Physica A 277 (2000) 106–114www.elsevier.com/locate/physaShort-time behaviour of the two-dimensionalhard-disk modelA. JasterFachbereich 7 - Physik, Universitat - GH Siegen, D-57068 Siegen, GermanyReceived 8 March 1999; received in revised form 12 October 1999AbstractStarting from the ordered state, we investigate the short-time behaviour of the hard-disk model.For the positional order, we determine the critical exponents and z from the dynamic relaxationof the order parameter and the cumulant with molecular dynamics simulations. The results arecompared with previous Monte Carlo (MC) simulations. The bond orientational order is studiedwith MC dynamics.c 2000 Elsevier Science B.V. All rights reserved.Keywords: Hard-disk model; Short-time dynamics; Critical phenomena[-3pt]PACS: 64.60.Ht; 82.20.Mj; 05.70.Jk; 64.70.Dv1. IntroductionFor a long time it was believed that universal scaling behaviour can be found onlyin the long-time regime. Therefore, numerical simulations were performed in the ther-modynamic equilibrium. Such simulations in vicinity of the critical point are aectedby the critical slowing down. However, recently Janssen et al. [1] showed that uni-versality exists already in the early time of the evolution. They discovered that asystem with non-conserved order parameter and energy (model A) quenched from ahigh-temperature state to the critical temperature shows universal short-time behaviouralready after a microscopic time scale tmic. Starting from an unordered state with a smallvalue of the order parameter m0, the order increases with a power law M (t) ∼ m0t,where is a new dynamic exponent. A number of Monte Carlo (MC) investigations[2–5]1support this short-time behaviour. These simulations can be also used to calcu-late the conventional (static and dynamic) exponents as well as the critical point [6].E-mail address: [email protected] (A. Jaster)1A review of short-time dynamics is given in Ref. [2].0378-4371/00/$ - see front matterc 2000 Elsevier Science B.V. All rights reserved.PII: S 0378-4371(99)00483-5A. Jaster / Physica A 277 (2000) 106–114 107This may eliminate critical slowing down, since the simulations are performed in theshort-time regime.First simulations of the dynamic relaxation in the short-time regime started froman unordered state. However, short-time dynamical scaling can be also found startingfrom the ordered state (M (t = 0) = 1). There exist no analytical calculations for thissituation, but several MC simulations were done [7,8,2]. Also, all critical exponents,except for the new exponent , can be calculated starting from the ordered state. Upto now, simulations of the dynamic relaxation have only been performed with MCdynamics. Normally, molecular dynamics (MD) simulations can cause ergodicy prob-lems, since the energy of the system is conserved. However, for the two-dimensionalhard-disk model the potential energy of the allowed congurations does not dependon the positions of the particles, but is constant. Therefore, the restriction of energyconservation does not lead to a reduction of possible congurations.The nature of the two-dimensional melting transition is a longstanding puzzle [9,10].The Kosterlitz–Thouless–Halperin–Nelson–Young (KTHNY) theory [11–14] predictstwo continuous phase transitions. The rst transition (dislocation unbinding) at themelting temperature Tmtransforms the solid with quasi-long-range positional orderand long-range orientational order into a new hexatic phase that posses short-rangepositional order and quasi-long-range orientational order. The disclination unbindingtransition at Titransforms this hexatic phase into an isotropic phase in which thepositional and orientational order are short range. There are several other theoreticalscenarios for the melting transition in two dimensions [9]. Most of them predict arst-order phase transition from the solid to the isotropic phase with a coexistenceregion instead of a hexatic phase.Even for the simple hard-disk system no consensus about the nature of the meltingtransition has been established. A large number of simulations of the two-dimensionalhard-disk model in the thermodynamic equilibrium have been performed. A meltingtransition was rst seen in a computer study by Alder and Wainwright [15]. Theyinvestigated 870 disks with MD methods (constant number of particles N , volumeV and energy E) and found the transition to be rst order. However, the results ofsuch a small system are aected by large nite-size eects. Recent simulations usedMC techniques either in the NVT ensemble (constant volume) [16–21] or the NpTensemble (constant pressure) [22–24]. Unfortunately, the results of these simulationsare not compatible.In this article, we study the short-time behaviour of the two-dimensional hard-diskmodel starting from the ordered state (perfect crystal). First, we examine the positionalorder parameter pos(t) and the cumulanteUpos(t) with MD simulations. The power-lawbehaviour of these observables is used to determine the critical exponents and z. Theresults are compared with those of a previous MC simulation [25]. In the second partwe study the bond orientational order parameter 6(t) and the cumulanteU6(t) withMC dynamics. All simulations are performed in a rectangular box with ratio 2 :√3,which is necessary for the ordered state, and with periodic boundary conditions. Thedisk diameter is set equal to one.108 A. Jaster / Physica A 277 (2000) 106–1142. Positional orderThe positional order parameter poscan be computed via pos(t)=1NNXi=1exp(i G · ri(t)); (1)where G denotes a reciprocal lattice vector and ri(t) is the position of particle i at timet. G has a magnitude of 2=a, where a =q2=(√3) is the average lattice spacing.The direction of G is xed to that of a reciprocal lattice vector of the perfect crystal(which are unique due to the boundary condition of a rectangular box of ratio 2 :√3).The reason for xing G is that large crystal tilting is not possible since we simulateonly the short-time behaviour of the system.MC simulations of the dynamic relaxation of systems with quasi-long-range orderwere performed for the 6-state clock model [26], the XY model [27,28], the fullyfrustrated XY model [3,28], the quantum XY model [29] and the hard-disk model[25]. However, no investigations exist for the relaxation with molecular dynamics.Independent
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