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
. The conditions
or
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 -spheres is only a matter of mathematical
convenience. It will turn out eventually that those properties of
-dimensional phase space which are of importance in Statistical
Mechanics are actually quite robust, and not reserved to -spheres.
For instance, the -rhomboid which pertains to the condition
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 , a variable that represents the
square of the radius:
.
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):
. It would be unphysical and misleading to keep
at some constant
value and at the same time raise the number of dimensions.
VOLUME
The volume of a -dimensional sphere with radius
is
(3.9)
where
and
.
The following recursion is useful:
(3.10)
EXAMPLES:
For large we have, using Stirling's approximation for ,
(3.11)
As soon as the Stirling approximation holds, i.e. for ,
the hypersphere volume may be written
(3.12)
Going to very large () we find for the logarithm of the volume
(the formula for proper is then difficult to handle due to
the large exponents)
(3.13)
EXAMPLE:
Let and . For the log volume we find
SURFACE AREA
For the surface area of a -dimensional sphere we have
(3.14)
EXAMPLES:
A very useful representation of the surface is this:
(3.15)
with
. This formula provides an
important insight; it shows that the ``mass'' of a spherical shell
is distributed along one sphere axis () as follows:
(The astute reader notes that for once we have kept , regardless of
the value of ;
this is permitted here because we are dealing with a normalized density
.)
APPLICATION:
Assume that a system has degrees of freedom of translatory motion.
(Example: particles moving on a line, or particles
in dimensions.)
Let the sum of squares of all velocities (energy!) be given, but apart
from that let any particular combination of the values
be equally probable. All
``phase space points''
are
then homogeneously distributed on the spherical surface
, and a single velocity 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 , 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 is near zero.
The case , meaning 3 particles on a line or one particle in
3 dimensions, is special: all possible values of occur with
equal probability.
Approach to the Maxwell-Boltzmann distribution: For very large
we have
(3.17)
with
.
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 of a -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
, then
is just the distribution density for
any single velocity component .
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]
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]
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
of a thin ``skin'' near the surface and the total volume of the
sphere is
(3.18)
or, using the quantities and :
(3.19)
For
this ratio approaches ,
regardless of how thin the shell may be!
At very high dimensions the entire volume of a sphere is concentrated
immediately below the surface:
(3.20)
EXAMPLE: ,
HYPERSPHERES AND HYPERCYLINDERS
The sphere volume may alternatively be written as
(3.21)
with
.
EXAMPLE:
(3.22)
Partitioning the integration interval in equ. 3.21
into small intervals of size we may write
(3.23)
where
.
Similarly. for the volume of the spherical shell we have
(3.24)
Remembering equ. 3.20 we may, for high enough
dimensions , and , always write in place of
. Therefore,
(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
(3.26)
From
(3.27)
we find for the argument that maximizes the integrand
3.21 or the summation term in 3.24 or
3.25,
(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 in and such that
also and
Find the maximum of the product
(or the maximum of the logarithm ) with respect to ;
this maximum is located at
(3.29)
With this, the following holds:
(3.30)
and similarly,
(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 is scanned in regular
steps. It is better to localize first, then compare with
slightly smaller and larger .)
The geometric interpretation of the relation 3.30
is amazing:
On a logarithmic scale the volume of an -sphere
equals the product of the volumes of two spheres in the
subspaces and . But this product may
be understood as the volume of a hypercylinder that is inscribed
in the -sphere and which has the ``base area'' in space and the ``height'' in space.
EXAMPLE:
, , : the maximum of the quantity
is located
at , and we have
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 , reserving the symbol
for the volume of a system in real space.
Thus,
is the -dimensional velocity
phase space volume of a -particle system,
refers to the positional subspace volume which usually will also have
dimensions, and denotes the full phase space (or volume
in phase space) with 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 -particle system to be divided into cells
of size
^{3.2}
(3.33)
Now let us consider a classical gas, or fluid, with
particle number and volume . In the -dimensional
velocity subspace of the -dimensional phase space
the condition
again defines a spherical shell whose volume is essentially
equal to the volume of the enclosed sphere. The number of
cells in (or below) this shell is
(3.34)
In the case of the ideal gas the contribution of the
position subspace may be included in a simple manner; we
write
(3.35)
Thus the total number of cells in space is
(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
by to find
(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 although quantum mechanics was
not yet known to him. Ho proposed the rule in an
ad hoc manner to solve a certain theoretical problem,
the so-called Gibbs Paradox.
The quantity is a measure of the available phase space volume,
given in units . The logarithm of has great
physical significance: it is - up to a prefactor - identical to
the entropy that was introduced in thermodynamics.
At present, this identification is no more than a hypothesis; the following
chapter will show how reasonable the equality
(3.38)
really is.
In the case of a classical ideal gas we find, using
and neglecting
in
equ. 3.36,
(3.39)
This is the famous Sackur-Tetrode equation, named for the
authors Otto Sackur and Hugo Tetrode.
The numerical value of , and therefore of , is obviously dependent
on the chosen grid size . This disquieting fact may be mitigated
by the following considerations:
As long as we are only comparing phase space volumes,
or entropies, the unit is of no concern
There is in fact a smallest physically meaningful gridsize
which may well serve as the natural unit of ; it is given by the
quantum mechanical uncertainty:
EXAMPLE:
, , . The average energy per particle is then
,and the mean squared velocity is
.
Assuming a cubic box with and a rather coarse grid
with
we find
(3.40)
Just for curiosity, let us determine
, where
is the phase space volume between
and :
(3.41)
and thus
(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!