We use the following special terminology in this article:
{} indicates superscript (including the power).
[] indicates subscript.
~A means the vector A.
^a means the unit vector a.
<four> means 4.
In a capital Greek letter the word "cap." is written.
(A)[av] means the average of A.

11          Wrong construing of the Boltzmann factor;
Ed 01.12.31 -----------------------------------------
                      E=h<nu> is wrong
                      ----------------
Abstract
--------
As we know probability of finding a system in one of its accessible
states is proportial to the Boltzmann factor. It is shown that
contrary to what is thought at present in this proportion the energy
appearing in the Boltzmann factor is not a variable but it is a
constant and the variable is the state accessible for the system
having this constant energy. So, what at present is accepted as
Boltzmann factor is not real. Deduction of the Maxwell velocity
distribution as an instance of the consequences of the real Boltzmann
factor, and the first deduction of the relation E=h<nu> as an
instance of the consequences of the wrong Boltzmann factor are
presented. A logical review of some of the fundamental elements of
the statistical mechanics, that also contains some new viewpoints,
has been necessary. A factor is introduced in a general expression
for molar specific heat which plays the role of partition (not
equipartition) of energy and giving suitable amounts to it all the
practical cases including ones related to ideal gases and crystalline
solids are covered.

I. Introduction
---------------
The Boltzmann factor is a famous factor upon which several important
relations have been obtained in physics. Unfortunately, carelessness
about the essence of the parameters used in this factor has caused
an untrue understanding of it. In other words the factor which at
present is being used under the name of Boltzmann factor, because of
using of some quantities other than what must be used in it, is a
wrong factor.

In this article the above material are proven in detail. In addition,
some samples of the valid results obtained from the correct form of
the Boltzmann factor and some samples of the current invalid results
obtained from the wrong construing of this factor, including Planck's
deduction of the famous relation E=h<nu>, are shown.

II. Necessary elements of the statistical mechanics
----------------------------------------------------
State of a system of some particles, which don't have any influence on 
each other (except probably collision), will be specified by knowing the
position and the momentum of each of these particles. We take into 
consideration a six-dimensional space named as phase space in which 
each point has six components x, y, z, p[x], p[y], and p[z]. We divide 
the phase space into small cells volume of each being 
<delta>x<delta>y<delta>z<delta>p[x]<delta>p[y]<delta>p[z]. If we prescribe
<delta>x<delta>p[x]=h' and <delta>y<delta>p[y]=h' and <delta>z<delta>p[z]=h',
then the volume of each of these cells will be h'{3}. Each particle in its
position and momentum is in fact placed in a point in the phase space and
in fact is set in a cell of volume h'{3}. We suppose that while a particle 
is being set in each point inside a cell, its energy change, ie its kinetic
energy change, is negligible. Thus, in fact to each cell an energy is 
connected which is related to a particle inside this cell. It is obvious 
that f particles, forming an isolated system, are occupying f cells (being
not necessarily different) in the phase space and their total energy will
be conserved. It is certain that it is possible that the positions of the
particles in the phase space will be changed, but this process will
proceed in a definite path, namely in some path in which the total energy 
of the particles will be conserved. Therefore, for a definite total energy
a definite path in the phase space will be distinguished in which the 
system can proceed. Attention to this fact is important that since a cell
volume of the phase space doesn't have a differential magnitude, ie zero
magnitude, although it is considered much small, it is possible that
for a definite total energy, E, the path will remain the same and won't be
changed in a much small energy interval about this energy, E. In fact since
each cell volume in the phase space doesn't have a differential, ie in fact
zero, magnitude, the path related to a definite energy in the phase
space is not a continuous path, but it is in fact in the form of a 
successive series of various configurations of a definite set of points 
in the phase space. For example if we suppose that the energies connected
to the cells of the phase space shown in the following figure are 
proportional to the numbers written in the cells, then the path related 
to the energy 6 of two particles will be the same shown at subsequent
figures. (Of course the displacement of these configurations by each other
is completely unimportant, just like the unimportance of the displacement
of the points of a curve by each other which will not at all change the
essence of the curve.)

