# The Andersen-Parrinello-Rahman Method Revised into a Scale Bridging Device

**Oct 28, 2015 at 11:00 AM - 1:00 PM**

202/204 Physics

**NOTE: Wednesday Seminar**

Around 35 years ago, Andersen, Parrinello and Rahman had the idea of letting the molecular-dynamics (MD) cell vary its volume (Andersen) and shape (Parrinello & Rahman) with time. The particle velocity was hence decomposed into the sum of a spatially tidy entrainment velocity, parameterized by the cell deformation rate, and a disordered streaming velocity. In order to govern the collective degrees of freedom associated with the cell, the Lagrangian functional was extended in a smart ad-hoc way. Whether the extended Lagrangian could be derived from “first principles” was a question left for further study, as Parrinello & Rahman themselves stated. In reality, since MD practitioners always considered the APR method just as an expedient trick for generating the desired particle statistics, this foundational issue remained latent until recently, when somebody with a background in continuum mechanics (CM) started looking at the APR method from an antipodal point of view. Here the idea is to bring the deforming computational cell to the fore, identifying it with an element - i.e., an infinitesimally small piece - of a continuous medium. Seen in this perspective, the appealing feature of the APR method is that it establishes a natural, explicit coupling between molecular and continuum DOFs. In conventional applications, dynamical quantities work-conjugate to these latter DOFs - namely, stress - are regarded as prescribed: as a matter of fact, Andersen’s original motivation was that of devising a barostat. In the novel multiscale implementation of the method, the stress is a priori unknown: to determine it, CM PDEs have to be solved concurrently with MD ODEs.

Roughly speaking, the solution strategy goes as follows. Imagine considering a material aggregate as either a molecular system or a continuous medium, and wishing to relate the two representations. Assume that the fields entering the continuum description, such as strain and stress, are adequately sampled on an array of positions (think of Gauss points in a finite element model), whose typical spacing H is enormously larger than the average intermolecular distance d. Associate with each of these macroscopic sampling positions an APR cell, whose reference size h is large enough with respect to d in order to allow for a decent sampling of the microscopic molecular states, and still much smaller than H: H >> h >> d (in practice, it is also essential to take full advantage of the fact that, typically, H/h >> h/d). Now, let the molecules in each cell interact directly with each other (and with their h-neighboring images), while being indirectly affected by those in the H-neighboring cells via the collective degrees of freedom of the deforming APR cell, governed by the force balance and compatibility equations of CM (sampled at the H scale). In turn, the elementwise stress-strain relation characterizing the response of the medium arises as an emergent property of MD (computed on the h scale).

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Contact Information:

Penny Davis, administration questions

shdavis@syr.edu

315-443-5960

**Antonio Di Carlo** Dipartimento di Matematica e Fisica UniversitÃ Roma Tre

Host: Matteo Paoluzzi