Talk:Sackur–Tetrode equation

message: Do other users think that we should include the derivation from the Schrodinger equation? It is relatively simple. Tiberius Curtainsmith  —Preceding undated comment added 18:55, 30 July 2009 (UTC)
thread_title: Derivation

message: The phrase, "Making sense of something using X" might imply to a neophyte that X is needed to make sense of something. Many can "make sense" of the equation through the thermodynamic or the information theory perspectives. The article does not signal that entropy has these different perspectives (even though it is obvious to experts that the equation was originally derived in the ... perspective). I will change the title and first sentence of the section on information theory to signal the two interpretations. Each perspective has its champions, but the enthusiasm of their POV should not be reflected in titles. Laburke (talk) 13:33, 18 September 2011 (UTC)
replies: text: I nominate that self/book-promotional section for deletion. "Interpretation of the equation through information theory" Kotika98 (talk) 03:24, 18 May 2015 (UTC)
replies: text:  I wrote the section, and have restored it, with modifications. I've changed the narrative to be factual, rather than a "he did this and that" kind of story. Point taken. This is not a "POV" kind of thing, its a clear mathematical derivation of the equation using information theoretic concepts. I have no desire to promote any book, and I am certainly not the author of the book. Never met or communicated with him. PAR (talk) 06:24, 29 December 2015 (UTC)
replies: text:  It is not clear how an "information theoretic concept" put together with a "mathematical derivation" could possibly produce the "quantum mechanical uncertainty principle" term with entropy equal to -3N ln(h). I could be convinced if such a term had an independent meaning in some other context outside of the S-T formula. AFAIK it does not have an independent meaning, and in fact it is nonsense altogether, since the S-T formula is incapable of producing S(T=0)=0 which would indeed fix the additive constants such as the 5/2 and/or ln(h) as it is claimed here. Kotika98 (talk) 15:51, 21 April 2016 (UTC)
replies: text: Once again, that section is total nonsense, I have had a lengthy exchange with the author of that section, "PAR" without a satisfactory resolution. He/she has since added "explanations" which amount to original research to try and reconcile the gibberish found in the crackpot book with common sense. Since I am a grownup and will not get dragged into a revert war, someone else should delete this section, preferably someone with a real name and identity. Kotika98 (talk) 06:35, 8 January 2018 (UTC)
replies: text: That section, as currently written, appears to be coherent and factually correct. All portions of the equation "make sense" and have the correct values. The algebraic manipulations appear to be correct. If there is something specifically incorrect or wrong, you have to point it out. If something feels incoherent, or sounds like gibberish, you have to point that out, too. As written, I see nothing wrong, and so levying accusations of "total nonsense" is uncalled for. 67.198.37.16 (talk) 20:59, 19 May 2024 (UTC)
thread_title: Information Theory