 ______________
| 1  | 2  | 3  |
|____|____|____|
| 4  | 5  | 6  |
|____|____|____|
| 7  | 0  | 3  |
|____|____|____|
 ______________     ______________     ______________
|    |    | ** |   | *  |    |    |   |    |    |    |
|____|____|____|   |____|____|____|   |____|____|____|
|    |    |    |   |    | *  |    |   |    |    | *  |
|____|____|____|   |____|____|____|   |____|____|____|
|    |    |    |   |    |    |    |   |    | *  |    |
|____|____|____|   |____|____|____|   |____|____|____|
 ______________     ______________     ______________
|    |    | *  |   |    | *  |    |   |    |    |    |
|____|____|____|   |____|____|____|   |____|____|____|
|    |    |    |   | *  |    |    |   |    |    |    |
|____|____|____|   |____|____|____|   |____|____|____|
|    |    | *  |   |    |    |    |   |    |    | ** |
|____|____|____|   |____|____|____|   |____|____|____|

In an isolated system, with a total energy E, we call each of these
configurations of the path related to E as an accessible state for the 
system. We call the number of the total of these configurations as
<cap. omega>(E). On definition we say that an isolated system is in the
equilibrium state when the probability of finding the system in each of its 
accessible states is independent of time, ie when always a definite fraction 
of a numerous number of these systems is related to an accessible state.

Now we express here two important statistical postulates:
a) If an isolated system is in equilibrium state, then all of its 
   accessible states will be equiprobable, and vice versa.
b) If an isolated system is not in equilibrium state, then it will 
   transform during the time till it will reach the eqilibrium state.

Now suppose that our isolated system consists only of a particle with the
energy <epsilon>. We know that this particle has three degrees of freedom.
Let's set at present this particle at freedom in only one of these degrees.
It is clear that in a specified time interval this particle can pass through 
some paths the number of which is proportional to the speed of the particle.
Then, if the particle (kinetic) energy becomes n fold, total number of
its accessible states will become n{1/2} fold, because in fact the speed
will have become n{1/2} fold. Therefore, in this case that the particle
has only one degree of freedom, the total number of its accessible states
is proportional to <epsilon>{1/2}; but since in practice the particle has 
three degrees of freedom, the total number of its accessible states is 
proportional to <epsilon>{1/2}<epsilon>{1/2}<epsiolon>{1/2}, ie
<cap. omega>(<epsilon>) is proportional to <epsilon>{3/2}.

Now suppose that we have an f-particle isolated system with a total 
energy E. Mean energy of each particle is <epsilon>=E/f. When
<cap. omega>(E) is the total number of accessible states of the system
with energy E, we say that <omega>(<epsilon>) is the total number of
accessible states of one particle of this system with energy <epsilon>.
As we indicated we have <omega>(<epsilon>) being proportional to
<epsilon>{3/2}. For evaluating <cap. omega>(E) we know that on the 
average there are <omega>(<epsilon>) accessible states for each particle
and then in general there will be <omega>(<epsilon>){f} accessible states
for f particles. Thus <cap. omega>(E) is proportional to 
<omega>(<epsilon>){f}, and since <omega>(<epsilon>) is proportional
to <epsilon>{3/2}, we conclude that <cap. omega>(E) is proportional to
<epsilon>{3f/2}.

Suppose that the two systems A and A' are in energetic contact with each 
other, but the set of these two systems is isolated. We name the set of 
these two systems as A{*}. Since A{*} is isolated, its total energy, E{*},
must remain constant. We suppose that <cap. omega>{*}(E{*}) is the total
number of accessible states of the isolated system A{*}, and
<cap. omega>{*}(E{*})[E] is the total number of accessible states of A{*}
in each of which the energy of the system A is E. Probability of finding
the energy of A equal to E is 
P(E)=<cap. omega>{*}(E{*})[E]/<cap. omega>{*}(E{*})=c<cap. omega>{*}(E{*})[E]
in which c=1/<cap. omega>{*}(E{*}) is a constant coefficient independent
of E.   <cap. omega>(E) is the number of accessible states for the system
A with the energy E, and <cap. omega>'(E{*}-E) is the number of accessible
states for the system A' with the energy E'=E{*}-E, and
<cap. omega>(E)<cap. omega>'(E{*}-E) is the number of accessible states 
for the system A{*} in each of which the energy of the system A is E, ie
<cap. omega>{*}(E{*})[E]=<cap. omega>(E)<cap. omega>'(E{*}-E), and 
therefore P(E)=c<cap. omega>(E)<cap. omega>'(E{*}-E).

