and small-world network 7 systems, etc. More recently, hierarchical lattices have been created for the study of satially anisotroic systems. The mutual reeated self-imbedding of two aroriately chosen grahs, with differentiated interactions, yields a uniaxially anisotroic system, whereas a higher number of grahs is needed to achieve higher satial anisotroy. These hierarchical systems must reduce to isotroy and/or lower satial dimensions when corresonding interactions are set equal to each other or to zero, as illustrated in ig.. n anisotroic hierarchical lattice has already been used to obtain the hase diagram of the uniaxially anisotroic d = tj model of electronic conduction. When imbedded into the hierarchical lattice of ig., the Hamiltonian yields a uniaxially anisotroic d = sin-glass system that is exactly soluble. III. EXT RENORMIZTION-GROU SOUTION: OWS O THE QUENHED DISTRIBUTIONS O THE NISOTROI SIN-GSS INTERTIONS The renormalization-grou solution roceeds in the direction oosite to the construction of a hierarchical model. Each grah is relaced by a renormalized bond via summation over the sins on the internal sites of the grah. This is achieved by a combination of two tyes of stes: the relacement, by a single bond K ij, of two bonds that are either in arallel, referred to as bond moving: K ij = K I ij + K II ij, or in series, referred to as decimation: HYSI REVIEW E 77, K ik = ln cosh K ij + K jk cosh K ij K jk. The quenched robability distribution K of the relacing bond is calculated by the convolution K = dk I dk II I K I II K II K R K I,K II, where R K I,K II is the right-hand side of Eq. and, K I and K II are the interactions entering the right-hand side of either of these equations, with quenched robability distributions I K I and II K II 9,9. ccordingly, the renormalization of xy is obtained as follows, following the uer ig. in the direction oosite to the arrow: i from the bond-moving of xy with itself, obtaining ; ii from the bond-moving of z with itself, ob- -

3 REENTRNT ND ORWRD HSE DIGRMS O THE TBE I. Sinks of the renormalization-grou flows in the different hases. These sinks are characterized here in terms of the average ositive and negative interactions of their limiting quenched robability distribution. hase K + xy K xy K + z K z erro + + ntiferro olumnar + ayered + Sin Glass + + ara taining ; iii from the decimation of and, obtaining ; iv from the decimation of and, obtaining ; v from the decimation of xy and xy, obtaining 5; vi from the decimation of and, obtaining ; vii from the decimation of and, obtaining 7; viii from the decimation of 5 and xy, obtaining ; ix from the bondmoving of and 7, obtaining 9; x from the bondmoving of 7 and 7, obtaining ; xi from the bondmoving of 9 and, obtaining xii finally, from the bond-moving of and, obtaining the renormalized quenched distribution xy. Thus, in each renormalizationgrou ste, the renormalized distribution xy is obtained from the convolutions of 7 unrenormalized distributions xy and z. The renormalized distribution z is similarly obtained from the convolutions of 7 unrenormalized distributions xy and z, but with a different sequencing dictated by the lower ig.. The renormalization-grou transformations of the quenched robability distributions xy and z, given in the receding aragrah, are imlemented numerically, resulting in a distribution of interaction-strength values and a robability associated with each value, namely, a histogram. Thus, the initial K u double- distribution functions, described after Eq., are of course not conserved under the scale coarsening of the renormalization-grou transformation. The number of histograms increases after each convolution. When a maximum number of histograms, set by us, is reached, a binning rocedure is alied 9,9 : Before each convolution, the range of interaction values is divided into bins, searately for ositive and negative interactions. The interactions falling into the same