Hydrodynamic and phoretic interactions between active particles are central to the understanding of their collective dynamics. Under experimentally relevant conditions, the motion of the fluid flow is governed by the Stokes equation and that of the phoretic field, if one is present, by the Laplace equation. The “activity” appears in these equations as boundary conditions on the particle surfaces that prescribe the slip velocity in the Stokes equation and flux of the phoretic field in the Laplace equation.
We have obtained a complete formalism to study phoresis and Stokesian hydrodynamics of colloidal spheres with slip. We use a grid-free method, combining the integral representation of Laplace and Stokes equations, spectral expansion, and Galerkin discretization. This formalism has been implemented in a free and open-source numerical library, PyStokes, and applied to study various experimental phenomena of recent interest in active matter.
The above animation is an application of our theoretical and numerical framework [Phys. Rev. Lett. 124(8), 088003, 2020]. Here, we show that the oscillatory dynamics of a pair of active particles near a boundary, best exemplified by the fascinating dance of the green algae Volvox, can be understood in terms of Hamiltonian mechanics, even though the system does not conserve energy. See also my blog post which provides more details of the theory and applications of the method described above.
Stokes traction on an active particle. Phys. Rev. E 106, 014601, 2022
Controlled optofluidic crystallization of colloids tethered at interfaces. Phys. Rev. Lett. 125(6), 068001, 2020
PyStokes: phoresis and Stokesian hydrodynamics in Python. J. Open Source Software 5(50), 2318, 2020
Periodic orbits of active particles induced by hydrodynamic monopoles. Phys. Rev. Lett. 124(8), 088003, 2020
Competing chemical and hydrodynamic effects in autophoretic colloidal suspensions J. Chem. Phys. 161, 044901, 2019
Flow-induced phase separation of active particles is controlled by boundary conditions. PNAS 115(21), 5403, 2018
Generalized Stokes laws for active colloids and their applications. J. Phys. Commun. 2, 025025, 2018
Fluctuating hydrodynamics and the Brownian motion of an active colloid near a wall. Eur. J. Comp. Mech. 26, 78-97, 2017
Universal Hydrodynamic Mechanisms for Crystallization in Active Colloidal Suspensions. Phys. Rev. Lett. 117, 228002, 2016
Many-body microhydrodynamics of colloidal particles with active boundary layers. J. Stat. Mech. P06017, 2015
See also my blog post which provides an introduction to the active scalar field theory described above.
Self-propulsion of active droplets without liquid-crystalline order. Phys. Rev. Research 2(3), 032024(R), 2020
Hydrodynamically interrupted droplet growth in scalar active matter. Phys. Rev. Lett. 123(14), 148005, 2019
In this section, I provide a brief description of my work on (a) Bayesian inference of stochastic process, and (b) Ritz method for finding most probable transition paths between meta-stable states of stochastic dynamical systems.
In probability theory and statistics, Gaussian processes has been an area of extensive research in the past few decades. See my blog post on Gaussian processes for more details. We have developed an O(N) method for inference of a stationary Gauss-Markov processes using four sufficient statistics matrices. This is in contrast to O(N^3) cost using the direct method of Gaussian processes, which makes approximate methods a necessity for large data-sets. Our method, on the other hand, exploits the Gauss-Markov property to obtain accurate and efficient inference. The formalism is then used to do inference on experimental data and we find that it outperforms fitting and sampling methods both in accuracy and speed. Read more: arXiv:1706.04961.
A recent interest is the application of mathematical and statistical tools to study the spread of infectious diseases. In particular, we seek to obtain the optimal intervention strategies which minimize the transmission of infection. For this, we have developed a numerical library, PyRoss, which provides an integrated platform for inference, forecasts, and control for epidemiological models.
Bayesian inference in PyRoss, on pre-defined or user-defined models, is performed using model-adapted Gaussian processes derived from functional limit theorems for Markov population process. Read more: arXiv:2010.11783.
