RHIC Physics and Kink/Vortex Dynamics      

Langevin solvers for the dynamics of nonlinear coherent structures are stochastic PDE solvers where the noise models thermal fluctuations. These solvers are also useful in understanding the dynamics of phase transitions and the role of nonlinear coherent structures in controlling equilibrium thermodynamics of field theories. Stochastic solvers require algorithms which are fundamentally different from the usual PDE solvers. We utilize a second-order stochastic Runge-Kutta scheme which has proven to be robust, accurate, and fast.

Our approach will focus on the study of effective field theories, such as the linear sigma and Gross-Neveu models, since at present (thermo)dynamical treatments of QCD at finite density are still in their infancy.

Effective theories such as the linear sigma model describe the lowest energy excitations of nuclear matter (viz. pions and sigma) which, in the spirit of the Landau theory of critical phenomena, control the long wavelength behavior of the whole system. In this regime (close to the critical temperature), one can therefore use an effective classical scalar theory in 3 spatial dimensions. Heavier degrees of freedom are included through the inclusion of stochastic noise and dissipation terms and the system as a whole can therefore be described by a set of Langevin equations (nonlinear stochastic PDEs) for scalar fields, with bulk quantities such as temperature and density varied either externally or effectively as a result of volume expansion.

In practice the scalar fields are placed on a cubic lattice, with appropriate boundary conditions. The fields are then evolved using stochastic update schemes (e.g. stochastic versions of RK and symplectic algorithms). We have several parallel implementations (f90+MPI) of these algorithms, allowing us to evolve simultaneously many realizations of the system or a single one in a very large computational volume.

An important point to which we are devoting particular effort is that these models can be made lattice spacing independent by the addition to the Hamiltonian of suitable temperature dependent counterterms. This procedure removes all (Rayleigh-Jeans) ultraviolet divergent perturbative diagrams at finite temperature resulting from the use of the classical thermal distribution implicit in the numerics and replaces them by their convergent quantum counterparts. The result is a renormalized effective theory that matches the behavior of the true underlying quantum theory in the ultraviolet (perturbatively) as well as in the infrared (fully nonperturbatively). The advantage of such a procedure is that fully dimensional quantities (in contrast to quantities such as critical exponents), such as the temperature and excitation frequencies and decay times, can now be extracted unambiguously from the numerics.

Other effective theories have the advantage of dealing directly with fermion fields (e.g. the Gross-Neveu model). We will treat these models dynamically in mean field approximations both in the spatially homogeneous case and in (the much more difficult) spatially inhomogeneous situations described elsewhere.

The nonequilibrium dynamics and nucleation/annihilation of nonlinear coherent structures remains an open problem in the study of dynamics of field theories. Perhaps the simplest model problem is the case of kinks in a Landau-Ginzburg theory in 1+1 dimensions. To attack this problem we have written a parallel Langevin solver (HPF and F90/MPI) which has already yielded several important results. The large size of problems that can be handled (millions of lattice sites) along with acceptable performance (a single run can take several hours on 128 PEs) has proven to be essential in obtaining the above results.

Aside from the Langevin dynamics we have also written new diagnostics codes that allow us to easily extract dynamical information such as the kink diffusion coefficient and nucleation rates. This has enabled us to settle some longstanding controversies in this field.

Two and three dimensional versions of these codes have been run previously. We have carried out theoretical convergence studies (which are strongly dimension dependent, now complete in the case of 1+1-dimensions) for these codes. The two/three dimensional codes need an understanding of convergence/UV divergence issues before they can attain production status. This is now under active investigation.

Our vortex dynamics codes are two and three dimensional Langevin solvers for complex scalar fields. Gauge fields will also be incorporated. This last step involves the use of a stochastic algorithm that maintains the gauge constraint. In addition we have written a new symplectic solver for microcanonical simulations of these systems.

The main thrust of production runs in FY01 will be the codes for investigating the physics of heavy-ion collisions at RHIC. Aside from a few test runs of the three-dimensional microcanonical solvers, we do not anticipate carrying out the research activity relating to kinks and vortices at NERSC in FY01.

Salman Habib / LANL / habib@lanl.gov / revised August 00