Aligning Active Particles Py Package

The package performs molecular-dynamics-like agent-based simulations for models of aligning self-propelled particles in two dimensions such as e.g. the seminal Vicsek model or variants of it. In one class of the covered models, the microscopic dynamics is determined by certain time discrete interaction rules. Thus, it is no Hamiltonian dynamics and quantities such as energy are not defined. In the other class of considered models (that are generally believed to behave qualitatively the same) Brownian dynamics is considered. However, also there, the forces are not derived from a Hamiltonian. Furthermore, in most cases, the forces depend on the state of all particles and can not be decomposed into a sum of forces that only depend on the states of pairs of particles. Due to the above specified features of the microscopic dynamics of such models, they are not implemented in major molecular dynamics simulation frameworks to the best of the authors knowledge. Models that are covered by this package have been studied with agent-based simulations by dozens of papers. However, no simulation framework of such models seems to be openly available. The program is provided as a Python package. The simulation code is written in C. In the current version, parallelization is not implemented.Comment: 23 pages, 7 figure

On commutatively dominated Op∗-algebras with Fréchet domains

AbstractSuppose that the maximal Op∗-algebra L+(D) on a Fréchet domain D contains a sequence of strongly commuting essentially self-adjoint operators which is cofinal in the set of all hermitian elements of L+(D). Then each positive linear functional on L+(D) decomposes in a unique way into a sum of an ultraweakly continuous positive linear functional, a uniformly continuous positive linear functional which vanishes on the finite rank operators, and a positive linear functional which vanishes on the dense subspace of the “very continuous” operators. For bounded linear functionals on certain subspaces of the completion L of L+(D), similar results are satisfied. These results are connected with properties of the uniform topology and the associated bornological topology of L