Let S be a set of n labeled points or vertices. The terminology "point" and "vertex" is to suggest the picture of graphical structures embedded in a geometrical space. Describe an interaction or binding between any two vertices by connecting them by a single line of a single length: any connecting line can only have this one length. The switches between "point" and "vertex", and between "line" and "edge" reinforces the double structure meanings: the object is to construct geometry by the statistical mechanics of graphs. Between any two vertices only one line may be drawn. With any such line associate to the set S an interaction energy E_I. There are additive contributions to energy E(S) from ech line.
That the points are labeled (distinguishable) signals a classical model, since the points of quantal model are indistiguishable. Classically, a permutation of points is a different state, while quantally, the state is invariant under any permutation of points.
Consider distributing p lines in S. While the points of S are distinguishable, the lines are not, hence are not labeled. Every such distribution defines a "microstate" of S, as a statistical mechanical system. The microstates of any statistical mechnaical system are the possible distinguishable distributions of the elements over which the statistics is carried out. The total energy of S
E(S) = pE_I
and since this can be done in binom( binom( n 2 ) p ) ways, this
number of ways is the degeneracy of a state of S with energy pE_I.
The degeneracy of the energy level E(S) = pE_I is the cardinality of
the set of the microstate equivalence class whose elements all have
the energy pE_I.
According to the Basic Assumption of statistical mechanics all the microstates of S with E(S) = pE_I are equally probable. The set of all microstates is partitioned by the value of E(S), or equivalently the value of p. For every allowed value of p, there is an equivalence class of microstates all of which are assigned equal probabilities. What is yet missing is a concept of the probability
P( E(S) )
of S having the energy E(S), which is assigned to the euivalence classes.
With S a closed and complete system, this concept has no sense,
since the probability for any equivalence class can,
mathematically be determined by fiat.
A collection of sets identical to S (replicants of S) together with all
its graphical microstates is customarily called a
microcanonical ensemble.
So imagine the lines to be mobile, moving as connectors among
the vertices, and that there is a microcanonical ensemble of S's
all in contact with a "thermal reservoir" at temperature T
that is a very large system much like itself, and with which it can
exchange lines.
The possibility of exchanging lines defines Thermal Contact.
The basic assumptions about this thermal reservoir is that it contains
lines which can be exchanged with S through its thermal contact and
that its temperature T represents an average line density.
Assume that the ensemble of S's have been in contact with the
thermal reservoir for a "long time" so that they are in thermal
equilibrium with it.
The number of lines in S can change as a result of its free exchanges
with the reservoir, and so its line density or temperature is in
fluctuation.
The concept of Thermal Equilibrium entails that a long
temporal average of the temperature of S defined by its number of
lines is equal to the temperature of the rervoir.
This defines the usual canonical ensemble.
I have in mind the consideration of S as being some arbitrarily
chosen subset or patch of the space of a large universal set,
which is the reservoir.
The Boltzmann factor P( E(S) )
Generally, not just for this model, consider any two macroscopic systems A and B that are both in contact with the same thermal reservoir. For a set S to be one of these systems, its cardinality n would have to be large. It follows immediately from the basic assumption of statistical mechanics that P_A( E_A ), the probability of finding system A in any microstate with energy E_A is just as implied by the notation a function only of E_A. Similarly, write and consider P_B( E_B ).
Now consider the system A + B, and the associated P_{A+B}( E_{A+B} ). The systems A and B are not assumed to be in direct thermal contact with each other, but are, of course, in an indirect interaction with each other by being in direct thermal contact with the same reservoir, so it is not unreasonable to take as true that
E_{A+B} = E_A + E_B
which is to say that any energy of indirect interaction
is being ignored, considering it to be negligible compared
to E_A and E_B.
NB: most textbook derivations of the Boltzman Factor allow
also that the systems A and B may also be in direct
thermal contact. The argument which acheives the above
linearity depends on a given a priori geometry
that enables the definition of an "interaction zone" which is
not present here.
Then,
P_{A+B}( E_{A+B} ) = P_{A+B}( E_A + E_B )
But the LHS of this equation may also be read as the joint
probability that A has energy E_A AND B has energy E_B.
Because, once again, A and B are not is direct interaction,
and any indirect interaction mediated by the reservoir is being ignored,
the probabilities P_A( E_A ) and P_B( E_B ) are independent
so that
P_{A+B}( E_{A+B} ) = P_A( E_A ) P_B( E_B )
so that
P_{A+B}( E_A + E_B ) = P_A( E_A ) P_B( E_B )
So far, we haven't needed recourse to the macroscopic assumption.
Use it now so that E_A and E_B can be sufficiently large that
a single line energy E_I is very small with respect them.
This enables the effective use of derivavtives with respect
to E_A and E_B as independent continuous variables to good
approximation.
For any differentiable F(x + y)
d/dx F(x + y) = d/dy F(x + y)
Then
P_A'( E_A ) P_B( E_B ) = P_A( E_A ) P_B'( E_B )
where the primes denote derivatives with respect to the
obvious energetic variables. Or,
P_A'( E_A )/P_A( E_A ) = P_B'( E_B )/P_B( E_B )
which is true independently of the choice of systems A and B.
This is to say that for any system A
P_A'( E_A )/P_A( E_A ) = K,
where . is independent of E_A. With tradition and foresight,
let K := - beta, so that the solution to the differential
equation is written
P( E ) = C exp( -beta E )
often referred to as the Boltzmann Factor.
The constant beta is necessarily independent of any system, and its
structure or composition by the preceding argument, while the
constant of integration C may, indeed, depend on the system's
structure or composition.
