- Applied Probablity and Network Science
Recent work on Network Science have focussed on social and mathematical questions on Tipping Fractions of Minority Opinion in Social Influencing: (1) Robustness wrt local rules in multi-agents models and smallness (5 - 10%) of tipping fractions, (2) Scalability of tipping points to large networks and different topologies including scale-free and small world, (3) Rigorous methods for calculating mean times to agreement (synchronization rates) and their fluctuations (Martingale estimates for variance of consensus times), (4) Diffusion (SDE) models for social influencing based on Dynkin formula and the relationship between SDE and semielliptic PDEs with Dirichlet BC. A highlight is the discovery of a 2D SDE system in the Naming game model which cannot be solved by traditional Feynman-Kac or Master eqns methods, and required the introduction of new discrete stochastic methods. Another highlight is the recent resolution of 2-Urns Models by Generating Function Methods which diagonalized large penta-diagonal stochastic transition matrices arising in several social opinion, genetics and ecological models, where the noise terms are dominant due to small populations.
 J. Xie, S. Sreenivasan, G. Korniss, W. Zhang, C. Lim, and B.K. Szymanski, "Social consensus through the influence of committed minorities", Physical Review E 84, 011130 (2011).
 W. Zhang, C. Lim, S. Sreenivasan, J. Xie, B.K. Szymanski, and G. Korniss, "Social Influencing and Associated Random Walk Models: Asymptotic Consensus Times on the Complete Graph", Chaos 21, 025115 (2011).
 W. Pickering and C. Lim, “Solution of the Voter Models by Spectral Analysis”, Physical Review E (Vol.91, No.1): DOI: 10.1103/PhysRevE.91.012812
 W. Pickering and Chjan Lim, ``Spectral solution of urn models for two particle systems”, submitted, arXiv:1503.03454, 2015
- Statistical physics and turbulence: Exactly-solvable models in 2D turbulence -- a long range spherical model for energy-enstrophy theories.
(I) Anomalous Expansion and Negative Specific Heat in quasi-2D trapped vortex filament bundles - this was discovered by Tim Andersen and Chjan Lim and applied to a large class of problems where wriggly vortex lines play dominant roles such as Electron Magneto-Hydrodynamics (EMH), London's theory of superconductivity and Plasma Physics. The original negative specific heat was discovered in gravo-thermal collapse of star clusters (cf. Lynden-Bell etal) where a sufficiently dense gravitational core transfers energy to a halo and increases in temperature (the halo gains energy but cools), leading to a run-away process. A similar process in vortex line bundles was found first by Path-Integral MC simulations and later confirmed by an elegant series of mean field and steepest descent calculations based squarely on applications of Kac's SPHERICAL CONSTRAINT (cf. preprints/papers by Andersen and Lim 2005 - ). Important scientific and technological consequences of these discoveries by the PI and co-workers will be discussed in a series of papers, talks and funding seminars, including applications to super-confinement of plasmas in EMH and other MHD systems considered to be significant for thermonuclear fusion.
(II) Orientation asymmetry from planetary spin is introduced naturally in a unified statistical mechanics for the Barotropic Vorticity Equation and the Shallow Water Equations on a rotating sphere and used to predict the phase transitions to super-rotating solid-body flows at high energy to enstrophy ratios, and the non-symmetrical phase transitions to antirotating solid-body flows when the planetary spin is large. This new statistical mechanics uses a canonical path-integral where the action is given by the Lagrangian L of the BVE model and microcanonical constraint on circulation and enstrophy. L = H + AM where H is the Hamiltonian that is conserved even over nontrivial topography and AM is the fluid's angular momentum that is not conserved except over trivial topography by the BVE. The analogy and connections are made clear in the following article which gives an exact non-mean field solution using Kac's spherical model method to the problem of the inverse cascade of energy to large scales while angular momentum is exchanged with a massive spinning sphere in a quasi-2d atmospheric flow. Several papers discuss further the significant results obtained by Xueru Ding and Chjan Lim using Monte-Carlo methods, including applications to Venusian and Titan super-rotation and the enigma of the absence of sub-rotating atmospheres amongst slowly-rotating terrestrial planets and to the cyclonic-anticyclonic debate around the large coherent spots on the Gas Giants - see Physica A 2007, Physics Fluids 2008.
- Computational Science and Vortex Dynamics:Monte Carlo simulations of vortex gas / lattice statistics --- computational results supporting inverse energy cascades and 2 - dim ideal turbulence on the plane and sphere via modern treatments of Kraichnan's energy-enstrophy theories.
- Vortex Dynamics:Relative equilibria, bifurcations, and stability
- Symmetric Dynamical Systems:Resersible Lyapunov Center Theorem, Reversible pitch-fork bifurcation, Applications to the Stokeslet Model