Microcanonical thermostatistics analysis without histograms: cumulative distribution and Bayesian approaches

TL;DR

This paper introduces two binning-free methods, based on cumulative distribution and Bayesian modeling, to estimate microcanonical inverse temperature and entropy from continuous energy data, improving analysis of complex systems.

Contribution

It presents novel binning-free techniques using CDF series expansion and Bayesian modeling to estimate thermodynamic quantities without histogram binning.

Findings

Both methods accurately estimate $eta(E)$ and $S(E)$ in protein models.

The approaches outperform traditional histogram-based methods in continuous energy systems.

They simplify analysis of complex systems with continuous spectra.

Abstract

Microcanonical thermostatistics analysis has become an important tool to reveal essential aspects of phase transitions in complex systems. An efficient way to estimate the microcanonical inverse temperature $\beta(E)$ and the microcanonical entropy $S(E)$ is achieved with the statistical temperature weighted histogram analysis method (ST-WHAM). The strength of this method lies on its flexibility, as it can be used to analyse data produced by algorithms with generalised sampling weights. However, for any sampling weight, ST-WHAM requires the calculation of derivatives of energy histograms $H(E)$, which leads to non-trivial and tedious binning tasks for models with continuous energy spectrum such as those for biomolecular and colloidal systems. Here, we discuss two alternative methods that avoid the need for such energy binning to obtain continuous estimates for $H(E)$ in order to evaluate $\beta(E)$ by using ST-WHAM: (i) a series expansion to estimate probability densities from the empirical cumulative distribution function (CDF), and (ii) a Bayesian approach to model this CDF. Comparison with a simple linear regression method is also carried out. The performance of these approaches is evaluated considering coarse-grained protein models for folding and peptide aggregation.