An investigation of molecular dynamics for simple liquids

This thesis contains work expanding the theoretical understanding of molecular dynamics used to aid the study of simple liquids. It does so by focusing on investigating forces, which govern the dynamics of manybody systems. We loosely address three questions: How can we categorise force distributions? What can we gauge from force data? When do forces obey Newton’s third law? The first of these questions is addressed using statistical mechanics to derive standardised moments of the force distribution for a simple LennardJones liquid in both 1d and 3d with the aid of molecular dynamics. To answer the second question, we introduce the notions of force spaces and configurations spaces, and look at equivalence of these. We begin the investigation using the harmonic potential, and develop homotopy continuation methods for non-linear forces like Lennard-Jones. Convergent behaviour and limitations are explored for many-body systems, and a general two-body direct inversion is developed and implemented. The final question is entrenched in classical potential theory, and approached through work focusing on understanding the functional dependence of the interatomic potential. We develop theorems and provide corresponding constructive proofs concluding that potentials which obey certain symmetries can be described by distances, as opposed to positions. This enables us to understand when forces display reciprocity

Symmetries of many-body systems imply distance-dependent potentials

Considering interatomic potential $U({\mathbf q})$ where ${\mathbf q} = [{\mathbf q}_1, {\mathbf q}_2, \dots, {\mathbf q}_N] \in {\mathbb R}^{3N}$ is a vector describing positions, $\mathbf{q}_i \in {\mathbb R}^3$, it is shown that $U$ can be defined as a function of the interatomic distance variables $r_{ij} = |{\mathbf q}_i - {\mathbf q}_j |$, provided that the potential $U$ satisfies some symmetry assumptions. Moreover, the potential $U$ can be defined as a function of a proper subset of the distance variables $r_{ij}$, provided that $N > 5$, with the number of distance variables used scaling linearly with the number of atoms, $N$

Data for 'On standardised moments of force distribution in simple liquids'

Data was created using self made FORTRAN 90 scripts to simulate simple fluids. We implemented a simple Velocity-Verlet integrator with a Nosé-Hoover thermostat (for figures 2,3,4,5,6,8) and a Langevin thermostat for figure (1). Data files uploaded are simple '.dat' files and can be read with any basic notepad software