PHYSICAL REVIEW B VOLUME 32, NUMBER 9 1 NOVEMBER 1985 Isothermal molecular-dynamics calculations Luis F. Rull Departamento Ffsica Teorica, Facultad de Fi'sica, Universidad de Sevilla, 41012 Sevilla, Spain Juan J. Morales and Francisco Cuadros Seccion de FI'sicas, Facultad de Ciencias, Universidad de Extremadura, 06071 Badajoz, Spain (Received 30 January 1985; revised manuscript received 28 May 1985) %e have performed long-time runs of molecular-dynamics computer simulations of a two-dimensional Lennard-Jones system, without any scaling procedure. The thermodynamic properties show spontaneous fluctuations except when the system is far from the melting zone. In a recent paper, ' Toxvaerd studied critically a special computational technique: "isothermal-isobaric" molecular dynamics ( T-p MD) (used by Abraham and Koch' to inves- tigate melting in a two-dimensional Lennard-Jones system). By using the calculation of the isothermal compressibility from the fluctuations of volume, Toxvaerd established that the scaling procedure was incorrect and cannot be used to obtain definitive statements about the nature of the phase transition. Toxvaerd noted that the T-p MD calculations, inspired by the N-p-T-ensemble Monte Carlo method, are obtained from Laplace transforms of the canonical partition function, and this transformation has to be performed for constant extensive state variables, excluding the volume. In an analysis of the ensembles used in molecular- dynamics simulations, Lado found that, in the classical lim- it, for interacting systems with hard-core interactions, Monte Carlo and molecular-dynamics calculations should yield the same equation of state without corrections, i.e. , BlnQ~ I (PP ) N v T+-- V ()V ( IS/ ) N- v-Ei where the configurational integral Q~ v is defined by Q~ r = V dr" 8'(r"), where ~( ~) 0, U(r ) =oo, 1, U(r )=0, and, as usual, P=(ksT) ', with the only assumptions be- ing the basic postulates of equilibrium statistical mechanics. The left-hand side (N-V-T) corresponds to the Monte Carlo calculation, that is, with the number of particles, volume, and temperature constant, and the right-hand side (N V E) correspo-nd-s to the molecular dynamics calcula- tion, with the number of particles, volume, and energy con- stant. Nose4 has recently made an exhaustive study of the en- sembles in molecular dynamics to reproduce both the canonical and the isothermal-isobaric probability densities in phase space. The physical system of interest consists of N particles, to which an external macroscopic variable and its LLI 1 LLI l— ~OQ ~ rl p++M+ 0 1 ~ +0 o e ++I ~ ~ ~ ~ ~ ~ ~ 0.7, . 20 40 60 '80 10 TIME STEPS FIG. 1. Reduced temperature vs time for a two-dimensional liquid at pr~=1. 00. 32 6050 1985 The American Physical Society