Research topics
My expertise is in the areas of applied and computational mathematics, computational materials science, and computational physics. More generally, I am interested in the development of widelyapplicable mathematical and algorithmic methodologies for the modeling of complex natural systems at the atomistic and mesoscopic levels, and their application to practical problems in fundamental and applied sciences and engineering. In particular, I work on developing algorithms that accelerate or systematically coarsegrain traditional methods such as Molecular Dynamics or (Kinetic) Monte Carlo, as well as multiscale (hybrid) methods combining particle with stochastic (fluctuating) coarsegrained models. My present focus is on fluid dynamics at small scales, and in particular, fluctuating hydrodynamics. Visit the Computer Codes page for publicdomain codes incorporating our work.
My research at the Courant Institute is funded by National Science Foundation grant NSF DMS1115341, the Air Force's Young Investigator Research Program AFOSR YIP2012, and the Department of Energy (DOE) Office of Science Early Career Research Program.
Specific details on several research subjects I have or am working on (with latest work first) can be found below. There are many interesting projects for undergraduate and graduate students in each of these areas, ranging from developing novel algorithms to applying computational methods to relevant problems in the sciences and engineering.
Hydrodynamics at Small Scales
This research focuses on fluid mechanics in regimes where thermal fluctuations are important. Notable examples include flows at micro and nano scales typical of new microfluidic, nanofluidic and microelectromechanical devices; biological systems such as lipid membranes, Brownian molecular motors, nanopores; as well as processes where the effect of fluctuations is amplified by strong nonequilibrium effects, such as combustion of lean flames, capillary dynamics, hydrodynamic instabilities, and others.
Here is, for example, an animation showing the development of an instability during diffusive mixing of a solution of salt on top of a solution of sugar, triggered entirely by thermal fluctuations, that is, starting with no initial interface perturbation (see below more details an a comparison to the case of including a white random initial perturbation):
My work in this field is joint with several collaborators, whose webpages contain additional information and publications of interest. In particular, I collaborate closely with John Bell (Lawrence Berkeley National Laboratory) and Alejandro Garcia (San Jose State University), Rafael DelgadoBuscalioni (UAM, Spain), Eric VandenEijnden (Courant), and Boyce Griffith (NYU Medical School). Our work builds on the mature field of deterministic computational fluid dynamics, and combines fundamental crossdisciplinary investigations with development of extensible yet efficient parallel computer codes (in particular, our algorithms are incorporated into the BoxLib and IBAMR codes, and presently also extended to run on GPUs).
There are several main avenues of research, described next, see also these presentation slides.
Numerical schemes for fluctuating hydrodynamics
Thermal fluctuations can be included in the classical NavierStokes fluid equations through stochastic forcing terms that are essentially the divergence of a whitenoise random field (stochastic flux), as first proposed by Landau and Lifshitz. The presence of nontrivial dynamics at all scales, as well as the necessity to maintain fluctuationdissipation balance in spatiotemporal discretizations, makes the continuum stochastic partial differential equations of fluctuating hydrodynamics difficult to solve using existing approaches.We have recently analyzed and implemented a specialized spatial discretization along with a threestage stochastic RungeKutta temporal integrator for the singlefluid compressible fluctuating equations, as described in this presentation. Recently we have developing numerical methods for solving the fluctuating incompressible and low Mach number NavierStokes equations. This work is reported in these papers:
1. "Temporal Integrators for Fluctuating Hydrodynamics", S. Delong and B. E. Griffith and E. VandenEijnden and A. Donev, to appear in Phys. Rev. E, 2013 [arXiv:1212.1033].
2. "Staggered Schemes for Fluctuating Hydrodynamics", F. Balboa and J. Bell and R. DelgadoBuscalioni and A. Donev and T. Fai and B. Griffith and C. Peskin, submitted, 2011 [arXiv:1108.5188].
3. "Low Mach Number Fluctuating Hydrodynamics of Diffusively Mixing Fluids", A. Donev and A. J. Nonaka and Y. Sun and T. Fai and A. L. Garcia and J. B. Bell, submitted to SIAM J. Multiscale Modeling and Simulation, 2013 [ArXiv:1212.2644].
Present work is focused on:
1. Developing algorithms for the fluctuating Low Mach number equations for multispecies fluid mixtures on adaptive Cartesian grids, see this presentation, see animation below.
2. Accounting for phase separation via the stochastic CahnHilliard equation and studying fluctuations at fluidfluid interfaces.
