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We present a reformulation of the distance metric learning problem as a penalized optimization problem, with a penalty term corresponding to the von Neumann entropy of the distance metric. This formulation leads to a mapping to statistical mechanics such that the metric learning optimization problem becomes equivalent to free energy minimization. Correspondingly, our approach leads to an analytical solution of the optimization problem based on the Boltzmann distribution. The mapping established in this work suggests new approaches for dimensionality reduction and provides insights into determination of optimal parameters for the penalty term. Furthermore, we demonstrate that the metric projects the data onto direction of maximum dissimilarity with optimal and tunable separation between classes and thus the transformation can be used for high dimensional data visualization, classification, and clustering tasks. We benchmark our method against previous distance learning methods and provide an efficient implementation in an R package available to download at: \urlhttps://github.com/kouroshz/fenn