3.2 From Hyperspheres to Entropy

Some model systems have the convenient property that their energy may be written as a sum over the squares of their microvariables. The most popular example is the classical ideal gas with $E \equiv E_{kin}$ $= (m/2)\sum v_{i}^{2}$. The conditions $E \leq E_{0}$ or $E \approx E_{0}$ then describe a high-dimensional sphere or spherical shell, respectively. Thus it will be worth the while to make ourselves acquainted with the geometrical properties of such bodies. It must be stressed, however, that the restriction to $n$-spheres is only a matter of mathematical convenience. It will turn out eventually that those properties of $n$-dimensional phase space which are of importance in Statistical Mechanics are actually quite robust, and not reserved to $n$-spheres. For instance, the $n$-rhomboid which pertains to the condition $\sum_{i} \sigma_{i} \approx const$ for simple spin systems could be (and often is) used instead.

VOLUME AND SURFACE OF HIGHDIMENSIONAL SPHERES
In the following discussion it will be convenient to use, in addition to the sphere radius $r_{0}$, a variable that represents the square of the radius: $z_{0} \equiv r_{0}^{2}$. In the phase space of interactionless many-particle systems, this quantity is related to an energy, as may be seen from equs. 3.7, 3.8 and Table 3.1); and in an isolated system it is just the energy which is given as a basic parameter.

Attention:
It should be kept in mind that the energy - and thus the square of the sphere radius - is an extensive quantity, meaning that it will increase as the number of degrees of freedom (or particles): $z_{0} \propto n$. It would be unphysical and misleading to keep $z_{0}$ at some constant value and at the same time raise the number of dimensions.

VOLUME
The volume of a $n$-dimensional sphere with radius $r_{0} = \sqrt{z_{0}}$ is

$$V_{n}(z_{0}) = C_{n} r_{0}^{n} = C_{n} z_{0}^{n/2} \; \; {\rm with} \; \; C_{n} = \frac{\pi^{n/2}}{(n/2)!}$$ (3.9)

where $(1/2)! = \sqrt{\pi}/2$ and $(x+1)!=x!(x+1)$. The following recursion is useful:

$$C_{n+2} = \frac{2 \pi}{ n+2} C_{n}$$ (3.10)

EXAMPLES: $C_{1}=2, \; \; C_{2}=\pi, \;\; C_{3}= \pi^{3/2}/(3/2)! = 4 \pi / 3, \;\; C_{4}... ...= 8\pi^{2}/15, \;\; C_{6}= \pi^{3}/6, \;\; \dots \; C_{12}= \pi^{6}/720, \dots$

For large $n$ we have, using Stirling's approximation for $(n/2)!$,

$$C_{n} \approx \left[ \frac{2 \pi e}{n} \right]^{n/2} \frac{1... ...pprox \frac{n}{2} ( \ln \pi + 1) - \frac{n}{2} \ln \frac{n}{2}$$ (3.11)

As soon as the Stirling approximation holds, i.e. for $n \geq 100$, the hypersphere volume may be written

$$V_{n}(z_{0}) \approx \frac{1}{\sqrt{\pi n}} \left( \frac{2 \... ... \ln \left( \frac{2 \pi e}{n} z_{0} \right) - \ln \sqrt{\pi n}$$ (3.12)

Going to very large $n$($\geq 500$) we find for the logarithm of the volume (the formula for $V$ proper is then difficult to handle due to the large exponents)

$$\ln V_{n}(z_{0}) \approx \frac{n}{2} \ln \left( \frac{2 \pi e}{n} z_{0} \right)$$ (3.13)

EXAMPLE: Let $n=1000$ and $z_{0}=1000$. For the log volume we find $\ln V_{1000}(1000) \equiv 500 \ln (2 \pi e) - \ln \sqrt{1000 \pi} = 882.76$

SURFACE AREA
For the surface area of a $n$-dimensional sphere we have

$$O_{n}(z_{0}) = \frac{d V_{n}(z_{0})}{dr_{0}} = n C_{n}z_{0}^{(n-1)/2}$$ (3.14)

EXAMPLES:

$$V_{1}(z_{0})=2 r_{0} , \;\; O_{1}(z_{0})= 2$$

$$V_{2}(z_{0})=\pi z_{0}, \;\; O_{2}(z_{0})= 2 \pi r_{0}$$

$$V_{3}(z_{0})=(4 \pi / 3) r_{0}^{3}, \;\; O_{3}(z_{0})= 4 \pi z_{0}$$

$$V_{4}(z_{0})=( \pi^{2} / 2) z_{0}^{2}, \;\; O_{4}(z_{0})= 2 \pi^{2} r_{0}^{3}$$

