# Physics with Propab

Simulations, observational data and theory worked through carefully, with error bars, finite-size checks and the literature value beside every result.

## Real runs

- **Statistical mechanics** — [replay](https://propabai.com/replay/share_UCErCeW5OVnexzEVKJNSWL3WbZz6OTmk): “Reproduce the 2D Ising model ferromagnetic phase transition from first principles. Run Monte Carlo (Metropolis, plus a Wolff cluster algorithm near criticality) on square lattices of several sizes, measure magnetisation, susceptibility, specific heat and the Binder cumulant, and estimate the critical temperature by finite-size scaling. Compare your estimate with Onsager's exact result and with values reported in the literature.” Made: Observables, Finite-size fit, PDF report.

## Workflows ready today

### Astrophysics

- **Known-exoplanet population study** — Mass-radius and period-radius diagrams (e.g. the radius valley) from the NASA Exoplanet Archive, with the selection cuts stated.
- **Redshifts and emission lines from SDSS spectra** — Redshift, line fluxes and a BPT classification for a galaxy sample, from SDSS line catalogues or the spectra themselves.
- **Hubble constant from Pantheon+ supernovae** — H0 / Omega_m posterior from the Pantheon+SH0ES distance moduli with the full covariance matrix, with a corner plot.
- **Gravitational-wave population from published posteriors** — Mass and spin distribution of merging black holes by hierarchical inference over the GWTC posterior samples.
- **Analyse my cosmological simulation output** — Halo profiles, projections and power spectra from a simulation snapshot the researcher uploads.

### Computational physics

- **Check an integrator's order and conservation** — Measured convergence order and energy/momentum drift of an integrator (symplectic vs Runge-Kutta) on a test problem.
- **Long-term stability of a planetary system** — MEGNO / Lyapunov map and survival time of a multi-planet system over millions of orbits.
- **Solve a PDE and prove the convergence order** — Finite-difference solution of a heat/wave/Poisson problem with a grid-refinement study against the exact solution.
- **Monte Carlo estimate with an honest error bar** — Monte Carlo integral or expectation with variance reduction and a convergence plot.
- **Spectrum of my time series** — Power spectral density (Welch/multitaper), peaks and their significance for a signal the researcher supplies.
- **Derive equations of motion and simulate them** — Lagrangian to equations of motion symbolically, then integrated and plotted (phase portraits, energy).
- **Fit a simulation's parameters by gradient** — Differentiable simulation fitted to data by automatic differentiation.
- **Solve a stiff ODE or DAE system** — Stiff kinetics or an index-reduced DAE solved with an implicit integrator, with a solver-tolerance study.

### Condensed matter

- **Tight-binding bands and density of states** — Band structure and DOS of a lattice model, checked against the analytic dispersion.
- **Topological invariant of a model** — Chern number or Z2 invariant of a Hamiltonian, with the Berry-curvature or Wilson-loop plot.
- **Ground state of an interacting 1D model by DMRG** — Energy, correlations and entanglement entropy of a spin or Hubbard chain with TeNPy or quimb.
- **Exact diagonalisation of a small interacting cluster** — Spectrum and correlation functions of a Heisenberg or Hubbard cluster by Lanczos.
- **Screen a materials database for a property** — Candidates ranked by band gap, stability or magnetism from open DFT databases.
- **Classical spin Monte Carlo for a magnetic transition** — Ordering temperature and susceptibility of a Heisenberg/Ising magnet on a lattice.

### Fluid and continuum mechanics

- **2D incompressible flow benchmark** — Lid-driven cavity or flow past a cylinder with drag, lift and Strouhal number against published reference values.
- **Linear stability of a shear flow** — Orr-Sommerfeld eigenvalues, neutral curve and critical Reynolds number.
- **Structural finite-element check against a reference** — Stress and deflection of a beam/plate/solid against an analytic or NAFEMS benchmark value.
- **Topology optimisation of a structure** — Minimum-compliance layout at a volume fraction (SIMP), with a mesh-independence check.
- **Bifurcations and chaos in a low-dimensional system** — Bifurcation diagram, Lyapunov exponent and period-doubling constants.
- **Recover governing equations from data** — Sparse identification (SINDy) of a dynamical system from measured trajectories, with a noise-robustness test.

### Nuclear physics

- **Decay an inventory forward in time** — Activities, decay heat and gamma lines of a nuclide inventory over time.
- **Binding energies against a mass model** — AME2020 binding energies vs the semi-empirical mass formula, with shell-closure residuals.
- **Measured cross sections against evaluated libraries** — EXFOR measurements plotted and compared against ENDF/B-VIII and JEFF evaluations for a reaction.
- **Nucleosynthesis reaction network** — Energy generation and abundance flow for a burning stage from a REACLIB network.
- **Astrophysical reaction rate from a cross section** — Thermonuclear rate vs temperature from an S-factor or narrow resonances, with its uncertainty.
- **Analyse my gamma-ray spectrum** — Peak fits, energy/efficiency calibration and nuclide identification for a detector spectrum.

