## Project Details

### Description

TECHNICAL SUMMARYThe Division of Materials Research, the Division of Mathematical Sciences, and the Physics Division contribute funds to this award. This award supports theoretical research and education aimed at advancing understanding of the properties of macroscopic systems originating in the collective behavior of their microscopic constituents. The methods used range from exact mathematical analysis to computer simulation. They provide a bridge between rigorous results and applications, for both equilibrium and nonequilibrium phenomena.The research is highly interdisciplinary, bringing together physicists, mathematicians, chemists and those working in theoretical areas of the biological and social sciences. Topics of study include:1) The effects of randomness in quantum systems will be elucidated. This will extend the recent proof that the addition of randomness to a quantum system rounds a first-order phase transition in the conjugate order parameter in 2 dimensions or less, or for cases involving the breaking of a continuous symmetry in 4 dimensions or less. New inequalities for random systems will be explored.2) The phase diagram of general systems with spatially asymmetric long range interactions will be investigated, extending the exact solution of the three species ABC model in 1D.3) Results on the phase diagram of lattice systems with multispin interactions will be extended.4) Pattern formation such as stripes, in equilibrium and nonequilibrium systems, due to short range attractions and long range repulsions will be elucidated.5) The evolution of the macrostate of an open system given by an autonomous equation willbe investigated. In analogy to the Boltzmann entropy in an isolated system, the large deviation function, with respect to the stationary measure of the nonequilibrium stationary state, is a Lyapunov function for this evolution. This yields new Lyapunov functions for nonlinear diffusion equations. The deviation function, considered as a relative free energy of an open system, will be explored from both a microscopic and macroscopic point of view.6) A rigorous determination of transport coefficients is an important objective of this research project. Domains of validity of Fourier's law in various systems, for example weakly anharmonic crystals with random masses, will be elucidated. When noise is added to the dynamics Fourier's law holds rigorously, but will a small amount of anharmonicity destroy phonon localization? Violations of Fourier's law for momentum conserving models in 1D and 2D will be investigated.7) Criteria for 'typicality' in realistic systems will be investigated: for a 'typical' large isolated quantum system every initial wave function from an energy shell evolves in such a way that, for most time it is macroscopically equivalent to the micro-canonical density matrix.This award supports the PI's efforts to organize two conferences every year in which both core subjects and new developments in statistical mechanics are discussed in a collegial atmosphere. Graduate students, postdocs and minority scientists are encouraged to present talks on their work and interact with leaders in the field. They also serve as a clearing house for positions and often lead to new collaborations. In addition to many invited lectures at scientific conferences the PI has given public lectures and has written an article for Scholarpedia on Time's Arrow.NONTECHNICAL SUMMARYThis award supports theoretical research and education in a variety of subfields of statistical mechanics. This work is jointly supported by the Division of Materials Research and the Division of Mathematical Sciences. The central theme of the effort is a better understanding of the properties of material systems originating in the collective behavior of their elementary atomic constituents. The methods used range from exact mathematical analysis to computer simulations. These approaches bridge the gap between rigorous results and applications. Topics of study range from classical physics to quantum physics, from highly formal to applications of technological relevance and even to dynamics of disease propagation. The project aims to advance the theory of systems that are far from the balance of equilibrium with an impact on biological systems, biomaterials, and materials more generally. The research activities are highly interdisciplinary, bringing together physicists, mathematicians, chemists and those working in theoretical areas of the biological and social sciences. The expected applications are in material science, complex fluids and in biological systems. The project also includes the organization of two conferences every year in which both core subjects and new developments in statistical mechanics are discussed in a collegial atmosphere. Graduate students, postdocs and minority scientists are involved and present talks on their work and interact with established researchers in the field. The conferences also serve as an opportunity for professional networking and can lead to new collaborations.

Status | Finished |
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Effective start/end date | 10/1/11 → 9/30/14 |

### Funding

- National Science Foundation (National Science Foundation (NSF))

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