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We consider in this work stochastic differential equation (SDE) model for particles in contact with a heat bath when the memory effects are non-negligible. As a result of the fluctuation-dissipation theorem, the differential equations driven by fractional Brownian noise to model memory effects should be paired with Caputo derivatives and based on this we consider fractional stochastic differential equations (FSDEs), which should be understood in an integral form. We establish the existence of strong solutions for such equations. In the linear forcing regime, we compute the solutions explicitly and analyze the asymptotic behavior, through which we verify that satisfying fluctuation-dissipation indeed leads to the correct physical behavior. We further discuss possible extensions to nonlinear forcing regime, while leave the rigorous analysis for future works.

A celebrated theorem of Kapranov states that the Atiyah class of the tangent bundle of a complex manifold X makes TX[−1] into a Lie algebra object in D+(X), the bounded below derived category of coherent sheaves on X. Furthermore, Kapranov proved that, for a Kähler manifold X, the Dolbeault resolution \Omega(T{1,0} X ) of TX[−1]is an L∞ algebra. In this paper, we prove that Kapranov’s theorem holds in much wider generality for vector bundles over Lie pairs. Given a Lie pair (L, A), i.e. a Lie algebroid L together with a Lie subalgebroid A, we define the Atiyah class αE of an A-module E as the obstruction to the existence of an A-compatible L-connection on E. We prove that the Atiyah classes αL/A and αE respectively make L/A[−1] and E[−1] into a Lie algebra and a Liealgebra module in the bounded below derived category D+(A), where A is the abelian category of left U(A)-modules and U(A) is the universal enveloping algebra of A. Moreover,we produce a homotopy Leibniz algebra and a homotopy Leibniz module stemming from the Atiyah classes of L/A and E, and inducing the afore said Lie structures in D+(A).

We derive macroscopic dynamics for self-propelled particles in a fluid. The starting point is a coupled Vicsek–Stokes system. The Vicsek model describes self-propelled agents interacting through alignment. It provides a phenomenological description of hydrodynamic interactions between agents at high density. Stokes equations describe a low Reynolds number fluid. These two dynamics are coupled by the interaction between the agents and the fluid. The fluid contributes to rotating the particles through Jeffery’s equation. Particle self-propulsion induces a force dipole on the fluid. After coarse-graining we obtain a coupled Self-Organised Hydrodynamics–Stokes system. We perform a linear stability analysis for this system which shows that both pullers and pushers have unstable modes. We conclude by providing extensions of the Vicsek–Stokes model including short-distance repulsion, finite particle inertia and finite Reynolds number fluid regime.

We introduce a new troubled-cell indicator for the discontinuous Galerkin (DG) methods for solving hyperbolic conservation laws. This indicator can be defined on unstructured meshes for high order DG methods and depends only on data from the target cell and its immediate neighbors. It is able to identify shocks without PDE sensitive parameters to tune. Extensive one- and two-dimensional simulations on the hyperbolic systems of Euler equations indicate the good performance of this new troubled-cell indicator coupled with a simple minmod-type TVD limiter for the Runge-Kutta DG (RKDG) methods.