Kawasaki dynamics with infinite-range spin exchange

The essential feature of the Kawasaki model is the conserved order parameter, which places the model in class B of the Halperin, Hoheberg, and Ma classification. We have studied the energy relaxation of this model in one and two dimensions with the added feature that spin exchange may take place between any pair of sites within the system. Our results for the dynamic exponentz are indistinguishable from those for class A models, in which the order parameter is not conserved.     

Presentation functions,fixed points,and a theory of scaling function dynamics

Presentation functions provide the time-ordered points of the forward dynamics of a system as successive inverse images. They generally determine objects constructed on trees, regular or otherwise, and immediately determine a functional form of the transfer matrix of these systems. Presentation functions for regular binary trees determine the associated forward dynamics to be that of a period doubling fixed point. They are generally parametrized by the trajectory scaling function of the dynamics in a natural way. The requirement that the forward dynamics be smooth with a critical point determines a complete set of equations whose solution is the scaling function. These equations are compatible with a dynamics in the space of scalings which is conjectured, with numerical and intuitive support, to possess its solution as a unique, globally attracting fixed point. It is argued that such dynamics is to be sought as a program for the solution of chaotic dynamics. In the course of the exposition new information pertaining to universal mode locking is presented.     

Spatially homogeneous random evolutions

Spatially homogeneous random evolutions arise in the study of the growth of a population in a spatially homogeneous random environment. The random evolution is obtained as the solution of a bilinear stochastic evolution equation. The main results are concerned with the asymptotic behavior of the solution for large times. In particular, conditions for the existence of a stationary random field are established. Furthermore space-time renormalization limit theorems are obtained which lead to either Gaussian or non-Gaussian generalized processes depending on the case under consideration.     

End‐to‐End Distances in Four‐Junction Polymers

The various end‐to‐end distances of four‐junction polymers are investigated. The sizes of the different kinds of equal length branches and the backbone of two different polymers, with either nine or eleven branches, are estimated by means of both renormalization‐group and MC calculations. The comparisons of first‐order ε = 4 − d predictions with the MC results are satisfactory. The same trends are present in both techniques. The excluded‐volume interactions from additional branches further expand the various parts of the chains so that internal branches are larger than external ones. The branch ratios in the eleven‐branch case are expanded even more than the corresponding ratios of the nine‐branch polymer.

     

Renormalized Polyakov loop in the fixed scale approach

I compute the deconfinement order parameter for the SU(2)$\mathit{SU}(2)$ lattice gauge theory, the Polyakov loop, using the fixed scale approach for two different scales and show how one can obtain a physical, renormalized, order parameter. The generalization to other gauge theories, including quenched or full QCD, is straightforward.     

Submanifolds in Cauchy Riemann Geometry

In this paper I would like to make a report on the results about hypersurfaces in the Heisenberg group and invariant curves and surfaces in CR geometry. The results are contained in the papers [8, 9, 16] and [14]. Besides, I would also report on the results about the strong maximum principle for a class of mean curvature type operators in [10].     

Train tracks with C 1+ self-renormalizable structures

We prove a one-to-one correspondence between C ¹⁺  conjugacy classes of diffeomorphisms with hyperbolic sets contained in surfaces and stable and unstable pairs of one-dimensional C ¹⁺  self-renormalizable structures.     

Fluctuations in the quark-meson model for QCD with isospin chemical potential

We study the two-flavor quark-meson (QM) model with the functional renormalization group (FRG) to describe the effects of collective mesonic fluctuations on the phase diagram of QCD at finite baryon and   isospin chemical potentials, _μB${\mu }_B$ and _μI${\mu }_I$. With only isospin chemical potential there is a precise equivalence between the competing dynamics of chiral versus pion condensation and that of collective mesonic and baryonic fluctuations in the quark-meson-diquark model for two-color QCD at finite baryon chemical potential. Here, finite _μB=3μ${\mu }_B=3\mu$ introduces an additional dimension to the phase diagram as compared to two-color QCD, however. At zero temperature, the (_μI,μ${\mu }_I,\mu$) plane of this phase diagram is strongly constrained by the “Silver Blaze problem.” In particular, the onset of pion condensation must occur at _μI=_mπ/2${\mu }_I=m_{\pi }/2$, independent of μ   as long as μ+_μI$\mu +{\mu }_I$ stays below the constituent quark mass of the QM model or the liquid-gas transition line of nuclear matter in QCD. In order to maintain this relation beyond mean field it is crucial to compute the pion mass from its timelike correlator with the FRG in a consistent way.     

Renormalization group second‐order approximation for singularly perturbed nonlinear ordinary differential equations

We consider a 2 time scale nonlinear system of ordinary differential equations. The small parameter of the system is the ratio ϵ of the time scales. We search for an approximation involving only the slow time unknowns and valid uniformly for all times at order O(ϵ²). A classical approach to study these problems is Tikhonov's singular perturbation theorem. We develop an approach leading to a higher order approximation using the renormalization group (RG) method. We apply it in 2 steps. In the first step, we show that the RG method allows for approximation of the fast time variables by their RG expansion taken at the slow time unknowns. Next, we study the slow time equations, where the fast time unknowns are replaced by their RG expansion. This allows to rigorously show the second order uniform error estimate. Our result is a higher order extension of Hoppensteadt's work on the Tikhonov singular perturbation theorem for infinite times. The proposed procedure is suitable for problems from applications, and it is computationally less demanding than the classical Vasil'eva‐O'Malley expansion. We apply the developed method to a mathematical model of stem cell dynamics.