Selected Scientific Discoveries upto 2015

funded by ARO


Exact solutions of an energy-enstrophy theory (based on Kac's spherical model) in the statistical physics of quasi-2D flows, using the newly discovered equivalence of the canonical and microcanonical ensembles for long-range interactions on compact oriented manifolds (via Hodge Theory), leads to results that are applicable to Planetary Atmospheres in extra-solar systems such as "Super-rotators are more common than Sub-rotators in Nearly Barotropic Planetary Atmospheres", and to the enigmatic super-rotation of Venus and Titan:

C. C. Lim, "Phase Transition to Super-rotating Atmospheres in a Simple Planetary Model for a Non-Rotating Massive Planet - Exact Solution", Physical Review E 86 (6)(2012) click here to download from APS.

For details on the rotating case, see submitted, 2013.

Anomalous Expansion and Negative Specific Heat in quasi-2D trapped vortex filament bundles in an unbounded horizontal domain was discovered by Tim Andersen and Chjan Lim and applied to the confinement and stability of hot Tokamak plasmas; see for example:

T. Andersen and C. Lim, "Negative Specific Heat in a Quasi-2D Generalized Vorticity Model'', Phys. Rev. Lett. 99, 165001, Oct 2007.

Tipping Points of committed minority fraction in Social Opinion Dynamics mapped to the Saddle-Node and Pitchfork Bifurcations with sociological significance in the robust smallness of the tipping fraction of 10 percent; application of monotone dynamical systems to the social interactions of tribes:

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).

Extending Kac's 1947 generating function solution of the Ehrenfest Dog Flea model, Pickering and Lim solved exactly a large family of 2 Urns-2 Particles stochastic models by diagonalizing pentadiagonal markov transition matrix Pij; this family includes variants of the Voter model, Moran's genetic drift model and all discrete-time birth death processes solved by the Karlin-McGregor (1955-1962) correspondence between these processes (all of which have tridiagonal Pij) and the Stieltjes Moment Problem of Orthogonal systems of Polynomials. Our method works for stochastic Urn models that are not the standard random walks or birth-death processes with tridiagonal Pij - we can diagonalize a large class of pentadiagonal Pij:

Pickering and C. Lim, "Solutions of Urn Models", submitted 2015

Time-Reversible Dynamical System, see for example:

M. Krupa, C. Lim, and M. Golubitsky, "Time-reversibility and particle sedimentation'', SIAM J. Applied Math. 51(1), 49-72 (1991).

Other results 0n Network Science and Applied Probability:

[1] W. Zhang and C.C. Lim, "The Concentration and Stability of the Community Detecting Functions on Random Networks", Internet Mathematics 9 (4), 360-383, 2013. click here to download

[2] W. Zhang, C.C. Lim and B. Szymanski, "Analytic Treatment of Tipping Points for Social Consensus in Large Random Networks", Phys Rev E 86 (6), 061134, 2012

[3] W. Zhang, Korniss, Szymanki, C.C. Lim, "Spatial Propagation of Opinions: Naming games on random geographic graphs", Sci Reports 2014, Scientific Reports 4, 5568 (2014) and doi: 10.1038/srep05568 Sci Reports

[4] W. Pickering, C.C. Lim, "Solution of Voter models by Spectral Analysis", click here to download

Physical Review E (Vol.91, No.1): DOI: 10.1103/PhysRevE.91.012812

[4a] C.C. Lim, W. Pickering, "Information sharing of strong neitrals in social forums - exact soln of 3 state voter model", arXiv preprint arXiv:1411.0530, 2014 click here to download

[4b] W. Pickering, B. Szymanski, C. Lim, "Analysis of the high dimensional naming game with committed minorities", arXiv preprint arXiv:1512.03390, 2015 click here to download

[5] C.C. Lim and W. Zhang, "Monotonicity of Social Opinion Dynamics on Large Networks", submitted July 31 2013, for the role of monotonicity in math. sociology where pure diffusion is secondary to drift.

[6] A. Thompson, B. Szymanski, C.C. Lim, "Propensity and Stickiness in the Naming Games", Phys Rev E 90, 042809, 2014 on robustness of tipping points in two para. family of social opinion models.

New Scientific Discoveries 2008-2011

funded by ARO


My recent work on Network Science in collaboration with SCNARC at RPI funded by the ARL 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.

[1] 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).

[2] 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).

[3] C. Lim, and W. Zhang, "Noisy Naming Games, Partial Synchonization and Coarse Graining", IEEE NSW Proc. West Point, NY, 25-29, DOI:10.1109/NSW.2011.6004654 (2011).

[4] Weituo Zhang, Chjan C. Lim, Noise in Naming Games, partial synchronization and community detection in social networks arXiv:1008.4115, 2010.

New Scientific Discoveries 2004-2008

funded by ARO and ARL


(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.