QC.174 · Complex Systems · Nonlinear Dynamics · Statistical Physics
Parth Bhargava
Physics undergraduate · National University of Singapore
A quiet research notebook on complex systems, nonlinear dynamics, and the habits of experimental and computational work.
About
I'm a second-year physics student building toward research in complex systems and nonlinear dynamics. This site is a public notebook of experiments, simulations, and ideas in progress.
I care about how simple rules generate structure across scales: nonlinear dynamics, emergence, and the geometry behind physical systems. I learn by building models, running measurements, and writing down what fails as much as what works.
I'm looking for research environments that value careful thinking, honest uncertainty, and iterative work with mentorship.
Projects
Active Matter: Vicsek Model + Run-and-Tumble
How does self-propulsion give rise to collective order and to anomalous transport? Two canonical models bracket the question from opposite ends. The Vicsek model takes the collective view — N aligning self-propelled particles on a periodic domain, swept over noise η to trace the order parameter φ = |⟨eiθ⟩| through its disorder-to-order nonequilibrium phase transition. Run-and-tumble dynamics take the single-particle view — one active walker whose Poisson reorientations set a ballistic-to-diffusive crossover in the mean-squared displacement, with the effective diffusion constant checked analytically and under harmonic confinement. Built in Julia as an interactive Pluto notebook.
Coupled Oscillators: Kuramoto Synchronization + Chimera States
When do coupled phase oscillators fall into step, and when do they fracture? The Kuramoto model gives the classic synchronization transition — followed here across all-to-all, Erdős–Rényi, and scale-free topologies, with onset matched to the analytic critical coupling Kc = 2σ√(2/π). Introducing a phase lag and finite interaction range (Kuramoto–Sakaguchi on a ring) opens a stranger possibility: chimera states, where synchronized and incoherent populations coexist on an identical ring. A sweep of the (K, R) plane then maps the full landscape from global synchrony through chimera to incoherence. Built in Julia as an interactive Pluto notebook.
Causal Inference: Transfer Entropy + CCM + Causal Emergence
What does it mean for one signal to drive another — and at what scale does the causation live? Three lenses, each built for a regime the others miss. Transfer entropy (histogram, KSG k-nearest-neighbour, and symbolic estimators) measures directed information flow in noisy, stochastic systems. Convergent cross-mapping recovers coupling direction in deterministic, weakly-coupled dynamics — exactly where correlation and Granger causality fail — through Takens delay embedding. Causal emergence (ΔEI over all 256 elementary cellular automata) turns to the orthogonal question of scale: when a coarse-grained description carries more causal power than the microdynamics beneath it. A shared benchmark shows where the three agree and where each sees something the others cannot. Built in Julia as an interactive Pluto notebook.
Show more projects
Physics Discovery: SINDy + Conservation Laws + Symbolic Regression
Can the governing physics be recovered from trajectory data alone? Three discovery methods, each answering a different form of that question. SINDy (sequentially-thresholded least squares) reconstructs the explicit equations of motion for Van der Pol, Duffing, and the driven pendulum from a sparse library of candidate terms. A kernel method asks instead what is conserved, extracting invariants through an RBF eigenproblem with no functional form assumed in advance. Symbolic regression by genetic programming rediscovers closed-form laws — kinetic energy, the pendulum period, Ohm’s law — as evolving expression trees. Run on a common benchmark, they separate cleanly into the three epistemic questions of dynamics, invariants, and phenomenological law. Built in Julia (shared custom RK4, no external ODE dependency) as an interactive Pluto notebook.
Spin Glass Analysis of Neural Network Training
Does a neural network's loss landscape behave like a spin glass? Training a small MLP on a spiral task, this reads the diagonal of the Hessian as a spectrum of parameter-axis curvatures and follows how it sharpens epoch by epoch. The inverse participation ratio (IPR) measures how concentrated that curvature becomes — a finite-size echo of replica-symmetry breaking — tracing a glassy, broad-curvature start into a crystallised minimum, and tying the spin-glass ideas behind the 2024 Physics Nobel to ordinary network training.
Physics-Constrained CD Spectral Inversion
Forward model maps protein secondary structure composition (helix/sheet/coil fractions) to CD spectra using reference basis spectra. A small physics-informed MLP inverts the problem — predicting composition from spectrum with a physics loss penalising spectral inconsistency and composition sum ≠ 1. Draws on hands-on CD experimental experience.
Percolation on Simplicial Complexes
Bond percolation on simplicial complexes of dimension k=1 (pairwise), k=2 (triangles), and k=3 (tetrahedra). Higher-order interactions produce sharper, near-discontinuous transitions vs the second-order transition of standard network percolation. Computes giant component fraction and susceptibility for finite-size scaling.
Minimal Reservoir Computing
How small can a reservoir be and still forecast chaos? A systematic phase diagram of Echo State Networks over the (sparsity, spectral-radius) plane, scoring each configuration by how many Lyapunov times its forecast stays valid — the horizon normalised by the largest Lyapunov exponent computed directly from the dynamics (Benettin method), not a tabulated constant. The same pipeline is benchmarked across two continuous flows (Lorenz-63, Rössler) and a discrete map (Hénon), locating the minimal reservoir that still tracks each attractor and the boundary beyond which prediction collapses. Built in Julia (interactive Pluto notebook) with a mirrored Python implementation.