Let's see under which conditions the curve of P(E) in terms of E will have
its maximum. Since E is greater than or equal to zero and then certainly
P(E) is greater than zero, for extermum we have:

dP(E)              d            1   dP(E)
----- = 0  <==>  (--- lnP(E) = ---- ------) = 0 ,
 dE               dE           P(E)  dE

and in the point of extremum, ie when we have dP(E)/dE=0, we shall also have:

d{2}P(E)  >            d{2}            1   d{2}P(E)   >
--------  <  0  <==>  (----- lnP(E) = ----- --------)  <  0 .
  dE{2}   =            dE{2}          P(E)  dE{2}     =

Therefore, the conditions that cause maximization of the curve P(E) in 
terms of E are the same conditions that cause maximization of the curve 
lnP(E) in terms of E, and we shall investigate this recent case. We have:

P(E) = c<cap. omega>(E)<cap. omega>'(E')  ==> 
lnP(E) = lnc + ln<cap. omega>(E) + ln<cap. omega>'(E{*} - E)  ==> 
dlnP(E)/dE = dln<cap. omega>(E)/dE + dln<cap. omega>'(E{*} - E)/dE =
dln<cap. omega>(E)/dE - dln<cap. omega>'(E{*} - E)/d(E{*} - E) =
dln<cap. omega>(E)/dE - dln<cap. omega>'(E')/dE' .

Therefore, it is seen that the condition under which P(E) is extremum is that
the expression dlnP(E)/dE=dln<cap. omega>(E)/dE-dln<cap. omega>'(E')/dE'
to be equal to zero, ie this condition is establishment of the equality
dln<cap. omega>(E)/dE=dln<cap. omega>'(E')/dE' or <beta>(E)=<beta>'(E')
where <beta>(E)=dln<cap. omega>(E)/dE and <beta>'(E')=
dln<cap. omega>'(E')/dE'.
We shall now prove that this condition for being extremum is the same
condition for maximization. We have:

(E=f<epsilon> & (<cap. omega>(E) is proportional to <epsilon>{3f/2})) ==>
(<cap. omega>(E) is proportional to E{3f/2}) ==>
ln<cap. omega>(E) = cte.+(3f/2)lnE ==>
(<beta>(E)=dln<cap. omega>(E)/dE)=(3f/2)(1/E) ==>
(d<beta>(E)/dE=d{2}ln<cap. omega>(E)/dE{2})=(-3f/2)(1/E{2})<0, and similarly
(E'=f'<epsilon>' & (<cap. omega>'(E') is proportional to <epsilon>'{3f'/2}))
==> (<cap. omega>'(E') is proportional to E'{3f'/2}) ==>
ln<cap. omega>'(E') = cte.'+(3f'/2)lnE' ==>
(<beta>'(E')=dln<cap. omega>'(E')/dE')=(3f'/2)(1/E') ==>
(d<beta>'(E')/dE'=d{2}ln<cap. omega>'(E')/dE'{2})=(-3f'/2)(1/E'{2})<0.

Therefore, P(E) will be maximum if and only if <beta>(E)=<beta>'(E').

Temperature of a system of particles (solid, liquid or gas) is an expression
proportional to the mean kinetic energy (excluding the potential one) of
each molecule (or particle) of the system. Proportion constant should be
selected properly one time for always. This act will be done soon. It 
should be emphasized that what the thermometers show as the temperature
of a system is the mean kinetic energy of the molecules of the system
(regardless of the kind of the system as solid, liquid or gas) not also
probably the (mean) potential energy of the molecules of the system. 
The reason of this statement is, in a quite intuitive manner, that it is
only the motion of the molecules that can have practical effect on the
thermometer, not probably their potential static states.

The system we have discussed so far in this article is a system which, 
as conditioned at first, its molecules don't have any influence on each
other (except probably collision) (ie it is practically a set of the 
molecules of a perfect gas), and then all the energies indicated by E or
<epsilon> are kinetic (not also potential which such an energy does not
exist in such a system). Therefore, the mean kinetic energy of this set
is <epsilon>=E/f. We saw that <beta>(E)=3f/(2E), and then 
<epsilon>=E/f=3/(2<beta>(E)). Therefore, temperature is an expression
proportional to the recent expression. We indicate the temperature by T
and the propotion constant (which we want to be positive) by 2/(3k).
In this manner we shall have:

     2               2        3            1
T = --- <epsilon> = --- . ---------- = ----------
     3k              3k   2<beta>(E)   k<beta>(E)

We see that the temperature of a system, which is propotional to the 
mean kinetic energy of its molecules, takes the form of T=1/(k<beta>(E))
in the case of the system under discussion. (k is named as Boltzmann
constant.)