3. Accounting for chemical reactions in complex fluid mixtures such as lean hydrogen fuels.
Here is a nice animation produced by collaborator Andy Nonaka showing the development of a mixedmode instability in a ternary mixture (water+sugar+salt) driven by diffusion and gravity, in good agreement with experimental observations reported in the bottom row of panels in Fig. 1 of the paper "Mixedmode instability of a miscible interface due to coupling between RayleighTaylor and doublediffusive convective modes" by the group of Anne De Wit:
Brownian Dynamics via Fluctuating Hydrodynamics
The study of complex fluids
such as micro and nanocolloidal or polymeric solutions is an
important application of fluctuating hydrodynamics. We have
previously used a particle method to study the dynamics of
polymer chains in solution. We are presently developing
methods to directly couple immersed
particles to a fluctuating fluid medium, based on
existing LatticeBoltzmann and Stochastic Immersed Boundary
techniques, see this presentation.
Initial work in this area is described here3. "Inertial Coupling Method for particles in an incompressible fluctuating fluid", F. Balboa Usabiaga and R. DelgadoBuscalioni and B. E. Griffith and A. Donev, submitted, 2013 [ArXiv:1212.6427], code available at https://code.google.com/p/fluam.
Present work is focused on:
1. Extending the temporal integrators to handle the overdamped (Brownian dynamics) limit.
2. Extending the physical fidelity of the blob models by adding stresslet terms.
3. Extending the work to rigid bodies. This work is in collaboration with the research group of Neelesh Patankar at Northwestern.
4. Understanding theoretical (applied stochastic analysis) issues related to the diffusive limit (in collaboration with Eric VandenEijnden).
Particle methods for hydrodynamics
Thermal fluctuations can be included in fluid dynamics by explicitly accounting for the particle nature of matter using particle methods. It is also important to perform efficient particle calculations in order to assess and expand the range of validity of continuum approximations. Our research focuses on developing coarsegrained stochastic particle models that build upon the Direct Simulation Monte Carlo (DSMC) method. This class of methods are also related to the dissipative particle dynamics (DPD) and the multiparticle collision (also called stochastic rotation) dynamics techniques. Further details can be found in this presentation or the following papers:1. "A ThermodynamicallyConsistent NonIdeal Stochastic Hard Sphere Fluid", by A. Donev and A. L. Garcia and B. J. Alder, J. Stat. Mech., P11008, 2009 [arXiv:0908.0510].
2. "Stochastic HardSphere Dynamics for Hydrodynamics of NonIdeal Fluids", by A. Donev, A. L. Garcia and B. J. Alder, Phys. Rev. Lett., 101:075902, 2008 [arXiv:0803.0359].
3. "Stochastic EventDriven Molecular Dynamics", by A. Donev, A. L. Garcia and B. J. Alder, J. Comp. Phys., 227(4):26442665, 2008, [arXiv:0708.0251].
Our present research is focused on extending these types of algorithms to multispecies reactive mixtures, implementing the algorithms in a public domain code that we will release in the future, as well as parallelizing the particle algorithms.
Hybrid particlecontinuum algorithms
Particle and continuum methods can be combined into hybrid multiscale methods that use the more expensive but accurate particle method only in regions where the continuum description fails or is difficult to implement (e.g., shocks, rarefication, singularities, near suspended structures). This can substantially lower the cost of particle methods while still keeping the advantages of particle methods in regions of interest. This image
illustrates how three different aspects of my
research are combined to study the dynamics of a
polymer chain in shear flow. The chain itself is
made up of hard disks (red, done using
eventdriven molecular dynamics) and suspended in
a coarsegrained particle solvent (green, done
using our IDSMC algorithm), further embedded in a
stochastic continuum fluid flow (purple arrows
show the fluctuating velocities in our RK3D
fluctuating hydrodynamics solver).

We have developed a bidirectional dynamic coupling between a stochastic particle fluid and a fluctuating continuum and demonstrated that thermal fluctuations have to be consistently included in the continuum component of hybrid calculations in order not to distort the thermal equilibrium in the particle solver. Here is a presentation (without the movies) on the subject, with details contained in this paper:
1. "A hybrid particlecontinuum method for hydrodynamics of complex fluids", by A. Donev and J. B. Bell and A. L. Garcia and B. J. Alder, SIAM J. Multiscale Modeling and Simulation 8(3):871911, 2010 [arXiv:0910.3968].
There are severe time step restrictions inherent to both particle and hybrid calculations, stemming from the need to resolve the dynamics of the fluid particles. In order to extend the time step, one needs to avoid the particle representation in the solver, as we do in more recent work described above.
EventDriven Asynchronous Algorithms
Rather broadly, I am interested in the development of efficient particle methods, specifically asynchronous eventdriven Molecular Dynamics and Kinetic Monte Carlo methods. My research in particle packings, reactiondiffusion systems, and coarsegrained solvents, all use an eventdriven framework to achieve substantial speedup over traditional timedriven methods, as described in this presentation and this review article:
1. "Asynchronous EventDriven Particle Algorithms", by A. Donev, SIMULATION: Transactions of the Society for Modeling and Simulation International, 85(4):229242, 2009.