bin are combined according to their relative robabilities. The convolution then restores the set maximum number of histograms. In this work, we have used the maximum number of 9 for histograms for each distribution xy and z. IV. HSE DIGRMS ND IXED DISTRIBUTIONS We have obtained the global hase diagram of the uniaxially anisotroic d= sin-glass system in terms of the original interactions and robabilities,,,. In each thermodynamic hase, quenched robability distributions / / xy xy = = = =.5 xy z HYSI REVIEW E 77,.... IG.. olor online Temerature-concentration hase diagrams for isotroically mixed uer left, transverse uer right, longitudinal lower left, and =.5 sin-glass systems. In all cases, / =.5. The uer left and right hase diagrams are seen to be, resectively, reentrant and forward, namely, with a ferromagnetic hase that, resectively, recedes from or roceeds toward the sin-glass hase as temerature is lowered, as clearly seen in the insets. There are no oints obeying Nishimori symmetry in the hase diagrams of this figure. Note the remarkably narrow singlass hase, reaching zero temerature, in the longitudinal singlass system, as also seen in the inset. ll hase transitions in this figure are second order. flow, under reeated renormalization-grou transformations, to a limiting behavior sink characteristic of that thermodynamic hase. hase boundary oints flow to their own characteristic unstable fixed distributions, shown below. nalysis at these unstable fixed distributions yields the order of the hase transitions 9,9. We find six different hases for this system, with corresonding sinks characterized in Table I in terms of the average ositive and negative interactions of the limiting distribution. These hases are the ferromagnetic, antiferromagnetic, layered, columnar, sin-glass ordered hases and the disordered aramagnetic hase. In the layered hase, the sins are mutually aligned in each xy lane; these lanes of mutually aligned sins form an antiferromagnetic attern along the z direction. In the columnar hase, the sins are mutually aligned along the z direction; these lines of mutually aligned sins form an antiferromagnetic attern along the xy directions. Both of these hases are thus distinct from the antiferromagnetic hase, which is antiferromagnetic in all three directions. There is a single sin-glass hase, extending to anisotroic systems.. hase diagrams with no Nishimori symmetry ross sections of the global hase diagram are given in igs.,, and. ll hase transitions in these figures are second order. igure shows constant-temerature cross sections of the global hase diagram as a function of and. / / -

4 GÜVEN et al. HYSI REVIEW E 77, / / = xy = z = =.... t low temeratures high, the central sin-glass hase searates the corner ferromagnetic, columnar, antiferromagnetic, and layered hases. The diagrams are twofold symmetric along each axis, but not fourfold symmetric, due to the difference between transverse = and longitudinal = sin glasses. s temerature increases, the aramagnetic hase aears at the central oint, first reaches the transverse sin-glass system and eliminates the sin-glass hase, then reaches the longitudinal sin-glass system and eliminates the sin-glass hase. igure shows temerature-concentration hase diagrams for isotroically mixed, transverse, longitudinal, and =.5 sin-glass systems. The uer left and right hase diagrams are seen to be, resectively, reentrant and forward, namely, with a ferromagnetic hase that, resectively, recedes from or roceeds toward the sin-glass hase as temerature is lowered, as clearly seen in the insets. The Nishimori symmetry see below is obeyed only at four isolated ordinary oints in each cross section in ig. and is not obeyed at any oint in the hase diagrams in igs. and, so that the forward behavior is not excluded by the rigorous results,. remarkably narrow sin-glass hase, reaching zero temerature, occurs in the longitudinal sin-glass system. Zerotemerature