Statistical mechanics of systems of unbounded spins

We develop the statistical mechanics of unboundedn-component spin systems on the latticeZ ^v interacting via potentials which are superstable and strongly tempered. We prove the existence and uniqueness of the infinite volume free energy density for a wide class of boundary conditions. The uniqueness of the equilibrium state (whose existence is established in general) is then proven for one component ferromagnetic spins whose free energy is differentiable with respect to the magnetic field.     

Time domain investigation on vibrational dephasing and spectral diffusion in CO-doped solid N2

Infrared picosecond accumulated photon echo experiments have been performed for the first time, using the Orsay Free Electron Laser, on the v = 0-->v = 1 transition of CO in solid nitrogen. The vibrational dephasing time is found to be exceptionally long ( T2>/=120 ns) at low temperature. The analysis of the observed spectral diffusion leads one to assume different energy transfer mechanisms depending on the CO concentration.     

Growth of Ordered Domains in a Computer Model Alloy with Lattice Misfit

We study via Monte Carlo simulations the influence of elastic interactions on the ordering and decomposition of a two-dimensional model binary alloy with antiferromagnetic nearest and ferromagnetic next nearest neighbor type interactions following a quench into the coexistence region. The elastic interaction leads to the development of a platelet morphology for the segregated ordered and disordered regions. A length scale characterizing the coarsening process follows a law of the type R=a+bt ^1/3 with the growth b decreasing with the amount of ordered phase; this appears to be due to the presence of anti-phase boundaries between neighboring domains ordered on different sublattices which are difficult to eliminate. The application of uniaxial external stress results in rafting of the domains. Many of the simulation results are in agreement with experimentally observed effects in nickel-base superalloys.     

Local Central Limit Theorem for Determinantal Point Processes

We prove a local central limit theorem (LCLT) for the number of points \(N(J)\) in a region \(J\) in \(\mathbb R^d\) specified by a determinantal point process with an Hermitian kernel. The only assumption is that the variance of \(N(J)\) tends to infinity as \(|J| \rightarrow \infty \) . This extends a previous result giving a weaker central limit theorem for these systems. Our result relies on the fact that the Lee–Yang zeros of the generating function for \(\{E(k;J)\}\) —the probabilities of there being exactly \(k\) points in \(J\) —all lie on the negative real \(z\) -axis. In particular, the result applies to the scaled bulk eigenvalue distribution for the Gaussian Unitary Ensemble (GUE) and that of the Ginibre ensemble. For the GUE we can also treat the properly scaled edge eigenvalue distribution. Using identities between gap probabilities, the LCLT can be extended to bulk eigenvalues of the Gaussian Symplectic Ensemble. A LCLT is also established for the probability density function of the \(k\) -th largest eigenvalue at the soft edge, and of the spacing between \(k\) -th neighbors in the bulk.     

Phase Transitions with Four-Spin Interactions

Using an extended Lee-Yang theorem and GKS correlation inequalities, we prove, for a class of ferromagnetic multi-spin interactions, that they will have a phase transition (and spontaneous magnetization) if, and only if, the external field h = 0 (and the temperature is low enough). We also show the absence of phase transitions for some nonferromagnetic interactions. The FKG inequalities are shown to hold for a larger class of multi-spin interactions.     

Approach to the Steady State in Kinetic Models with Thermal Reservoirs at Different Temperatures

We continue the investigation of kinetic models of a system in contact via stochastic interactions with several spatially homogeneous thermal reservoirs at different temperatures. Considering models different from those investigated in Carlen et al. (Braz J Probab Stat 29:372–386, 2015), we explicitly compute the unique spatially uniform non-equilibrium steady state (NESS) and prove that it is approached exponentially fast from any uniform initial state. This leaves open the question of whether there exist NESS that are not spatially uniform. Making a further simplification of our models, we then prove non-existence of such NESS and exponential approach to the unique spatially uniform NESS (with a computably boundable rate). The method of proof relies on refined Doeblin estimates and other probabilistic techniques, and is quite different form the analysis in Carlen et al. (Braz J Probab Stat 29:372–386, 2015) that was based on contraction mapping methods.