Counting the number of configurations (Entropy)

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    Counting Entropy
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Discussion Overview

The discussion revolves around the concept of entropy in statistical mechanics, specifically focusing on the method of counting configurations in a gas system as described by Roger Balian. Participants explore the implications of using a discrete model for positions while potentially ignoring velocities, and the relationship between quantum mechanics and classical phase space.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant questions how Balian can replace continuous volume with a discrete number of sites (Q = 10^100) while counting possible positions for gas molecules.
  • Another participant suggests that Balian may be considering fast processes where changes in the wall do not affect the velocities of the gas particles, implying that only the number of available positions matters for entropy variation.
  • A different participant raises a concern about the validity of ignoring velocities in the counting process, noting that the distance between sites is comparable to Planck's length.
  • One participant proposes that if compression occurs at high velocities relative to the mean velocity of gas particles, the method of counting positions without considering speeds could be justified.
  • A later reply introduces a general principle from statistical mechanics, mentioning that the number of quantum states can be derived from classical phase space volume divided by Planck's constant raised to the power of the number of degrees of freedom.

Areas of Agreement / Disagreement

Participants express differing views on the appropriateness of ignoring velocities in the context of counting configurations for entropy. There is no consensus on the validity of Balian's approach or the implications of the assumptions made.

Contextual Notes

Participants highlight potential limitations in the assumptions regarding the relationship between position and velocity in the context of entropy calculations, as well as the implications of quantum mechanics on classical models.

naima
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Hi all,

Entropy uses the 6N dimensional phase space. But ...
Roger Balian in "Scientific American" takes one liter gas in a cube and he writes:
I can replace the continuous volume by Q = 10^100 sites after having
evacuated the speeds (he says this is possible with quantum mechanics)
He then counts the number of possible places for the N molecules and get the entropy.

How can he do that?
 
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I guess he is considering very fast processes in which the modifications on the wall of this cube does not introduce modifications in the pattern of velocities, just the number of positions available will change, so, in order to evaluate the entropy variation one has only to monitor the number of available positions.

I am not sure.

Best wishes

DaTario
 
Surprisingly, the distance between 2 sites is like Planck's length!.
Is it really correct to ignore speeds in that counting?
 
IMO, if you are doing the compression with high velocity compared with the mean velocity of the particles in the gas and by small steps, it seems one can well defend this procedure.

Best wishes

DaTario
 
naima said:
Hi all,

Entropy uses the 6N dimensional phase space. But ...
Roger Balian in "Scientific American" takes one liter gas in a cube and he writes:
I can replace the continuous volume by Q = 10^100 sites after having
evacuated the speeds (he says this is possible with quantum mechanics)
He then counts the number of possible places for the N molecules and get the entropy.

How can he do that?

Is the article you're referring to available online? There is a general principle in statistical mechanics that you can get the number of quantum states in a given energy range by calculating the classical phase space volume and dividing by h^M, where h is Planck's constant, M is the number of degrees of freedom (3N for N atoms in a monatomic gas); perhaps that's what he's talking about?
 

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