Probing the partition function for temperature-dependent potentials with nested sampling

TL;DR

The paper addresses computing thermodynamic properties when the potential depends on temperature, which breaks the single‑run efficiency of traditional nested sampling. It introduces an extended partition function method that samples an auxiliary temperature β̃, enabling a single exploration to recover Z_c(β) across all temperatures, and applies it to path‑integral quantum effects in harmonic potentials and Lennard‑Jones clusters. The authors demonstrate that the extended method can match or surpass direct per‑temperature nested sampling while dramatically reducing energy evaluations, though it requires careful tuning of smearing parameters, priors, and sampling bounds. This approach offers a practical route to incorporating nuclear quantum effects in complex many‑body systems with reduced computational cost, with potential extensions to more realistic materials.

Abstract

Thermodynamic properties can be in principle derived from the partition function, which, in many-atom systems, is hard to evaluate as it involves a sum on the accessible microscopic states. Recently, the partition function has been computed via nested sampling, relying on Bayesian statistics, which is able to provide the density of states as a function of the energy in a single run, independently of the temperature. This appealing property is lost whenever the potential energy that appears in the partition function is temperature-dependent: for instance, mean-field effective potential energies or the quantum partition function in the path-integral formalism. For these cases, the nested sampling must be carried out at each temperature, which results in a massive increase of computational time. Here, we introduce and implement a new method, that is based on an extended partition function where the temperature is considered as an additional parameter to be sampled. The extended partition function can be evaluated by nested sampling in a single run, so to restore this highly desirable property even for temperature-dependent effective potential energies. We apply this original method to compute the quantum partition function for harmonic potentials and Lennard-Jones clusters at low temperatures and show that it outperforms the straightforward application of nested sampling for each temperature within several temperature ranges.

Figures (19)

• Figure 1: Sampling frequency for the harmonic potential with $N=1$ and $P=2$ (a) before the change of variables using $\tilde{\beta}$ as the explored parameter, (b) after the change of variables using $1/\tilde{\beta}$ as the explored parameter and (c) after the change of variables using $\tilde{\beta}$ as the explored parameter. We have $\tilde{T}=1/(k_\textrm{B}\tilde{\beta})$. The prior is indicated by the black curve.
• Figure 2: Representation of the parameter transformation given by Eqs \ref{['eq_lambda_p']}--\ref{['eq_potential_PI_transf']} (for $P=5$). Before the transformation, the polymer is in a box of fixed size (the same for all temperatures). In that case, the extension of the polymer depends on the temperature: the polymer contracts with increasing temperature. After the transformation, the extension of the polymer is the same for all temperatures (see first term of Eq. \ref{['eq_potential_PI_transf']}), however the size of the box is then proportional to $\lambda_P$, hence to the temperature.
• Figure 3: Comparison between $Z$ and the smeared-delta function approximation for (a) the Gaussian function (Eq. \ref{['eq_harm_gauss']}) and (b) the rectangular function (Eq. \ref{['eq_harm_rect']}). The curves corresponding to $Z$, $\alpha=10$ and $\alpha=100$ almost coincide at the graph scale.
• Figure 4: Heat capacity for the quantum harmonic potential with the direct method. The analytical results for the classical and quantum cases are plotted as full lines, as well as the formulas for two (Eq. \ref{['eq_cv_harm_p_2']}) and four (Eq. \ref{['eq_cv_harm_p_4']}) replicas ($P=2$ and $P=4$, respectively). Also shown are the numerical results obtained with nested_fit (nf in the legend). The points correspond to the temperatures at which the heat capacities were computed. Taking $\nu=\omega/(2\pi)=100$cm$^{-1}$, which is a typical vibration frequency in solids, the temperature range considered here would correspond to [290K,5000K]. The error bars were computed from four runs: for $P=1,2,4$, they are smaller than the symbols for most temperatures.
• Figure 5: MAPE as a function of $\alpha$ for different values of $K$ with the Gaussian (full lines) and rectangular (dashed lines) windows.