Direct Boltzmann inversion method from particle configurations at arbitrary state points

Abstract

We introduce a direct Boltzmann inversion method to infer the interaction potential in particle systems using as input particle configurations generated at an arbitrary state point of the system. Unlike iterative Boltzmann inversion, the proposed method does not require performing a new Monte Carlo simulation at each step of the iteration process. It relies instead on enforcing consistency between two independent estimates of the pair correlation function, respectively obtained from interparticle distances and from pairwise forces. As a result, the approach is computationally inexpensive and straightforward to implement. Because it relies on the sole expression of interparticle forces, our method naturally applies to any state point, including when the density is large and alternative methods may fail. Here we present the basic principles of the method and benchmark its performance on a diverse set of test potentials studied using computer simulations. Practical aspects and detailed implementation of the method are also discussed. Owing to its simplicity and generality, the method should be broadly applicable, from the construction of coarse-grained interaction potentials to the inference of effective interactions in non-equilibrium systems.

Figures (4)

• Figure 1: Sketch of the direct Boltzmann inversion method. The available data set (series of images) produces a reference RDF $g_{\rm ref}(r)$. A guess potential $u_t(r)$ is used to generate the RDF $g_t(r)$ using the force formula, and the difference between $g_{\rm ref}(r)$ and $g_t(r)$ is used to make an improved guess for the potential $u_{t+1}(r)$. The fixed point provides the correct inverted potential $u(r)$.
• Figure 2: (a) The RDF measured by the distance-histogram $g_{DH}(r)$ is smoothed and finely discretized to produce the reference $g_{\rm ref}(r)$. (b) Weights used for the spline interpolation. Data taken for the shoulder potential at $\rho\sigma^2=0.56$ and $\beta \epsilon =0.5$.
• Figure 3: Convergence of the iterative procedure for an inverse cubic potential at $\rho\sigma^2 = 0.80$ and $\beta \epsilon = 10/3$, using $\alpha=0.5$. (a) Evolution of $g_t(r)$ toward the reference (red line). (b) Effective potentials $\beta u_t(r)$ reconstructed on the window $[0.663\sigma, 5\sigma]$; dashed line shows the analytical potential and the thick blue line indicates the initial guess $\beta u_0 = -\log g_{\text{ref}}(r)$. (c) Convergence metrics as a function of the iteration number $t$.
• Figure 4: Potential reconstruction results for various interaction types. The top row displays the reconstructed pair potentials $\beta u(r)$, while the bottom row shows the corresponding radial distribution functions $g(r)$. In all panels, the solid black lines represent the reference and the colored scatter points denote the final reconstruction results (a) Lennard-Jones (LJ) at $\rho\sigma^2 = 0.56, \beta\epsilon = 1.0, \alpha = 0.4$, reconstructed over the window $[0.915\sigma, 2.5\sigma]$; (b) Weeks-Chandler-Andersen (WCA) at $\rho\sigma^2 = 0.56, \beta\epsilon = 1.0, \alpha = 0.2$, reconstruction window $[0.920\sigma, 2^{1/6}\sigma]$; (c) LJ at $\rho\sigma^2 = 0.92, \beta\epsilon = 1/2, \alpha = 0.5$, reconstruction window $[0.863\sigma, 2.5\sigma]$; (d) Shoulder potential at high density $\rho\sigma^2 = 0.28, \beta\epsilon = 1.0, \alpha = 0.2$, reconstruction window $[0.911\sigma, 2.8\sigma]$.