hase diagrams are shown in ig. for the longitudinal left column and transverse right column singlass systems. With the aroriate reversal in variables, the longitudinal and transverse sin-glass hase diagrams are seen in this figure to be qualitatively similar, but quantitatively different. The sin-glass hase is more extensive in the transverse case. This can be understood from the more extensive intermixing of the ferromagnetic and antiferromagnetic bonds. IG.. olor online Zero-temerature hase diagrams of the longitudinal left column and transverse right column sin-glass systems. With the aroriate reversal in variables, the transverse and longitudinal sin-glass hase diagrams are seen here to be qualitatively similar, but quantitatively different. The sin-glass hase is more extensive in the transverse case. ll hase transitions in this figure are second order. / / / / / IG. 5. olor online hase diagrams with Nishimori symmetry lines dashed for different anisotroy arameters: The ratio / is,, and.5 from to to bottom. In the left column, satisfies the Nishimori condition. In the right column, satisfies the Nishimori condition. ll hase transitions in this figure are second order. B. Temerature-concentration hase diagrams with Nishimori symmetry curved lines The Nishimori symmetry condition, for isotroic systems = e K 5 generalizes, for uniaxially anisotroic sin-glass systems, to = e and = e. or Nishimori symmetry to obtain, both equations have to be satisfied, but the signs in the exonents can be chosen indeendently. The Nishimori condition, in its general form u K u = e K u 7 u K u for each histogram air of each distribution, is invariant closed under our renormalization-grou transformation. If one of the two conditions in Eq. is fixed, hase diagram cross sections are obtained, in which Nishimori symmetry holds along a line. Thus, throughout the three hase diagrams on the left in ig. 5, the condition on, is fixed. The condition on,, and therefore Nishimori symmetry, is satisfied along the dashed lines on the left in ig. 5. In these temerature versus concentration hase diagrams, it is seen that the multicritical oints between the ferromagnetic, sin-glass, and aramagnetic hases lie on the Nishimori symmetry line. urthermore, it has been roven, that a forward hase diagram cannot occur below such a multicritical oint that is on the symme- -

5 REENTRNT ND ORWRD HSE DIGRMS O THE HYSI REVIEW E 77,... =.7 =.7 Nishimori Surface....9 (a) K/<K> K (b) IG.. olor online The Nishimori condition for is held throughout the leftmost figure and for throughout the center figure. The comlimentary Nishimori condition, for and resectively, is held along the dashed straight lines, which intersect the ordered,,, or sin-glass aramagnetic multicritical oints. In the rightmost figure both conditions are satisfied throughout the figure. In this figure, the hase boundaries around the aramagnetic hase are actually lines of the multicritical oints where the aramagnetic, ordered,,, or, and sin-glass not seen in this cross section hases meet. In the side figures, first-order boundaries dotted occur between the ferromagnetic and layered hases, and between the antiferromagnetic and columnar hases, terminating at d= critical oints. ll other hase transitions full lines in this figure are second order. try line. On the left in ig. 5, this is indeed the case, with reentrant hase diagrams, as also seen in isotroic sin glasses 9,. Recall that in Sec. IV, multicritical oints, between the same hases as here, that do not lie on Nishimori symmetry occur with both reentrant and forward hase diagrams. However, the latter nonsymmetric multicritical oints flow, under renormalization-grou transformations, to the doubly unstable fixed distribution of the symmetric multicritical oints, therefore being in the same universality class and having the same critical exonents. In the three hase diagrams on the right of ig. 5, the condition on, is