Lab

The same sims as a ladder of scales →

• Solar system

  ▶⟲Speed (log scale)

  the maths

  $${\ddot{\mathbf{r}}}_i=-\sum _{j\ne i}\frac{G{M}_j}{|{\mathbf{r}}_i-{\mathbf{r}}_j{|}^3}({\mathbf{r}}_i-{\mathbf{r}}_j)$$

  Velocity Verlet integration of the full gravitational N-body problem using J2000.0 ephemeris. Each body moves under the combined pull of all others. The simulation runs at ~100 years per frame; set your own speed.

• Multi-electron atoms

  ▶⟲Atomic number

  the maths

  $${\psi }_{n\ell m}(r,\theta ,\varphi )=R_{n\ell }(r)Y_{\ell m}(\theta ,\varphi ),Z_{\text{eff}}=Z-\sigma$$

  Effective nuclear charge model with Slater screening. The electron cloud is built up shell-by-shell, sampling from |ψ|² for each valence configuration. The shimmering effect shows continuous re-measurement of electron position.

• Schwarzschild black hole

  ▶⟲Zoom

  the maths

  $$d{s}^2=-\left(1-\frac{r_s}{r}\right)c^{2}d{t}^2+{\left(1-\frac{r_s}{r}\right)}^{-1}d{r}^2+r^{2}d{\Omega }^2$$

  Null geodesics traced backward from the observer. The photon sphere at r=1.5 r_s marks the edge of stable light orbits. The accretion disk temperature follows T∝r^(-3/4), glowing bright white as material spirals inward.

• Quantum wavepacket

  ▶⟲MomentumBarrier heightModeBarrierDouble slit

  the maths

  $$i\hslash {\partial }_t\psi =\left(-\frac{{\hslash }^2}{2m}{\nabla }^2+V\right)\psi$$

  Solved by split-step Fourier: half a step of the potential, a Fourier-space half step of the kinetic part, half a step of the potential again. The barrier sits just above the packet's mean energy, so you watch tunnelling and reflection at once.

• Hydrogen orbital

  ▶⟲Orbital1s2s2p3p3d(z²)3d(xy)4f(z³)

  the maths

  $${\psi }_{n\ell m}=R_{n\ell }(r)Y_{\ell m}(\theta ,\varphi ),P(x)\propto |\psi {|}^2$$

  Exact analytic orbitals sampled as |ψ|² point clouds: the radius comes from an inverse-CDF of r²R², the direction from rejection sampling on |Y|². Gold and violet mark the two signs of the real ψ — the nodal plane between lobes is where the wavefunction crosses zero.

• Lennard-Jones melting

  ▶⟲Temperature

  the maths

  $$u(r)=4\epsilon \left[(\sigma /r{)}^{12}-(\sigma /r{)}^6\right]$$

  Velocity-Verlet dynamics in reduced units, the potential cut and shifted at 2.5σ, and a Berendsen thermostat. The order readout is ⟨|ψ₆|⟩, the local six-fold bond-orientation order parameter — near 1 for a crystal, collapsing when it melts.

• Kinetic gas

  ▶⟲TemperaturePiston positionParticles

  the maths

  $$f(v)=\frac{v}{{\sigma }^2}{e}^{-v^2/2{\sigma }^2},{\sigma }^2=\frac{kT}{m}$$

  Every collision is elastic; nothing is drawn from the curve. Starting all discs at the same speed, the histogram still relaxes to the thin cream line — equilibrium is emergent. The piston wall does real work, so compressing the gas genuinely heats it.

• Particles → Probability

  ▶⟲Dots ↔ cloudTemperature

  the maths

  $$\dot{x}=-{\partial }_{x}V+\sqrt{2k_{B}T}\xi (t),V=a(x^2-1{)}^2+\frac{y^2}{2}$$

  Euler–Maruyama overdamped Brownian dynamics — nothing about the equilibrium shape is drawn in by hand. The cloud is a Gaussian-kernel density estimate of where the particles actually are, which relaxes to the Boltzmann distribution on its own.

• Shear-flow fluid

  ▶⟲Shear speedViscosityDots

  the maths

  $${\partial }_t\mathbf{u}+(\mathbf{u}\cdot \nabla )\mathbf{u}=-\nabla p+\nu {\nabla }^2\mathbf{u},\nabla \cdot \mathbf{u}=0$$

  Stam's stable-fluids scheme: semi-Lagrangian advection, a Jacobi pressure projection to keep the flow divergence-free, and a little vorticity confinement so the swirls survive numerical damping. Seeded with a tanh shear layer along the diameter.

• Pendulum phase space

  ▶⟲Damping γ

  the maths

  $$\ddot{\theta }=-\frac{g}{L}\sin \theta -\gamma \dot{\theta },H=\frac{p^2}{2}+\frac{g}{L}(1-\cos \theta )$$

  Leapfrog integration — symplectic without damping, so the orbit neither spirals nor blows up. The ember curve on the phase plane is the separatrix: inside it the pendulum swings, outside it loops over the top. Drag the bob, or tap the plane to drop the state anywhere.

• Wave Lab

  Ten more animated tools — hydrogen orbitals, spherical harmonics, Bose–Einstein condensates, rainbows — the chalkboard side of the lab.

  Open Wave Lab →