message: Volume and energy are not the direct source of the states that generate entropy, so I wanted to express it in terms of x*p/h' number of  states for each N.  Someone above asked for a derivation from the uncertainty principle ("the U.P.") and he says it's pretty easy. S-T pre-dates U.P., so it may be only for historical reasons that a more efficient derivation is not often seen.
The U.P. says x'p'>h/4pi where x'p' are the standard deviations of x and p.  The x and p below are the full range, not the 34.1% of the standard deviation so I multiplied x and p each by 0.341. It is 2p because p could be + or - in x,y, and z. By plugging the variables in and solving, it comes very close to the S-T equation. For Ω/N=1000 this was even more accurate than S-T, 0.4% lower.
Sackur-Tetrode equation:
S = k_B \ln\left(\frac{\Omega^N}{N!}\right) \approx k_B N \left(\ln\left(\frac{\Omega}{N} \right) +1\right)
where
\Omega= \left(\frac{px}{\hbar/2}\right)^{3} = \left(\frac{p_x x_x}{\hbar/2}\right)\left(\frac{p_y x_y}{\hbar/2}\right)\left(\frac{p_z x_z}{\hbar/2}\right)
\sigma=0.341, x=\sigma V^\frac{1}{3},  p = \sigma\left(\frac{2mU*b}{N!}\right)^\frac{1}{2}, U*b=K.E.=3/2 k_B T
Stirling's approximation N!=(N/e)^N is used in two places that results in a 1/N^(5/2) and and e^(5/2) which is where the 5/2 factor comes from.  The molecules' internal energy U is kinetic energy for the monoatomic gas case for which the S-T applies. b=1 for monoatomic, and it may simply be changed for other non-monoatomic gases that have a different K.E./U ratio. The equation for p is the only difficult part of getting from the U.P. to the S-T equation and it is difficult only because the thermodynamic measurements T (kinetic energy per atom) and V are an energy and a distance where the U.P. needs x*p or t*E. This strangeness is where the 3/2 and 5/2 factors come from. The 2m is to get 2*m*1/2*m*V^2 = p^2. Boltzmann's entropy assumes it is a max for the given T, V, and P which I believe means the N's are evenly distributed in x^3 and assumes all are carrying the same magnitude p momentum.
[edit: see information theory entropy talk page for a better way]
To show how this can come directly from information theory, first remember that Shannon's H function is valid only for a random variable. In this physical case, there are only 2 possible values that each phase space can have: with an atom in it carrying energy, or not, so it is like a binary file. But unlike normal information entropy, some or many of the atoms may have zero momentum. The only requirement is that the total energy be the same, so the physical system has more freedom of choice than you would expect. Physical entropy can use anywhere from N to 1 symbols (atoms) to carry the same message (the energy), whereas information entropy is typically stuck with N.  The modified Shannon H shown below is the sum of the surprisals and is equal to the information (entropy) content in bits. The left sum is the information contributions from the empty phase space slots and the right side are those where an atom occurs. The left sum is about 0.7 without regard to N (1 if ln() had been used) and the right side is about 17*N for a gas at room temperature (for Ω/N ~ 100,000 states/atom):  
H*N = \sum_{i=1}^{\Omega-N} \log_2\left(\frac{\Omega}{\Omega-i N/(\Omega-N)}\right) + \sum_{j=0}^{N-1} \log_2\left(\frac{\Omega}{N-j}\right) \approx \log_2\left(\frac{\Omega^N}{N!}\right)
Physical entropy S then comes directly from this Shannon entropy:
S = H*N k_B \ln(2)
A similar procedure can be applied to phonons in solids to go from information theory to physical entropy. For reference here is a 1D oscillator in solids:
S = k_B \left[ \ln \left( \frac{k_B T}{hf}\right) +1 \right]
Ywaz (talk) 02:37, 15 January 2016 (UTC)
replies: text: Have you read the Ben-Naim book? - it is excellent. I looked at the above and the approximations in σ worry me. Ben-Naim did it quite straightforwardly, the article only states the results. PAR (talk) 06:58, 25 January 2016 (UTC)
replies: text: I looked at U.P. and saw it covers only 34.1% of the possible positions for each of x and p, so I naively applied σ to each, and surprisingly it allowed the (4*pi/3)^3/2 to be removed from the equation. Apparently the π and sqrt(2) in the normal distribution that determine σ already have the necessary "spherical" math in it.  In any event, I used a shorter reasoning process, had fewer variables in the result (smaller Kolmogorov complexity), and it gave the right answer to a high degree of precision. If I had used more instead of fewer constants, my reasoning would be be much more suspect.  In whatever way it deviates from the exact proper physics, it appears to be an OK mythology with only some minor loss in the simplification of the ideas.  I am applying the de Broglie wavelength to get the number of possible states.  I found a reference that refers Ben_Naim doing the same. Ywaz (talk) 13:06, 25 January 2016 (UTC)
replies: text: In the references, the Ben-Naim reference links to a preview of his book, which might be useful. PAR (talk) 15:38, 27 January 2016 (UTC)
thread_title:  Derivation from uncertainty principle and relation to information entropy 

message: Would be nice to see the equivalent eqn in arbitrary dimension d, instead of just d=3. 67.198.37.16 (talk) 21:06, 19 May 2024 (UTC)
thread_title:  In arbitrary dimensions?