We saw that the condition of maximization of the curve P(E) in terms of
E was the establishment of the equality <beta>(E)=<beta>'(E'). This means,
according to the definition of temperature, that the condition of 
maximization of P(E) is that T and T', which are in turn the temperatures 
of the systems A and A', to become equal to each other. It is obvious that
if E is such that T isn't equal to T', then the systems A and A' will be
in states of little probability, and then will interchange so much energy 
that at last the result T=T' will be obtained. Since
dT/dE=-1/(k<beta>{2})d<beta>/dE and we saw d<beta>/dE<0, we have dT/dE>0.
(In a similar manner we can see that also dT'/dE'>0.) This means that,
as expected, the (kinetic) energy will flow toward that system which its
temperature is less. (It can be seen from the relation
T=1/(k<beta>)=1/(k(3f/(2E)))=2E/(3kf) that if E=0, we shall have T=0, and
if E=<infinity>, we shall have T=<infinity>.) 

Considering the relation which we now have between the temperature and 
the mean kinetic energy of the molecules (T=2<epsilon>/(3k)), let's see
what more we can say about the molecules of a perfect gas.

As performed in most of the preliminary textbooks, we can consider a 
molecule of the gas and calculate difference between its momentums before
and after its hitting against a wall of the container, and consider the
time of its travel between the walls and the number of the molecules of the 
gas. (For example see Physics by Halliday and Resnick, John Wiley & Sons, 
1978.) In this manner we finally shall obtain p=<rho>(v{2})[av]/3 in which
p is the gas pressure, <rho> is the (mass) density of the gas, and
(v{2})[av] is the mean value of the squares of the speeds of the molecules.
Indicating the total mass of the gas molecules by M, the mass of each 
moleculer by m, the volume occupied by the molecules by V, and the 
number of the molecules by N, we have

p=(1/3)<rho>(v{2})[av]=(1/3)(M/V)(v{2})[av] ==>
pV=(2/3)M((v{2})[av]/2)=(2/3)Nm((v{2})[av]/2)=
(2/3)N((1/2)m(v{2})[av])=(2/3)N<epsilon>

in which, as before, <epsilon> is the mean kinetic energy of each molecule
ie is equal to (1/2)m(v{2})[av]. We saw, beforehand, that T=2<epsilon>/(3k)
or <epsilon>=(3/2)kT, and then pV=2/3N(3/2kT)=NkT=nN[0]kT in which n is the 
number of moles of the molecules and N[0] is Avogadro's number. Since both
N[0] and k are constant, the factor N[0]k is shown by a single constant, R
(named as gas constant). Therefore, we have  pV = nRT .

III. The real Boltzmann factor
------------------------------
Now we proceed to the more fundamental part of this article. So far we have
been discussing about P(E), which was the probability of finding the energy
of the system A equal to E. Now we want to know the probability of finding 
the system A with energy E in the quite special state r, ie we want to find
the probability of finding the system A in a special one of its accessible
states while the energy of A is the constant value E. This probability is
obviously equal to 
P[r]=<cap. omega>{*}(E{*})[rE]/<cap. omega>{*}(E{*}) = 
c<cap. omega>{*}(E{*})[rE] in which 
<cap. omega>{*}(E{*})[rE] = 1<cross><cap. omega>'(E{*}-E) and then 
P[r] = c<cap. omega>'(E{*}-E) = c<cap. omega>'(E'). (See Statistical
Physics, Berkeley physics course, by Reif, McGraw-Hill, 1975.)
We suppose that E<<E{*}. Let's expand ln<cap. omega>'(E') in the form of
a Taylor series about the point E'=E{*}:
ln<cap. omega>'(E') = ln<cap. omega>'(E')|[E'=E{*}] +
dln<cap. omega>'(E')/dE'|[E'=E{*}].(-E) +
(1/2!)d{2}ln<cap. omega>'(E')/dE'{2}|[E'=E{*}].(-E){2} +
(1/3!)d{3}ln<cap. omega>'(E')/dE'{3}|[E'=E{*}].(-E){3} + ....
We know that ln<cap. omega>'(E')=cte.'+(3f'/2)lnE' because

                                     E' = f'<epsilon>'  )
                                                        >  ==>
<cap. omega>'(E') is proportional to <epsilon>'{3f'/2}  )

<cap. omega>'(E') is proportional to E'{3f'/2}.

