Monte Carlo Simulator

Initializing stochastic sampling engine…

◎Method

Algorithm ?Metropolis–Hastings is the standard choice. Hamiltonian MC is more efficient for continuous distributions with gradients.

Chain length ?Total number of MCMC samples to draw. Burn-in samples are discarded.

Burn-in

Thinning

Proposal step σ0.50

0.50

Posterior sampling · parameter fitting · Bayesian UQ · free energy

Algorithm

Simulation time (s) ?Physical simulation time. kMC naturally advances in variable physical time steps, so rare events are correctly captured.

Temperature (K)

Lattice type

Attempt frequency ν₀ (Hz)

Activation barrier E_a (eV)

0.80

Rare events · surface diffusion · vacancy migration · catalysis

Number of replicas ?More replicas → broader temperature coverage and better exchange acceptance. 8–32 is typical.

T_min (K)

T_max (K)

Spacing

Swap every

Steps/replica

Replica exchange · escape local minima · free energy surfaces

Chemical potential μ (eV) ?Controls particle number. More negative μ → fewer particles. At μ = μ_coexistence you get the equilibrium density.

-0.50

Temperature (K)

MC cycles

Move probabilities

Insert

Delete

Displace

Fixed μ,T,V · adsorption isotherms · open-system equilibrium

⬡System

Model ?Ising: binary spins ±1. Potts: q-state generalisation. Lattice gas: particle on/off. Heisenberg: 3-component spin vectors.

Lattice size N4³ = 64

Temperature kT/J ?Critical temperature for 2D Ising: kT_c/J ≈ 2.269. Below T_c → ordered phase. Above → disordered.

2.00

External field H/J

0.00

Periodic boundary conditions

Parallelise (checkerboard decomp.)

◈Observables

Sample every N steps

Total MC steps

↗HPC Execution

Command

● Idle

step 0 · 0.0% accepted

⬡

No system initialized

Choose a model and click ▶ Start to begin simulation

T— · H0.00 · N—

E/N— · ⟨m⟩— · acc—

⟨E⟩ / N

—

⟨m⟩ order

—

χ suscept.

—

C_V spec. heat

—

Accept rate

—

Step · T

— · —

〰️

MCMC chain trace

Shows raw chain values and running mean — use to diagnose convergence and mixing

⬡

Order parameter ⟨m⟩

Magnetization per site vs simulation step — watch for spontaneous symmetry breaking

✓

Acceptance rate

Running acceptance rate — target ~23–40% for Metropolis. Too high → step too small

▬

Energy histogram

Distribution of sampled energies — bimodal at T_c indicates first-order transition

♻

Replica temperatures (PT)

Temperature assignment of each replica vs swap step — good mixing → all replicas visit all T