  We formulate a class of velocity-free finite-particle methods for mass
transport problems based on a time-discrete incremental variational principle
that combines entropy and the cost of particle transport, as measured by the
Wasserstein metric. The incremental functional is further spatially discretized
into finite particles, i.e., particles characterized by a fixed spatial profile
of finite width, each carrying a fixed amount of mass. The motion of the
particles is then governed by a competition between the cost of transport, that
aims to keep the particles fixed, and entropy maximization, that aims to spread
the particles so as to increase the entropy of the system. We show how the
optimal width of the particles can be determined variationally by minimization
of the governing incremental functional. Using this variational principle, we
derive optimal scaling relations between the width of the particles, their
number and the size of the domain. We also address matters of implementation
including the acceleration of the computation of diffusive forces by exploiting
the Gaussian decay of the particle profiles and by instituting fast
nearest-neighbor searches. We demonstrate the robustness and versatility of the
finite-particle method by means of a test problem concerned with the injection
of mass into a sphere. There test results demonstrate the meshless character of
the method in any spatial dimension, its ability to redistribute mass particles
and follow their evolution in time, its ability to satisfy flux boundary
conditions for general domains based solely on a distance function, and its
robust convergence characteristics.
