Paste your data as lines with four fields: set, param, x, y. Delimiters may be commas, semicolons, or tabs. Use # for comments.
These rows mimic two parameter values with similar scaled behavior.
| Set | Param | x | y |
|---|---|---|---|
| A | 10 | 1.0 | 2.50 |
| A | 10 | 2.0 | 3.55 |
| A | 10 | 4.0 | 5.02 |
| B | 20 | 1.0 | 3.52 |
| B | 20 | 2.0 | 5.02 |
| B | 20 | 4.0 | 7.10 |
A common scaling-collapse hypothesis assumes a one-parameter form: y(param, x) = param^b · f(x / param^a).
The calculator produces scaled variables: x' = x / param^a and y' = y / param^b.
If the hypothesis is consistent, curves from different sets overlap when plotted as y' vs x'.
- Enter trial exponents a and b from theory or fits.
- Paste your measurements as set, param, x, y lines.
- Submit to compute x' and y' for each row.
- Review the collapse metric; smaller values imply better overlap.
- Export CSV or PDF for plotting and lab notes.
1) Purpose of scaling collapse
Scaling collapse checks whether datasets taken at different parameter values can be mapped onto one universal curve. It is widely used in finite-size scaling, critical phenomena, and transport studies where raw curves differ mainly by scale, not by shape. A convincing collapse helps compare models, estimate exponents, and communicate trends clearly.
2) Scaling form and transformed variables
The working hypothesis is y(param, x) = param^b · f(x / param^a). The calculator computes x' = x / param^a and y' = y / param^b. If the hypothesis is consistent, plots of y' versus x' overlap.
3) Choosing param and preparing data
Use a strictly positive param that labels each set, such as system size, field, or a reduced control variable. Keep units consistent across sets before rescaling. Provide enough points per set to compare shapes across the same x' range.
4) Selecting trial exponents
Start from theory when available, then refine with data. A practical approach is a coarse grid search for a and b, followed by small adjustments that minimize mismatch in overlapping regions. Record each trial so your final choice is reproducible.
5) Interpreting the overlap score
The optional quality metric bins x' and measures variability of y' inside each bin. It reports an average coefficient of variation across bins with at least two points. Lower values indicate tighter overlap, but always inspect the scaled table too. For best comparisons, keep the same bin count while testing different exponent pairs.
6) Crossovers, noise, and outliers
Many systems show crossovers where one scaling form holds only over part of the domain. If a single pair of exponents works only for a subset, analyze ranges separately and justify exclusions. Outliers can dominate variability; check measurement errors, saturation, or regime changes.
7) Exporting for plots and documentation
Export CSV to plot y' vs x' in your plotting workflow and to compare multiple exponent trials. Export PDF to keep a compact record of inputs, exponents, and the quality score for lab notebooks, reports, or collaboration.
8) Reporting results responsibly
A good collapse supports the scaling assumption, but it is not definitive evidence by itself. Report uncertainty, sensitivity to exponent changes, and any filtering rules. Combine collapse results with theory, error bars, and independent checks for robust conclusions. If possible, validate exponents using a second observable that scales differently.
FAQs
1) What does “param” represent in the input?
It is the scaling parameter that changes between datasets, such as system size, field, or reduced temperature scale. It must be positive because the rescaling uses powers of param.
2) How should I choose the exponents a and b?
Use theoretical expectations when possible, then refine by checking overlap and the quality score. Try small increments and keep notes so you can justify the final choice.
3) What does the collapse metric mean?
It averages within-bin variability of scaled y values across binned scaled x. Lower numbers indicate tighter overlap. It is a diagnostic for comparison, not a definitive statistical proof.
4) Why is the metric sometimes “Unavailable”?
If scaled x has no spread or bins lack enough points, the calculator cannot compute variability reliably. Add more rows, broaden the x range, or adjust bin count.
5) Can I mix different x units across sets?
Avoid mixing units. Convert raw x and y to consistent units before rescaling. Otherwise, apparent collapse may be misleading and exponent tuning may compensate for unit errors.
6) Should I include all data points?
Include points from the intended scaling regime. If crossovers or saturation occur, analyze ranges separately and report selection criteria. Sensitivity checks help validate conclusions.
7) How do I use the exported CSV?
Import the CSV into your plotting tool and graph scaled y versus scaled x. Overlay sets by color or marker. Keep the chosen exponents with the exported file for traceability.
1) Purpose of scaling collapse
Scaling collapse checks whether datasets taken at different parameter values can be mapped onto one universal curve. It is widely used in finite-size scaling, critical phenomena, and transport studies where raw curves differ mainly by scale, not by shape. A convincing collapse helps compare models, estimate exponents, and communicate trends clearly.
2) Scaling form and transformed variables
The working hypothesis is y(param, x) = param^b · f(x / param^a). The calculator computes x' = x / param^a and y' = y / param^b. If the hypothesis is consistent, plots of y' versus x' overlap.
3) Choosing param and preparing data
Use a strictly positive param that labels each set, such as system size, field, or a reduced control variable. Keep units consistent across sets before rescaling. Provide enough points per set to compare shapes across the same x' range.
4) Selecting trial exponents
Start from theory when available, then refine with data. A practical approach is a coarse grid search for a and b, followed by small adjustments that minimize mismatch in overlapping regions. Record each trial so your final choice is reproducible.
5) Interpreting the overlap score
The optional quality metric bins x' and measures variability of y' inside each bin. It reports an average coefficient of variation across bins with at least two points. Lower values indicate tighter overlap, but always inspect the scaled table too. For best comparisons, keep the same bin count while testing different exponent pairs.
6) Crossovers, noise, and outliers
Many systems show crossovers where one scaling form holds only over part of the domain. If a single pair of exponents works only for a subset, analyze ranges separately and justify exclusions. Outliers can dominate variability; check measurement errors, saturation, or regime changes.
7) Exporting for plots and documentation
Export CSV to plot y' vs x' in your plotting workflow and to compare multiple exponent trials. Export PDF to keep a compact record of inputs, exponents, and the quality score for lab notebooks, reports, or collaboration.
8) Reporting results responsibly
A good collapse supports the scaling assumption, but it is not definitive evidence by itself. Report uncertainty, sensitivity to exponent changes, and any filtering rules. Combine collapse results with theory, error bars, and independent checks for robust conclusions. If possible, validate exponents using a second observable that scales differently.
FAQs
1) What does “param” represent in the input?
It is the scaling parameter that changes between datasets, such as system size, field, or reduced temperature scale. It must be positive because the rescaling uses powers of param.
2) How should I choose the exponents a and b?
Use theoretical expectations when possible, then refine by checking overlap and the quality score. Try small increments and keep notes so you can justify the final choice.
3) What does the collapse metric mean?
It averages within-bin variability of scaled y values across binned scaled x. Lower numbers indicate tighter overlap. It is a diagnostic for comparison, not a definitive statistical proof.
4) Why is the metric sometimes “Unavailable”?
If scaled x has no spread or bins lack enough points, the calculator cannot compute variability reliably. Add more rows, broaden the x range, or adjust bin count.
5) Can I mix different x units across sets?
Avoid mixing units. Convert raw x and y to consistent units before rescaling. Otherwise, apparent collapse may be misleading and exponent tuning may compensate for unit errors.
6) Should I include all data points?
Include points from the intended scaling regime. If crossovers or saturation occur, analyze ranges separately and report selection criteria. Sensitivity checks help validate conclusions.
7) How do I use the exported CSV?
Import the CSV into your plotting tool and graph scaled y versus scaled x. Overlay sets by color or marker. Keep the chosen exponents with the exported file for traceability.