Welcome to the web site of Chjan C. Lim, Professor
Department Links	RPI Mathematics Department Chjan Lim's CV word doc Fellow Chappaquarian, Bill Clinton's Best Wishes in neighborhood bookstore, recorded on Weinberg's Volume 2, March 2001 Network Science at ARO - applications of recent results on spectra of Laplacian of weighted graphs, Perron-Frobenius estimates of spectral gaps and random perturbations to clustering in social-cognitive networks, to coalition and community-formation, and to emergence of trust in competitive agent-based models of conflict and cooperation via Mean-variance optimization methods. Errata For World Scientific Monographs (December 2008) TextBook by Chjan Lim, Xueru Ding and Joseph Nebus Vortex dynamics, Statistical Mechanics and Planetary atmospheres To order Springer-Verlag Monographs on Math TextBook by Chjan Lim and Joseph Nebus Vorticity, Statistical Mechanics and Monte Carlo Simulations To order Financial Math and Simulations Recent Graduate Alumni of My Research Programme (in reverse chrono order) · Dr. Xueru Ding, PhD 2008 currently postdoc at Florida Inst Tech · Dr. Timothy Andersen, PhD 2007 currently senior scientist at Wagner Assoc · Dr. Joseph Nebus, PhD 2002 currently scientific writer and programmer, Vital Comp Services Recent MSc Alumni and Co-authors Rajinder S. Mavi, currently a PhD student at UC Irvine Math Zhu Da, currently a PhD student at UChicago Geophysics Sean's Piano Chopin's Fantasie Impromptu with Sean on Piano, m4v go NEW SCIENTIFIC DISCOVERIES funded by ARO and DOE 2004 - 2008. (I) A Unified Statistical mechanics Theory of the Formation of Large-scale Coherent Structures in Planetary Atmospheres has been constructed recently motivated by exciting new discoveries on Phase Transitions in spin-lattice models for planetary atmospheres made by Monte-Carlo simulations performed at RPI - Math by Xueru Ding and Chjan Lim. Based on a Gibbs ensemble that is canonical in the Lagrangian for the Shallow Water Equations on a rotating sphere and microcanonical constraints on circulations and enstrophies, we obtained a statistical mechanics formulation of the Coherent Spots phenomena on the Gas Giants which distinguishes between anticyclonic and cyclonic-dominated regimes, thus providing a possible explanation of the enigmatic observations that 90% of Jovian spots are anticylconic. In particular, the coherent cyclonic (anticyclonic) spots were stable only for very high (low) total energy and large planetary spins and we observed that the formation of spots is related to first-order phase transitions. Indeed, there is a sharp and large increase in total mechanical energy at the formation of cyclonic spots and a correspondingly sharp decrease in mechanical energy at the formation of anticyclonic spots. The key idea behind this ability to explain the anticyclonic – cyclonic dichotomy is the Orientation Asymmetry in the Lagrangian due to large planetary spin. A similar effect works in the case of the Barotropic Vorticity Model which is a non-divergent approximation that has been used by us to explain the absence of sub-rotation in slowly-rotating terrestrial planets. (II) Classical Einstein-de Haas effect - the original effect displayed by a twisting ferromagnetic rod in an EM field continues to play a key role in mesoscopic or nanosize quantum systems where tunneling and thermal overbarrier processes compete and polarization of random electron spins into net macroscopic angular momentum is expressed by a torque on the substrate's elastic matrix. A related and completely classical Einstein-de Haas Effect was first observed in MC simulations performed by the PI and co-workers on small LINUX clusters at the National U. of Singapore and RPI. Rigorous analysis and exact solutions of this family of classical 2D problems show that the global angular momentum in the classical systems arise from a condensation of random spins into a symmetry-breaking directed macroscopic SPIN as the kinetic energy (or effective T) of the system is lowered. A related variant of this effect was found when the systems have a distinguished axis of symmetry such as in a rotating frame. The Bohr-Van Leeuwen theorem which normally blocks magnetization of a classical EM gas does not hold for lattice systems with finite spins. The Mermin-Hohenberg-Wagner theorem which blocks phase transitions in 2D classical lattice systems with continuous symmetry group does not hold here due to the discrete Z^2 symmetry of the continuous real-valued local spins. DOE Research Accomplishment Highlights July 2007 pdf format (III) 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. Negative specific heat in quasi-2D generalized vorticity model, accepted by PRL pdf format Preprints on Mathematics and Physics of Spherical constraints and Mean field models for macroscopic flows with complex boundaries and quasi - 2D turbulence - recent experiments select most suitable statistical equilibrium models. 