fixed. In these concentrationconcentration hase diagrams, the multicritical oints between the ordered ferromagnetic, antiferromagnetic, layered, or columnar, sin-glass, and aramagnetic hases again lie on the Nishimori symmetry lines.... (d) K/< K > (c). oncentration-concentration hase diagrams with Nishimori symmetry straight lines In the hase diagrams in ig. 5, the ratio / is held constant. On the left and center of ig., again the condition in Eq. on one interaction is fixed and the other interaction strength is held constant. Thus, the Nishimori symmetry lines become straight lines. The multicritical oints between the ordered ferromagnetic, antiferromagnetic, layered, or columnar, sin-glass, and aramagnetic hases again lie on the Nishimori symmetry lines. In the left hase diagram, due to the enforced Nishimori symmetry condition, = along the line =.5 and the system reduces to d=. long this line, first-order transitions between ferromagnetic and layered hases and between antiferromagnetic and columnar hases terminate at d= critical oints. rom.5, d= secondorder boundaries between each ordered hase and the aramagnetic hase terminate on the d= critical oints. In the center hase diagram, due to the enforced Nishimori symmetry condition = along the line =.5 and the system reduces to d=. ccordingly, the system is disordered aramagnetic along the entire length of this line. D. The hase diagram entirely imbedded in Nishimori symmetry In the rightmost ig., both conditions of Eq. are satisfied throughout the figure. With two symmetry constraints, this is a unique surface in the global hase diagram of our model. The system reduces to d= and d=,asexlained above, for =.5 and =.5, resectively. The hase boundaries around the aramagnetic hases are actually lines of the multicritical oints where the aramagnetic, ordered ferromagnetic, layered, antiferromagnetic, or columnar, and sin-glass not seen in this cross section hases meet. No sin-glass hase occurs within the Nishimorisymmetric subsace. The hase transitions seen in the right- K... IG. 7. olor online ixed distributions, with circles and crosses showing one renormalization-grou transformation and thereby by their exact suerosition attesting to the fixed nature of the distributions. The distributions have been binned for exhibition uroses. a or the ferromagnetic sin-glass hase boundary, a runaway to infinite couling; b for the aramagnetic sin-glass hase boundary. Both of these fixed distributions are satially isotroic, attracting isotroic and anisotroic boundaries, and do not obey Nishimori symmetry. c or the ferromagnetic sin-glass aramagnetic multicritical oint. This fixed distribution is satially isotroic and obeys Nishimori symmetry. This fixed distribution attracts the isotroic multicritical oint, which obeys Nishimori symmetry, and anisotroic multicritical oints, which obey and do not obey Nishimori symmetry. The fixed distributions for the antiferromagnetic sin-glass, columnar sin-glass, layered singlass hase boundaries and for the antiferromagnetic sin-glass aramagnetic, columnar sin-glass aramagnetic, layered singlass aramagnetic multicritical oints are as shown here in a and c, resectively, but with the aroriate and/or reflections. d ixed distribution for the sin-glass hase. This hase sink is an isotroic runaway, attracting both satially isotroic and anisotroic sin-glass hase oints, and does not obey Nishimori symmetry. -5

