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• ## Re:Inconsistency in classical theory of gases

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• ... This sharpy Roger Bagula has pointed out a problem between the classical theory of diffusion by a Gaussian mechanism and current relativity theory? Does
Message 1 of 1 , Jul 2, 2007
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Roger Bagula wrote:

I was a graduate student in 1968 and I found an inconsistency in
classical theory of gases
in my text book.

The Maxwell - Boltzman distribution law in my Chemical Thermodynamics text book read like this: ( for velocities vx,vy,vz in the 3d coordinate directions)
f[vx_, vy_, vz_] = Exp[-m*(vx^2 + vy^2 + vz^2)/(2*k*T) ]
N=C1*Integrate[ Integrate[ Integrate[ f[x,y,z], {x, -Infinity, Infinity}],{ y, -Infinity, Infinity}], {z, -Infinity, Infinity}]

But we had just had relativity and quantum mechanics. I was reading a book on field quantization on the side.
The actual law in terms of modern physics had to look like this:
N=C2*Integrate[ Integrate[ Integrate[ f[x,y,z], {x, -c, c}],{y, -c, c}], {z, -c, c}]
I realized even then that these two were radically different!

In the first case using a mass scaled temperature:
T=m*c^2/k
The integration normalization constant is:
1/C1=Sqrt[2* Pi]*c=7.51468357 0223141*10^ 10
In the second case the integration constant is:
1/C2=(Sqrt[2* Pi]*c*Erf[ 1/Sqrt[2] ])^3=1.350211332 4962547*10^ 32
In the first case the constant is in the "quantum" wavelength range.
In the second case it is more like a quantum volume
and is in actuality very near the Planck radius.

This constitutes a large scale discrepancy between the theory in the book and it's derivations of diffusion distributions with Infinite limit velocities and velocities limited to the speed of light.
But the thermodynamics laws work fine using infinite velocities.
No one told them they had a speed limit?
The secret I learned later was it called "asymptotic containment" .
A lot of modern open ended hyperbolic 4 and 5 space theories
use that for keeping particles from decaying immediately
along the hyperbolic time axis.

That kind of concept was a little beyond a first year graduate thermodynamics course in the late 1960's.
The point of all this is that the special relativity constraints
in velocity aren't all that obvious in classical Gaussian distribution theories.

We have since found out that this second self-similar scale is relevant
when you get very very small at very high energy:
the Plank scale. Along came second order uncertainty and quantum black holes... we got a new limiting doctrine.
We also got a glimpse of the idea of quantum gravity, unifying gauge fields and Lie algebras in a new vacuum outlook
with a quantum lattice of space itself.
Respectfully, Roger L. Bagula
11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 :http://www.geocitie s.com/rlbagulatf tn/Index. html
alternative email: rlbagula@sbcglobal. net

This sharpy Roger Bagula has pointed out a problem between the classical theory
of diffusion by a Gaussian mechanism and current relativity theory?
Does there  exist a way out of this in the standard model in 5 dimensions?
The answer after some thought and calculation seems to be yes.
Our function was:
f[vx_, vy_, vz_] = Exp[-m*(vx^2 + vy^2 + vz^2)/(2*k*T)]
such that:
Integrate[Integrate[ Integrate[f[x, y, z], {x, -c, c}], {y, -c, c}], {z, -c, c}]
is a constant different than when the velocities are Infinite.
Suppose we create a new function in hyperbolic five space:
g[vx_, vy_, vz_, vt_, vw_] = Exp[-m*(vx^2 + vy^2 + vz^2 -  vt^2 - vw^2)/(2*k*T)]
and we integrate that with velocity limits  -a to a for the bloody time velocities.
Integrate[Integrate[Integrate[Integrate[Integrate[g[x, y, z, t, w], {x, -c, c}], {y, -c, c}], {z, -c, c}], {t, -a, a}], {w, -a, a}]
such that the resulting constant is the same as the Infinite velocity result.
We an equation like this:
C5*Erfi[ka*a]^2-C1=0
where the constants are approximately:
C5=7.624706741199758*10^53
ka=2.3586540063095223*10^(-11)
C1=7.514683570223143*10^10
Solve[7.624706741199758*10^53 *(2.3586540063095223*10^(-11)*  a)^2 - 7.514683570223143*10^10 == 0, a]
gives an estimate of a as:
a~1.3310045443281747*10^(-11)
Using that estimate in FindRoot in Mathematica gives:
FindRoot[7.624706741199758*10^53 *Erfi[2.3586540063095223*10^(-11)* a]^2 - 7.514683570223143*10^10 == 0, {a, 0,
1.3310045443281747*10^(-11)}]
{a -> 1.1795720650836038*10^-11}
What does this give us?
Well first we have this C5 constant:
r5=1/C5=1.3115258513439542*10^-54
I'm not claiming all the decimal place accuracy ( that is produced automatically by Mathematica)
since I used a 4 place Boltzman constant to start!
Right here your string fellows are going to cry foul,
because that involves a distance smaller than the Planck radius ( substantially).
But there is evidence for this in the observations of a particle in a supernova
by Dr. Parker.
Where the original quantum distances are in the range ( Infinite velocity in 3d)
r~k/c
The Planck distances are in the range: ( velocity c in 3d)
r~k/c^3
This new range is ( velocity c in 3d and a in 2d time)
r~k/c^5
In fractal terms we have an odd power self-similarity:
r(i)=k(i)/r^(2*i-1)
for i={0,1,2,...,m}
This model is a quantum model.
The existence of the exceptional Lie Algebra E11 suggests that a scale like:
r~k/r^11
m=6 and a total 7 levels to the power 11.
Since c is a very large scale, such a fractal universe
converges very rapidly.
You will notice that no where have I claimed a velocity greater than light:
I actually calculate a velocity "a" very close to no motion at all!
But the result is that I have an "equivalence" to an Infinite velocity.
I haven't fooled with causality at all
,but just with geometry in higher dimensions.
This kind of multidimensional diffusion model isn't really all that new,
but is is applied here in a unique new way.
The result  makes the problem of Infinite velocity go away
and also gives a model for a
fractal scaled universe.

```Respectfully, Roger L. Bagula
11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 :http://www.geocities.com/rlbagulatftn/Index.html
alternative email: rlbagula@...

```
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