Then d{n}ln<cap. omega>'(E')/dE'{n} = 3(-1){n+1}(n-1)!f'/(2E'{n}) in which
n = 1, 2, 3, .... Therefore, the general term of the above expansion is
-3f'/(2n)(E/E{*}){n} (of course for n=00 we have the first term of the 
expansion, ie ln<cap. omega>'(E')|[E'=E{*}]), and since (E/E{*})<<1, we can
write with a very good approximation
ln<cap. omega>'(E') = ln<cap. omega>'(E')|[E'=E{*}] - (3f'/2)E/E{*}.
Now we assert that we can write (-3f'/2)(E/E') instead of
(-3f'/2)(E/E{*}) with a very good approximation if (E/E{*})<<1, because
<delta>(E/E{*})/<delta>E{*}=-E/E{*}{2} and since here <delta>E{*}=E'-E{*}=-E,
we have <delta>(E/E{*})=(E/E{*}){2}, and since (E/E{*})<<1, we conclude that
<delta>(E/E{*})=(E/E{*}){2} is, with a very good approximation, equal to 
zero. Namely with replacing E{*} by E' in the second term of the above 
relation the produced change is negligible because of excessive smallness.
Therefore, we can write
ln<cap. omega>'(E')=ln<cap. omega>'(E')|[E'=
E{*}]-dln<cap. omega>'(E')/dE'.E
=ln<cap. omega>'(E')|[E'=E{*}]-<beta>'(E').E ==>
<cap. omega>'(E') = cte.exp(-<beta>'(E').E)
in which cte. is independent of E'. Therefore, 
P[r] = c<cap. omega>'(E') = c.cte.exp(-<beta>'(E').E) or

P[r] = c'exp(-<beta>'(E')E)                                        (1)

in which c'=c.cte. is a constant independent of E and E'. It must be 
emphasized again that the expression (1) is the probability of finding the 
system A, with constant energy E, in the quite special accessible state r,
under the condition that A is in contact with the source A' and 
E<<E+E'=E{*} and E{*} will remain constant because of the isolation of
the set of A and A'.

Unfortunately, there is a wrong construing of this equation (ie (1)) in the
current statistical physics books. This wrong construing is that after
replacing E in Eq.(1) by E[r], this equation is interpreted as the 
probability of this matter that the system A, which is in contact with  
the much more energetic system A', has an energy between the special
energy E[r] and the energy E[r]+<delte>E[r], while the set of A and A'
is isolated. It is obvious that this interpretation is not at all real.

Indeed, P[r] in Eq.(1) is not a function of E, but E in this relation 
plays the role of a constant. P[r] is a function of different states 
accessible for the system A which has the constant energy E. In the wrong 
interpretation, explained just now, P[r] is taken as a function of E[r];
then one can integrate it over some extent of energy. Such an integration
will be done over (E[r] in) the exponent of e in the equation (because
wrongly E is replaced by E[r] in this equation).

But truly the only integration which can be done on Eq.(1) is over the 
different states accessible for the system with the constant energy (E),
ie in fact over the variable r which itself doesn't appear as a mathematical
factor. In such an integration, E plays only the role of a constant.

But what is really the cause of this wrong construing of Eq.(1)? The 
existence reason of this wrong construing is that for obtaining Eq.(1),
instead of the explicit method of this article the implicit method of
Lagrangian multipliers is used often, which eventually results in a 
relation like
n[i] = g[i]exp(-<alpha>)exp(-<beta>u[i])
in which n[i] is the number of molecules each having the energy u[i], and
g[i] is the probability of that a molecule has the energy u[i] 
(g[i]=n[i]/n), and <beta> and <alpha> are the Lagrangian multipliers which
must be determined (eg see Perspectives of Modern Physics by Beiser,
McGraw-Hill, 1969). It is obvious that dividing each side of the recent
relation by n (the number of all the molecules) we have 
n[i]/n=(g[i]exp(-<alpha>)/n)exp(-<beta>u[i]) the left side of which gives 
the probability of that the molecule has the energy u[i], and we can say
that this probability has linear propotion to exp(-<beta>u[i]) only when 
the coefficient g[i]exp(-<alpha>)/n is constant (ie does not depend on the
energy u[i]), and it is quite obvious that this is not the case (because
as in Eq.(1), <beta> depends on E' and then on E, or in other words as in
the recent relation <beta> depends on u[i], <alpha> can also depend on 
u[i]). But, unfortunately, to this fact is not paid attention, and it is
assumed that this probability is propotional to exp(-<beta>u[i]). This is
just the same current wrong construing of Eq.(1). The correct is that we 
say that in the true relation n[i]=g[i]exp(-<alpha>)exp(-<beta>u[i]) the
factor exp(-<beta>u[i]) is what that in this article appears in the form
of Eq.(1).

