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Lectures / Reading

Instructor

Aleksandar Donev, 909 Warren Weaver Hall
E-mail: donev@courant.nyu.edu
Phone: (212) 992-7315

Course description

Describing the full atomistic structure of materials at relevant time and length scales is typically not computationally feasible.  A well-known alternative is to use classical continuum models such as the Navier-Stokes equations or linear elasticity.  At time and length scales in-between the atomistic and macroscopic, however, lies a continuum of mesoscopic scales at which is is necessary to use coarse-grained models.  These mesoscopic models try to capture some of the microscopic details but at a fraction of the cost of a full molecular-dynamics simulation.  These include particle models such as the Direct Simulation Monte Carlo method for gas flows or Dissipative Particle Dynamics for fluid flows, stochastic continuum models such as the fluctuating Navier-Stokes equations for small-scale fluid flows, hybrid models such as the quasi-continuum method, geometric models such as hard-particle packings for granular materials or graph models for magnetic materials, and others. 

In this course, we will discuss the fundamental ideas behind coarse-grained models, as well as computational algorithms for mesoscopic modeling of gases, liquids, solids, and granular materials.  Students will study a review or seminal paper and discuss what they have learned in class.

3 points per term

Topics covered include:

• Coarse graining by eliminating degrees of freedom
  
• Hamiltonian and statistical mechanics
  
• Molecular dynamics, including symplectic integrators and hard-sphere dynamics
  
• Markov Chain models and Kinetic Monte Carlo algorithms
  
• Direct Simulation Monte Carlo and variants for gas dynamics
  
• Reaction-diffusion Monte Carlo
• Langevin and Fokker-Planck equations
• Zwanzig-Mori formalism
• Fluctuating hydrodynamics
• Fluctuation-dissipation balance in continuum and discrete models
• Hybrid particle-continuum methods
• Quasi-continuum models for solid elasticity
• Discrete particle models of granular materials and jamming
  

The schedule for the lectures is flexible and will be created as we go along, depending on students' background and interests.

Reading Materials

There is no textbook for this course: We will mostly use review articles and sometimes chapters from books. Under the Lectures tab I will post links to the corresponding PDFs, which will either be available on the open web, or accessible via the NYU/Courant library. Depending on your background you may need to supplement / substitute the recommended readings.

Prerequisites

This is a special topics seminar and therefore a lot of background will be assumed. I will try to provide a quick review of key background material. However, the students should have some familiarity with statistical mechanics, numerical analysis, molecular dynamics, PDEs. and stochastic analysis. Start by reading sections 1-3 of "Statistical Mechanics of Coarse-Graining" by Pep Espanol (official published version on SpringerLink), and see if you quickly get lost. We will continue with the remaining sections later in the course.

There will be no actual computing in this class, however, I will spend considerable amount of time discussing how to actually use the coase-grained mathematical models in computational science, including some discussion of algorithms and numerical analysis.

Assignments and grading

This is a seminar course and the focus will be on learning new things. There will not be graded assignments, however, there will be regular readings and each student will present one paper / topic  of choice in class.

As a first assignment, please submit the answers to this questionnaire via email as soon as possible:

1. Your name, degree you are working on (if any) and class/year, and thesis advisor and topic if any.
2. Are you taking this course for credit?
   
3. List your previous academic degrees or relevant educational experience.
4. Explain in words (e.g., relevant courses, research, lack of exposure) your background in the following areas: (a) statistical mechanics, (b) computational modeling of materials, (c) numerical analysis, (d) PDEs, and (e) stochastic analysis.
   
5. Why did you choose this course, and which of the topics listed in the course description interest you most (in particular, do you know what subject you would like to present on in class)?
6. Are there any additional subjects you want to suggest?