It is shown that the Ising model in two dimensions reveals the intermittent behaviour at the critical temperature. We conjecture that intermittency exists generally at the second order phase transition points

Wosiek, Jacek

Acta Physica Polonica B

English

Jagiellonian Univeristy Repository

Vol. B19 (1988) ACTA PHYSICA POLONICA NO 10LETTERS TO THE EDITORINTERMITTENCY IN THE ISING SYSTEMS** The work supported in part by the CPBP 01.03. and CPBP 01.09.** Address: Instytut Informatyki UJ, Reymonta 4, 30-059 Kraków, Poland.By J. WosiekInstitute of Computer Science, Jagellonian University, Cracow**(Received August 15, 1988)It is shown that the Ising model in two dimensions reveals the intermittent behaviour at the critical temperaturę. We conjecture that intermittency exists generally at the second order phase transition points.PACS numbers: 05.70.JkUntil recently intermittency was studied in a variety of phenomena described by the classical physics [1]. Some time ago it was suggested that the spectra of particles produced in the high energy collisions may also show intermittent beħaviour [2-4], Several models of the multiparticle production with the intermittency were proposed and the specific experi- mental tests suggested [5, 3, 6]. Very interestingly, the Kraków-Louisiana-Minnesota grouρ has found an evidence for the intermittency in the high energy proton-nucleus and nucleus-nucleus collisions [7].In generał intermittency is defined as the occurrence of the large fluctuations which may dominate the average characteristics of the system [1]. In the case of the multiparticle production the good averages to look at are the factorial moments [2] (1)where km is the number of particles emitted in the m-th bin of rapidity in a given event, and <>ev denotes averaging over all events in the sample; n is the multiplicity of the event.ΔYMdetermines the size of the rapidity bin δY = —into which the relevant phase space (863)864Δ Y has bcen divided. Tlιe working definition of the intermittency employed in Refs [2-4] is that thc factorial moments, Eq. (1), grow with M, i.e. with thc decreasing bin size δY. It was argucd that this can happen only if the system undcr coπs'deration revcals the strong fluctuations appearing over all rapidily rangcs. If below certain scalę <5F*  say, the system is “smooth”, then the moments, Eq. (1), would flatten for δY < δY*.In this letter we point out that the intermittent behaviour is naturally exρected in a varicty of statistical systems at the phase transition point of the second order tyρe. The reason for the requirement of the second order phase transition is simple. Only in this case the fluctuations of arbitrary sizes appear in the system [9] (the correlation length diverges), and consequently the factorial momenty Eq. (1), would grow with the decreasing scalę. We illustrate this conjecture by the well known examρle of the Ising system in two dimensions. The model is defined by thc action(2)with β being the inverse temperaturę, and the summation extends over all pairs of nearest neighbouring spins.We shall work in the framework of the Lattice Gauge Theories. It is well known that in the Ising 2 system, close to the critical temperaturę, the domains of the ordered spins of arbitrary sizes appear [10]. Thcrefore, the natural definition of the factorial moments rcmains as in Eq. (1) with km denoting now the number of spins oriented in the positive direction in the m-th “celi”, and the whole lattice is divided into the M(M ≡ 22(!-1)) cells (m = (m1, m2)', mi = 1, 2, ..., 2i_1; z = 1,2). The average over the events becomes the usual LGT average over the configurations. Monte Carlo s⅛ulations were performed in a standard fashion [11,12], Thc configurations wcre generated with the aid of the Metropolis algorithm [13]. After the first 2000 thermalizing sweeps we ħave measured the moments, Eq. (1) at every 30-th sweep in order to reduce the sweep-to-sweep correlations. Altogether we ħave run 10000 sweeps for each β value. All computations were done on the IBM Personal Computer.Fig. 1 summarizes our results for the factorial moments as a function of the celi s’ze (the whole 32x32 lattice has been divided into 22(!