### Optics and electromagnetism

- **Mie scattering of particles** — Extinction, scattering and backscatter efficiencies vs size parameter and wavelength.
- **Thin-film stack reflectance** — Reflectance/transmittance spectra of a multilayer coating from real dispersion data.
- **Diffraction pattern or telescope PSF** — Fraunhofer/Fresnel propagation of an aperture and the PSF with aberrations.
- **Resonator modes and Q factor (2D FDTD)** — Cavity resonances, Q factors and transmission of a 2D photonic structure.
- **Photonic-crystal band structure** — Band diagram and band gaps of a photonic crystal slab.
- **Grating diffraction efficiency (RCWA)** — Diffraction efficiencies of a periodic grating vs wavelength and angle.
- **Lens design and ray tracing** — Sequential ray trace of a lens with real glasses: spot diagrams, aberrations and a tolerance check.
- **Analyse my measured S-parameters** — Touchstone network data de-embedded, fitted and plotted (Smith chart, return loss).

### Particle physics

- **Rediscover the Z and J/psi in collision data** — Dimuon invariant-mass spectrum from CMS open data with fitted resonance masses and widths.
- **Bump hunt with local and global significance** — Search for a resonance in a mass spectrum with a look-elsewhere-corrected significance.
- **Upper limits with a HistFactory model** — 95% CL CLs limits on a signal from binned templates.
- **Jet clustering and substructure** — Anti-kt jets and substructure observables of boosted objects on open data or supplied events.
- **Standard Model prediction vs the world average** — Literature synthesis of a precision observable (e.g. muon g-2): each value, source and tension in sigma.
- **Efficiency and scale factor by tag-and-probe** — Reconstruction/ID efficiency vs pT/eta in data and simulation, and their ratio.

### Plasma physics

- **Characterise a plasma regime** — Debye length, plasma frequency, beta, collisionality and the regime they imply, for stated conditions.
- **Kinetic dispersion relation** — Roots of the plasma dispersion function (Landau damping, ion-acoustic waves) against analytic limits.
- **1D electrostatic particle-in-cell** — Two-stream instability growth rate from a PIC run against linear theory.
- **Solar-wind driving of geomagnetic storms** — OMNI solar-wind coupling functions against Dst/AE over a storm set, with lag correlations.
- **Turbulence spectrum from spacecraft magnetometer data** — Spectral index and break frequency of solar-wind or magnetosheath turbulence (Wind, MMS).
- **Free-boundary tokamak equilibrium** — Grad-Shafranov equilibrium and the coil currents that produce a target shape.
- **MHD benchmark problem** — Brio-Wu shock tube or Orszag-Tang vortex with a finite-volume MHD solver, checked against reference solutions and div B.
- **Analyse my Langmuir-probe sweep** — Electron temperature, density and floating/plasma potential from an I-V characteristic.

### Quantum physics

- **Decoherence of a driven qubit** — T1/T2 and Rabi dynamics from a Lindblad master equation.
- **Variational algorithm on a spin Hamiltonian** — VQE or QAOA with gradients, converged energy and comparison with exact diagonalisation.
- **Noisy circuit simulation at scale** — Circuits under a device noise model, up to the memory limit of a statevector/density matrix.
- **Surface-code error-correction threshold** — Logical error rate vs physical error rate across code distances, and the crossing point.
- **Tomography and randomised benchmarking** — State/process tomography and RB decay on a simulated device.
- **Bound a Bell scenario by semidefinite programming** — NPA-hierarchy bound on a Bell expression with the SDP certificate.
- **Tensor-network simulation of many qubits** — MPS/tensor-network simulation of a low-entanglement circuit beyond statevector size.
- **Molecular Hamiltonian and VQE** — Qubit Hamiltonian of a small molecule and a VQE energy against full CI.
- **Optimal control of a quantum gate** — GRAPE/CRAB pulse that implements a gate at target fidelity under constraints.

### Relativity and mathematical physics

- **Curvature of a metric** — Christoffel, Riemann, Ricci and Einstein tensors, and whether the metric solves the field equations.
- **Geodesics in a curved spacetime** — Orbits, perihelion precession, light bending or a black-hole shadow by geodesic integration.
- **Random-matrix spectral statistics** — Level-spacing distribution and spectral form factor against GOE/GUE predictions.
- **Asymptotic expansion of an integral or special function** — Series/asymptotic forms derived symbolically and checked numerically to high precision.
- **Small lattice gauge simulation** — Pure-gauge U(1)/SU(2) Monte Carlo with Wilson loops and the string tension.
- **Bethe ansatz vs exact diagonalisation** — Bethe-ansatz energies of an integrable chain checked against exact diagonalisation.

### Statistical mechanics

- **Critical point and exponent by finite-size scaling** — Critical temperature and an order-parameter exponent with statistical and systematic errors separated.
- **Cluster Monte Carlo that beats critical slowing down** — Wolff/Swendsen-Wang updates with measured autocorrelation times vs Metropolis.
- **Density of states by Wang-Landau** — g(E) of a lattice model and thermodynamics at any temperature from one run.
- **Error bars for correlated Monte Carlo data** — Integrated autocorrelation time and binning/blocking error for a Monte Carlo series.
- **Percolation threshold and cluster scaling** — p_c and cluster-size exponents by finite-size scaling.
- **Lennard-Jones fluid against NIST reference data** — Energy and pressure of LJ configurations compared with the NIST Standard Reference Simulation values.
- **Stochastic-process simulation** — Exclusion processes, kinetic Monte Carlo or Gillespie simulations with scaling exponents.