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We formulate a class of velocity-free finite-particle methods for masstransport problems based on a time-discrete incremental variational principlethat combines entropy and the cost of particle transport, as measured by theWasserstein metric. The incremental functional is further spatially discretizedinto finite particles, i.e., particles characterized by a fixed spatial profileof finite width, each carrying a fixed amount of mass. The motion of theparticles is then governed by a competition between the cost of transport, thataims to keep the particles fixed, and entropy maximization, that aims to spreadthe particles so as to increase the entropy of the system. We show how theoptimal width of the particles can be determined variationally by minimizationof the governing incremental functional. Using this variational principle, wederive optimal scaling relations between the width of the particles, theirnumber and the size of the domain. We also address matters of implementationincluding the acceleration of the computation of diffusive forces by exploitingthe Gaussian decay of the particle profiles and by instituting fastnearest-neighbor searches. We demonstrate the robustness and versatility of thefinite-particle method by means of a test problem concerned with the injectionof mass into a sphere. There test results demonstrate the meshless character ofthe method in any spatial dimension, its ability to redistribute mass particlesand follow their evolution in time, its ability to satisfy flux boundaryconditions for general domains based solely on a distance function, and itsrobust convergence characteristics.

An optimal-transport finite-particle method for mass diffusion