Formula used
For a q-state Potts system, let pi be the fraction in state i, and pmax = max(pi). A widely used scalar order parameter is:
This gives m = 0 for a perfectly mixed configuration (pi=1/q) and m = 1 for complete order (all weight in one state).
How to use
- Set q to the number of Potts states in your model.
- Choose Counts for raw occupation numbers, or Probabilities for normalized fractions.
- Enter exactly q values in your state order.
- Click Calculate to compute m, entropy diagnostics, and the breakdown table.
- Download outputs using the CSV or PDF buttons.
Example data table
| q | Input mode | Values | pmax | m | Interpretation |
|---|---|---|---|---|---|
| 3 | Counts | 70, 20, 10 | 0.700000 | 0.550000 | Moderate ordering toward one dominant state. |
| 4 | Probabilities | 0.25, 0.25, 0.25, 0.25 | 0.250000 | 0.000000 | Completely mixed across all states. |
| 2 | Counts | 98, 2 | 0.980000 | 0.960000 | Strong alignment, near full order. |
Potts order parameter in phase studies
The Potts order parameter m condenses multi-state occupancy into a single alignment score. It ranges from 0 for uniform mixing (pi=1/q) to 1 when one state dominates. In Monte Carlo or agent-based simulations, tracking m(T) or m(h) helps identify symmetry breaking and locate transition regions with minimal post-processing.
Choosing q and interpreting scale
Common choices include q=2, q=3, and q=4. Because the mapping uses pmax, the same pmax produces a slightly different m when q changes. With pmax=0.70, you get m≈0.55 at q=3 but m≈0.60 at q=4, reflecting stronger contrast against baseline 1/q.
Counts versus probabilities
Counts ni are converted to fractions using pi=ni/N. This suits sweep-by-sweep tallies, time-binned measurements, or categorical samples. Probability mode fits when you already computed pi from histograms or analytic models, and auto-normalization corrects small drift from rounding.
Uncertainty and sampling effects
With finite samples, pmax fluctuates. When a sample size N is available, the calculator reports SE(p)≈sqrt(p(1−p)/N) and propagates to SE(m)≈(q/(q−1))·SE(p). At p=0.70 and N=10,000, SE(p)≈0.0046, producing a small SE(m) useful for error bars.
Entropy as a complementary diagnostic
Shannon entropy H=−∑piln pi and normalized entropy H/ln q provide an orthogonal view of disorder. Low entropy aligns with ordered phases, while values near one indicate strong mixing. Using both m and entropy helps distinguish “dominance” from “broad coexistence” patterns.
Finite-size scaling workflow
For lattice size L, compute m across many sweeps to estimate <m> and its variance. Comparing sizes such as L=16, 32, 64 reveals sharpening transitions and shifting pseudo-critical points. CSV exports simplify merging runs and fitting relations like m∼L−β/ν.
Simulation checks and sanity tests
For a symmetric case, pi=1/q should produce m=0 and normalized entropy near one. For polarized samples like counts 98,2 at q=2, the tool yields m≈0.96, confirming near-complete alignment.
Reporting and reproducibility
For reproducible reports, record q, input mode, sample size, and whether auto-normalization was applied. The PDF export captures computed summaries and per-state values, supporting consistent reporting across parameter scans and collaboration reviews.
FAQs
1) What does m = 0 mean physically?
It indicates no preferred state: all q states are equally populated, so the system is maximally symmetric and disordered under this measure.
2) Why is pmax used instead of a vector order parameter?
Using pmax yields a simple scalar that works for any q and is easy to compare across runs, while still reflecting dominant-state alignment.
3) Can I use this for non-lattice data?
Yes. Any categorical data with q labels works, such as clustering assignments, agent states, or discrete phase labels, as long as you can provide counts or probabilities.
4) What if my probabilities do not sum to one?
Enable auto-normalization in probability mode. The calculator rescales all inputs by their sum, which is helpful when rounding or truncation introduced small drift.
5) How reliable are the uncertainty estimates?
They are quick approximations based on binomial sampling of the dominant fraction. For correlated samples, effective N is smaller, so treat SE as a lower bound unless you thin or block-average.
6) Why include entropy if m already measures order?
Entropy detects broader mixing patterns. Two configurations can share the same pmax yet differ in how the remaining weight spreads, which entropy captures.
7) What ranges of q are practical here?
The interface supports q from 2 to 50 for readability and export size. Larger q is possible conceptually, but the per-state table and inputs become harder to manage.