This 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 macroscopic systems originating in the collective behavior of their microscopic constituents. The methods used range from exact mathematical analysis to computer simulations. These approaches bridge the gap between rigorous results and applications.
Research topics employed by PI and collaborators are varied. Using mesoscopic free energy functionals allows investigation of periodic states, wetting in mixtures and droplet formation in supersaturated vapor. An ongoing effort continues on fluctuations and large deviations in nonequilibrium stationary states and partial currents. This includes cases where the hydrodynamic scaling is inadequate. Significant advance continue in fundamental studies such as establishing Fourier?s law of heat conduction in open systems with anharmonic interactions. In quantum statistical physics, the study of subsystems of large quantum systems leads to understanding of when these systems have density matrices given by canonical Gibbs measures. In some cases, such as the study of ionization of model quantum systems in time periodic fields, there are applications to laser induced transitions in atoms and molecules. Beyond physics, researchers also apply statistical mechanical methods to the mathematical study of epidemics taking into account correlations as well as saturation effects on networks. Extension of these techniques to models of population dynamics and ecology involves derivation of reaction-diffusion equations via scaling limits and these are planned investigations.
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.
NONTECHNICAL SUMMARY: This 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 range from classical physics to quantum physics and highly formal to applications of technological relevance and even dynamics of disease propagation.
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.
The aim of our research is a better understanding of the properties of macoscopic systems originating in the collective behavior of theor microscopic constituents. The methods used range from exact mathematical analysis to computer simulations. We strive to bridge the gap between rigorous results and applications. Below I've listed some specific results from the research supported by this NSF grant. We proved that the addition of an arbitrarily small random perturbation to a quantum spin system rounds a first-order phase transition in the conjugate order parameter in d ≤ 2 dimensions, or for cases involving the breaking of a continuous symmetry in d ≤ 4. this establishes rigorously for quantum systems the existence of the Miry-Ma phenomenon which for classical systems was proven by Aizenman and Wehr. We obtained the full phase diagram of the three species asymmetric ABC model on a one-dimensional lattice of N sites with closed (zero flux) boundaries. We proved that in the limit N → ∞ the scaled density profiles, i.e., the existence of a single phase, in all regions of the parameter space (of average densities and temperature) except at low temperature with all densities equal; in this case our results apply also to the system on the ring, and there extend previous results. Using an extended Lee-Yang theorem and correlationinequalties, we proved, for a class of ferromagnetic multispin interactions, that they will have phase transition if, and only if, the external field h = 0 (and the temperature is low enough). We also showed the absence of phase transitions for some nonferromagnetic interactions. Useful inequalities were shown to hold for a new class of multi-spin interactions. We derived new type inequalities fro random classical and quantum systems. These prove the monotone approach for the free energy to it thermodynamic limit and give information about the surface tension of such systems. We have generalized the use of entropy as a Lyapunov function for equations describing the time evolution of an isolated macroscopic system to situations where the microstates dynamics is described effectively as a stochastic (Markov) process, e.g. Hamiltonian systems interacting with thermal reservoirs at the boundaries and lattice systems with Glauber or Kawasaki evolutions. When the evolution of this systems macrostate at time t, Mt, is given by an autonomous equations then in analogy with what heppens in the isolated system, the large deviations function (LDF) of Mt with respect to the stationary measure will again be a Lyapunov function for this evolution. The LDF of the macrostate M may thus be considered as a relative free energy mathcal{F}(M) of an open system. Using analytic methods as well as extensive computer simulations we studied the transport of energy in a variety of microscopic model systems. We found that: a) For harmonic crytsals with random masses Fourier's law holds in three dimensions (3D) only when the atoms are pinned/ it never holds in 2D (or 1D) due to localization. When an (arbitrary small) amount of noise is added to the dynamics Fourier's law does hold (rigorously). This is similar to what we found from simulation (but nor proven) that a small amount of anharmonicity destroys localization in classical systems. b) Violation of Fourier's law for a deterministic nonmomentum conserving 1D model for which the Boltzmann equation (solved exactly) predicts its validity. Our research is highly interdisciplinary, bringing together physicists, mathematicians, chemists and those working in theoretical areas of the biological and social services. The expected applicationss are in material science, complex fluids and in biological systems. Our program also includes the organization of two conferences every year in which both core subjects and new developments in statistical mechanics are dicussed in a collegial atmosphere. Graduate students. postdocs and minority scientists are encouraged to present talks on theor work and interact with leaders in the field.
