Lecture materials
I will post here some hand-written notes or presentation slides and links to relevant reading materials. The subject is vast and still an active area of research. The selected topics and readings are a biased selection focusing on my own research interests and background.Some of the links below may require being on the NYU network, which you can do from off-campus via a proxy server as well.
1. (Sept 4th and 11th) Coarse-Graining of Microscopic Models
The lecture notes consists of three parts: Review of coase graining, microscopic dynamics, and molecular dynamics. Here are some more recent notes on molecular dynamics and a demo MD code.We will start class by jumping directly into coarse graining in order to understand what this class will be about and what coarse-graining is and why it is important. Start by reading sections 1-3 of "Statistical Mechanics of Coarse-Graining" by Pep Espanol (official published version on SpringerLink). We will continue with the remaining sections later in the course and fill in the missing background and details.
We begin the technical material at the bottom of the coarse-graining hierarchy by discussing microscopic models of materials, notably, classical Hamiltonian and statistical mechanics of molecular systems, as well as numerical methods for solving them. Good background readings for this:
- Sections 5.1-5.3 of the chapter on Statistical Mechanics from the book Stochastic tools in mathematics and science by Alexandre J. Chorin and Ole H. Hald.
- Sections 1.9-1.11 in the lecture notes on Numerical methods for molecular
and continuum dynamics by Tony Ladd. We will cover
the sections on Langevin dynamics a little later in the course.
- Some additional but optional reading can be found in Sections 1-4 and 6 of the review article "Understanding modern molecular dynamics: techniques and applications", by Mark E. Tuckerman and Glenn J. Martyna (official published version).
- A good and systematic textbook on Hamiltonian systems and symplectic integrators is Simulating Hamiltonian Dynamics by Benedict Leimkuhler and Sebastian Reich (NYU ebook).
2. (Sept 18th and 25th and Oct 2nd) Markov Chain Monte Carlo (MC) Models
Guest lecture on Sept. 18th by Monte Carlo expert Professor Jonathan Goodman.The lecture notes consist of three parts: Equilibrium MC and Kinetic MC, and finally DSMC.
Suggested background readings:
- Section 5.5 of the chapter on Statistical Mechanics from the book Stochastic tools in mathematics and science by Alexandre J. Chorin and Ole H. Hald.
- A tutorial on Monte Carlo and Kinetic Monte Carlo Methods by Peter Kratzer (backup copy).
- A quick but optional
review article on the Direct
Simulation Monte Carlo by F. Alexander and A. Garcia.
- Some examples of
asynchronous computational algorithms for KMC are reviewed in
the optional article Asynchronous
Event-Driven Particle Algorithms by Aleksandar Donev. This
includes the First-Passage
KMC algorithm for reaction-diffusion systems.
4. (Oct 9th and 16th) The Langevin Equation
The lecture notes consist of several parts, starting from an introduction to Diffusion Processes and stochastic ODEs, and finally a discussion of Langevin Equations. We will also take a brief but more general look at Kolmogorov equations for Markov Processes to unify some of the different topics (MD, MCMC and SODEs) we have discussed so far.Suggested readings:
- My lecture is closely based on these lecture
notes on Langevin methods by my colleague Burkhard Duenweg.
- Sections 6.1-6.3 (see also the chapter on Brownian Motion) of the chapter on Time Dependent Statistical Mechanics from the book Stochastic tools in mathematics and science by Alexandre J. Chorin and Ole H. Hald.
- We will complete going through the material nicely covered in the lecture notes on Numerical methods for molecular and continuum dynamics by Tony Ladd.
- Mathematically more rigorous (technical) lecture notes on Applied Stochastic Processes have been posted by G. A. Pavliotis.
5. (Oct 23rd) Overdamped Langevin Equation
This is a continuation of the lectures on Langevin dynamics to
consider the overdamped limit and then generalizations thereof as well
as numerical schemes. We first discuss the theory behind Brownian
Dynamics and then some related Numerical
Methods (we will skip some parts of these older notes and devote
a separate lecture to the GENERIC formalism). The Fixman and
metropolized temporal integrators are briefly discussed in these
lecture notes on numerics
for overdamped Langevin equations.We will also discuss the notion of time reversibility for Markov processes in preparation for the next lecture.
