Explicit Symplectic Integrators for Non-Separable Hamiltonians in Molecular Dynamics

Published in KTH Royal Institute of Technology (Bachelor's thesis), 2019

In molecular dynamics, mathematical models of metallic systems should in general have the temperature of the system as a dependent variable [1]. In particular, the potential energy term of the Hamiltonian function of the interaction model should be dependent on temperature in addition to interparticular distances. This puts the Hamiltonian function on a form which is generally non-separable. Conventional explicit numerical methods which are symplectic when used to integrate the equations of motion of systems with separable Hamiltonians are not in general symplectic when used to integrate the equations of motion of systems with a non-separable Hamiltonian. Hence, an integrator which sustains symplecticity when used in a system with non-separable Hamiltonian is sought. A family of explicit integrators which are symplectic when integrating systems with a non-separable Hamiltonian are shown to exhibit similar or superior performance to the Velocity Verlet and fourth-order Runge-Kutta schemes, albeit with the drawback of numerical instability when used on a system where forces depend exponentially on the inverted interparticular distances. To the knowledge of the authors, this study is the first time this family of integrators is applied in the context of molecular dynamics. The results of this study provide a first indication that a comprehensive solution to the problem of integrating the equations of motion of a system with a non-separable Hamiltonian explicitly and symplectically is not provided by the considered family of integrators. However, further investigations into using this family of integrators in other molecular dynamics systems than those investigated here are needed to provide a more definitive conclusion.

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