Setup

We will present a simplified version of the calculation. It differs from the full calculation in three ways:

1. The ideal gas consists of particles confined to one spatial dimension.
2. We keep only the terms of order ${\displaystyle n\log(n)}$, dropping all terms of size n or less, where n is the number of particles. For our purposes, this is enough, because this is where the Gibbs paradox shows up and where it must be resolved. The neglected terms play a role when the number of particles is not very large, such as in computer simulation and nanotechnology. Also, they are needed in deriving the Sackur–Tetrode equation.
3. The subdivision of phase space into units of the Planck constant (h) is omitted. Instead, the entropy is defined using an integral over the "accessible" portion of phase space. This serves to highlight the purely classical nature of the calculation.

We begin with a version of Boltzmann's entropy in which the integrand is all of accessible phase space:

${\displaystyle S=k\ln \Omega =k\ln \oint \limits _{H({\vec {p}},{\vec {q}})=E}(d{\vec {p}}\,)^{n}(d{\vec {q}}\,)^{n}}$

The integral is restricted to a contour of available regions of phase space, subject to conservation of energy. In contrast to the one-dimensional line integrals encountered in elementary physics, the contour of constant energy possesses a vast number of dimensions. The justification for integrating over phase space using the canonical measure involves the assumption of equal probability. The assumption can be made by invoking the ergodic hypothesis as well as the Liouville's theorem of Hamiltonian systems.

(The ergodic hypothesis underlies the ability of a physical system to reach thermal equilibrium, but this may not always hold for computer simulations (see the Fermi–Pasta–Ulam–Tsingou problem) or in certain real-world systems such as non-thermal plasmas.)

Liouville's theorem assumes a fixed number of dimensions that the system 'explores'. In calculations of entropy, the number dimensions is proportional to the number of particles in the system, which forces phase space to abruptly change dimensionality when particles are added or subtracted. This may explain the difficulties in constructing a clear and simple derivation for the dependence of entropy on the number of particles.

For the ideal gas, the accessible phase space is an (n − 1)-sphere (also called a hypersphere) in the n-dimensional ${\displaystyle {\vec {v}}}$ space:

${\displaystyle E=\sum _{j=1}^{n}{\frac {1}{2}}mv_{j}^{2}\,,}$

To recover the paradoxical result that entropy is not extensive, we integrate over phase space for a gas of ${\displaystyle n}$ monatomic particles confined to a single spatial dimension by ${\displaystyle 0<x<\ell }$. Since our only purpose is to illuminate a paradox, we simplify notation by taking the particle's mass and the Boltzmann constant equal to unity: ${\displaystyle m=k=1}$. We represent points in phase-space and its x and v parts by n and 2n dimensional vectors:

${\displaystyle {\vec {\xi }}=[x_{1},...,x_{n},v_{1},...,v_{n}]=[{\vec {x}},{\vec {v}}\,]}$   where   ${\displaystyle {\vec {x}}=[x_{1},...,x_{n}]}$   and   ${\displaystyle {\vec {v}}=[v_{1},...,v_{n}]\,.}$

To calculate entropy, we use the fact that the (n-1)-sphere, ${\displaystyle \Sigma v_{j}^{2}=R^{2}\,,}$ has an (n − 1)-dimensional "hypersurface volume" of

${\displaystyle {\tilde {A}}_{n}(R)={\frac {n\pi ^{n/2}}{(n/2)!}}R^{n-1}\,.}$

For example, if n = 2, the 1-sphere is the circle ${\displaystyle {\tilde {A}}_{2}(R)=2\pi R}$, a "hypersurface" in the plane. When the sphere is even-dimensional (n odd), it will be necessary to use the gamma function to give meaning to the factorial; see below.

Gibbs paradox in a one-dimensional gas

Gibbs paradox arises when entropy is calculated using an ${\displaystyle n}$ dimensional phase space, where ${\displaystyle n}$ is also the number of particles in the gas. These particles are spatially confined to the one-dimensional interval ${\displaystyle \ell ^{n}}$. The volume of the surface of fixed energy is

${\displaystyle \Omega _{E,\ell }=\left(\int dx_{1}\int dx_{2}...\int dx_{n}\right)\left(\underbrace {\int dv_{1}\int dv_{2}...\int dv_{n}} _{\Sigma v_{i}^{2}=2E}\right)}$

The subscripts on ${\displaystyle \Omega }$ are used to define the 'state variables' and will be discussed later, when it is argued that the number of particles, ${\displaystyle n}$ lacks full status as a state variable in this calculation. The integral over configuration space is ${\displaystyle \ell ^{n}}$. As indicated by the underbrace, the integral over velocity space is restricted to the "surface area" of the n − 1 dimensional hypersphere of radius ${\displaystyle {\sqrt {2E}}}$, and is therefore equal to the "area" of that hypersurface. Thus

${\displaystyle \Omega _{E,\ell }=\ell ^{n}{\frac {n\pi ^{n/2}}{(n/2)!}}(2E)^{\frac {n-1}{2}}}$

