Setup

We plot speed distribution for N point particles subject to a common Gaussian thermostat on various billiard tables with finite horizon. The average energy per particle is always set to 1.0.

If you bring two (or more) plots on the screen, then the Kolmogorov-Smirnov test will be automatically applied to each pair and results reported on the right margin.

Notations

A B C

Standard table, obstacles are two disks:
at (0, 0) of radius 0.4 and
at (0.5, 0.5) of radius 0.2

A little bit modified standard table: position of smaller obstacle is (0.5, 0.42)

Simple table made to be different from standard table. Consists of four arcs of a circles.

Mean free paths:

  • "discrete": 0.31
  • "continuous": 0.245

Conductivity matrices:

  • 1 particle: $$ \left( \begin{matrix} 0.0893 & -0.0007 \\ -0.0007 & 0.0892 \end{matrix} \right) $$
  • 5 particles: $$ \left( \begin{matrix} 0.136 & -7.62 \times 10^{-5} \\ -7.62 \times 10^{-5} & 0.136 \end{matrix} \right) $$
  • 10 particles: $$ \left( \begin{matrix} 0.139 & -0.0001 \\ -0.0001 & 0.139 \end{matrix} \right) $$

Mean free paths:

  • "discrete": 0.31
  • "continuous": 0.25

Conductivity matrices:

  • 1 particle: $$ \left( \begin{matrix} 0.0978 & 9.97 \times 10^{-5} \\ 9.97 \times 10^{-5} & 0.0692 \end{matrix} \right) $$
  • 5 particles: $$ \left( \begin{matrix} 0.146 & -3.09 \times 10^{-5} \\ -3.09 \times 10^{-5} & 0.104 \end{matrix} \right) $$
  • 10 particles: $$ \left( \begin{matrix} 0.152 & 0.0002 \\ 0.0002 & 0.107 \end{matrix} \right) $$

Mean free paths:

  • "discrete": 0.431
  • "continuous": 0.357

Conductivity matrices:

  • 1 particle: $$ \left( \begin{matrix} 0.461 & -0.159 \\ -0.159 & 0.0783 \end{matrix} \right) $$ e-values: 0.0209 and 0.518
  • 5 particles: $$ \left( \begin{matrix} 0.705 & -0.242 \\ -0.242 & 0.12 \end{matrix} \right) $$ e-values: 0.033 and 0.792
  • 10 particles: $$ \left( \begin{matrix} 0.718 & -0.25 \\ -0.25 & 0.124 \end{matrix} \right) $$ e-values: 0.0327 and 0.809

Graphs

Graph below is interactive. On the right margin, the Kolmogorov-Smirnov test results are shown.

For each pair of distributions, we give D (the geometric distance, not quite meaningful) and the p-value. A common interpretation of the p-value is as follows: if it is greater than 10%, then the two distributions are close; if it is less than 1%, then they are substantially different. Between 10% and 1% we cannot make a definite conclusion, more simulations are perhaps necessary...

Each empirical graph was built from (almost independent) observations. This number will grow as programs continue running (thanks to Alabama Super Computer for resources!).

Compasison of distributions:

comparison goes here

Data will be here...