Energy, temperature, and heat capacity in discrete classical dynamics

Area of Science:

• Computational Physics
• Thermodynamics
• Statistical Mechanics

Background:

• Classical dynamics simulations are typically performed using discrete algorithms like the leapfrog or Verlet method.
• The Verlet algorithm, first formulated by Newton and later rederived by Verlet, includes approximations for velocity and kinetic energy.
• These approximations can lead to significant thermodynamic errors in molecular dynamics (MD) simulations under certain conditions (high density, temperature, strong forces, large time steps).

Purpose of the Study:

• To derive exact expressions for discrete dynamics in classical simulations.
• To address and correct the thermodynamic inaccuracies caused by approximated discrete dynamics algorithms.
• To validate the derived exact expressions through simulations of a Lennard-Jones system.

Main Methods:

• Derivation of exact mathematical expressions for discrete dynamics.
• Implementation of these exact expressions in molecular dynamics (MD) simulations.
• Simulation of a Lennard-Jones system to test the new expressions.
• Comparison of temperatures derived from kinetic energy and configurational temperature (Landau-Lifshitz).

Main Results:

• Exact expressions for discrete dynamics were successfully derived.
• Simulations using the exact expressions demonstrated equality between kinetic and configurational temperatures.
• The derived expressions resolve thermodynamic errors observed with approximated methods in MD simulations.

Conclusions:

• The derived exact discrete dynamics expressions provide accurate thermodynamic properties in simulations.
• This advancement corrects errors in molecular dynamics (MD) simulations, particularly under challenging conditions.
• The findings offer a more reliable approach for classical dynamics simulations in natural sciences.