An important remaining challenge for future research is efficient parallelization of asynchronous eventdriven algorithms.
DiffusionReaction Systems
Together with collaborators at Lawrence
Livermore National Labs we developed an eventdriven Kinetic Monte Carlo algorithm for
diffusionreaction systems that is far superior to
traditional algorithms. This method has the potential to be
applied in a variety of problems in material science and
biology. Notably, we have used it to study radiation damage in
metals, and the method has been implemented for biochemical
reactive systems in the eGFRD
code. Further details can be found in this presentation and
these papers:1. "Firstpassage Kinetic Monte Carlo method", by T. Oppelstrup, V. V. Bulatov, A. Donev, M. H. Kalos, G. H. Gilmer and B. Sadigh, Phys. Rev. E, 80(6):066701, 2009 [arXiv:0905.3575].
2. "A FirstPassage Kinetic Monte Carlo Algorithm for Complex DiffusionReaction Systems", by A. Donev, V. V. Bulatov, T. Oppelstrup, G. H. Gilmer, B. Sadigh and M. H. Kalos, J. Comp. Phys., 229(9):32143236, 2010 [arXiv:0905.3576].
Future work will extend these methods to enable a mixed timedriven with eventdriven framework that combines the advantages of each approach, without the inefficiency of timedriven algorithms or the complexity of purely eventdriven handling.
Packing of Hard Particles
My Ph.D. dissertation (thesis) and my Final Oral Exam (FPO) presentation (June 9th, 2006) give a rather extensive overview of my research on Jammed Packings of Hard Particles, performed under the supervision of Dr. Salvatore Torquato as part of the Complex Materials Theory Group. Here are some MNG and GIF animations related to my thesis research.
I have also developed dynamic VRML models to render and animate packings of spheres and ellipsoids. These are barely documented but you can get them here if you are brave. 
Rigidity Theory and Jamming
By using mathematical results from rigidity theory, in collaboration with Robert
Connelly, we were able to develop a rigorous framework
for defining and testing for jammed
packings of hard spheres and ellipsoids, as described
in these papers:1. "Underconstrained Jammed Packings of Hard Ellipsoids", by A. Donev, R. Connelly, F. H. Stillinger and S. Torquato, Phys. Rev. E, 75:051304, 2007 [condmat/0608334].
2. "A Linear Programming Algorithm to Test for Jamming in HardSphere Packings", by A. Donev, S. Torquato, F. H. Stillinger, and R. Connelly, J. Comp. Phys., 197(1):139166, 2004.
3. "Comment on "Jamming at zero temperature and zero applied stress: The epitome of disorder", by A. Donev, S. Torquato, F. H. Stillinger, and R. Connelly, Phys. Rev. E, 70:043301, 2004.
4. "Jamming in Hard Sphere and Disk Packings", by A. Donev, S. Torquato, F. H. Stillinger, and R. Connelly, J. Appl. Phys., 95(3):989, 2004.
5. "Breakdown of Elasticity Theory for Jammed HardParticle Packings: Conical Nonlinear Constitutive Theory", by S.Torquato, A. Donev, and F. H. Stillinger, Int. J. Solids Structures, 40(25):71437153, 2003.
Further work is necessary to develop codes that can analyze the rigidity properties of largescale packings.
Packing of Hard Ellipsoids
We further developed sophisticated eventdriven molecular
dynamics algorithms and codes to generate jammed packings of hard spheres and
also, for the first time, hard ellipsoids and hard
superellipsoids, in two, three and, for spheres, in higher
dimensions. Together with the experimental research
group of Paul
Chaikin, we made some surprising discoveries about
random packings of ellipsoids. Firstly, random packings of
moderately aspherical ellipsoids are substantially denser that
that of spheres, specifically, the density grows linearly with
the aspect ratio, even though the packings were found to be
hypostatic and not isostatic as commonly expected. An
astounding packing fraction of 74% was found for a specific
ellipsoid shape that we named ollipsoids (optimal ellipsoids). These results
are detailed in the following publications:1. "Improving the Density of Jammed Disordered Packings using Ellipsoids" by A. Donev, I. Cisse, D. Sachs, E. A. Variano, F. H. Stillinger, R. Connelly, S. Torquato and P. M. Chaikin, Science, 303:990993, 2004.
2. "Neighbor List CollisionDriven Molecular Dynamics Simulation for Nonspherical Particles. I. Algorithmic Details II. Applications to Ellipses and Ellipsoids", by A. Donev, F. H. Stillinger, and S. Torquato, J. Comp. Phys, 202(2):737764 (part I) and 202(2):765793 (part II), 2005, [physics/0110034].
