Boltzmann–Gibbs random fields with mesh-free precision operators based on smoothed particle hydrodynamicsBoltzmann–Gibbs random fields with mesh-free precision operators based on smoothed particle hydrodynamics Peer-Reviewed Journal Publication Δημοσίευση σε Περιοδικό με Κριτές 2024-02-082022enBoltzmann–Gibbs random fields are defined in terms of the exponential expression, where is a suitably defined energy functional of the field states. This paper presents a new Boltzmann–Gibbs model which features local interactions in the energy functional. The interactions are embodied in a spatial coupling function which uses smoothed kernel-function approximations of spatial derivatives inspired from the theory of smoothed particle hydrodynamics. A specific model for the interactions based on a second-degree polynomial of the Laplace operator is studied. An explicit, mesh-free expression of the spatial coupling function (precision function) is derived for the case of the squared exponential (Gaussian) smoothing kernel. This coupling function allows the model to seamlessly extend from discrete data vectors to continuum fields. Connections with Gaussian Markov random fields and the Matérn field with are established.http://creativecommons.org/licenses/by/4.0/Theory of Probability and Mathematical Statistics10737-60 Christopoulos Dionysios Χριστοπουλος Διονυσιος Taras Shevchenko National University of Kyiv Random fields Kernel functions Precision matrix Smoothed particle hydrodynamics