Differentiable Pendulum: Parameter Inference
A double pendulum is chaotic, yet its physical parameters can still be read back out of the motion it traces. Treating the RK4 simulator as a differentiable forward model, gradient descent on a trajectory-matching loss (with finite-difference gradients in the parameters) recovers the bob masses and arm lengths from noisy observed trajectories — to within a few percent even at realistic noise. It is the inverse of the driven-pendulum chaos study: not predicting motion from parameters, but inferring parameters from motion.
Gō Model: Protein Folding Free Energy
Off-lattice Gō model for a simplified Trp-cage protein. Replica-exchange Monte Carlo samples the folding free energy landscape F(Q) as a function of the fraction of native contacts. Identifies the folded basin, transition state, and folding temperature — bridging computational protein physics with spectroscopic observables from CD/fluorescence experiments.
Driven Quadruple Pendulum
Simulated chaotic dynamics of a driven, damped quadruple pendulum; custom RK4 integrator with LU decomposition, Poincaré sections, bifurcation diagrams, and Lyapunov divergence. Co-authored with Soham Bhar.
Quantum Wavepacket — Schrödinger in Any Potential
One solver for every one-dimensional scattering and bound-state problem in the textbook. A Crank-Nicolson integrator — unitary, O(N) via the Thomas algorithm, and norm-preserving to machine precision — evolves a minimum-uncertainty Gaussian wavepacket through any potential: six are built in (free propagation, rectangular barrier, finite square well, harmonic oscillator, double barrier, potential step) alongside a custom expression field for arbitrary V(x). Sliders set the potential strength (in units of the packet's kinetic energy), the barrier or well width, the wavenumber k₀, and the animation speed. A 3D Argand diagram renders ψ as a complex helix whose pitch is the de Broglie wavelength, while a synchronized panel tracks |ψ|² against the potential with live norm, transmission/reflection coefficients, and centre-of-mass readouts.
Gray-Scott Reaction-Diffusion — Turing Patterns
Finite-difference solver for the Gray-Scott PDE system on a 200×200 grid (5-point Laplacian, explicit Euler). Sweeping feed rate F and kill rate k maps the full Pearson diagram of morphologies — spots, stripes, worms, solitons, labyrinthine patterns, and uniform steady states. The progressive animation reveals how the spatial pattern grows from random initial conditions. Implemented in Julia with a Pluto interactive applet.
Coursework
Experimental Work
Hall Effect in Semiconductors
Measured Hall voltage and magnetoresistance across temperature to extract carrier type, mobility, and the intrinsic transition; includes uncertainty analysis.
Raman Spectroscopy
Characterised carbon allotropes via D/G band ratios, estimated graphene layer count, and identified an unknown semiconductor wafer from its 521 cm⁻¹ mode.
X-Ray Diffraction Analysis
Used Bragg peaks to determine lattice constants and identify an unknown crystal; report documents calibration limits.
Show more experiments
Magnetic Moment in Helmholtz Field
Calibrated Helmholtz field constant and tested scaling laws; corrected a model assumption during analysis.
X-Ray Fluorescence Analysis
Qualitative elemental identification of unknowns and quantitative brass composition via Gaussian deconvolution of overlapping K-lines.
Paschen Curve & Gas Discharge
Mapped breakdown voltage vs. electrode gap at three pressures; extracted the second Townsend coefficient and identified the avalanche-to-streamer transition.
Laue Diffraction & X-Ray Crystallography
Indexed Laue back-reflection patterns and determined lattice parameters; combined with powder XRD for crystal identification.
Propagation of Laser Light (PLL)
Measured Gaussian beam parameters and beam quality; report focuses on fitting and measurement limits.
Fluorescence Spectroscopy: Protein Unfolding
Tracked acid denaturation of BSA and lysozyme via intrinsic Trp/Tyr fluorescence; observed emission shifts and intensity changes tied to tertiary structure loss.
Circular Dichroism: Protein Secondary Structure
Monitored α-helical content of BSA and lysozyme under acid denaturation using far-UV CD; lysozyme's disulfide bonds preserved its secondary structure.
Electron Spin Resonance (ESR)
Extracted g-factor from frequency–field measurements; analyzed Zeeman splitting and systematics.
TRIM Ion Transport Simulation
Monte Carlo simulations of ion range, damage, and sputtering across nuclear microscopy, semiconductor implantation, and proton therapy applications.
Fluorescence Microscopy: Cellular Structure
Imaged Hoechst-stained onion epidermal and cheek epithelial cells; compared brightfield and fluorescence contrast for nuclear localisation.
Notes and Assignments
Assignments
Classical Mechanics II (PC3261)
- CMA 1
- CMA 2
- CMA 3
- CMA 4
- CMA 5
- CMA 6
- CMW 2
- CMW 4
- CMW 5
- CMW 6
- CMW 7
- CMW 8
- CMW 9
- CMW 10
- CMW 11
- CMW 12
- CMW 13
- CMW 14
- CMW 15
- CMW 16
- CMW 17
- CMW 18
- CMW 19
- CMW 20
- CMW 21
- CMW 22
- CMW 23
- CMW 24
Thermodynamics & Statistical Mechanics (PC2135)
- Assignment 1
- Assignment 2
- Assignment 3
- Assignment 4
- Assignment 5
- Assignment 6
- Assignment 7
- Assignment 8
- Assignment 9
- Assignment 10
- Assignment 11
- Assignment 12