**A Combinatorial Perspective of Error Analysis in Molecular Dynamics** - Reginald Paul

In this talk I will compare two approaches that have evolved for the computation of general time displacement operators:
(1) Based on the Taylor expansion (moment expansion) and re-summation of the terms in an approximate fashion leading to such methods as truncated continued fractions.
(2) The more modern approach based on molecular dynamics for which I will derive an analytical form (Analytical Molecular Dynamics or AMD) that can be used to calculate the implicit moments for comparison with the results in (1).
Using very simple combinatorics I will derive a formula that will show the precise manner in which a finite step approximation based on AMD converges to the exact result. I will also show that the method commonly used in numerical simulations (Conventional Molecular Dynamics) lacks this convergence and, in fact, diverges. I will conclude that by using modern computational methods such as Mathematica it is possible to use the moment expansion methods to get better results than MD.