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In Huang's book on Statistical mechanism and Statistical interpretation of Entropy, it is not mentioned that $\Omega$ is the phase space volume, but it is the states of the system. So, how does entropy be called as logarithm of phase space volume? A detailed explanation and reference where it is shown that Entropy as phase space volume will be appreciated. Where can I find the derivation that entropy is the phase space volume?

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Boltzmann's formula states that for an isolated system in thermal equilibrium $S=k_B \log \mathrm{deg}$, where deg is the degeneracy (i.e., number of) accessible microstates. A semi-classical argument (not reproduced here) shows that this degeneracy can be estimated as $M:=\frac{\Omega}{h^N}$ where $\Omega$ is the volume of the accessible phase space and $N$ is the number of degrees of freedom (i.e., the phase space is $2N$ dimensional). An example that clearly illustrates the counting in a simple example is here.

Edit: People usually drop the $h^N$ factor (it is needed, at the very least, on grounds of dimensionality). However, since it appears inside a log and leads to an additive constant, it can and is dropped. Famously, Boltzmann's tomb has S = k log W written on it. So the original reference is Boltzmann himself. If you want to cite someone, might as well cite the great man himself. The connection with Shannon information entropy $S_{info}=-\sum_{i=1}^M p_i \log p_i$ can be seen if you realise that for $M$ accessible microstates, the maximum value of the information entropy is $S_{info}=\log M$ which is Boltzmann's answer up to the multiplicative factor of $k_B$. (I have nothing to say about Kolmogorov entropy.)

Edit2: I don't think there is a proof for Boltzmann's formula. It is usually a fundamental postulate in statisical mechanics.

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Thank you for your reply.Can you provide a reference where it is clearly mentioned that $\Omega$ is the phase space volume. In the link here books.google.ca/… Eq(1.27) mentions about the phase space volume but it is not clear and without any reference.Actually,I need to show and cite that entropy is logarithm of phase space volume. –  Srishti M Jan 22 at 4:17
The thing is I have used entropy as phase sapce volume being equal to Kolmogorov entropy but I cannot justify how these two are linked and if at all I can use this phase sapce in communication theory. COuld you provide some insights please as to whether Entropy = logarithm of phase space volume can be used in information theory or communication. –  Srishti M Jan 22 at 4:28
Here is an article by Lebowitz (an important name in the mathematical foundations of statistical mechanics) that discusses Boltzmann's entropy: scholarpedia.org/article/… –  suresh Jan 22 at 5:07
Thank you for the link but no where it is mentioned that Tau is the phase space volume (eq (2))! Also, could you kindly provide a link where the connection between Shannon entropy is mentioned.I am not from physics background and find it verry hard to find a connection.Under what conditions can I apply the equivalence to Shannon in information theory? –  Srishti M Jan 22 at 5:30
Yes, the connection between Shannon's information entropy and Boltzmann's entropy is mentioned in the 1948 paper of Shannon (see the section titled "Choice, Uncertainty and Entropy"). I would strongly recommend reading that paper, if you haven't done so. –  suresh Jan 22 at 7:18

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