Explore thermodynamic averages from a spin lattice quickly. Tune temperature, field, and coupling with confidence. Get energy, magnetization, and acceptance statistics in seconds today.
| Case | L | J | h | T | Equil | Meas | Stride | Typical outputs |
|---|---|---|---|---|---|---|---|---|
| Low temperature ordering | 20 | 1.0 | 0.0 | 1.5 | 300 | 1200 | 5 | Low E/N, |M|/N near 1, low acceptance. |
| Near critical region | 30 | 1.0 | 0.0 | 2.269 | 800 | 3000 | 10 | Large fluctuations, higher Cv and Chi. |
| High temperature disorder | 20 | 1.0 | 0.0 | 4.0 | 200 | 1000 | 5 | E/N closer to 0, M/N near 0, high acceptance. |
Energy: E = -J Σ<ij> sisj - h Σi si, where si ∈ {+1, -1}. The neighbor sum uses the four nearest neighbors on a square lattice.
Metropolis update: pick a random site i and propose si → -si. The energy change is ΔE = 2 si(J Σnn snn + h). Accept if ΔE ≤ 0, otherwise accept with probability exp(-ΔE/T).
Fluctuation observables (per spin, N=L²): Cv = (⟨E²⟩-⟨E⟩²)/(N T²) and χ = (⟨M²⟩-⟨M⟩²)/(N T). If you enable |M|, the average uses |M| for reporting, while χ still uses the measured M values stored.
This calculator runs a Metropolis Markov chain on a 2D square-lattice Ising model to estimate equilibrium averages. After equilibration sweeps are discarded, measurement sweeps provide energy per spin, magnetization per spin, heat capacity, susceptibility, and an acceptance-rate diagnostic.
The Hamiltonian is E = -J Σ⟨ij⟩ sisj - h Σi si with spins si ∈ {+1, -1} and kB=1. For J=1 and h=0, a widely used reference point is Tc ≈ 2.269 for the 2D square lattice.
Each proposal flips one random spin. The energy change is ΔE = 2 si(J Σnn snn + h). Moves with ΔE ≤ 0 are accepted, while uphill moves are accepted with probability exp(-ΔE/T). Low T tends to lower acceptance; high T raises it.
Equilibration removes memory of the initial state, but near Tc the chain mixes slowly. For L=20 to 40, a practical start is 300 to 1000 equilibration sweeps, then 1500 to 5000 measurement sweeps. Use a stride (for example 5 to 20) to reduce correlation between recorded samples.
With ferromagnetic coupling (J>0) and h=0, lowering T makes E/N more negative and ordering increases, so |M|/N approaches 1 on finite lattices. Raising T disorders the system and drives M/N toward 0. A nonzero field h biases magnetization and smooths the apparent transition.
The calculator estimates Cv = (⟨E2⟩ - ⟨E⟩2)/(N T2) and χ = (⟨M2⟩ - ⟨M⟩2)/(N T). Both often peak near the critical region. Longer runs improve peak stability because autocorrelation inflates noise.
Finite lattices round sharp phase transitions into crossovers, so peaks broaden and shift with L. Runtime scales with N=L2 attempts per sweep, so increasing L costs more. Periodic boundaries usually reduce edge artifacts compared with open boundaries for small and medium sizes.
Fix the random seed to reproduce a run exactly and to compare parameter changes fairly. A common workflow is scanning T from 1.5 to 4.0 at fixed J and L, then plotting E/N, |M|/N, Cv, and χ. Export CSV for analysis and PDF for reporting.
It is the fraction of proposed spin flips that were accepted. Very low values often indicate low temperature or strong ordering. Very high values usually indicate high temperature or weak constraints. Use it as a fast health check.
Near the critical region, magnetization can flip sign between samples, making the average M close to zero. Averaging |M| reduces sign cancellation and better reflects the amount of ordering in a finite lattice.
For L around 20 to 40, start with 300 to 1000 equilibration sweeps and 1500 to 5000 measurement sweeps. Increase both near Tc or when repeated runs show noticeable variability.
Stride records observables every few sweeps instead of every sweep. This reduces correlation between saved samples and can make averages and fluctuation-based quantities more stable, especially close to criticality.
Periodic boundaries wrap the lattice and typically reduce edge effects, so small systems behave more like bulk material. Open boundaries can be useful for surface studies but can bias energy and magnetization for small L.
Monte Carlo estimates have statistical noise and samples can be correlated. Keep the seed fixed for exact repeatability, or change the seed to sample another trajectory. Increasing sweeps and using a larger stride reduces variability.
Yes. Set J to a negative value to favor alternating spins. Expect different ordering patterns and, for some temperatures, slower convergence. Use more equilibration sweeps and consider larger L to see clearer behavior.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.