# Maximum Likelihood Inference of the Symmetric Double-Well Potential

Post date: Nov 11, 2019 12:19:47 AM by Solmaz Azimi

The double-well potential serves as an optimal, one-dimensional model for exploring physical phenomena. This project aimed to estimate the probability distribution of the double-well potential fitted to a Gaussian mixture model by maximum likelihood inference. Hypothetical datasets of the quadratic function were generated using smart-darting Monte-Carlo simulations under the Metropolis acceptance criterion. A major challenge in studying free energy landscapes is sampling efficiency, therefore a larger displacement was implemented in the Monte-Carlo simulations that are generally done at smaller displacements. Although a conventional Monte-Carlo approach can accomplish displacement from one free energy minimum to the next, this is done at the expense of accuracy, such that the acceptance rate of the simulation deviates from an optimal 50%. The generated datasets demonstrated close resemblance to normalized Boltzmann Distribution functions at two temperatures. TensorFlow was utilized to obtain the most optimal parameters for observing the generated datasets based on the Gaussian Mixture Model. Our results demonstrated that TensorFlow approximates a set of optimal parameters for the Gaussian Mixture Model that accurately resemble the Boltzmann Distribution at 300 K, however the method is unable to do so with similar accuracy at 2000 K.