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. br> Text of Plenary Talk IUTAM Moscow 2006 pdf format Exact solution for the partition function of a coupled barotropic flow - rotating sphere system under a microcanonical constraint on relative enstrophy in which angular momentum of the fluid is the key order parameter indicating Bose-Einstein condensation to self-organized super- and sub- rotating flows from random initial states pdf format of Como Proceedings Monte-Carlo results for spin-lattice models of energy-enstrophy theory with microcanonical / spherical constraint on relative enstrophy showing clearly the low entropy of the free energy extremizers - super and sub-rotating states Positive critical temperature in barotropic fluid coupled to massive rotating sphere Ding pdf format Mean field derivation of a fixed point result for the statistical equilibrium of barotropic flows coupled to a massive rotating sphere emphasizing the fact that enstrophy is only implicitly constrained by an inequality, shows by single-step block spin renormalization that a discrete-spin Ising lattice model is in the same universality class as the continuous coupled barotropic vorticity model with Raj cond-mat/0611430 accepted by Physica A Variational energy-enstrophy-circulation theory for barotropic flow coupled to massive rotating sphere - companion to stat mech papers shows contrained kinetic energy extremizers are super and sub-rotating flows - complements the discovery (proved in the spherical model exact soln) that the free energy at small-absolute-valued temperatures is well approximated by internal energy because the most probable states at these T are minimal entropy solid-body flows, JMP june 07 pdf format Simple mean field theory for transitions in barotropic flow coupled to massive rotating sphere DCDS A 07 pdf format Exact mean field predictions of vortex filaments containment radius via quantum theoretic form of the spherical constraint and comparisons with path-integral Monte Carlo results accepted 2006 pdf format Dr. Tim Andersen is now a senior scientist at Wagner Associates. For more information on Tim Andersen's work on Statistical equilibrium of vortex filaments - PIMC results and free energy based on QFT form of Spherical constraint go to Tim's page Dr. Xueru Ding is now a postdoc working on computational material science at Florida Inst. Tech. For more information on Xueru Ding's projects on Statistical equilibrium of barotropic and shallow water flows coupled to massive rotating spheres go to Xueru's page Talks on Mathematics and Physics of Spherical constraints and Mean field models for macroscopic flows with complex boundaries and quasi - 2D / geophysical turbulence of coupled fluid - massive rotating sphere systems Powerpoint for Institute Seminar Math Indiana U and CCAM Seminar Math Purdue U April 9 and 11 2008, Powerpoint for Plenary Talk IUTAM, Steklov Institute, Moscow, August 24-31 2006 IUTAM Symposium 'Hamiltonian dynamics, Vortex structures, Turbulence' website AMS meeting New Orleans Jan 5-7, 2007, Special Session on Analysis and Numerics of Geophysical flows Phase transitions in vortex filaments and geophysical flows, Yale University Mathematics Department Analysis Seminar April 17 2006 Exact solution and Monte-Carlo simulations of a energy-enstrophy theory for geo/astro - physical flows - Venus Super-rotating atmosphere at Collective Phenomena in Macroscopic Systems, Lake Como, Italy December 4-6 2006 Summary pdf format Two dimensional turbulence, Lorentz Center, Netherlands, March 19-23 2007 ICIAM Zurich, July 07, Special session on Recent advances in Vortex Dynamics, organized by Blackmore with help from Bontoux and Lim
Vortex Dynamics, Variational Problems and Turbulence Research	Vortex Dynamics, Computational Mathematics and Variational Problems Sponsors: ARO and DOE 2004 - 2008 for a combined total of $540,000.00