Suggested readings:
- The kinetic interpretation of SDEs associated with Langevin
equations and the associated kinetic stochastic integral are
described in the paper "Fluctuation-dissipation
theorem, kinetic stochastic integral and efficient simulations"
by Hutter and Ottinger. The Fixman scheme is derived from the
kinetic integral.
- Metropolization for overdamped Langevin equations is proposed in the paper "Metropolis Integration Schemes for Self-Adjoint Diffusions", N. Bou-Rabee and A. Donev and E. Vanden-Eijnden [ArXiv:1309.5037].
- Fast multipole techniques for Brownian dynamics with hydrodynamic interactions is described in the article Fast Computation of Many-Particle Hydrodynamic and Electrostatic Interactions in a Confined Geometry by Juan Hernandez-Ortiz et al.
5. (Oct 30th) The GENERIC formalism
We will discuss the notion of entropy
in coarse-grained descriptions and then discuss a general class of
Langevin equations for coarse-grained models called the GENERIC
formalism which is a two-generator (energy and entropy) approach
suitable for isolated systems. For the case of a generalized canonical
ensemble (isothermal system) you are encourated to study these lecture
notes on a one-generator (free energy) formalism leading to augmented
Langevin equations.
6. (Nov 6th) Large Deviations and Transition State Theory
We will review the mathematical foundations of classical thermodynamics and equilibrium statistical mechanics from the perspective of large deviation theory. Then we will briefly discuss how LDT can be used to justify and obtain rates of rare events for Markov Chain models.Suggested readings:
- A brief summary can be found in this unfinished review article by Weinan E and Eric Vanden-Eijnden on Modeling Rare Transition Events.
- A long but pedagogical physics-based introduction to the foundations of large-deviation theory and its relation to statistical mechanics is found in The large deviation approach to statistical mechanics by Hugo Touchette (official published version).
7. (Nov 13th) Mori-Zwanzig Formalism
Guest lecture by Mori-Zwanzig expert Professor Pep Español. Here are detailed lecture notes based on a book he is writing. Also see the Applied Math Seminar he is giving on how to actually compute the Green-Kubo expressions from molecular dynamics. Here are brief notes on how the microscopic foundations of the GENERIC formalism via the Markovian Mori-Zwanzig formalism.Here are older lecture notes of mine outlining the Mori-Zwanzig formalism and the Markovian assumption for Hamiltonian dynamics.
Suggested readings:
- Sections 6.4-6.5 of the chapter on Time Dependent Statistical Mechanics from the book Stochastic tools in mathematics and science by Alexandre J. Chorin and Ole H. Hald.
- Sections 4-5 of "Statistical Mechanics of Coarse-Graining" by Pep Espanol (official published version on SpringerLink or on Citeceer).
- A recent article on how to actually compute the coefficients in
the coarse-grained equation from microscopic simulations titled "Mori-Zwanzig
formalism as a practical computational tool" by Hijon,
Espanol, Vanden-Eijnden and Delgado-Buscalioni (official published
version).
8. (Nov 20th, 25th and Dec 4th) Fluctuating Hydrodynamics
Special time and date during Thanksgiving week:Monday Nov 25th 3:30PM- 4:45PM in WWH 517
We start our discussion of fluctuating hydrodynamics by discussing how to formulate a stochastic diffusion equation for a system of particles jumping on a lattice. Then we will move on to stochastic advection-diffusion equations. Finally, we will write down the fluctuating Navier-Stokes equations and finish with a discussion of some numerical methods.
Suggested readings:
- Notes by A. Donev on the
fluctuating Burgers and incompressible Navier-Stokes equations,
as well as more detailed notes on the Langevin
formalism for fluctuating hydrodynamics.
- A recent article "Temporal Integrators for Fluctuating Hydrodynamics", by S. Delong and B. E. Griffith and E. Vanden-Eijnden and A. Donev.
- The review article An algorithmic introduction to numerical simulation of stochastic differential equations by D. J. Higham.