Markov Steady State Calculator

A Markov chain with transition matrix P evolves probabilities by p(t+1) = p(t) P, where each row sums to 1. The steady state (stationary distribution) π satisfies π = πP and Σπ = 1.

This tool uses power iteration: πₖ₊₁ = normalize(πₖ P) until convergence.

1. Set the number of states n.
2. Enter an n×n transition matrix with nonnegative entries.
3. Ensure each row sums to 1, or enable auto-normalization.
4. Choose tolerance and maximum iterations for convergence control.
5. Select uniform or custom initial distribution π₀.
6. Click Compute Steady State. Results appear above the form.
7. Export CSV or PDF for documentation.

If your matrix is nearly reducible, increase iterations and review residual.

Markov Steady State in Physics

1) Why Steady State Matters

Many physical systems switch between discrete regimes: energy levels, conformations, metastable wells, or operational modes. A Markov chain models those switches with probabilities. The steady state distribution π estimates the long run fraction of time spent in each state under stable conditions.

2) Transition Matrix as Measured Data

In experiments, entries often come from counts. If state i is followed by state j a total of C_ij times, then P_ij ≈ C_ij/Σ_kC_ik. Rows must sum to 1. Auto-normalize helps when rows are proportional to counts.

3) Power Iteration and Convergence

The calculator uses power iteration, multiplying a probability vector by P until the L1 change is below tolerance. Ergodic chains typically converge quickly. Nearly decomposable chains mix slowly, requiring more iterations. The residual ‖πP−π‖₁ checks solution quality.

4) Diagnostics You Should Report

A professional report includes tolerance, iterations used, and residual. When comparing scenarios, report π and confirm Σπ≈1. Large residuals can indicate reducibility, periodic structure, or an unreliable estimate from limited data.

5) Example: Two-State Relaxation

For P = [[0.80, 0.20],[0.30, 0.70]], the steady state is π = [0.60, 0.40]. If states represent low and high energy, the model predicts 60% occupancy in the low state. Increasing the return probability from high to low shifts equilibrium further.

6) Example: Three-State Regime Switching

For the 3×3 example above, π concentrates near state 1 because it has strong self-persistence and receives inflow from other states. This pattern is consistent with a dominant basin of attraction with occasional transitions to secondary regimes.

7) How Settings Affect Results

Tighter tolerance yields more precise π but may require more steps. The initial distribution usually does not change the final π for ergodic chains, but it can affect convergence speed. When P is estimated from short records, uncertainty dominates extra decimal places.

8) Practical Workflow

Build P from transition counts, normalize rows, compute π, and export CSV/PDF. Then run sensitivity checks by perturbing rows within measurement uncertainty. If π stays stable, the interpretation is robust. If not, collect more data or refine state definitions.

FAQs

1) What does π represent physically?

π gives long-run occupancy of each state. Under repeated transitions and stable conditions, it approximates the fraction of time the system spends in each regime.

2) Why must each row of P sum to 1?

Row i lists probabilities of moving from state i to all next states. Those mutually exclusive outcomes must add to 1 to define a valid conditional distribution.

3) When should I enable auto-normalize?

Enable it when rows come from counts or unnormalized rates. The calculator scales each row to sum to 1. Keep it off for strict validation of your probabilities.

4) What does the residual mean?

The residual ‖πP−π‖ measures how closely the computed π satisfies the steady-state equation. Smaller residual indicates a more consistent stationary solution.

5) Why is convergence slow?

Slow convergence suggests weak mixing, near-reducibility, or very high self-transition probabilities. Increase max iterations, reassess state definitions, or gather more transitions.

6) Can there be multiple steady states?

Yes. Reducible chains can have multiple stationary distributions depending on communicating classes. In such cases, results can depend on π₀ and observed transitions.

7) How many decimals should I report?

Match precision to data quality. For experimental matrices, 3–6 decimals is often enough. Always include tolerance and residual to justify numerical accuracy.