Vortex Statistics, Computational Mathematics and Variational Problems People Chjan C. Lim, Professor at RPI, PhD Brown U. Applied Math 1987, BSE Princeton 1982. Joseph F. Nebus, PhD has a book with me on Vorticity, Stat Mech and Simulations for Springer-Verlag - order online at Springer Syed M. Assad, Physics Department at NUS works on quantum computation and cryptography · Statistical equilibrium of baroclinic vortices in a rotating two-layer system, appeared GAFD 100(6) 1-22, 2006 pdf format Tim Andersen, Graduate Student at RPI works on reverse applications of quantum path-integral Monte Carlo Methods to New Vortex Matter - tangled vortex lines in BEC · Exact mean field predictions of vortex filaments containment radius via quantum theoretic form of the spherical constraint and comparisons with path-integral Monte Carlo results submitted 2006 pdf format · Poster for Tim's and Lim's presentation To ARO: pdf format Ding Xueru Graduate Student at RPI works on the statistical mechanics of barotropic and shallow water flows on rotating spheres · Poster for Xueru's presentation at AMS conference 2006 Atlanta pdf format paper pdf format · numerical evidence for positive critical temperature in barotropic vortex dynamics pdf format · Poster for M.Sc student Raj Mavi - presentation at AMS conference 2006 Atlanta ppt format AMS proc. pdf format submitted paper dvi format · GAMM 2006 Berlin paper: pdf format Research Notes · Application 1 of EM - Vorticity - Elasticity Trinity to barotropic flows dvi format abstract dvi format Conferences and Announcements of New Results Plenary talk at IUTAM, Steklov Institute, Moscow, Russia, Aug 26 2006: Stability and Phase transitions to super-rotation in coupled barotropic vortex dynamics on a rotating sphere --IUTAM Symposium 'Hamiltonian dynamics, Vortex structures, Turbulence' proc paper pdf format talk ppt format GAMM symposium on Vortex Dynamics, Berlin, Germany, March 2006: Phase transitions in new vortex matter in superfluid He4, superconductors, BEC in magnetic and laser traps and ideal fluids, a new form of chaos that arise from fluctuations along vortex lines, and vortex tangles (Path Integral Monte Carlo simulations of Tim Andersen provide crucial numerical evidence of these new forms of phase transitions and chaos). Yale Math Department Analysis Seminar April 17 2006: Phase transitions in new vortex matter and in the Shallow Water Equations (Joint work with Tim Andersen and Xueru Ding). --Yale Math Department Analysis Seminar Conference Paper: 4th Theoretical Fluid Mech., AIAA Toronto June 2005 invited hour talk on Variational Theory for Super-rotation · Barotropic Vortex Dynamics and Statistics - application to Super-rotation Preprint 2005 dvi format Conferences: AIAA Toronto June 2005 invited hour talk on Vortex Dynamics and Statistics · Recent Advances in 2D and 2.5D Vortex statistics and Dynamics Preprint 2005 pdf format Conferences: 3rd MIT Conference, Cambridge June 2005 talks on Heton model MC simulations Math Sciences Colloquium Friday April 29th 2005: David Sattinger, Yale University. · Tsunamis and Barge Canals - article for David Sattinger's talk Friday April 29 th dvi format
Vortex Dynamics, Variational Problems and Turbulence Research	Research Grants Currently Funded by the DOE and ARO Previously Funded by National U. of Singapore under grants R-151-000-015-112 R-151-000-024-112
Vortex Statistics, Computational Math and Variational Problems	Introduction Statistical Physics, 2D and 3D Vortex Dynamics (of round Jets, super-rotating planetary flows, etc) Damodaran-Hashimoto-Klein-Majda-Lions-Ting Model of nearly parallel vortex filaments and a statistical ensemble are used to confirm (via Path-Integral Monte-Carlo simulations in joint work with grad student Tim Andersen) the close to solid-body rotation flow in the inner core of axisymmetric jets (as in Prandtl's work), the self-confinement in ground state of cylindrical electron plasmas, the Feynman-Onsager theory of quantized vortex lines in rotating superfluid He4. Exactly-solvable models in 2D turbulence -- a long range spherical model for energy-enstrophy theories on the plane and sphere based on a Ising model like spin-lattice hamiltonian - correct exact solns available. Extended this stat mech spin-lattice hamiltonian to geophysical (nondivergent and divergent) flows coupled to a massive rotating sphere where the effects of planetary rotation appear as a non-uniform external "magnetic" field - barotropic and shallow water models. The minimization of this free energy done via Monte Carlo simulations by Ding Xueru, gives a minimalist theory for both super-rotating states at high energy and sub-rotating flows at lower energy and low relative enstrophy The stat mech simulation results agree with my recent variational theory for super-rotation in the barotropic flows coupled to a rotating sphere and with a simple mean field theory - J Math Phys 2007 issue.