The same result can be had easily using
the fact that given the functional equation
F ( x + y ) = G ( x ) H ( y )
and very mild conditions on the functions, there is a general solution
G ( x ) = F ( x ) / H ( 0 ) , H ( x ) = F ( x ) / G ( 0 ) ,
and
F ( x ) = F ( 0 ) exp ( - K * x ) .
The only thing that the two systems A and B had in common was a
thermal reservoir of temperature T. Since thermal reservoirs
are featureless except for temperature, it is not unreasonable
to expect that beta is a function of T.
For the set S as above, if we sum over the probabilities of all the microstates of S, the sum must equal 1. Using the Boltzman factor, the energy expression and the degeneracy of energy:
binom( n 2 )
SUM binom( binom( n 2 ) p ) C exp( - beta E_I p ) = 1
p=0
binom( n 2 ) = n(n-1)/2
C and E_I are constants
If one ignores a kinetic energy attributed to the points of S and takes
C := [Z_S( beta )]^(-1)
where
Z_S( beta ) is called the partition function for S,
then the probability that S in contact with a reservoir characterized
by beta will have energy (p * E_I) is given by
P_S( p ) = [Z_S( beta )]^(-1) binom( n(n-1)/2 p ) exp( -beta E_I p )
The summation of the partition function can be performed since as a binomial expansion,
binom( n 2 )
SUM binon( binom( n 2 ) p ) exp( - beta E_I p ) =
p=0
[1 + exp( -beta E_I )]^[n(n-1)/2]
Then finally, for the probability of S having energy (E_Ip) where
there are only vertex interaction energies:
P_S( p ) =
exp( -beta E_I p )
binom( n(n-1)/2 p ) -----------------------------------
[1 + exp( -beta E_I )]^[n(n-1)/2]
with the probability requirement
SUM P_S(p) = 1
p
being satisfied.
The internal energy U of S is defined as usual, as the expected
value of E(S) = p * E_I = p E_I
U( S ) := SUM p E_I P_S( p )
p
= E_I SUM p P_S( p )
p
If one uses the simple isometric graph dimension
(2p/n), treating S as a graph, the expected dimension of S is given by
DIM( S ) := SUM (2p/n) P_S( p )
p
= (2/n) SUM p P_S( p )
p
so
(n/2) DIM( S ) = (1/E_I) U( S ) = SUM p P_S( p )
p
and
U( S ) = (n * E_I)/2 DIM( S )
a statement akin to the equipartition of energy from the usual
statistical mechanics of particles, and solids.
The significance of beta
In standard models of statistical mechanics, beta = (kT)^{-1}, where T is the temperature, a macroscopic intensive parameter associated with an average kinetic energy density of the thermal reservoir. In this model, we have not included a kinetic energy; energy is exclusively interaction or bonding energy.
In the extreme case beta -&gr; infinity, U( S ) and DIM( S ) approach zero, which describes the condition of no interaction lines. All vertices are isolated from one another and do not interact (have no contact with one another).
In the extreme case beta -> 0, maximal interaction line density
P_S( p ) →
1
binom( n(n-1)/2 p ) --------------
2^[n(n-1)/2]
then,
DIM( S ) = (2/n) SUM p P_S( p )
p
SUM p binom( n(n-1)/2 p )
= (2/n) --------------------------
2^[n(n-1)/2]
[n(n-1)/2] 2^[n(n-1)/2] (1/2)
= (2/n) ------------------------------
2^[n(n-1)/2]
n(n-1)/2
= ---------- = (n-1)/2
n
The full n-vertex graph is an (n-1)-simplex, with dimension (n-1), yet we compute the dimension of S as (n-1)/2. We have done so under the assumption that the vertices of S are not in direct interaction with its reservoir of lines. S merely exchanges lines with the reservoir.
Envision S as an artifically delimited subset of the reservoir with which it is in equilibrium. The vertex dimension expression (2p/n) is a counting of lines attached to a vertex.
In equilibrium, "inside of S" is just like "outside of S". To correct the isolation of vertices of S, we must double the lines of connection to each vertex, thereby doubling DIM( S ). Conceptually extending S into the reservoir, the entire reservoir becomes a rigid (N-1)-dimensional simplex, where N is an exceedingly large number of vertices in the reservoir with N(N-1)/2 lines. So, beta is a parameter the expresses the average density of lines available in the reservoir. If N is finite, the maximal density, lines per vertex is (N-1)/2. For low line densities in the reservoir, the calculated DIM( S ) is an expression of the line density within S, and as seen proportional to the internal energy U( S ).
Assuming a fictious universe to be of a particulate and set theoretic nature, with a fundamental interaction length associated with a fundamental binding energy, a statistical model has been constructed where the dimension of the space is a function of its internal energy. The dimension is a statistical feature of this universe, dependent on the energy contained within it. The "colder" the universe, the lower the dimension. Local statistical fluctuations of the energy (increase) could increase the local dimension. The concept a magnitude of dimension arises as a result of the statistics of the underlying dynamics much as pressure of a gas arises from the statistics of momentum transferred to the container walls.
From a cosmological viewpoint, the average temperature of the universe is cold, a few degrees above absolute zero. One would expect the fundamentally empty space of the universe from such a model to have a low dimension. Highly "localized" objects like planets, stars and galaxies are, of course, being neglected, as are any energies of motion that may be associated with the points of any local set S. I expect to try including intrinsic point selfenergies shortly, to obtain a more reasonable and complex model.
Email me, Bill Hammel at