4. "Experiments on Random Packings of Ellipsoids", W. Man, A. Donev , F. H. Stillinger, M. T. Sullivan, W. B. Russel, D. Heeger , S. Inati, S. Torquato and P. M. Chaikin, Phys. Rev. Lett., 94:198001, 2005.
At the time we discovered random packings of ellipsoids as dense as the FCC hardsphere crystal, it was thought that the FCC crystal structure and maximal volume fraction of 74% applies to ellipsoids as well. However, using hardellipsoid molecular dynamics and some geometry we soon discovered that there are denser ellipsoid crystals, as dense as 77% in volume fraction:
1. "Unusually Dense Crystal Packings of Ellipsoids", by A. Donev, F. H. Stillinger, P. M. Chaikin and S. Torquato, Phys. Rev. Lett., 92:255506, 2004, [condmat/0110034].
Since then, this work has been extended to superellipsoids (generalized ellipsoids), which you can learn more about from the homepage of Dr. Salvatore Torquato.
Random Sphere Packings
We also performed detailed studies of large random packings of
hard spheres in three and higher dimensions and found some
surprising results, including that maximally random jammed packings of hard spheres
are hyperinform (incompressible) and potentially exhibit
longranged correlations, unlike hardsphere fluids. Further
details can be found in these papers:1. "Pair Correlation Function Characteristics of Nearly Jammed Disordered and Ordered HardSphere Packings", by A. Donev, F. H. Stillinger, and S. Torquato, Phys. Rev. E, 71:011105, 2005, [condmat/0408550].
2. "Unexpected Density Fluctuations in Jammed Disordered Sphere Packings", by A. Donev, F. H. Stillinger, and S. Torquato, Phys. Rev. Lett., 95:090604, 2005, [condmat/0506406].
3. "Packing Hyperspheres in HighDimensional Euclidean Spaces", by M. Skoge, A. Donev, F. H. Stillinger and S. Torquato, Phys. Rev. E, 74:041127, 2006 [ibid 75:029901, 2007], [condmat/0608362].
Glass Transition
One of the most fascinating open questions in condensed matter physics is the glass transition. Two central questions are whether there are thermodynamicallystable disordered (noncrystalline) solid phases and whether there is an ideal glass transition (vanishing configurational entropy in a disordered phase). By using our hardsphere molecular dynamics algorithms we found convincing evidence that there can exist disordered solid phases and that there is no ideal glass transition in certain hard particle systems, as described in the following papers:1. "Do Binary Hard Disks Exhibit an Ideal Glass Transition?", by A. Donev, F. H. Stillinger, and S. Torquato, Phys. Rev. Lett., 96:225502, 2006, [condmat/0603183].
2. "Configurational Entropy of Binary HardDisk Glasses: Nonexistence of an Ideal Glass Transition", by A. Donev, F. H. Stillinger and S. Torquato, J. Chem. Phys., 127:124509, 2007.
3. "Calculating the Free Energy of Nearly Jammed HardParticle Packings Using Molecular Dynamics", by A. Donev, F. H. Stillinger, and S. Torquato, J. Comp. Phys., 225:509–527, 2007.
4. "Tetratic Order in the Phase Behavior of a HardRectangle System", by A. Donev, J. Burton, F. H. Stillinger, and S. Torquato, Phys. Rev. B, Vol. 73:054109, 2006, [condmat/0508550].
These results point to a kinetic origin of the glass transition, and further study is needed to understand the geometrical origin of the configurational trapping experienced near the kinetic glass transition.
Multifunctional Composites
During my Ph.D. studies I also joined an effort in the group of Dr. Salvatore Torquato concerning the computational design of binary composites (mixtures of two materials) that optimize transport properties such as conductivity. This research led to the discovery that triplyperiodic minimal surface structures are optimal for several transport properties, as illustrated in this figure for the Schwartz simple cubic surface bicontinuous structure:

Further details can be found in these papers and more recent publications by the Torquato group:
1. "Multifunctional Optimal Composite Microstructures: Simultaneous Transport of Heat and Electricity", by S.Torquato, S. Hyun and A. Donev, Phys. Rev. Lett., 89(26):266601, 2002.
2. "Manufacturable extremal lowdielectric, highstiffness porous materials", S. Torquato, A. Donev, A. G. Evans, and C. J. Brinker, J. Appl. Phys., 97:124103, 2005.
3. "Minimal Surfaces and Multifunctionality", by S.Torquato and A. Donev, Proceedings of the Royal Society of London: Mathematical, Physical and Engineering Sciences, 460(2047):1849  1856, 2004.
4. "Optimal design of manufacturable threedimensional composites with multifunctional characteristics", by S.Torquato, S. Hyun and A. Donev, J. Appl. Phys., 94(9):57485755, 2003.
Subsequent work in the group of Dr. Salvatore Torquato has also studied fluid permeability in such composite structures.