Vortex Dynamics, Computational Math, Variational Problems	Movies and Plots Spherical Models Three modes seen in the energy-enstrophy model at different temperatures can be easily simulated. Data sheets for those modes are available. The modes represent: Negative temperatures, in which the system organizes itself into hills of positive and negative vorticity Checkerboard patterns, in which an antiferromagnetic pattern arises, as sites try to align their vorticity opposite their neighbors Banded regions, in which a row of positive vorticity lies next to a row of negative vorticity. Monte Carlo Results for energy-enstrophy theory on the Sphere An animated movie of superrotation from a random initial state at negative temperatures is available. The movie shows the evolution of absolute vorticity into a solid body rotation pattern with red colors representing positive vorticities and blue colors representing negative vorticities. Superrotation of the vorticity can be observed in downloadable plots. Compared this to theoretical calculations for the superrotation in the atmosphere of Venus as submitted for publication. The alternate mode of wavy banded activity can also be found by experiments. An animated movie of wavy band formation, seen when temperature is small but positive, can be downloaded.
Laboratory for Modern Applied and Computational Math	2D Simulation Applets Ising Model
Laboratory for Applied and Computational Math	3D Simulation Applets Monte Carlo Simulations of 3D Turbulence (Under construction.)
Vortex Dynamics, Variational Problems and Computational Science	Invited Talks International Conference on Dynamical Systems Beijing, China, June 18 - 22, 2001 Title: Logarithmic Interations, 2-D Turbulence, and Random Matrices
Vortex Dynamics, Variational Problems and Turbulence Research	References Books
Computational Vortex Dynamics	Monte Carlo Approaches to Vortex Dynamics Current working results, in progress, joint research with Joseph Nebus.
Recent Publications	Variational problems, Statistical Physics and computational Vortex dynamics Monte Carlo simulation of Baroclinic vortices in the Heton Model - low Froude number case, (with Assad), Geophysical and Astrophy. Fluid Dynamics, 100 (6), 1-22, 2006 pdf format Free energy Extremal in a simple Mean Field Theory of a coupled Barotropic flow - Rotating Sphere system pdf format Energy extremals and Nonlinear stability in an Energy - relative enstrophy theory of a coupled Barotropic flow - Rotating Sphere system accepted by J. Math Phys to appear June 07 pdf format Energy maximizers, Negative Temp, and Robust symmetry breaking on non-rotating sphere SIAM J Applied Math 2005 appeared ps format Monte Carlo simulation of Baroclinic Vortices in the Heton Gas submitted 2004, appeared in the Third MIT Conf. Proc 2005 PDF format The Spherical Model of Logarithmic potentials - MC simulations appeared Physics Fluids Oct 2004 PDF format Self-containment radius of the single species vortex gas / electron plasma submitted for publication 2003, appeared in Reg Chaotic Dyn 2005 PDF format Statistical equilibrium of the Coulomb / Vortex gas in the unbounded plane DCDS B 5(1) 1-14, 2005 PDF format Coherent structures in an energy-enstrophy theory for axisymmetric flows Physics of Fluids 15, 478 - 487 (2003) xdvi format | Postscript format | PDF format Monte-Carlo and Polyhedron-based Simulations I: Extremal States of the Logarithmic N-Body Problem on a Sphere Appeared, Discrete and Continuous Dynamical Systems B xdvi format | Postscript format | PDF format Monte Carlo simulations of the spherical model for 2D Euler turbulence on the infinite plane unpublished xdvi format | Postscript format | PDF format A Monte Carlo Algorithm for Free and Coaxial Ring Extremal States of the Vortex N-Body Problem on a Sphere Appeared Physica A xdvi format | Postscript format | PDF format Nonexistence of Lyapunov Functions and the Instability of the von Karman Vortex Street Physics of Fluids A 5(9), 2229-2223, 1993 xdvi format | Postscript format Mean Field Theory and Coherent Structures for Vortex Dynamics on the Plane Physics of Fluids, 1999 pages 1201-1207 Postscript format Relative Equilibria of Symmetric n-body Problems on the Sphere: Inverse and Direct Results CPAM 1998 pages 341-371 xdvi format | Postscript format A Long Range Spherical Model and Exact Solutions for an Energy-Enstrophy Theory for the Euler Equations Phys. Fluids July 2001 pages 1961-1973 xdvi format | Postscript format | PDF format Point Vortex Dynamics for a Coupled Surface / Interior QG and Propagating heton Clusters in Models for Ocean Convection (with