Validity of Molecular Dynamics Simulations for Soft Matter

In this work, we analytically examine the validity of molecular dynamics for a soft potential system by considering a simple one-dimensional system with a piecewise continuous linear repulsive potential wall having a constant slope $a$. We derive an explicit analytical expression for an inevitable energy change $\mathrm{\Delta }E$ due to the discrete process, which is dependent on two parameters: 1) $\alpha$, which is a fraction of time step $\tau$ immediately after the collision with the potential wall, and 2) $\mu \equiv \frac{a \tau}{p_0}$,where$p_0$ is the momentum immediately before the collision. The whole space of parameters $\alpha$ and $\mu$ can be divided into an infinite number of regions, where each region creates a positive or negative energy change $\mathrm{\Delta }E$. On the boundaries of these regions, energy does not change, $\mathit{\text{i.e}}$, $\mathrm{\Delta }E=0$. The envelope of $\mathrm{\mid }\mathrm{\Delta }E\mathrm{\mid }$ $\mathit{\text{vs.}}$ $\mu$ shows a power law behavior $\mathrm{\mid }\mathrm{\Delta }E\mathrm{\mid }\propto {\mu }^{\beta }$, with the exponent $\beta \approx 0.95$. This implies that the round-off error in energy introduced by the discreteness is nearly proportional to the discrete time step $\tau$.