2

Suppose I have N ideal gas particles of unknown types, but they have the same mass. I then sample a negligible number of particles and conclude I have actually have 3 different colors of particles. That is, my detector measures the color of a random sample of particles (and nothing else) and by chernoff bound from a small sample size I conclude (with high probability) that I have some mixture $ p_{1} + p_{2} + p_{3} = 1 $ so that I have $ p_{1} N $ red particles, $ p_{2} N $ blue particles, and $ p_{3} N $ green particles. Then from Sackur-Tetrode: $$ S_{ \text{before sampling} } = k \, N \left( ln \left[ \frac{V}{N} \left( \frac{ 4 \pi m }{ 3 h^{2} } \frac{U}{N} \right)^{3/2} \right] + \frac{5}{2} \right) $$

$$ S_{ \text{after sampling} } = \sum_{i=1}^{3} p_{i} k \, N \left( ln \left[ \frac{V}{p_{i} N} \left( \frac{ 4 \pi m }{ 3 h^{2} } \frac{U}{ p_{i} N} \right)^{3/2} \right] + \frac{5}{2} \right) = S_{ \text{before sampling} } - \frac{5}{2} k \, N \left( \sum_{i=1}^{3} p_{i} ln \left( p_{i} \right) \right) $$

So recognizing some feature of my particles, regardless of how relevant to any of the physical properties of the particles, seems to change the entropy. If entropy has physical consequences, shouldn't there be a way to know if to regard or disregard properties of relevance or irrelevance in a system? Otherwise it would seem that inconsequential features may change my physical understanding.

valerio
  • 16,231
user73236
  • 151
  • Precisely what are the states of the system before and after? – Chet Miller Jun 23 '17 at 11:29
  • before and after I "discover" the color-ness of the particles. – user73236 Jun 23 '17 at 16:13
  • Why not an expression like this: $$ S ( \vec{N}, \vec{V}, \vec{U} ) = S_{Sackur-Tetrode} + \frac{5}{2} k , N \left( \sum_{i}^{ \dim{N} } \frac{ N_{i} }{ N } \ln \left( \frac{ N_{i} }{ N } \right) + \ln \left( \frac{ V_{i} }{ V } \right) + \ln \left( \frac{ U_{i} }{ U } \right) \right) $$ which would be invariant to this sort of behavior it seems to me. – user73236 Jun 23 '17 at 16:33
  • Related: https://physics.stackexchange.com/questions/315104/does-entropy-depend-on-the-observer – CDCM Jun 23 '17 at 17:46
  • Is it necessarily the case that we cannot find entropy expressions that are invariant under irrelevant additional information? Is it contradictory to the axioms of entropy that $ S(N_{1}+N_{2}, V, U) = S(N_{1}, V, U) + S(N_{2}, V, U) $, or do we just need to find clever expressions to capture this? – user73236 Jun 23 '17 at 17:59

0 Answers0