We present a Ritz method for finding most probable transition paths between meta-stable states of stochastic dynamical systems. Our method provides a route to constructing quasipotentials (non-equilibrium potentials) by repeatedly sampling transition paths asymptotically in time. In our direct method, utilizing the Ritz discretization, the most-probable path (instanton) is computed by minimizing the Freidlin-Wentzell action. Analyzing the paths in a spectral basis of Chebyshev polynomials, nonlinear optimization is used to obtain coefficients that give the least action from which the instanton is synthesized in the spectral basis.
Ritz method for transition paths and quasipotentials of rare diffusive events. Phys. Rev. Research 2(3), 033208, 2020
Fast Bayesian inference of the multivariate Ornstein-Uhlenbeck process. Phys. Rev. E 98, 012136, 2018
Direct verification of the fluctuation-dissipation relation in viscously coupled oscillators. Phys. Rev. E 96, 050102(R), 2017
Fast Bayesian inference of optical trap stiffness and particle diffusion. Sci. Rep. 7, 41638, 2017
Efficient Bayesian inference of fully stochastic epidemiological models with applications to COVID-19. R. Soc. Open Sci. 8, 211065, 2021
Inference, prediction and optimization of non-pharmaceutical interventions using compartment models: the PyRoss library. arXiv:2005.09625, 2020
Age-structured impact of social distancing on the COVID-19 epidemic in India. arXiv:2003.12055, 2020
Along with my collaborators, I develop state-of-the-art numerical libraries for research in theoretical physics and applied mathematics. See my Github profile for a full list of open-source projects and the list of contributors to these libraries. A summary of the selected libraries follows.
PyStokes is a numerical library for phoresis and Stokesian hydrodynamics in Python. The PyStokes library has been specifically designed for studying phoretic and hydrodynamic interactions in suspensions of active particles. It uses a grid-free method, combining the integral representation of Laplace and Stokes equations, spectral expansion, and Galerkin discretization, to compute phoretic and hydrodynamic interactions between spherical active particles with slip boundary conditions on their surfaces. The library has been used to model suspensions of microorganisms, synthetic autophoretic particles and self-propelling droplets. The current implementation includes unbounded volumes, volumes bounded by plane walls or interfaces, and periodic volumes.
PyRoss is a numerical library that offers an integrated platform for inference, forecasts and non-pharmaceutical interventions in structured epidemiological compartment models. Generative processes can be formulated stochastically (as Markov population processes) or deterministically (as systems of differential equations). Population processes are sampled exactly by the Doob-Gillespie algorithm or approximately by the tau-leaping algorithm while differential equations are integrated by both fixed and adaptive time-stepping. A hybrid algorithm transits dynamically between these depending on the magnitude of the compartmental fluctuations. Bayesian inference on pre-defined or user-defined models is performed using model-adapted Gaussian processes derived from functional limit theorems for Markov population process.
PyRitz is a Python package, using the Ritz method, for computing transition paths and quasipotentials in Python. The most-probable path (instanton) is computed by minimizing the Freidlin-Wentzell action. Analysing the paths in a spectral basis of Chebyshev polynomial, nonlinear optimisation is used to obtain coefficients that give the least action from which the instanton is synthesised in the spectral basis.
PyGL is a numerical library for statistical field theory in Python. The library has been specifically designed to study field theories without time-reversal symmetry. The library can be used to study models of statistical physics of various symmetries and conservation laws. In particular, we allow models with mass and momentum conservations. The library constructs differentiation matrices using finite-difference and spectral methods. To study the role of momentum conservation, the library also allows computing fluid flow from the solution of the Stokes equation.
Exclusion processes refer to a family of models of transport where particles have hard-core interaction. A randomly chosen particle on a 1D lattice will move to its neighboring site if it is empty. These models can be studied under different kinds of boundary conditions. We consider open-chain exclusion processes where particles are injected from the left boundary at a rate α and are absorbed in the right boundary at rate β. We study the role of interaction between several such "lanes". In this blog post, I provide an introduction to the asymmetric exclusion process and describes our work on multi-lane exclusion models.
Phase-plane analysis of driven multi-lane exclusion models. J. Stat. Mech. P04004, 2012