Statistical mechanics

The Ising model

demoAIStatistical mechanicswritten with an agent#7746705aea

Adam Zettel@adamSep 25, 2026

Metropolis sampling finds Onsager's magnetization

M (T) = (1 - sinh^{- 4} (2 J / k_{B} T))^{1/8}, T < T_{c} = \frac{2 J}{k _{B} ln ( 1 + 2 )}

A 16 by 16 grid of spins, each flipped at random by the Metropolis rule, settles at $T = 1.8$ to an average magnetization of $0.9567 \pm 0.0013$ . The exact infinite-lattice answer, from Onsager and Yang, is $0.95686$ . The critical temperature $2/ ln (1 + 2) = 2.26919$ is where that formula reaches zero.

Onsager and Yang, infinite latticeMetropolis, 16 x 16 gridone run's spread, from 256 reference runs

The 16 x 16 grid below, near and above T_c

The grid at T = 1.8, below T_c

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column

16 × 16 grid

The grid at T = 2.3, just above T_c

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16 × 16 grid

The grid at T = 3.2, well above T_c

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16 × 16 grid

A 32 x 32 grid at T_c, every two sweeps

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40/40

32 × 32 grid · 40 frames

|m| of the 32 x 32 grid at T_c

|m|

✓ 6 tests passchecked when the site was built#ising-model #monte-carlo #phase-transitions

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Random walks and rates

demoAIStatistical mechanicswritten with an agent#fb6a40dd60

Adam Zettel@adamSep 25, 2026

A random walk spreads as 2Dt and turns Gaussian

⟨ x^{2} ⟩ = 2 D t, D = \frac{σ ^{2}}{2 Δ t}, \frac{⟨ x ^{4} ⟩}{⟨ x ^{2} ⟩ ^{2}} - 3 = - \frac{6}{5 n}

4000 walkers each take 200 random steps, drawn evenly from an interval. Their mean squared distance from the start grows in a straight line, $2 D t$ , with the diffusion constant $D = 0.5$ set by the step size; a line through the measurements gives $D = 0.5004 \pm 0.0098$ . After one step the walkers are spread flat, and by step 200 they are spread as the Gaussian that solves the diffusion equation: the flatness measure called excess kurtosis falls as $- 6/ (5 n)$ , from $- 1.2$ to zero within the noise.

Mean squared displacement of 4000 walkers

measured2 D t, D = 0.5one standard error

exact, -6 / (5 n)measured, 4000 walkers

Where the walkers are, in units of their standard deviation

Gaussianwalkers

✓ 8 tests passchecked when the site was built#random-walk #diffusion #central-limit-theorem #monte-carlo #notebook

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demoAIStatistical mechanicswritten with an agent#45c11d9591

Adam Zettel@adamSep 25, 2026

Kinetic Monte Carlo against the master equation

\frac{d p _{i}}{d t} = j \sum (k_{j \to i} p_{j} - k_{i \to j} p_{i})

A molecule hops between three states, A to B to C and back, at fixed rates. The master equation gives the exact probability of each state at every time; kinetic Monte Carlo instead follows 10000 molecules one random jump at a time. At 25 times the fraction of molecules in each state matches the exact probability within four standard errors, the intermediate B rises to 0.460 and falls back, and the mean time to first reach C, exactly 1.75, is measured as $1.721 \pm 0.014$ .

Three KMC trajectories

molecule 1

state

molecule 2

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molecule 3

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A, exactB, exactC, exactA, KMCB, KMCC, KMC

KMC, four standard errorsmaster equation

✓ 7 tests passchecked when the site was built#kinetic-monte-carlo #gillespie #master-equation #markov-chains #notebook

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