Markov Chain Monte Carlo Calculator

Calculator Inputs

Example Data Table

Formula Used

Potential energy: U(x) = 0.5 × k × (x − x_eq)²

Inverse temperature: β = 1 / (k_B × T)

Metropolis acceptance rule: α = min(1, exp(−β × ΔU))

Observable estimate: <x^p> = (1 / N) × Σ x_i^p

Sample variance: s² = Σ(x_i − x̄)² / (N − 1)

Standard error: SE = √(s² / ESS)

Theoretical harmonic variance: Var(x) = k_BT / k

Theoretical mean potential energy: <U> = 0.5 × k_BT

This tool uses a one dimensional harmonic oscillator. The chain proposes a new position near the current position. Accepted proposals form the Markov chain. Saved samples are then used for statistics.

How to Use This Calculator

1. Enter the spring constant, temperature, and Boltzmann constant.
2. Set the equilibrium position and starting position.
3. Choose a proposal step width for trial moves.
4. Enter total iterations, burn-in steps, and thinning interval.
5. Set the observable power to estimate x, x², or higher moments.
6. Optionally enter a random seed for repeatable runs.
7. Click the calculation button to simulate the Markov chain.
8. Review acceptance, variance, ESS, confidence bounds, and sample preview.
9. Download CSV or PDF files to save the results.

Markov Chain Monte Carlo in Physics

What this calculator does

This Markov Chain Monte Carlo calculator helps physics users sample a one dimensional harmonic oscillator. It applies the Metropolis algorithm. You enter stiffness, temperature, equilibrium position, proposal width, burn in, thinning, and total iterations. The script then builds a Markov chain and estimates observable averages from the accepted states. It also reports acceptance rate, lag one autocorrelation, effective sample size, confidence bounds, and variance error against theory.

Why MCMC matters

Markov Chain Monte Carlo matters because many physics distributions are difficult to integrate directly. Sampling offers a practical route. A chain can visit states according to the Boltzmann weight without evaluating a hard normalization constant. That is useful in statistical mechanics, thermodynamic averaging, inverse modeling, and uncertainty studies. The method is simple, but setup choices still matter. Proposal width changes acceptance. Burn in removes early transients. Thinning reduces stored correlation, although it should be used carefully.

The physical model

This calculator uses the harmonic potential U(x)=1/2 k(x-x_eq)². The exact equilibrium variance is k_BT/k. The exact mean potential energy is 1/2 k_BT. These closed form results provide a strong reference for validation. If your simulated variance stays close to theory, the chain is likely mixing well. Large differences can point to short runs, poor proposal width, or an overly aggressive burn in choice.

How to read the outputs

Read the outputs together. Acceptance rate shows how often proposed moves enter the chain. Very low values usually mean steps are too large. Very high values can mean moves are too small and exploration is slow. Lag one autocorrelation measures dependence between neighboring saved samples. Lower values often indicate better mixing. Effective sample size converts correlated draws into an approximate count of independent information.

Practical workflow

A practical workflow is straightforward. Start with a moderate proposal width. Run the calculator. Check acceptance, variance error, and the confidence interval. Then repeat with more iterations and a different seed. Stable summaries across runs increase trust. Exporting CSV and PDF outputs also makes it easier to document settings, compare runs, and share results with colleagues. Because the observable power is configurable, you can estimate moments like x, x squared, or higher order responses from the same simulation run. This improves quick model exploration.

Frequently Asked Questions

1. What does this MCMC calculator simulate?

It simulates a one dimensional harmonic oscillator with the Metropolis method. The chain samples positions using the Boltzmann distribution and estimates equilibrium statistics from saved states.

2. Why is acceptance rate important?

Acceptance rate shows whether the proposal step is sensible. Too low means rejected moves dominate. Too high can mean the chain moves in tiny steps and explores the state space slowly.

3. What is burn-in?

Burn-in is the early part of the chain that you discard. It reduces the effect of the chosen starting point and helps focus the analysis on equilibrium behavior.

4. Should I always use thinning?

Not always. Thinning reduces stored correlation and file size, but it also discards data. Use it when memory or reporting simplicity matters, not as a universal fix.

5. What does effective sample size mean?

Effective sample size estimates how many nearly independent samples your correlated chain contains. A larger ESS usually means more reliable averages and smaller standard errors.

6. Why compare variance with theory?

The harmonic oscillator has a known equilibrium variance. Comparing simulated and exact values is a quick validation check for mixing quality, run length, and parameter choices.

7. What does observable power do?

Observable power lets you estimate moments such as x, x², x³, or higher powers. This helps study symmetry, spread, and nonlinear response from one simulation setup.

8. Can I export my results?

Yes. After a successful run, you can download a CSV file for spreadsheet work and a PDF file for quick reporting or documentation.