Click to view the algebraic steps

We begin with: ${\displaystyle \Omega _{E,\ell }=\ell ^{n}{\frac {n\pi ^{n/2}}{(n/2)!}}(2E)^{\frac {n-1}{2}}}$ ${\displaystyle \ln \Omega _{E,\ell }=\ln \left(\ell ^{n}n\pi ^{n/2}(2E)^{\frac {n-1}{2}}\right)-\ln \left[(n/2)!\right]}$ Both terms on the right hand side have dominant terms. Using the Stirling approximation for large M, ${\displaystyle \ln M!\approx M\ln M}$ ${\displaystyle -M+\ln {\sqrt {2\pi M}}}$, we have: ${\displaystyle \ln \left(\ell ^{n}n\pi ^{n/2}(2E)^{\frac {n-1}{2}}\right)=\underbrace {\ln \left(\ell ^{n}E^{\frac {n}{2}}\right)} _{\text{important}}+\underbrace {\ln \left({\frac {n(2\pi )^{\frac {n}{2}}}{\sqrt {2E}}}\right)} _{\text{drop}}}$ ${\displaystyle {\begin{aligned}-\ln[(n/2)!]&\approx -{\frac {n}{2}}\ln {\frac {n}{2}}+{\frac {n}{2}}-\ln {\sqrt {n\pi }}\\&=\underbrace {-{\frac {n}{2}}\ln n} _{\text{keep}}+\underbrace {{\frac {n}{2}}\ln 2+{\frac {n}{2}}-\ln {\sqrt {n\pi }}} _{\text{drop}}\\\end{aligned}}}$ Terms are neglected if they exhibit less variation with a parameter, and we compare terms that vary with the same parameter. Entropy is defined with an additive arbitrary constant because the area in phase space depends on what units are used. For that reason it does not matter if entropy is large or small for a given value of E. We instead to seek how entropy varies with E, i.e., we seek ${\displaystyle \partial S/\partial E}$: An expression such as ${\displaystyle E^{\frac {1}{2}}}$ is much less important than an expression like${\displaystyle E^{\frac {n}{2}}}$ An expression like ${\displaystyle \pi ^{n}}$ is much less important than an expression like ${\displaystyle n^{n}}$. Note that the logarithm is not a strongly increasing function. The neglect of terms proportional to n compared with terms proportional to n ln n is only justified if n is extremely large. Combining the important terms: ${\displaystyle {\begin{aligned}\ln \Omega _{E,\ell }&=\ln \left(\ell ^{n}E^{\frac {n}{2}}\right)-{\frac {n}{2}}\ln n\\&=n\ln \ell +{\frac {n}{2}}\ln \left({\frac {E}{n}}\right)\\\end{aligned}}}$

After approximating the factorial and dropping the small terms, we obtain

${\displaystyle {\begin{aligned}\ln \Omega _{E,\ell }&\approx n\ln \ell +n\ln {\sqrt {\frac {E}{n}}}+{\text{const.}}\\&=\underbrace {n\ln {\frac {\ell }{n}}+n\ln {\sqrt {\frac {E}{n}}}} _{\text{extensive}}+\,n\ln n+{\text{const.}}\\\end{aligned}}}$

In the second expression, the term ${\displaystyle n\ln n}$ was subtracted and added, using the fact that ${\displaystyle \ln \ell -\ln n=\ln(\ell /n)}$. This was done to highlight exactly how the "entropy" defined here fails to be an extensive property of matter. The first two terms are extensive: if the volume of the system doubles, but gets filled with the same density of particles with the same energy, then each of these terms doubles. But the third term is neither extensive nor intensive and is therefore wrong.

The arbitrary constant has been added because entropy can usually be viewed as being defined up to an arbitrary additive constant. This is especially necessary when entropy is defined as the logarithm of a phase space volume measured in units of momentum-position. Any change in how these units are defined will add or subtract a constant from the value of the entropy.

Two standard ways to make the classical entropy extensive

As discussed above, an extensive form of entropy is recovered if we divide the volume of phase space, ${\displaystyle \Omega _{E,\ell }}$, by n!. An alternative approach is to argue that the dependence on particle number cannot be trusted on the grounds that changing ${\displaystyle n}$ also changes the dimensionality of phase space. Such changes in dimensionality lie outside the scope of Hamiltonian mechanics and Liouville's theorem. For that reason it is plausible to allow the arbitrary constant to be a function of ${\displaystyle n}$.^[7] Defining the function to be, ${\displaystyle f(n)=-{\frac {3}{2}}n\ln n}$, we have:

${\displaystyle {\begin{aligned}S=\ln \Omega _{E,\ell }&\approx n\ln \ell +n\ln {\sqrt {E}}+{\text{const.}}\\&=n\ln \ell +n\ln {\sqrt {E}}+f(n)\\\ln \Omega _{E,\ell ,n}&\approx n\ln {\frac {\ell }{n}}+n\ln {\sqrt {\frac {E}{n}}}+{\text{const.}},\\\end{aligned}}}$

which has extensive scaling:

${\displaystyle S(\alpha E,\alpha \ell ,\alpha n)=\alpha \,S(E,\ell ,n)}$

Swendsen's particle-exchange approach

Following Swendsen,^[5][6] we allow two systems to exchange particles. This essentially 'makes room' in phase space for particles to enter or leave without requiring a change in the number of dimensions of phase space. The total number of particles is ${\displaystyle N}$:

• ${\displaystyle n_{A}}$ particles have coordinates ${\displaystyle 0<x<\ell _{A}}$.