(Under the condition E<<E{*} we showed that P[r](E)=c'exp(-<beta>'(E')E).
It is obvious that we must have <summation over r>P[r](E)=1, in which 
the summation is over all the accessible states in energy E. Therefore,
c' = 1/(<summation over r>exp(-<beta>'(E')E)) = 
1/(<cap. omega>(E)exp(-<beta>'(E')E)) and then, as it is expected, we shall
have P[r](E)=1/<cap. omega>(E). If y is a quantity that assumes the amount
y[r] when the system A, with the constant energy E, is in the state r, then
the mean value of y will be obtained from the relation
(y)[av] = <summation over r>P[r](E)y[r] = 
(<summation over r>y[r])/<cap. omega>(E)
which could be written in the unsimplified form of 

          <summation over r>exp(-<beta>'(E')E)y[r]
(y)[av] = ---------------------------------------- .)
            <summation over r>exp(-<beta>'(E')E)

IV. Maxwell velocity distribution
---------------------------------
Before more continuing the discussion it is necessary to say that the
relation (1) can yield the fundamental relations of the statistical 
mechanics. For showing this, we shall obtain the Maxwell velocity 
distribution:

We saw that if E<<E{*}, then P[r] will be proportional to exp(-<beta>'(E')E).
It is obvious that this probability (ie P[r]) is also proportional to the
volume of the cell in the phase space, h'{3} =
<delta>x<delta>y<delta>z<delta>p[x]<delta>p[y]<delta>p[z]. Thus P[r] is
ptoportional to 
exp(-<beta>'(E')E)<delta>x<delta>y<delta>z<delta>p[x]<delta>p[y]<delta>p[z].

P[r]/h'{3} is the density of the probability of finding the system A, with 
the energy E, in the special state r, and we indicate it by <cursive P>[r].
Thus <cursive P>[r]h'{3}=c[1]exp(-<beta>'(E')E)h'{3} in which c[1] is the
proportion constant. We have <summation over r><cursive P>[r]<delta>x<del 
ta>y<delta>z<delta>p[x]<delta>p[y]<delta>p[z]=1, which in differential and
integral form we can write it as 
<integral over r><cursive P>[r]dxdydzdp[x]dp[y]dp[z]=1. (It is important
to mind that the integration must be done over the different r-states not 
over any special volume in the phase space.) Therefore, c[1]=1/(<summation
over r>exp(-<beta>'(E')E)<delta>x<delta>y<delta>z<delta>p[x]<delta>p[y]<del
ta>p[z]) or c[1]=1/(<integral over r>exp(-<beta>'(E')E)dxdydzdp[x]dp[y]dp[z]).

In the case of a particle of a monoatomic perfect gas we have 
E=1/2mv{2}=p{2}/(2m); so the expression exp(-<beta>'(E')mv{2}/2)d{3}rd{3}p,
in which we have d{3}r=dxdydz and d{3}p=dp[x]dp[y]dp[z], is proportional to 
the probability of finding the location of the particle in the cubic range
between ~r and ~r+d~r, and the momentum of the particle in the cubic range
between ~p and ~p+d~p. Now since m~v=~p and so md{3}v=d{3}p, we conclude that
exp(-<beta>'(E')mv{2}/2)d{3}rd{3}p is proportional to the probability of
finding the location of the particle in the cubic range between ~r and
~r+d~r and the velocity of the particle in the cubic range between ~v and
~v+d~v. If the proportion constant of this proportion is c[2], then this
probability will be c[2]exp(-<beta>'(E')mv{2}/2)d{3}rd{3}v. If we integrate
this probability over the unit volume about a definite point, the result, ie
<integral over V>c[2]exp(-<beta>'(E')mv{2}/2)d{3}rd{3}v =
c[2]exp(-<beta>'(E')mv{2}/2)d{3}v in which V is the same unit volume about     
the point, will be the probability of this matter that in a unit volume 
about this point, the end of the velocity vector of a particle, which has
the absolute speed v, is in the cubic range between ~v and ~v+d~v (ie its
velocity components are between v[x] and v[x]+dv[x], v[y] and v[y]+dv[y],
and v[z] and v[z]+dv[z]). If we show this probability as a percentage, eg in
the form of m%, this will mean that if, in the unit volume about the given
point, we evaluate simultaneously a hundred particles, each of which 
having the absolute speed v, from the totality of the particles which are
similar to and independent of each other, we shall see that on the
average only m particles of these one hundred particles will have the
velocity vectors the end of each being in the cubic range between ~v and 
~v+d~v. So, if we have n[v] particles with absolute speed v in the unit
volume about that definite point, and evaluate them simultaneously, we 
shall see that n[v]<cross>(m/100) particles of them will have the above
characteristic. Therefore, since we said that
(m/100)=c[2]exp(-<beta>'(E')mv{2}/2)d{3}v, the expression
n[v]<cross>(m/100)=n[v]c[2]exp(-<beta>'(E')mv{2}/2)d{3}v is in fact the
volume density of those particles which the end of the velocity of each
of them is in the cubic range between ~v and ~v+d~v, in a definite point
~r (or in fact in a desirable point ~r, because there isn't any dependence 
on ~r on the right side of the recent relation). Since the range of the 
velocity is a differential range (d{3}v), this volume density is also
in fact differential, and if we divide it by d{3}v, the velocitical
density of the volume density of those particles which the end of the 
velocity vector of each of them is in the cubic range between ~v and
~v+d~v will be obtained that we show it by f(~v). Therefore, we reach the
Maxwell velocity distribution:
f(~v)d{3}v=c[3]exp(-<beta>'(E')mv{2}/2)d{3}v  in which c[3]=n[v]c[2].