A very useful representation of the surface is this:

$$O_{n}(r_{0}) = \int_{-r_{0}}^{r_{0}} dr_{1} \frac{r_{0}}{r_{2}} O_{n-1}(r_{2})$$ (3.15)

with $r_{2} \equiv \sqrt{r_{0}^{2}-r_{1}^{2}}$. This formula provides an important insight; it shows that the ``mass'' of a spherical shell is distributed along one sphere axis ($r_{1}$) as follows:

$$p_{n}(r_{1}) = \frac{r_{0}}{r_{2}} \frac{O_{n-1}(r_{2})}{O_{... ...= \frac{(n-1)C_{n-1}}{n C_{n}} \frac{r_{2}^{n-3}}{r_{0}^{n-2}}$$ (3.16)

EXAMPLES: Let $r_{0}=1$; then we find (see Fig. 3.3)

$$p_{2}(r_{1}) = \frac{1}{\pi} (1-r_{1}^{2})^{-1/2}$$

$$p_{3}(r_{1}) = \frac{1}{2} \;\; (constant!)$$

$$p_{4}(r_{1}) = \frac{2}{\pi} (1-r_{1}^{2})^{1/2}$$

$$p_{5}(r_{1}) = \frac{3}{4} (1-r_{1}^{2})$$

$$\dots$$

$$p_{12}(r_{1}) = \frac{256}{63 \pi} (1-r_{1}^{2})^{9/2}$$

(The astute reader notes that for once we have kept $r_{0}=1$, regardless of the value of $n$; this is permitted here because we are dealing with a normalized density $p_{n}$.)

APPLICATION: Assume that a system has $n$ degrees of freedom of translatory motion. (Example: $n$ particles moving on a line, or $n/3$ particles in $3$ dimensions.) Let the sum of squares of all velocities (energy!) be given, but apart from that let any particular combination of the values $v_{1}, v_{2}, \dots$ be equally probable. All ``phase space points'' $\vec{v} \equiv \{ v_{1} ... v_{n} \}$ are then homogeneously distributed on the spherical surface $O_{n}(\vert v\vert^{2})$, and a single velocity $v_{1}$ occurs with probability density 3.16.

As we increase the number of dimensions, the character of this density function changes dramatically at first (see Fig. 3.3). If just two particles on a line (or the two d.o.f. of a pin ball) share the total energy $mv^{2}/2$, then the velocity of one of them is most probably near the possible maximal value while the other has only a small speed. In contrast, for many dimensions (or particles) the maximum of the probability density $p_{n}(v_{1})$ is near zero. The case $n=3$, meaning 3 particles on a line or one particle in 3 dimensions, is special: all possible values of $v_{1}$ occur with equal probability.

Approach to the Maxwell-Boltzmann distribution: For very large $n$ we have

$$p_{n}(v) \approx \frac{1}{\sqrt{2 \pi \langle v^{2} \rangle}} \exp \{-v^{2}/2 \langle v^{2} \rangle \}$$ (3.17)

with $\langle v^{2} \rangle = 2 E_{0}/nm$.
The Maxwell-Boltzmann distribution may thus be derived solely from the postulate of equal a priori probability and the geometric properties of high-dimensional spheres.

Figure 3.3: Mass distribution $p(r_{1})$ of a $(n-1)$-dimensional spherical surface along one axis. With increasing dimension the mass concentrates more and more at the center of the axis. If we interprete the surface as the locus of all phase space points with given total energy $E_{kin} = (m/2) \sum v_{i}^{2}$, then $p(r_{1}) \equiv p(v_{i})$ is just the distribution density for any single velocity component $v_{i}$.

Applet Stadium: Start

Simulation: Bunimovich's stadium billard. Distribution of flight directions, p(phi)
and of a single velocity component, p(vx)
[Code: Stadium]

Applet Harddisks: Start

Simulation: $N=1$ to $3$ hard disks in a 2D box with reflecting walls.
Distribution of flight directions, p(\phi), and of a single velocity component, p(v1x). Comparison with the theoretical distribution. [Code: Harddisks]

Applet Hspheres: Start

Simulation: One hard sphere in a box with reflecting walls. Distribution of one velocity component; demonstration of the special case n=3 (where p(v1x)=const). [Code: Hspheres]

Applet LJones: Start

Simulation: N Lennard-Jones particles in a 2D box with periodic boundary conditions. Distribution of a velocity component, p(v1x). [Code: LJones]

TWO AMAZING PROPERTIES OF HYPERSPHERES
We have seen that spheres in high-dimensional spaces exhibit quite unusual geometrical properties. In the context of Statistical Physics the following facts are of particular relevance:

• Practically the entire volume of a hypersphere is assembled in a thin shell immediately below the surface
• The volume of a hypersphere is - at least on a logarithmic scale - almost identical to the volume of the largest inscribed hyper-cylinder

We will consider these two statements in turn.

