Long-Run Accuracy of Variational Integrators in the Stochastic Context

This paper presents a Lie-Trotter splitting for inertial Langevin equations (Geometric Langevin Algorithm) and analyzes its long-time statistical properties. The splitting is defined as a composition of a variational integrator with an Ornstein-Uhlenbeck flow. Assuming the exact solution and the splitting are geometrically ergodic, the paper proves the discrete invariant measure of the splitting approximates the invariant measure of inertial Langevin to within the accuracy of the variational integrator in representing the Hamiltonian. In particular, if the variational integrator admits no energy error, then the method samples the invariant measure of inertial Langevin without error. Numerical validation is provided using explicit variational integrators with first, second, and fourth order accuracy.Comment: 30 page

Solution of some problems in the arithmetical complexity of first-order fuzzy logics

This short paper addresses the open problems left in a previous paper by Franco Montagna and Carles Noguera. Besides giving solutions to these two problems, some clarification concerning the role of the full vocabulary (including functional symbols) in the proofs there given is also discussed.Comment: 5 page

Parasurface groups

A residually nilpotent group is \emph{$k$-parafree} if all of its lower central series quotients match those of a free group of rank $k$. Magnus proved that $k$-parafree groups of rank $k$ are themselves free. We mimic this theory with surface groups playing the role of free groups. Our main result shows that the analog of Magnus' Theorem is false in this setting.Comment: 6 page

Ballistic Transport at Uniform Temperature

A paradigm for isothermal, mechanical rectification of stochastic fluctuations is introduced in this paper. The central idea is to transform energy injected by random perturbations into rigid-body rotational kinetic energy. The prototype considered in this paper is a mechanical system consisting of a set of rigid bodies in interaction through magnetic fields. The system is stochastically forced by white noise and dissipative through mechanical friction. The Gibbs-Boltzmann distribution at a specific temperature defines the unique invariant measure under the flow of this stochastic process and allows us to define ``the temperature'' of the system. This measure is also ergodic and weakly mixing. Although the system does not exhibit global directed motion, it is shown that global ballistic motion is possible (the mean-squared displacement grows like t squared). More precisely, although work cannot be extracted from thermal energy by the second law of thermodynamics, it is shown that ballistic transport from thermal energy is possible. In particular, the dynamics is characterized by a meta-stable state in which the system exhibits directed motion over random time scales. This phenomenon is caused by interaction of three attributes of the system: a non flat (yet bounded) potential energy landscape, a rigid body effect (coupling translational momentum and angular momentum through friction) and the degeneracy of the noise/friction tensor on the momentums (the fact that noise is not applied to all degrees of freedom).Comment: 33 page