_1) cells). The smallest (Z = 5) celi contains 4 spins while the largest one (/ = 1) constitutes the whole lattice. By definition all moments are normalized to unity at Z = 1. The highest computed moment (z = 5) is necessarily equal to zero for Z = 5. Statistical errors are of the order of the size of thc displayed symbols. The solid lines are hand drawn just to guide the eye.It is seen from Fig. 1 that close to the Curie temperaturę (βc ≈ 0.44 for the infinite system) the moments grow rapidly with Z. That is the system reveals its intermittent behav- iour. This is in agreement with the physical picture of the phase transition being accompa- nied by the creation of the scalę invariant domains of ordered spins [9], The fluctuations, though weaker, persist also in the high temperaturę region while in the ordered phase they vanish rather abruptly.It is unclear, at present, if the rapid change in the behaviour of F,s between β = 0.45 and β = 0.46 is the genuine property of the factorial moments, or if it results from some865Fig. 1. Monte Carlo results for the factorial moments: F(1), i = 2(+), 3(Δ), 4(*), 5(0); as the function of the celi size, and for various temperatures T = -j-kind of the numerical instability. The former case would be very intriguing, it implies that we have found a new family of the order parameters which is very effective in locating the critical points of the system. Morę extensive calculations on bigger machines must be done in order to verify∕reject the latter possibility.The lower value of all moments for l = 5 results from the finite multiplicity effect [2]. At Z = 5 we have km ≤ 4, and consequently all moments, especially the higher ones, are “kinematically suppressed” for large l. Due to the finite size of our lattice the available l rangę is rather smali, hence the asymptotic form of the dependence of the moments on the celi size did not develop yet. However, Fig. 1 suggests that, in the vicinity of the phase transition, the dependence is power like — typical to the intermittent phenomena.Our calculations may be improved in many details. One should go for larger lattice, sweep-to-sweep correlations should be treated morę carefully etc. However, the relation between the growth of the factorial moments and the existence of the fluctuations occurring over all scales is undoubtedly demonstrated.Many interesting questions arise. For example, we predict that the growth of F(,)’s is closely related to the order of the phase transition. In the systems with the first order 866phase transition, factorial moments should not rise indefinitely sińce the correlation length there is finite, and hence the size of the fluctuations is also limited by this scalę.The Ising model is the prototype of the Quantum Field Theory, hence it is interesting to extend our study also to the Yang-Mills Lattice Gauge Theories. Since the factorial moments appear to be rather efficient probe of the fluctuations occurring in the system, they may ofier a good approach to study the structure of the Yang-Mills vacuum in both: the high temperaturę and confining phases.I would like to thank A. Białas for the continuous interest in this work.REFERENCES[1] Ya. B. ZeΓdovich, S. A. Molchanov, A. A. Ruzmaikin, D. D. Sokolov, Usp. Fiz. Nauk 152, 3 (1987); [Sov. Phys. Usp. 30 (5), 353 (1987)].[2] A. Białas, R. Peschanski, Nuci. Phys. B273, 703 (1988).[3] A. Białas, R. Peschanski, Jagellonian University preprint, TPJU-4/88.[4] A. Białas, R. Peschanski, Jagellonian University preprint, TPJU-5/88.[5] J. Dias de Deus, Phys. Lett. 194B, 297 (1987).[6] W. Ochs, J. Wosiek, Max-Planck Institute preprint, MPI-PAE/PTh 28/88, May 1988.[7] R. Hołyński et al., Kraków-Louisiana-Minnesota Collaboration, Institute of Nuclear Physics Raport No 1410/PH, Kraków, June 1988.[8] J. B. Kogut, Rev. Mod. Phys. 51, 659 (1979).[9] L. P. Kadanoff, in Phase Transitions andCriticalPhenomena, vol. 5A, edited by C. Domb and M. S. Green, Academic Press, London 1983.[10] J. B. Kogut, Rev. Mod. Phys. 55, 775 (1983).[11] M. Creutz, Quarks, Gluons and Lattices, Cambridge University Press, 1983.[12] N. Metropolis, A. W. Rosenbluth, A. H. Teller, E. Teller, J. Chem. Phys. 21, 1087 (1953).

Intermittency in the ising systems

https://ruj.uj.edu.pl/xmlui/bitstream/handle/item/313439/wosiek_intermittency_in_the_ising_systems_1988.pdf?sequence=1&isAllowed=y

Intermittency in the ising systems

Abstract

Similar works

Full text

Available Versions

Jagiellonian Univeristy Repository