VOLUME OF A SPHERICAL SHELL
The ratio of the volume $\Delta V_{n}(r_{0}-\Delta r, r_{0})$ of a thin ``skin'' near the surface and the total volume of the sphere is

$$\frac{\Delta V_{n}}{V_{n}} = \frac{r_{0}^{n}-(r_{0}-\Delta r... ...}{r_{0}})^{n} \longrightarrow 1-\exp \{-n(\Delta r / r_{0}) \}$$ (3.18)

or, using the quantities $z_{0}$ and $\Delta z$:

$$\frac{\Delta V_{n}}{V_{n}} \longrightarrow 1-\exp [-\frac{n}{2}\frac{\Delta z}{z_{0}} ]$$ (3.19)

For $n \rightarrow \infty$ this ratio approaches $1$, regardless of how thin the shell may be!



At very high dimensions the entire volume of a sphere is concentrated immediately below the surface:

$$\Delta V_{n}(z_{0},\Delta z) \longrightarrow V_{n}(z_{0}), \;\;\; n » 1$$ (3.20)

EXAMPLE: $n=1000$, $\Delta r = r_{0}/100$ $\rightarrow$ $\Delta V / V = 1 - 4.3 \cdot 10^{-5}$

HYPERSPHERES AND HYPERCYLINDERS
The sphere volume $V_{n}(z_{0})$ may alternatively be written as

$$V_{n}(z_{0}) = \int_{0}^{r_{0}} dr_{1} O_{n_{1}}(z_{1}) V_{n-n_{1}}(z_{2})$$

$$\equiv \int_{0}^{z_{0}} \frac{dV_{n_{1}}(z_{1})}{dz_{1}} dz_{1} V_{n-n_{1}}(z_{0}-z_{1})$$

$$\equiv \int_{0}^{z_{0}} dV_{n_{1}}(z_{1}) V_{n-n_{1}}(z_{0}-z_{1})$$ (3.21)

with $r_{1} \equiv \sqrt{z_{1}}$.

EXAMPLE:

$$V_{6}(z_{0})= \int_{0}^{r_{0}} dr_{1} O_{3}(z_{1}) V_{3}(z_{... ...rac{4 \pi}{3} (z_{0}-z_{1})^{3/2} = \frac{\pi^{3}}{6}z_{0}^{3}$$ (3.22)

Partitioning the integration interval $[0,z_{0}]$ in equ. 3.21 into small intervals of size $\Delta z$ we may write

$$V_{n}(z_{0}) \approx \sum_{k=1}^{K} \Delta V_{n_{1}}(z_{k}) V_{n-n_{1}}(z_{0}-z_{k})$$ (3.23)

where $\Delta V_{n_{1}}(z_{k}) \equiv V_{n_{1}}(k \Delta z) - V_{n_{1}}\left( (k-1) \Delta z \right)$.

Similarly. for the volume of the spherical shell we have

$$\Delta V_{n}(z_{0}) = \int_{0}^{z_{0}} dV_{n_{1}}(z_{1}) ... ...{K} \Delta V_{n_{1}}(z_{k}) \Delta V_{n-n_{1}}(z_{0}-z_{k})$$ (3.24)

Remembering equ. 3.20 we may, for high enough dimensions $n$, $n_{1}$ and $(n-n_{1})$, always write $V$ in place of $\Delta V$. Therefore,

$$V_{n}(z_{0}) \approx \sum_{k=1}^{K} V_{n_{1}}(z_{k}) V_{n-n_{1}}(z_{0}-z_{k})$$ (3.25)

Now, the terms in these sums are strongly varying. For high dimensions there is always one single term that dominates the sum; all other terms may safely be neglected. To find this term we evaluate

$$\frac{d}{dz} \left[ V_{n_{1}}(z) V_{n-n_{1}}(z_{0}-z) \right] = 0$$ (3.26)