V. Consequences of the wrong construing of the Boltzmann factor
----------------------------------------------------------------
In this section we want to see how the first deduction of the relation
E=h<nu> has been based on the wrong construing of the Boltzmann factor.
So we accept the relation (1) in the form of 
P[E[r]]=c'exp(-<beta>'(E')E[r]) deliberately temporarily and then we shall
evaluate its consequences: Since <summation over r>P[E[r]]=1, we have 
c'=1/(<summation over r>exp(-<beta>'(E')E[r])) and so 
P[E[r]]=exp(-<beta>'(E')E[r])/(<summation over r>exp(-<beta>'(E')E[r])).
Now we divide each side of the recent relation by <delta>E[r]:

  P[E[r]]                    exp(-<beta>'E[r])
----------- = -------------------------------------------------
<delta>E[r]   (<summation over r>exp(-<beta>'E[r]))<delta>E[r]

in which the expression on the left side is the density of probability, and
in limit will result in the relation <cursive P>[E[r]]=exp(-<beta>'E[r])/<in
tegral from 0 to <infinity>>exp(-<beta>'(E')E[r])dE[r] and since 
<integral from 0 to <infinity>>exp(-<beta>'E[r])dE[r]=1/<beta>', we
shall have <cursive P>[E[r]]=<beta>'exp(-<beta>'E[r]) which is the density
of probability of finding the system A with some energy between E[r] and
E[r]+<delta>E[r]. If we have a big system consisting of some numerous number
of smaller systems (eg each similar to A) such that all of them are in 
contact with a thermal source which is much more energetic than each of 
them, then the mean value of the energy of each of these small systems 
will be (E)[av]=<integral from 0 to <infinity>>E[r]<cursive P>[E[r]]dE[r]
or (E)[av]=<integral from 0 to <infinity>><beta>'exp(-<beta>'E[r])E[r]dE[r] 
=1/<beta>'=kT' in which T' is the source temperature and k is the 
Boltzmann constant. (Just here we see another contradiction, because while
it is claimed that (E)[av]=kT' is the mean energy of the small systems 
which are in contact with a thermal source which its temperature is T',
the limit of the smallness of these systems is not at all distinct, and so
if we suppose that the source is the whole universe for example, we can 
suppose that the big system, mentioned above, to be a set of some separate
molecules in a turn, and in another turn to be a set of some separate
blocks of stone for example, while in each turn we shall obtain the same
result (E)[av]=kT' for the mean value of the energy of each molecule or
each block! It is obvious that this is quite irrational.) Since according 
to the current equipartition theorem the mean value of the energy of each
particle of a system of particles having both kinetic and elastic potential
energies, is equal to 1/2kT'+1/2kT'=kT', the obtained (E)[av]=kT'
can be interpreted as the mean value of the energy of these particles.

