Grigorios A. Pavliotis

Two-dimensional droplet spreading over random topographical substrates.

We examine theoretically the effects of random topographical substrates on the motion of two-dimensional droplets via statistical approaches, by representing substrate families as stationary random functions. The droplet shift variance at both early times and in the long-time limit is deduced and the droplet footprint is found to be a normal random variable at all times. It is shown that substrate roughness inhibits wetting, illustrating also the tendency of the droplet to slide without spreading as equilibrium is approached. Our theoretical predictions are verified by numerical experiments.

STOCHASTIC PROCESSES AND APPLICATIONS

Stochastic Processes.- Diffusion Processes.- Introduction to Stochastic Differential Equations.- The Fokker-Planck Equation.- Modelling with Stochastic Differential Equations.- The Langevin Equation.- Exit Problems for Diffusions.- Derivation of the Langevin Equation.- Linear Response Theory.- Appendix A Frequently Used Notations.- Appendix B Elements of Probability Theory.

Parameter Estimation for Multiscale Diffusions

We study the problem of parameter estimation for time-series possessing two, widely separated, characteristic time scales. The aim is to understand situations where it is desirable to fit a homogenized single-scale model to such multiscale data. We demonstrate, numerically and analytically, that if the data is sampled too finely then the parameter fit will fail, in that the correct parameters in the homogenized model are not identified. We also show, numerically and analytically, that if the data is subsampled at an appropriate rate then it is possible to estimate the coefficients of the homogenized model correctly.The ideas are studied in the context of thermally activated motion in a two-scale potential. However the ideas may be expected to transfer to other situations where it is desirable to fit an averaged or homogenized equation to multiscale data.

Periodic Homogenization for Hypoelliptic Diffusions

We study the long time behavior of an Ornstein–Uhlenbeck process under the influence of a periodic drift. We prove that, under the standard diffusive rescaling, the law of the particle position converges weakly to the law of a Brownian motion whose covariance can be expressed in terms of the solution of a Poisson equation. We also derive upper bounds on the convergence rate in several metrics.

Periodic homogenization for inertial particles

We study the problem of homogenization for inertial particles moving in a periodic velocity field, and subject to molecular diffusion. We show that, under appropriate assumptions on the velocity field, the large scale, long time behavior of the inertial particles is governed by an effective diffusion equation for the position variable alone. To achieve this we use a formal multiple scale expansion in the scale parameter. This expansion relies on the hypo-ellipticity of the underlying diffusion. An expression for the diffusivity tensor is found and various of its properties studied. In particular, an expansion in terms of the non-dimensional particle relaxation time τ (the Stokes number) is shown to co-incide with the known result for passive (non-inertial) tracers in the singular limit τ→0. This requires the solution of a singular perturbation problem, achieved by means of a formal multiple scales expansion in τ Incompressible and potential fields are studied, as well as fields which are neither, and theoretical findings are supported by numerical simulations.

General Dynamical Density Functional Theory for Classical Fluids

We study the dynamics of a colloidal fluid in the full position-momentum phase space. The full underlying model consists of the Langevin equations including hydrodynamic interactions, which strongly influence the non-equilibrium properties of the system. For large systems, the number of degrees of freedom prohibit a direct solution of the Langevin equations and a reduced model is necessary, e.g. a projection of the dynamics to those of the reduced one-body distribution. We derive a generalized dynamical density functional theory (DDFT), the computational complexity of which is independent of the number of particles. We demonstrate that, in suitable limits, we recover existing DDFTs, which neglect either inertia, or hydrodynamic interactions, or both. In particular, in the overdamped limit we obtain a DDFT describing only the position distribution, and with a novel definition of the diffusion tensor. Futhermore, near equilibrium, our DDFT reduces to a Navier-Stokes-like equation but with additional non-local terms. We also demonstrate the very good agreement between the new DDFT and full stochastic calculations, as well as the large qualitative effects of inertia and hydrodynamic interactions.

Asymptotic analysis for the generalized Langevin equation

Various qualitative properties of solutions to the generalized Langevin equation (GLE) in a periodic or a confining potential are studied in this paper. We consider a class of quasi-Markovian GLEs, similar to the model that was introduced in Eckmann J-P et al 1999 Commun. Math. Phys. 201 657–97. Ergodicity, exponentially fast convergence to equilibrium, short time asymptotics, a homogenization theorem (invariance principle) and the white noise limit are studied. Our proofs are based on a careful analysis of a hypoelliptic operator which is the generator of an auxiliary Markov process. Systematic use of the recently developed theory of hypocoercivity (Villani C 2009 Mem. Am. Math. Soc. 202 iv, 141) is made.

Noise induced state transitions, intermittency and universality in the noisy Kuramoto-Sivashinksy equation

Consider the effect of pure additive noise on the long-time dynamics of the noisy Kuramoto-Sivashinsky (KS) equation close to the instability onset. When the noise acts only on the first stable mode (highly degenerate), the KS solution undergoes several state transitions, including critical on-off intermittency and stabilized states, as the noise strength increases. Similar results are obtained with the Burgers equation. Such noise-induced transitions are completely characterized through critical exponents, obtaining the same universality class for both equations, and rigorously explained using multiscale techniques.

Variance Reduction using Nonreversible Langevin Samplers

A standard approach to computing expectations with respect to a given target measure is to introduce an overdamped Langevin equation which is reversible with respect to the target distribution, and to approximate the expectation by a time-averaging estimator. As has been noted in recent papers [30, 37, 61, 72], introducing an appropriately chosen nonreversible component to the dynamics is beneficial, both in terms of reducing the asymptotic variance and of speeding up convergence to the target distribution. In this paper we present a detailed study of the dependence of the asymptotic variance on the deviation from reversibility. Our theoretical findings are supported by numerical simulations.

Optimal non-reversible linear drift for the convergence to equilibrium of a diffusion

We consider non-reversible perturbations of reversible diffusions that do not alter the invariant distribution and we ask whether there exists an optimal perturbation such that the rate of convergence to equilibrium is maximized. We solve this problem for the case of linear drift by proving the existence of such optimal perturbations and by providing an easily implementable algorithm for constructing them. We discuss in particular the role of the prefactor in the exponential convergence estimate. Our rigorous results are illustrated by numerical experiments.