  The total energy of these particles is ${\displaystyle E_{A}}$

• ${\displaystyle n_{B}}$ particles have coordinates ${\displaystyle 0<x<\ell _{B}}$.

  The total energy of these particles is ${\displaystyle E_{B}}$

• The system is subject to the constraints, ${\displaystyle E_{A}+E_{B}=E}$ and ${\displaystyle n_{A}+n_{B}=N}$

Taking the integral over phase space, we have:

${\displaystyle {\begin{aligned}\Omega _{E,\ell ,N}&=\left(\underbrace {\int dx_{?}...\int dx_{?}} _{n_{A}\;{\text{terms}}}\underbrace {\int dx_{?}...\int dx_{N}} _{n_{B}\;{\text{terms}}}\right)\left(\underbrace {\int dv_{1}\int dv_{2}...\int dv_{N}} _{\Sigma v^{2}=2E_{A}\;{\text{or}}\;\Sigma v^{2}=2E_{B}}\right)\\&=\left(\ell _{A}\right)^{n_{A}}\left(\ell _{B}\right)^{n_{B}}\left(\underbrace {\frac {N!}{n_{A}!n_{B}!}} _{\text{combination}}\right)\left(\underbrace {{\frac {n_{A}\pi ^{n_{A}/2}}{(n_{A}/2)!}}(2E_{A})^{\frac {n_{A}-1}{2}}} _{n_{A}-{\text{sphere}}}\right)\left(\underbrace {{\frac {n_{B}\pi ^{n_{B}/2}}{(n_{B}/2)!}}(2E_{B})^{\frac {n_{B}-1}{2}}} _{n_{B}-{\text{sphere}}}\right)\end{aligned}}}$

The question marks (?) serve as a reminder that we may not assume that the first n_A particles (i.e. 1 through n_A) are in system A while the other particles (n_B through N) are in system B. (This is further discussed in the next section.)

Taking the logarithm and keeping only the largest terms, we have:

$${\displaystyle S=\ln \Omega _{E,\ell ,N}\approx n_{A}\ln \left({\frac {n_{A}}{\ell _{A}}}{\sqrt {\frac {E_{A}}{\ell _{A}}}}\right)+n_{B}\ln \left({\frac {n_{B}}{\ell _{B}}}{\sqrt {\frac {E_{B}}{\ell _{B}}}}\right)+N\ln N+{\text{const.}}}$$

This can be interpreted as the sum of the entropy of system A and system B, both extensive. And there is a term, ${\displaystyle N\ln N}$, that is not extensive.

Visualizing the particle-exchange approach in three dimensions

A three particle ideal gas partitioned into two parts

The correct (extensive) formulas for systems A and B were obtained because we included all the possible ways that the two systems could exchange particles. The use of combinations (i.e. N particles choose N_A) was used to ascertain the number of ways N particles can be divided into system A containing n_A particles and system B containing n_B particles. This counting is not justified on physical grounds, but on the need to integrate over phase space. As will be illustrated below, phase space contains not a single n_A-sphere and a single n_B-sphere, but instead

${\displaystyle {N \choose n_{A}}={\frac {N!}{n_{A}!n_{B}!}}}$

pairs of n-spheres, all situated in the same (N + 1)-dimensional velocity space. The integral over accessible phase space must include all of these n-spheres, as can be seen in the figure, which shows the actual velocity phase space associated with a gas that consists of three particles. Moreover, this gas has been divided into two systems, A and B.

If we ignore the spatial variables, the phase space of a gas with three particles is three dimensional, which permits one to sketch the n-spheres over which the integral over phase space must be taken. If all three particles are together, the split between the two gases is 3|0. Accessible phase space is delimited by an ordinary sphere (2-sphere) with a radius that is either ${\displaystyle {\sqrt {2E_{1}}}}$ or ${\displaystyle {\sqrt {2E_{2}}}}$ (depending which system has the particles).

If the split is 2|1, then phase space consists of circles and points. Each circle occupies two dimensions, and for each circle, two points lie on the third axis, equidistant from the center of the circle. In other words, if system A has 2 particles, accessible phase space consists of 3 pairs of n-spheres, each pair being a 1-sphere and a 0-sphere:

${\displaystyle v_{1}^{2}+v_{2}^{2}=2E_{A}\qquad v_{3}^{2}=2E_{B}}$

${\displaystyle v_{2}^{2}+v_{3}^{2}=2E_{A}\qquad v_{1}^{2}=2E_{B}}$

${\displaystyle v_{3}^{2}+v_{1}^{2}=2E_{A}\qquad v_{2}^{2}=2E_{B}}$

Note that

${\displaystyle {3 \choose 2}=3.}$