# Connection between Different Kinds of Entropy (Boltzmann, Volume, and Surface Entropies)

$\omega$ is said to be an energy width. I am not entirely sure what that means, but I believe it means that it is a variable stuck in to annihilate the units (like $h$).

I am only familiar with the Boltzmann Entropy, but by further reading, it seems that only the volume entropy reproduces thermodynamics.

The volume entropy Sv and associated Tv form a close analogy to thermodynamic entropy and temperature. It is possible to show exactly that

$dE=T_{\rm {v}}dS_{\rm {v}}-\langle P\rangle dV,$

Ignoring the warning that preferred solution to these problems is avoid use of the microcanonical ensemble, I am concerned at where these entropies come from. I have a reasonable intuition for the Boltzmann entropy as is is a special case of entropy from information theory, but these other entropies seem to not be based on information theory. Furthermore, I have trouble finding additional material on them.

Can you explain the origin of volume entropy and surface entropy? If possible explain their origin from information theory so I can understand them better.

• You should clarify what $\omega$ signifies. Jul 10, 2017 at 22:58
• @RubenVerresen Did my best, but I am not entirely sure. Perhaps you can do better.
– user92177
Jul 10, 2017 at 23:09
• The relevance of the width of the energy shell is addressed in p38 of James Sethna's book, maybe that is helpful. The book is freely available: pages.physics.cornell.edu/~sethna/StatMech/… Jul 10, 2017 at 23:30
• @RubenVerresen Are you referring to the $\delta E$ on page 38? I think the $\omega$ is different. I could be wrong. Its used like $W=\int \ldots \int {\frac {1}{h^{n}C}}f({\tfrac {H-E}{\omega }})\,dp_{1}\ldots dq_{n}$ where $f$ is a rectangle function. Seemed to be to just be a choice of normalization
– user92177
Jul 11, 2017 at 0:26

You can find a very good discussion in K. Huang, Statistical Mechanics, chapter 6. I will use the notation of Wikipedia instead of that of Huang.

In the microcanonical ensemble, you are considering a system with energy between $E$ and $E+\omega$, with $\omega \ll E$.

The total phase space volume with energy between $E$ and $E+\omega$ is

$$W(E)=\int_{E<H(p,q)<E+\omega} d^{3N} p d^{3N} q$$

If you define the quantity

$$v(E)=\int_{H<E} d^{3N} p d^{3N} q$$

you can see that

$$W(E) = v(E+\omega)-v(E)$$

and in the limit in which $\omega/E \rightarrow 0$, we can write

$$W(E) = \frac{\partial v}{\partial E} \omega$$

Now, the Boltzmann entropy is defined as

$$S= k \log W(E)$$

It can be shown that the following definitions

\begin{align} S & = k \log W(E)\\ S & = k \log \left( \frac{\partial v}{\partial E} \right)\\ S & = k \log v(E) \end{align}

are equivalent up to a constant of order $\log N$ or smaller. Since $S$ is usually of order $N$, it means that we can consider the above expressions as practically equivalent.

Basically, $W$ is an "hypershell", $v$ an hypervolume and $\frac{\partial v}{\partial E}$ an hypersurface. We are basically saying that it doesn't matter if we consider the shell, the volume or the surface: the result will be the same. The reason is the large number of degrees of freedom of the system: $6N$, where usually $N \simeq 10^{23}$.

For an ideal gas,

$$v(E) = c_N V^N E^{3N/2}$$

where $c_N$ is a constant and $V$is the volume. Therefore, in this simple case you can verify yourself that the above expressions are equal up to a constant of order $\log N$ or less. In the more general case, this is a more complicated task.

• I think, it is important to note that the volume and surface entropy differ for specific, small systems. Several examples are listed in the Negative Temperature Wikipedia article. Jul 11, 2017 at 8:13
• @JulianHelfferich Yes, of course. The above considerations are valid in the limit of large N. Jul 11, 2017 at 9:13
• Does information theory have similar problems?
– user92177
Jul 11, 2017 at 17:16
• @aidan.plenert.macdonald Information theory uses a more abstract definition of entropy: $S= -\sum_i p_i \log p_i$. This definition is quite unambiguous. Anyway, I would't really call these "problems": as long as $N$ is large enough (which anyway is the assumption behind all of statistical mechanics), you can calculate the entropy in the microcanonical ensemble in three different ways. That's all, there is no real problem. Jul 11, 2017 at 17:42

This is how I picture the different entropies: The microcanonical ensemble states that the total energy E is constant. This condition selects only specific states from the phase space, forming a surface in phase space (this sketch from wikipedia is a nice illustration). Here, $\omega$ is the (infinitesimal) width of this surface. This is the basis of the Boltzmann and surface entropy, which differ only by a constant factor. The Gibbs or volume entropy on the other hand considers all states with an energy smaller than $E$, which forms a volume in phase space.

In the formulas, the difference is that the volume entropy uses $v(E)$, the phase space volume with all states of energy smaller than $E$. This is the cumulative function of the number of states at a given $E$. Thus, the surface entropy uses the derivative $\frac{dv}{dE}$, where $\omega$ is the infinitesimal width of the surface.

If the canonical ensemble is used or if the system is large enough, both definitions of entropy give the same results. However, in small systems they can differ significantly. For example, the volume entropy always has a positive temperature $T_s > 0$, whereas the surface entropy can lead to negative temperatures if the number of accessible states decrease with increasing energy.

Which form of entropy is appropriate in which situation is still heavily discussed in the scientific community. See for example  vs  and 

 J. Dunkel and S. Hilbert, Inconsistent thermostatistics and negative absolute temperatures, Nature Physics 10, 67-72 (2014)

 R. Swendsen and J.-S. Wang, Gibbs volume entropy is incorrect, Phys. Rev. E 92, 020103 (2015)

 J. Poulter, In defense of negative temperature, Phys. Rev. E 93, 032149 (2016)