Boltzmann distribution

Summary

In the context of MD simulations, the Boltzmann distribution in configurational space ($\nu$) and in phase space ($\mu$) is defined as:

$$\nu (\mathbf{x})=\int \mu (\mathbf{x},\mathbf{p})d\mathbf{p}=\frac{1}{Z}{e}^{-\beta U(\mathbf{x})}\text{\hspace{1em}(1)}$$

Details

In phase space, which considers both positions and momenta of atoms:

$$\mu (\mathbf{x},\mathbf{p})=\frac{1}{Q}{e}^{-\beta (U(\mathbf{x})+K(\mathbf{p}))}\text{\hspace{1em}(2)}$$

• $\mathbf{x}$: positions of 3D coordinates, $\mathbf{x}\in {\mathbb{R}}^{3N}$ ($N$: number of particles)
• $\mathbf{p}$: momenta, $\mathbf{p}\in {\mathbb{R}}^{3N}$
• $U(\mathbf{x})$: potential energy function
• $K(\mathbf{p})$: kinetic energy function
• $\beta$: inverse temperature ($\frac{1}{k_{B}T}$)
• $Z$: configurational partition function, $Z=\int e^{-\beta U(\mathbf{x})}d\mathbf{x}$
• $Q$: partition function, $Q=\int e^{-\beta (U(\mathbf{x})+K(\mathbf{p}))}d\mathbf{x}d\mathbf{p}$

See Mathematical terms in MD for notation. Extensively cites (1)

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