Potts Model Order Calculator

Calculator

Formula used

For a q-state Potts configuration with state fractions pᵢ and p_max = max(pᵢ), a common scalar order parameter is:

m = (q·p_max − 1) / (q − 1)

This maps uniform disorder (pᵢ = 1/q) to m = 0, and perfect ordering (p_max = 1) to m = 1. If you enter counts nᵢ, the calculator converts using pᵢ = nᵢ / N with N = Σ nᵢ.

We also report normalized entropy H = −Σ pᵢ ln(pᵢ) / ln(q) and effective states N_eff = exp(−Σ pᵢ ln(pᵢ)).

How to use this calculator

1. Select the number of states q that your model uses.
2. Choose whether you will enter counts nᵢ or fractions pᵢ.
3. Fill the fields for each state; leave unused ones as zero.
4. Press Compute order to get m, entropy, and dominance.
5. Use the download buttons to export the latest result.

Example data table

Professional article

1) Why an order parameter matters

In lattice spin models, a compact order metric helps separate disordered configurations from symmetry-broken ones. For the q-state Potts family, the distribution of state occupancies carries this information even before you compute energies or correlation lengths, making it ideal for quick diagnostics.

2) From counts to fractions

Simulations typically store raw counts n_i for each state. The calculator converts these to fractions p_i=n_i/N where N=Σn_i. Using fractions allows consistent comparisons across runs with different system sizes, sampling windows, or thinning strategies.

3) Potts ordering metric used here

The reported scalar order parameter is m=(q·p_max−1)/(q−1), where p_max is the largest fraction among all states. If all states are equally populated, p_max=1/q and m=0. If one state dominates completely, p_max=1 and m=1.

4) Interpreting typical ranges

In finite systems, m rarely hits the extremes. Near high temperature, random mixing yields p_max slightly above 1/q, giving small m values. In ordered phases, one state becomes prevalent and m rises sharply, while remaining below 1 due to domain walls, interfaces, or incomplete equilibration.

5) Entropy as a complementary indicator

Normalized entropy H=−Σp_iln(p_i)/ln(q) quantifies how evenly probability mass is spread. H≈1 indicates near-uniform mixing, while smaller H indicates concentration. Reporting both m and H helps distinguish “one-state dominance” from more nuanced multi-domain patterns.

6) Effective number of states

The calculator also reports N_eff=exp(−Σp_iln(p_i)), which can be read as the number of equally weighted states that would produce the same entropy. For example, N_eff close to q suggests high mixing, while values near 1 suggest near-complete ordering.

7) Practical workflow for simulation data

After each sweep block, record counts per state and compute m and H to build time series. Use moving averages to reduce noise, then compare across temperatures or couplings. Exporting CSV makes it easy to plot m(T) or H(T) and locate transition regions with consistent preprocessing. These summary metrics are lightweight, enabling rapid screening before finite-size scaling or histogram reweighting studies.

8) Reporting and reproducibility

When publishing, include q, the sampling method, and the computed p_i distribution. Provide both m and H, plus N_eff to summarize mixing. The built-in PDF export produces a compact snapshot suitable for lab notes, peer review, and quick comparisons across parameter scans.

FAQs

1) What does m measure in this tool?

It measures dominance of the most populated Potts state using m=(q·p_max−1)/(q−1). Values near 0 indicate uniform mixing; values near 1 indicate strong ordering into one state.

2) Should I enter counts or fractions?

Use counts if you have raw tallies n_i. Use fractions if you already have p_i values. Fractions are normalized automatically, so imperfect sums still produce consistent results.

3) Why report entropy H along with m?

m only uses p_max. Entropy H uses all p_i, so it captures whether probability is broadly spread or concentrated. Together, they provide a clearer picture of mixing and ordering.

4) What is N_eff and how do I use it?

N_eff=exp(S) with S=−Σp_iln(p_i). It is the “effective” number of populated states. It helps summarize partial ordering, especially when several states retain weight.

5) Can m detect phase transitions by itself?

It is a useful indicator, but finite-size effects can blur transitions. For stronger evidence, track m(T), susceptibility-like fluctuations, and compare with Binder-type analyses or correlation-length estimates.

6) What if two states are equally dominant?

m uses the largest fraction, so ties give the same p_max and the same m. In that case, entropy H and N_eff better reflect the shared dominance across states.

7) How are the CSV and PDF downloads generated?

After you compute results, the page stores the latest output in session memory. The download buttons export that saved result immediately, including q, m, H, N_eff, and the full state breakdown.