From

$$\frac{d \left[ z^{n_{1}/2} (z_{0}-z)^{(n-n_{1})/2} \right]}{dz} = 0$$ (3.27)

we find for the argument $z^{*}$ that maximizes the integrand 3.21 or the summation term in 3.24 or 3.25,

$$z^{*} = \frac{n_{1}}{n} z_{0}$$ (3.28)

We have thus found a very surprising property of high-dimensional spheres which we may summarize by the following calculation rule:

• Divide the given dimension $n$ in $n_{1}$ and $n-n_{1}$ such that also $n_{1}»1$ and $n-n_{1}»1$
  
  
• Find the maximum of the product $f(z) \equiv V_{n_{1}}(z) \cdot V_{n-n_{1}}(z_{0}-z)$ (or the maximum of the logarithm $\ln f(z)$) with respect to $z$; this maximum is located at
  

  $$z^{*} = \frac{n_{1}}{n} z_{0}$$ (3.29)

  
  

• With this, the following holds:
  

  $$\ln V_{n}(z_{0}) \approx \ln V_{n_{1}}(z^{*}) + \ln V_{n-n_{1}}(z_{0}-z^{*})$$ (3.30)

  
  
  and similarly,
  

  $$\ln \Delta V_{n}(z_{0}) \approx \ln \Delta V_{n_{1}}(z^{*}) + \ln \Delta V_{n-n_{1}}(z_{0}-z^{*})$$ (3.31)

  
  

(Remark: in a numerical verification of these relation it must be remembered that the maximum of the function at the r.h.s. of 3.31 or 3.30 is very sharp indeed; it may be overlooked when the interval $[0,z_{0}]$ is scanned in regular steps. It is better to localize $z^{*}$ first, then compare with slightly smaller and larger $z$.)

The geometric interpretation of the relation 3.30 is amazing:
On a logarithmic scale the volume of an $n$-sphere equals the product of the volumes of two spheres in the subspaces $n_{1}$ and $n-n_{1}$. But this product may be understood as the volume of a hypercylinder that is inscribed in the $n$-sphere and which has the ``base area'' in $n_{1}$ space and the ``height'' in $(n-n_{1})$ space.

EXAMPLE: $n=1000$, $n_{1}=400$, $z_{0}=1000$: the maximum of the quantity $\ln f(z) \equiv \ln V_{400}(z) + \ln V_{600}(1000-z)$ is located at $z^{*}= 400$, and we have

$$500 \ln (2 \pi e)- \ln \sqrt{1000 \pi} \approx 200 \ln (2 \pi e) -\ln \sqrt{400 \pi} + 300 \ln (2 \pi e) -\ln \sqrt{600 \pi}$$

$$1418.94 - 4.03 \approx 567.58 -3.57 + 851.36 -3.77$$ (3.32)

Applet Entropy1: Start

Simulation: Hyperspheres and -cylinders. In a given hypersphere of dimension n we inscribe hypercylinders whose "base areas" and "heights" have n1 and n-n1 dimensions, respectively. The hypercylinder with the maximum volume is identified: its log volume is almost equal to that of the circumscribed sphere. [Code: Entropy1]

DISCRETISATION OF PHASE SPACE; ENTROPY
Returning from geometry to physics, we will from now on denote phase space volumes by $\Gamma$, reserving the symbol $V$ for the volume of a system in real space. Thus, $\Gamma_{\vec{v}}$ is the $3N$-dimensional velocity phase space volume of a $N$-particle system, $\Gamma_{\vec{r}}$ refers to the positional subspace volume which usually will also have $3N$ dimensions, and $\Gamma$ denotes the full phase space (or volume in phase space) with $6N$ dimensions.

Depending on which model system we are considering, the microvariables are either continuous (classical manybody system) or discrete (quantum models, spin lattices.) In order to be able to ``enumerate'' the states in phase space it is opportune to introduce a raster even in the case of continuous microvariables. Thus we imagine the phase space of a $N$-particle system to be divided into cells of size ^3.2

$$g \equiv \Delta x \Delta v$$ (3.33)

Now let us consider a classical gas, or fluid, with particle number $N$ and volume $V$. In the $3N$-dimensional velocity subspace of the $6N$-dimensional phase space the condition $E_{kin} \epsilon [E,\Delta E]$ again defines a spherical shell whose volume is essentially equal to the volume of the enclosed sphere. The number of $\Delta v$ cells in (or below) this shell is

$$\Sigma_{\vec{v}} (E,\Delta E) \equiv \frac{\Delta \Gamma_{\v... ...N}} = C_{3N}\left( \frac{2E}{m}\right)^{3N/2} /(\Delta v)^{3N}$$ (3.34)

