I was reading about the illumination problem:

"In 1958, a young Roger Penrose used the properties of the ellipse to describe a room with curved walls that would always have dark (unilluminated) regions, regardless of the position of the candle. Penrose's room, illustrated above, consists of two half-ellipses at the top and bottom and two mushroom-shaped protuberances (which are in turn built up from straight line segments and smaller half-ellipses) on the left and right sides. The ellipses and mushrooms are strategically placed as shown, with the red points being the foci of the half-ellipses. There are essentially three possible configurations of illumination. In this figure, lit regions are indicated in white, unilluminated regions are indicated in gray, and the position of the light source is indicated by the black cross-hairs. As can be seen, the entire room (the space within the blue border) can never be fully illuminated. " - http://mathworld.wolfram.com/IlluminationProblem.html

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This made me wonder of the consequences for an classical ideal gas: Suppose we place a box with a gas in a vaccum version of the upper half of this room. Now after some time imagine if the box breaks and the gas begins to spread. No matter how much time has elapsed the gas molecules can never bounce reach the dark regions of the room. Hence, that part of the room will experience $0$ pressure. But this seems rather counterintuitive to me to happen in reality. Is there anything that would stop this effect from being measured in reality? (besides the imperfections of the wall being curved?)


1 Answer 1


Molecules of gas collide with one another whereas photons do so very rarely. Another way of putting that is that light travels in straight lines (for the Penrose analysis) whereas gas molecules do not.

  • $\begingroup$ okay, so at time $ t \to \infty $ the box will have uniform pressure (seems like I asked a silly question, now) . But the time taken to reach the dark spots is some function of the mean free path? If so, any idea about what the function will be? $\endgroup$
    – drewdles
    Commented Mar 13, 2017 at 11:04
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    $\begingroup$ @AnantSaxena Actually I thought it was a great question, bringing out a succinct and illuminating answer. $\endgroup$ Commented Mar 13, 2017 at 11:09
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    $\begingroup$ @AnantSaxena I thought that it was a good question which made me realise that Penrose et al made a key assumption about light travelling in straight lines with prefect reflectors so it then becomes more of a Mathematics problem. According to this table the mean free path at an "ambient" pressure and room temperature is 68 nm. Given how many molecules per cubic metre there are and the speed of the molecules there are and the speed of the molecules, ~500 m/s, it will not take very long to reach an equilibrium state. en.wikipedia.org/wiki/Mean_free_path $\endgroup$
    – Farcher
    Commented Mar 13, 2017 at 11:25

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