Now we suppose that these particles are the contents of a blackbody cavity
in the temperature T'. Experience shows that the mean energy of the 
cavity radiation is about kT' for the frequencies approaching zero, but
is about zero for the frequencies approaching infinity. As an attempt to
justify this matter, we can draw the curve of E[r]<cursive P>[E[r]] in 
terms of E[r] (see Quantum Physics of atoms, molecules, solids, nuclei and
particles by Eisberg and Resnick, John Wiley & Sons, 1974). Since
<cursive P>[E[r]]=(1/(kT'))exp(-E[r]/(kT')), the maximum of the curve 
will occur in E[r]=(E)[av]=kT', also we have 
(E)[av] = <integral from 0 to <infinity>>E[r]<cursive P>[E[r]]dE[r] = kT'
which is equal to the area under the curve. If we suppose that E[r] can
exist in discontinuous forms of 0, <cap. delta>E[r], 2<cap. delta>E[r],
3<cap. delta>E[r], ..., then we shall see that (E)[av] is approximately 
equal to kT' when <cap. delta>E[r] is approximately equal to zero, and
(E)[av] is approximately equal to zero when <cap. delta>E[r] is 
approximately equal to <infinity>. Comparison of these results with the
previous results obtained by the experience indicates that we must have:
( <cap. delta>E[r] is proportional to <nu> ).

Above reasoning is Planck's reasoning in obtaining the relation (<cap.
delta>E is proportional to <nu>) or E=h<nu>; but as we said, this 
reasoning is on the basis of a quite wrong construing of Eq.(1).
Therefore, the equation <cap. delta>E=h<nu> or simply E=h<nu> may be
valid only if it can be derived uniquely from the results obtained from
the photoelectric effect (which is not of course the case as shown in 
another article of this book).

The Langevin formula for orientational polarizability has also been 
obtained in a similar manner by using the above mentioned wrong construing
of Eq.(1), and then can not be valid unless it can be obtained by another
way.

VI. Specific heat of solids and gases
-------------------------------------
Maybe it is said that but the equipartition theorem has been obtained by
using the very current construing of the Boltzmann factor which according
to this article is wrong. (To prove this theorem, in fact, integration is
done over different energies not over different forms of a constant energy.)
And important is that the result of this theorem can predict the empirical 
relation c[V]=3R valid for hot crystalline metals. Thus, let's see what the 
actuality is: 

As we said, temperature of a set of molecules is an expression proportional 
to the mean kinetic energy of them. Indicating this mean kinetic energy
by <epsilon>, the total kinetic energy (not including the potential one)
of the molecules of the set by E, and the number of the molecules of the
set by f, we saw that for the temperature of the system (or set), T, we have

     2              2    E      2      E        2E        2
T = --- <epsilon> = --- --- = ------- --- = ---------- = --- E     (2)
     3k             3k   f    3R/N[0]  f    3(f/N[0])R   3nR 

in which R is the gas constant, k is the Boltzmann constant, N[0] is
Avogadro's number and n is the number of moles of the molecules of the set.

Notice a point again. Temperature is the mean "kinetic" energy of each 
molecule, not its mean total energy regardless of its kind (including 
probably its potential energy). Therefore, if we should calculate the
expression 

        1   dU
c[V] = --- ----                                                    (3)
        n   dT

for the molar specific heat of a crystalline solid by giving it the energy
dU and measuring its increment of dT in temperature, we must be attentive
that U in this relation (ie in Eq.(3)) is other than the term E in Eq.(2).
In other words all the energy given to the set won't necessarily be
conserved as the kinetic energy of the molecules of the set, but a part of 
the given energy may be conserved, eg, in the form of potential energy
of the molecules of the set if possible.

Then, we suppose that the conditions of the molecules of the set are such 
that only 1/z of the energy given to the set conserves in the form of 
kinetic energy of the molecules of the set and the rest of it conserves 
as the potential energy of these molecules. Then we must write E=U/z which
considering Eq.(2) results in T=2U/(3nRz) which leads to U/n=3/2RzT
whereby we shall have n{-1}dU/dT=3/2Rz. Therefore, we have

       3Rz
c[V] = ---                                                         (4)
        2

for Eq.(3).

For ideal gases that all the energy given to the gas molecules are 
conserved as their kinetic energy and z is equal to one for them, we have
c[V]=3/2R. But for a crystalline solid which, on average, half of the 
energy of each of its molecules, like an oscillating spring, is in the
form of kinetic energy and the other half is in the form of potential
energy, z is equal to 2. In this case we shall have c[V]=3R for Eq.(4).
As we see the empirical equation c[V]=3R for hot crystalline metals 
has been obtained without any necessity to use the current wrong construing
of the Boltzmann factor.

The mechanism presented above by introducing the factor z merits the name
of partition of energy instead of equipartition of energy.

Hamid V. Ansari