In the case of the ideal gas the contribution of the position subspace may be included in a simple manner; we write

$$\Sigma_{\vec{r}} = \frac{V^{N}}{\Delta x^{3N}}$$ (3.35)

Thus the total number of cells in $6N$ space is

$$\Sigma' (N,V,E) = \Sigma_{\vec{r}} \Sigma_{\vec{v}} = C_{3N} \left[ \frac{V}{g^{3}} \left(\frac{2E}{m} \right)^{3/2} \right]^{N}$$

$$= \frac{1}{\sqrt{3 N \pi}} \left( \frac{2 \pi e}{3 N}\right)^{3N/2} \left[ \frac{V}{g^{3}} \left(\frac{2E}{m} \right)^{3/2} \right]^{N}$$ (3.36)

Now we have to prematurely introduce a result of quantum statistics. One characteristic property of quantum objects is their indistinguishability. It is evident that the number of distinct microstates will depend on whether or not we take into account this quantum property. A detailed analysis which must be postponed for now leads up to the simple result that we have to divide the above quantity $\Sigma' (N,V,E)$ by $N!$ to find

$$\Sigma(N,V,E) \equiv \Sigma' (N,V,E)/N!$$ (3.37)

which is now indeed proportional to the total number of physically distinct microstates. This rule is known as the rule of correct Boltzmann enumeration.

It is characteristic of the physical ``instinct'' of J. W. Gibbs that he found just this rule for the correct calculation of $\Sigma$ although quantum mechanics was not yet known to him. Ho proposed the $1/N!$ rule in an ad hoc manner to solve a certain theoretical problem, the so-called Gibbs Paradox.

The quantity $\Sigma(N,V,E)$ is a measure of the available phase space volume, given in units $g^{3N}$. The logarithm of $\Sigma(N,V,E)$ has great physical significance: it is - up to a prefactor $k$ - identical to the entropy $S(N,V,E)$ that was introduced in thermodynamics. At present, this identification is no more than a hypothesis; the following chapter will show how reasonable the equality

$$S(N,V,E)=k \ln \Sigma(N,V,E)$$ (3.38)

really is.

In the case of a classical ideal gas we find, using $\ln N! \approx N \ln N$ and neglecting $\ln \sqrt{3N\pi}$ in equ. 3.36,

$$S(N,V,E) \equiv k \ln \Sigma(N,V,E) = Nk \ln \left[ \frac{V}{N} \left( \frac{4 \pi E e}{3 N m g^{2}} \right)^{3/2} \right]$$ (3.39)

This is the famous Sackur-Tetrode equation, named for the authors Otto Sackur and Hugo Tetrode.

The numerical value of $\Sigma$, and therefore of $S$, is obviously dependent on the chosen grid size $g$. This disquieting fact may be mitigated by the following considerations:

$\bullet$

As long as we are only comparing phase space volumes, or entropies, the unit is of no concern

$\bullet$

There is in fact a smallest physically meaningful gridsize which may well serve as the natural unit of $\Sigma$; it is given by the quantum mechanical uncertainty: $g_{min} = h/m$

EXAMPLE: $N=36$, $m=2$, $E=N=36$. The average energy per particle is then $= 1$,and the mean squared velocity is $\langle v^{2}\rangle =1$. Assuming a cubic box with $V=L^{3}=1$ and a rather coarse grid with $\Delta x = \Delta v = 0.1$ we find

$$S(N,V,E)/k = 36 \ln \left[ \frac{1}{36} \left( \frac{4 \cdot... ...\cdot 36 \cdot 2 \cdot 10^{-4}} \right)^{3/2} \right] = 462.27$$ (3.40)

Just for curiosity, let us determine $\ln (\Delta \Sigma)$, where $\Delta \Sigma$ is the phase space volume between $E=35.5$ and $36.0$:

$$\Delta \Sigma = \Sigma(36.0)-\Sigma(35.5) = \Sigma(36.0) \le... ...( \frac{35.5}{36}\right)^{54}\right] = \Sigma(36.0) \cdot 0.53$$ (3.41)

and thus

$$S(N,V, \Delta E) / k = 462.27 - 0.63 = 461.64$$ (3.42)

We see that even for such a small system the entropy value is quite insensitive to using the spherical shell volume in place of the sphere volume!