A.J. Majda) Appeared Geophysical and Astrophys. Fluid Dynamics, August 2001 xdvi format | Postscript format Equilibrium Energy Density Spectrum for Two-Dimensional Flows in the Zero Viscosity Limit and Random Matrices xdvi format | Postscript format Referenced figures: Figure 1 | Figure 2a | Figure 2b | Figure 3 Symmetric Dynamics Stability of equilibria for a class of time-reversible equivariant vector fields SIAM J. Math Anal. 24(4), 1009-1029, 1993 xdvi format | Postscript format Stability of normal modes and subharmonic bifurcations in the 3-body Stokeslet problem J. Diff. Eqn. 121(2), 384-405, 1995 xdvi format | Postscript format Time-reversible and Equivariant Pitchfork Bifurcations Proc. Warwick MRC Workshop on Reversible Problems, December 1996, U. of Warwick, UK, pages 117-121 xdvi format | Postscript format Relative Equilibria of symmetric n-body Problems on the Sphere: Inverse and Direct Results CPAM 1997 xdvi format | Postscript format Relative equilibria of Point Vortices on a Sphere (with M. Roberts and J. Montaldi) Physica D 148 (2001) pages 97 - 135 xdvi format | PDF format Discrete Math On Noneven Digraphs and Symplectic Pairs Bull. Malaysian Math. Soc., December 1995 Postscript format Full-Sign Invertibility and Symplectic Matrices Linear Algebra and its Applic. 1995, pages 97-110 xdvi format | Postscript format Kastelyn's Theorem and Symmetric Noneven Digraphs Proc. Southeastern Comf. on Comb., Graph Theory and Computing, Baton Rouge, 1996, pages 231-242 xdvi format | Postscript format New Proofs of the Uniqueness of Noneven Digraphs with the Most Number M(n) of Arcs and M(n) - 1 Arcs Proc. Southeastern Conf. on Comb., Graph Theory and Computing, Boca Raton, 1997, pages 129-142 xdvi format | Postscript format New Proofs of the Uniqueness of Extremal Noneven Digraphs Appeared Bull. Malaysian Math. Soc., 2001. xdvi format | Postscript format
Invited Talks	International Conference on Dynamical Systems Beijing, China, June 18 - 22, 2001 Title: Logarithmic Interations, 2-D Turbulence, and Random Matrices Program on Physics of Hydrodynamic Turbulence, Miniworkshop on Mathematics and Fluids Santa Barbara, California, USA, June 2 - 3, 2000 Title: Exact solution of a three constraints equilibrium statistical mechanics model for 2-D turbulence RealAudio streaming media recording of the talk is available; the full talk may also be downloaded if desired.
Research interests	Statistical physics and turbulence Exactly-solvable models in 2D turbulence -- a long range spherical model for energy-enstrophy theories. 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 Combinatorial Matrix Theory
Graduate Students	Ph.D.'s Supervised I-Heng McComb, Reversible and Equivariant Vector Fields, 1993 David Schmidt, Non-Even Digraphs, Sign-Invertibility and Symplectic Pairs, 1995 Greg van Patten, Non-Even Digraphs and Associated Sign-Nonsingular Matrices, 2001 Joseph Nebus, Monte Carlo Simulations, Statistical Mechanics, and Ground States of the Logarithmic Potential, 2003 M.Sc. Supervised Carlos Salazar, Planar Extensions and Noneven Digraphs, 1998 Frank Turner, Uniqueness of Extremal digraphs, 1997 Current Students Tim Andersen, Path-Integral Monte Carlo Methods Ding, Xueru, Analysis and Simulations of Geophysical Flows on Rotating spheres
Links	Laboratory for Turbulence Research, under construction. RPI Mathematics Department Department of Computational Science, National U. of Singapore, where I am visiting for a year. United States Army Research Office, Durham, North Carolina, USA My paper in the ESIAM Proceedings of the St. Saveur Canada Meeting 1995. IMA postdoc and visitors research accomplishment page The Deep Ocean Convection Experiment in the Labrador and Greenland seas Paul K. Newton's The N-Vortex Problem: Analytical Techniques, Springer, NY, a textbook on vortex dynamics.
Research Grants	R-151-000-015-112 Amount for two years, Singapore $62,000.00 Used to support Joseph Nebus as a research assistant at the National University of Singapore.
Awards	Institute for Mathematics and its Applications, Minneapolis Postdoctoral Fellowship, 1988 - 1989 Malaysian Mathematical Olympiad Silver Medal, 1977
Teaching	Short Courses Vortex Dynamics Regular Courses Mathematics of Finance RPI, Spring 2001. An options pricer applet from this course is available. Monte Carlo Simulations and Molecular Dynamics NUS, Fall 2001. Thursdays 10 a.m. to Noon, S17-07-41; biweekly laboratory session alternate Tuesdays 2 p.m. to 4 p.m, S17-07-11 (Honours Year Laboratory). Page under construction.
Web Master	Web page designed by Joseph Nebus ( nebusj@rpi.edu ). Last updated 14 February 2003. | |