A teaching tape for LQ-05, Independent Configurations and Boltzmann Entropy
The locked positions
Demonstrates that the probability is (V/V0)^n only when the points are independent; rigidly locked points give V/V0.
Open LQ-05 with these settings
Where it starts
| Setting | Value |
|---|---|
| Number of Particles (n) | 10 |
| Subvolume Fraction (f) | 0.5 |
| Sampling Trials (trials) | 10000 |
What it changes, and what to expect
A question to answer before the next change
Ten independent particles in half a volume
With 10 independent points, W = (1/2)^10 = 1/1024.
Ten independent points move in the box. At a moment picked at random, what is the chance that all ten are in the left half?
- About one in two The chance for a single point, unchanged by adding points.
- About one in twenty The chance falls with n, but only in proportion to it.
- About one in a thousand Each point independently halves the chance, so ten of them give 1/2^10 = 1/1024.
The answer this tape's author recorded
About one in a thousand
What the author recorded that a reader should see here. Quantity Expected Under W (independent) 9.76563 × 10⁻⁴ modern-si-2019 Set Number of Particles (n) to 60 changes what the run starts from
Ten locked particles moving rigidly
When the 10 points are rigidly locked, W = 1/2.
What the author recorded that a reader should see here. Quantity Expected Under W (locked) 0.5 modern-si-2019
These are the values the tape's author recorded, not a result this page computed. Nothing here runs the instrument. Open LQ-05 with these settings: the link opens it with them already in place. Compare what it gives you with what is recorded here. The settings travel in the link and the laboratory reads them as it loads, so without JavaScript it opens at its worked example instead.
What these numbers were recorded under
| Identity | Recorded as |
|---|---|
| Model | lq05-independent-configurations-v1 version 1 |
| Seed | 12345 |
| Random-stream semantics | version 1 |