Maxwells distribution model has to be linked to the ideal gas laws, the Euler fluid dynamic equations, the Bernoulli principle , and to the Navier stokes equations and the Ed Lorenz equations. But it seems prudent to start with Newtons fluid dynamics to gain a better understanding of how this model displays information for cogitation. We also need to look at Gauss derivation and Boltzmans to see what the original models were.
I also want to radicalise the notion of mercury temperature. Mercury is used to measure both pressure and temperature. We are quite able to measure radiant light as temperature, and perhaps should move to this frequency based measure. Thermometer temperature should be retailed to measure heat pressure or expansion of a material in proportion to the heat it has retained. This would be a ratio of the amount of pressure the mercury exudes compared against the amount of pressure the material exudes and proportioned by the amount of heat given to the material.
I would like to do a Bernoulli like heat flow experiment using this concept.mheat pressure can be measured st uSing heat pressure concept at one end, in a thiner portion of material in the middle and a large lump at the other end.
Now Maxwells distribution is a simple relationship. I expect that there will be a variety of velocities in space. How do I find out what they are!
I stick a kite in the wind and se how it flies! Or I watch a snowflake move .
So if I want to model the velocities in a region I do the same thing.
Today I can get a pretty much instantaneous velocity distribution by radar measurement of test probes in my given space. Maxwell of course did not have that so he tried for a general method that would allow him to take boundary measurements to generate a fictional model of what the internal distribution was like. All he could do was calculate a 3d:distribution from a set of 2 d distributions. He could not be specific, only general. At the same time it imposed a restrictive probability distribution condition on the measurements. The only use of such a method is to inspire others on to tackle the more complex situations. This analysis would not help give specific results only very general relationships that might be next to useless.
However, tables of data now had a design principle which provided a paradigm for compiling data. On the way quantitative data changed, how it was collected, presented and interpreted. Thus Data piled up while mathematics struggled. The demand for computational machines was growing exponentially.
The only note able success of this distribution was in the Kelvin kinetic theory of Gases
One of the reasons the distribution became so influential is because of Fouriers Transform. Kelin had fallen in love with it and applied it as Isley as he could. He could apply it to Maxwells distribution, greatly complicating it's essential simplicity, but making it more accessible to computation and data input! The actual data could be arranged to provide the probability distributions required to calculate the solutions. This introduced frequency and amplitude into the data description and laid the ground for psi the state function of quantum mechanics. In the mean time fluid dynamics was ignored for tis kind of mathematical research and physical treatment.
I have to point out again, this distribution only models an existing "structural" relationship of uantities , it does not allow extrapolation, only interpolation. The Fourier transform emphasises this because along with polynomial interpolation it was one of the standard methods of interpolation.
This flurry of ideas and mathematical advance belies the fact that physically No ADVANCE was being made. Mathematicians were advancing there understanding of these interpolation forms and relationships , getting used to using these strange restrictions to model relationships, but no one was actually understanding how fluids flow, gases flow, materials behave as a system of particles or light , electricity and or magnetism were incorporated.
The only advance came through empirical data and observation. Faraday provided a rich source of this and theoreticians like Maxwell tried to use the new Mathematical techniques to Model the data. Maxwell went back to fluid mechanics and the Navier Stokes Equations. In the meantime Helmholtz had tried to model the perfect fluid?
The models were becoming increasingly sophisticated, but something was missing.
Maxwell decide to study electric phenomenon as Faraday suggested it was like some kind of elastic fluid, providing electric tension but the discharging it through " pipes" like a water stream or current.
Maxwell decided to work with Equipotential surfaces. He spent considerable time understanding these solutions to differential equations. This gave him a model of equipotentials in space, slices through his distribution formulae. What he could no model was the discharge.
His equipotentials were time consuming to compute so he never developed a dynamic representation of them. Rather he talked over the 2 d representations of them. He however was convinced they represented vorticular flow and his discussions always carry that basic meaning.
He did hope Hamilton's Quaternions would make his ideas plain. However Kelvin and others took a dislike to Hamilton and discredited his creation. Maxwell was forced by the " club" to denounce Hamilton, and he rewrote his work in terms of the new vector algebra Gibbs had purloined from Grassmann and twisted out of shape. Maxwell could never get the sign to go right consistently!
The capacitor was the key to an experimental problem. When Maxwell pooled all the research together, and especially in the light of Faradays law, he realised no one had looked at the battery!
The source of power in any electrical system could be represented by a capacitor. Today we obscure this simple fact. The capacitor was not about preventing current flowing around a circuit, it was about providing the source of flow in a circuit!
So, in all the formulae nobody had remembered to put the battery in the circuit. Maxwell corrected this and called it the displacement current for displacement current read " Battery"
What maxwell had found was the tautology of electric circuits. They appear to self generate if a capacitor is placed in it with another charged capacitor. The discharge was audible and visible and tuneable. Oscillating circuits could now be made.
Maxwell predicted oscillating circuits, and oscillating discharges travelling through the dielectric medium as sparks.
It is not that these had not been observed already, it was simply that others had not put it into the formulary!
Faraday had created the displacement flux, that is the changing flux that displaced a galvanometer briefly, so it seemed fitting to call his creation the displacement current, that is the changing electrostatic field that displaced a magnetometer . Besides Oersted had already shown a steady current established a "steady" magnetic field circular to a wire.
It turns out that Faraday and Maxwell had got it wrong. Heaviside and Now Ivor Catt refuted with empirical evidence the notion of current flowing in a wire. Instead the capacitors communicate with each other by what Catt notes as a" step" wave. He defines it as an energy current ExH that flows in the dielectric around and between the wires. One capacitor flows its energy into another capacitor. It flows in the dielectric and so interacts with distant capacitors according to an inverse square law. This flowing " step" wave of course is only a step between the two wave guides. Outside of the wave guides it falls away according to an inverse square law.
This step wave I want to refer back directly to a fluid dynamic model, in hich typically it will be considered as as shear wave. The model of this shear wave I will relate to Maxwells distribution. The distribution may model any quantities and in this case it will modl a variation of voltage potentials. The conducting wave guides will selecta narrow range of voltage potentials as stored in the charged capacitor and released into the dielectric.
It will be noted that this is not a sinusoidal wave model, it is a voltage shear potential model. It goes without saying that it can also be modelled by Fourier transforms