Mathematical terms in MD

Summary

Notation follows the convention of (1).

• $\mathbf{x}$: positions of 3D coordinates, $\mathbf{x}\in {\mathbb{R}}^{3N}$ ($N$: number of particles)
• $\mathbf{p}$: momenta, $\mathbf{p}\in {\mathbb{R}}^{3N}$
• $\beta$: inverse temperature ($\frac{1}{k_{B}T}$)
• $Q$: partition function or normalization factor, $Q=\int e^{-\beta (U(\mathbf{x})+K(\mathbf{p}))}$
• $M$: diagonal mass matrix
• $W_t$: Brownian motion in dimension $3N$

Energy terms

• $U(\mathbf{x})$: potential energy function
• $u_i(\mathbf{X})$: reduced energy function for state $i$, defined as $u_i(\mathbf{x})={\beta }_i(U_i(\mathbf{x})+p_{i}V(\mathbf{x}))$
• $p_{i}V(\mathbf{x})$: pressure-volume term, only included in constant-pressure ensembles (e.g., NPT ensemble); $p_i$ is the external pressure
• $K(\mathbf{p})$: kinetic energy function
• $F$: Helmholtz free energy
• $S$: entropy, $S=k_B\ln \Omega$
• $\Omega (E)$: density of states with specific total energy $E$, $\int \delta (E-H(\mathbf{x},\mathbf{p}))d\mathbf{x}d\mathbf{p}$
• $\Omega (U)$: configurational density with specific potential energy $U$, $\int \delta (U-U(\mathbf{x}))d\mathbf{x}$

1.

Hénin J, Lelièvre T, Shirts MR, Valsson O, Delemotte L. Enhanced Sampling Methods for Molecular Dynamics Simulations [Article v1.0]. Living Journal of Computational Molecular Science. 2022;4(1):1583. Available from: https://doi.org/10.33011/livecoms.4.1.1583