Calculator input form
The page uses a single-column flow. The calculator fields switch to three columns on large screens, two on smaller screens, and one on mobile.
Formula used
Core estimate:
TR ≈ C × τ × Ω
Entropy relation:
Ω = e^(S/kB), so TR ≈ C × τ × e^(S/kB)
Bit-based relation:
Ω = 2^H, so TR ≈ C × τ × 2^H
Logarithmic form used for numerical stability:
log10(TR) = log10(C) + log10(τ in seconds) + log10(Ω)
Poincaré recurrence arguments apply to finite, bounded, measure-preserving systems. This calculator gives a useful theoretical estimate, not a universal prediction for every real-world physical process.
How to use this calculator
- Choose the mode matching your available data.
- Enter a base interval τ and its unit.
- Set the model multiplier C if your model needs correction.
- Provide entropy, microstates, or bits depending on mode.
- Press the calculate button to view the result above.
- Review logarithmic outputs for extremely large magnitudes.
- Download CSV for data capture or PDF for reporting.
Example data table
| Example | Mode | Inputs | log10 TR (s) | Approximate result |
|---|---|---|---|---|
| Small entropy ratio | S/kB | τ = 1 ns, C = 1, S/kB = 10 | -4.657 | 2.20 × 10^-5 s |
| Bit-based model | Bits | τ = 1 s, C = 1, H = 128 | 38.532 | 3.40 × 10^38 s |
| Finite state count | Microstates | τ = 1 µs, C = 1, Ω = 1 × 10^12 | 6 | 1 × 10^6 s |
Frequently asked questions
1) What is Poincaré recurrence time?
It is the estimated time for a bounded dynamical system to return arbitrarily close to a previous state. In practice, it is often unimaginably large.
2) Is this result exact?
No. It is a model-based estimate built from a chosen base interval, state-space size, and optional correction factor. Real systems can deviate strongly.
3) Why does the calculator show logarithms?
Recurrence times commonly exceed normal floating-point limits. Logarithms let the calculator stay stable and still report meaningful magnitudes.
4) When should I use the S/kB mode?
Use it when entropy is already expressed as a dimensionless ratio. It is especially helpful for very large theoretical systems where raw entropy input can overflow.
5) What does the model multiplier C do?
It rescales the estimate. You can use it to represent a different mixing time, coarse-graining choice, or a custom correction from your model.
6) Why are some results far bigger than the age of the universe?
Because the number of accessible microstates often grows exponentially with entropy. Even modest entropy ratios can produce enormous recurrence estimates.
7) Can I use this for open or dissipative systems?
Only with caution. The classical recurrence argument assumes an effectively closed, bounded, measure-preserving system, which many real systems do not satisfy.
8) Which mode should I choose?
Use microstates if Ω is known, bits if information content is known, entropy units for thermodynamic data, and S/kB for cleaner large-scale theory work.