6 GÜVEN et al. most ig., namely, ordered sin-glass aramagnetic multicritical and ferromagnetic-layered, antiferromagneticcolumnar first-order transitions, are the only hase transitions of the system that occur under Nishimori symmetry. E. ixed distributions The fixed distributions underinning the hase diagrams of this system are given in ig. 7. The fixed distributions for the ferromagnetic sin-glass boundary, aramagnetic singlass boundary, and the ferromagnetic sin-glass aramagnetic multicritical oints are satially isotroic, but attract both satially isotroic and anisotroic hase transitions. The fixed distribution for the ferromagnetic singlass aramagnetic multicritical oints obeys Nishimori symmetry, but attracts multicritical oints that obey and do not obey Nishimori symmetry. In the latter cases, as seen above, both reentrant and forward hase diagrams occur. The fixed distributions for the antiferromagnetic sin-glass, columnar sin-glass, layered sin-glass hase boundaries and for the antiferromagnetic sin-glass aramagnetic, HYSI REVIEW E 77, columnar sin-glass aramagnetic, layered sin-glass aramagnetic multicritical oints are as shown in igs. 7 a and 7 c, resectively, but with the aroriate and/or reflections. V. ONUSION The exact solution of the satially uniaxially anisotroic sin glass on a d= hierarchical lattice yields different hase diagrams. In view of the semiquantitative agreement between satially isotroic sin-glass results on cubic and hierarchical lattices 9, it would certainly be worthwhile to investigate on cubic lattices the henomena found in the resent study. urthermore, the exact study of sin glasses on fully anisotroic d= hierarchical lattices may yield additional hase transition henomena. KNOWEDGMENTS This research was suorted by the Scientific and Technological Research ouncil TÜBİTK and by the cademy of Sciences of Turkey. H. Nishimori, Statistical hysics of Sin Glasses and Information rocessing Oxford University ress, Oxford,. H. Nishimori, J. hys., 7 9. H. Nishimori, rog. Theor. hys., 9 9. H. Nishimori, J. hys. Soc. Jn. 55, Y. Ozeki and H. Nishimori, J. hys., H. Nishimori, hysica 5, H. Nishimori and K. Nemoto, J. hys. Soc. Jn. 7, 9. J.-M. Maillard, K. Nemoto, and H. Nishimori, J. hys., K. Takeda and H. Nishimori, Nucl. hys. B, 77. K. Takeda, T. Sasamoto, and H. Nishimori, J. hys., M. Hinczewski and. N. Berker, hys. Rev. B 7, 5. H. Nishimori, J. Stat. hys., K. Binder and K. Schroder, hys. Rev. B, 97. I. Morgenstern and K. Binder, hys. Rev. ett., Brangian, W. Kob, and K. Binder, J. hys., 7. H. G. Katzgraber, M. Körner, and.. Young, hys. Rev. B 7,. 7 M. Hasenbusch,. elissetto, and E. Vicari, J. Stat. Mech.: Theory Ex.. H. G. Katzgraber,. K. Hartmann, and.. Young, in omuter Simulation Studies in ondensed Matter hysics, Vol. XXI, edited by D.. andau, S.. ewis, and H. B. Schuttler Sringer Verlag, Heidelberg,, e-rint arxiv:.7v. 9 G. Migliorini and. N. Berker, hys. Rev. B 57, 99.. D. Nobre, hys. Rev. E,. M. Ohzeki, J. hys. Soc. Jn. 7, 7. M. Hinczewski and. N. Berker, Eur. hys. J. B 5,. M. Hinczewski and. N. Berker, e-rint arxiv:cond-mat/ 77v.. N. Berker and S. Ostlund, J. hys., R. B. Griffiths and M. Kaufman, hys. Rev. B, R5 9. M. Kaufman and R. B. Griffiths, hys. Rev. B, 9. 7 S. R. McKay,. N. Berker, and S. Kirkatrick, hys. Rev. ett., Monthus and T. Garel, hys. Rev. B 77,. 9. alicov,. N. Berker, and S. R. McKay, hys. Rev. B 5, 995. E. Domany, S. lexander, D. Bensimon, and.. Kadanoff, hys. Rev. B, 9. J.-M. anglois,.-m. S. Tremblay, and B. W. Southern, hys. Rev. B, 9. R. B. Stinchcombe and.. Maggs, J. hys. 9, R.. ngulo and E. Medina, J. Stat. hys. 75, T.. S. Haddad, S. T. R. inho, and S. R. Salinas, hys. Rev. E,. 5 J.-X. e and Z. R. Yang, hys. Rev. E 9, 7. B. Derrida and R. B. Griffiths, Eurohys. ett., R.. da Silveira and J.-. Bouchaud, hys. Rev. ett. 9, 59..-H. Tang and H. haté, e-rint arxiv:cond-mat/75v. 9. Monthus and T. Garel, J. Stat. Mech.: Theory Ex.. M. Hinczewski and. N. Berker, hys. Rev. E 7,. M. Hinczewski, hys. Rev. E 75, 7. Z. Z. Zhang,.. Rong, and S. G. Zhou, hysica 77,

7 REENTRNT ND ORWRD HSE DIGRMS O THE Z. Z. Zhang, S. G. Zhou, and T. Zou, Eur. hys. J. B 5, Z. Z. Zhang, Z. G. Zhou, and.. hen, Eur. hys. J. B 5, H.D. Rozenfeld, S. Havlin, and D. ben-vraham, New J. hys. 9, HYSI REVIEW E 77, H.D. Rozenfeld and D. ben-vraham, hys. Rev. E 75, 7. 7 E. Khajeh, S. N. Dorogovtsev, and J... Mendes, hys. Rev. E 75, 7.. Erbaş,. Tuncer, B. Yücesoy, and. N. Berker, hys. Rev. E 7,