Hard sphere system molecular dynamic computations

Although, the notion of molecular dynamics was known in the early turn of the century, the first conscious effort in the use of computer for molecular dynamics simulation was made by Alder and Wainright, who in their paper [1] reported the application of molecular dynamics to realistic particle systems. Using hard spheres potential and fastest computers at the time, they were able to simulate systems of 32 to 108 atoms in 10 to 30 hours. Since the work of Alder and Wainright, interests in MD have increased tremendously, see... 

Molecular dynamics computations for a system of N(= 108) hard spheres were first performed by Alder and co-workers. In the case of hard spheres the motion of each particle is determined by the laws of elastic collisions. When a force on a particle / can be represented by the negative gradient of a given potential function, then the classical equations of motion may be written in the following form... 

The equation of state of a hard-sphere fluid has been the subject of considerable research, and far better approximate equations than Eq. (9.8-4) have been obtained. One such equation of state is found in Problem 9.64. Much additional research on gases and liquids has used the technique of molecular dynamics, in which solutions to the classical equations of motion for a system of several hundred particles are numerically simulated by a computer program. Energies, pressures, etc., are then calculated by averaging over the particles positions and velocities. As we would expect, the molecular dynamics calculations indicate that there is no gas-liquid condensation in the hard-sphere system. However, there is considerable evidence from these calculations that a gas-solid phase transition occurs. This result was originally somewhat surprising because of the absence of attractive forces. 

The first molecular simulations were performed almost five decades ago by Metropolis et al. (1953) on a system of hard disks by the Monte Carlo (MC) method. Soon after, hard spheres (Rosenbluth and Rosenbluth, 1954) and Lennard-Jones (Wood and Parker, 1957) particles were also studied by both MC and molecular dynamics (MD). Over the years, the simulation techniques have evolved to deal with more complex systems by introducing different sampling or computational algorithms. Molecular simulation studies have been made of molecules ranging from simple systems (e.g., noble gases, small organic molecules) to complex molecules (e.g., polymers, biomolecules). 

Discontinuous molecular dynamics (DMD) simulations can be used to investigate large systems efficiently with moderate computational resources. DMD simulations were designed to be applicable to systems that interact via discontinuous potentials (square-well/square-shoulder and hard-sphere). They proceed by analytically calculating the next collision time. Several papers [26-28] describe the details of DMD simulations. The algorithm complexity of DMD simulations is O (Mog N). (One paper by Paul [29] even claims a realization of the DMD method... 

Fig. 19. A plot of T// as a function of density for hard-sphere molecules. Here 17 is the actual viscosity as determined by computer-simulated molecular dynamics, 17 the Enskog theory value of the viscosity at the same density, Vq the volume of the system of spheres at close packing, and V the actual volume of the system. The computer data on which this curve is based are good to about 5%. [From B. J. Alder, D. M. Gass, and T. E. Wainwright, J. Chem. Phys. 53, 3813 (1970).]...

Computer simulations also constitute an important basis for the development of the molecular theory of fluids. They could be regarded as quasiexpeiimental procedures to obtain datasets that connect the fluid s microscopic parameters (related mainly to the structure of the system and the molecular interactions) to its macroscopic properties (such as equation of state, dynamic coefficients, etc.). In particular, some of the first historical simulations were performed using two-dimensional fluids to test adaptations of commonly used computer simulation methods [14,22] Monte Carlo (MC) and molecular dynamics (MD). In fact, the first reliable simulation results were obtained by Metropolis et al. [315], who applied the MC method to the study of hard-sphere and hard-disk fluids. 

The second simulation technique is molecular dynamics. In this technique, which was pioneered by Alder, initial positions of theparticles of a system of several hundred particles are assigned in some way. Displacements of the particles are determined by numerically simulating the classical equations of motion. Periodic boundary conditions are applied as in the Monte Carlo method. The first molecular dynamics calculations were done on systems of hard spheres, but the method has been applied to monatomic systems having intermolecular forces represented by the square-well and Lennard-Jones potential energy functions, as well as on model systems representing molecular substances. Commercial software is now available to carry out molecular dynamics simulations on desktop computers. ... 

The goal of extending classical thermostatics to irreversible problems with reference to the rates of the physical processes is as old as thermodynamics itself. This task has been attempted at different levels. Description of nonequilibrium systems at the hydrodynamic level provides essentially a macroscopic picture. Thus, these approaches are unable to predict thermophysical constants from the properties of individual particles in fact, these theories must be provided with the transport coefficients in order to be implemented. Microscopic kinetic theories beginning with the Boltzmann equation attempt to explain the observed macroscopic properties in terms of the dynamics of simplified particles (typically hard spheres). For higher densities kinetic theories acquire enormous complexity which largely restricts them to only qualitative and approximate results. For realistic cases one must turn to atomistic computer simulations. This is particularly useful for complicated molecular systems such as polymer melts where there is little hope that simple statistical mechanical theories can provide accurate, quantitative descriptions of their behavior. 

A large body of the computer simulation work has been reported on model systems such as hard discs, spheres or Lennard-Jones particles. Here the interparticle potential is known and can be used to rapidly calculate the configurational energy of the system as required for Monte Carlo studies or the configurational force on a particle as required for molecular dynamics. 

With their strength tied to available computer speed, simulations continue to become a more powerful tool. A letter to the Journal of Chemical Physics by B. J. Alder and T. E. Wainwright in 1957 was the first work that reported results from molecular dynamics simulations. The Lawrence Radiation Laboratory scientists studied two different sized systems with 32 and 108 hard spheres. They modeled bulk fluid phases using periodic boundary conditions. In the paper they mention that they counted 7000 and 2000 particle collisions for 32 and 108 particle systems, respectively. This required one hour on a UNIVAC computer. Incidentally, this was the fifth such commercial computer delivered out of the 46 ever produced. The computer cost around 200 000 in 1952 and each of ten memory units held 100 words or bytes. Nowadays, a 300 personal computer with a memory of approximately 500000000 bytes can complete this simulation in less than 1 second. And Moore s empirical law that computer power doubles every 18 months still holds. 

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