Hypothesis of Temporal Homogeneity
//One Dragon in Ten Tampan,// Paminggir people. Lampung region of Sumatra, circa 1900, 45 x 41 cm. From the library of Darwin Sjamsudin, Jakarta. Photograph by D Dunlop.
One Dragon in Ten Tampan, Paminggir people. Lampung region of Sumatra, circa 1900, 45 x 41 cm. From the library of Darwin Sjamsudin, Jakarta. Photograph by D Dunlop.

The hypothesis of temporal homogeneity is an assumption that all Newtonian particles are endlessly splashing about in a bath of photons and other ethereal particles. The Newtonian particles supposedly absorb and emit these ethereal particles in a way that moderates extreme motions, scatters energy values and causes regression toward mean temperatures.

To be more exact, consider a Newtonian particle P described by a chain of events $\Psi$ where about one in every couple hundred events involves an interaction with an ethereal particle. Let these processes of absorption or emission occur independently of each other, as if particles were randomly bumping into each other. The changes in P are described by a coefficient of variation $\delta E / E$. The hypothesis of temporal homogeneity is an assumption that the coefficient of variation is a constant $k_{S}$ given by

$\begin{align} k_{S} \equiv \frac{1 }{2} \sqrt{ \ 1 + \frac{1\vphantom{1^{2}} }{ \pi } \ } - \frac{1 }{2} \end{align}$

This number $k_{S}$ is approximately 7%. Interactions might be minimized for an insulated particle, or increased if P is in an accelerator. But the hypothesis asserts that as a statistical inevitability, the energies of Newtonian particles will be dispersed by $k_{S}$. The conjecture is plausibly justified anywhere near the Earth because a flux of sunlight and tides moving through the solar system presumably keeps things well-stirred. Any particle that has its energy measured must interact with macroscopic laboratory apparatus and this logically precludes being isolated. So the coefficient of variation in its energy is presumably at least as large as usual. Thus for energy measurements on Newtonian particles, the hypothesis of temporal homogeneity implies that

$\begin{align} \frac{ \delta E }{ E } \ge k_{S} \end{align}$

sun.png
Sensory interpretation: The quantity $k_{S}$ is just another way of expressing Archimedes'Xlink.png well-known constant $\pi$. Traditionally $\pi$ is defined from the geometry of circles. But such a treatment is not consistent with the premise of WikiMechanics. So the hypothesis of temporal homogeneity provides another interpretation of $\pi$ that is based on a description of sensation. The number $k_{S}$ can be understood as a signal to noise ratio of about 11 decibels in the stream of sensory consciousness described by $\Psi$. This quantity is presumably a characteristic of the Sun.
Right.png Next step: dispersion.




Notice: this page is under construction
Notice: this page is under construction

To do: Let P be described by a repetitive chain of historically ordered events

$\Psi ^{\sf{P}} = \left( \ldots \, \sf{\Omega}_{\it{i}} \, \ldots \, \sf{\Omega}_{\it{f}} \, \ldots \right)$

where each orbit $\sf{\Omega}$ is characterized by $K$, $\mathcal{U}$ and the total number of quarks it contains $N$. Changes between some arbitrary initial and final events are noted as

$\begin{align} &\tilde{ N } = \left( N_{\it{f}} + N_{\it{i}} \right) /2 &\ &\ &\ &\tilde{ K } = \left( K_{\it{f}} + K_{\it{i}} \right) /2 \\ &\Delta N = N_{\it{f}} - N_{\it{i}} &\ &\ &\ &\tilde{ \mathcal{U} } = \left( \mathcal{U}_{\it{f}} + \mathcal{U}_{\it{i}} \right) /2 \end{align}$

show that temporal homogeneity implies that these quantities are all related as

$\begin{align} \frac{ \Delta N }{ \tilde{ N } }=\frac{\tilde{ K } - \tilde{ \mathcal{U} } }{ \tilde{ K } + \tilde{ \mathcal{U} } } \end{align}$

So that if P is in dynamic equilibrium, then on average there is no change to the total number of quarks in P.

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