  Boltzmann-Gibbs random fields are defined in terms of the exponential
expression exp(-H), where H is a suitably defined energy functional of the
field states x(s). 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 Mat\'{e}rn field with $\nu=1$ are established.
