Small Negative, Big Positive

From the post "No Experimental Proof" dated 29 Jul 2016,

$\cfrac{a_{\psi\,\pi}}{a_{\psi\,c}}=\cfrac{cosh^{-1}(e)}{cosh^{-1}(e^{1/4})}$

$\cfrac{a_{\psi\,\pi}}{a_{\psi\,c}}=2.24921$

From the post "Deep Blue Deeper" dated 01 Jun 2016,

$\left.\cfrac { dq }{ dx }\right|_{ a_{ \psi  } }  =3k-2\left.\cfrac { \partial \, T }{ \partial \, x }\right|_{a_{\psi}}$

where,

$\cfrac{\partial\psi}{\partial x}=\cfrac{\partial V}{\partial x}+\cfrac{\partial T}{\partial x}=k$

a constant and, $x\le a_{\psi\,\pi}$.

and the post "Why A Positron And Deep Blue..." dated 01 Jun 2016,

$\cfrac { dq }{ dx } =\left[3 \cfrac { \partial V\,  }{ \partial \, x } +\cfrac { \partial \, T }{ \partial \, x } \right]_{x=a_{\psi}}$

But we know that $a_{\psi\,c}$ is at a higher frequency than $a_{\psi\,\pi}$.   And intuitively,

$\left.\cfrac { \partial \, T }{ \partial \, x }\right|_{a_{\psi\,\pi}}\lt \left.\cfrac { \partial \, T }{ \partial \, x }\right|_{a_{\psi\,c}}$

$\because a_{\psi\,c}\lt a_{\psi\,\pi}$ and both particles are waves wrap around a center at light speed.

If $\cfrac { \partial \, T }{ \partial \, x }\propto \cfrac{1}{a_{\psi}}$, as in the case of gravitational force under earth's surface (a force being the rate of change of energy with distance), that,

$\left.\cfrac { \partial \, T }{ \partial \, x }\right|_{a_{\psi\,c}}=\cfrac{a_{\psi\,\pi}}{a_{\psi\,c}} \left.\cfrac { \partial \, T }{ \partial \, x }\right|_{a_{\psi\,\pi}}$

$\left.\cfrac { \partial \, T }{ \partial \, x }\right|_{a_{\psi\,c}}=2.24912 \left.\cfrac { \partial \, T }{ \partial \, x }\right|_{a_{\psi\,\pi}}$

So,

$\left.\cfrac { dq }{ dx }\right|_{ a_{ \psi \,c } } =3k-2\left.\cfrac { \partial \, T }{ \partial \, x }\right|_{a_{\psi\,c}}$

$\left.\cfrac { dq }{ dx }\right|_{ a_{ \psi \,c } } =3k-2*2.24912\left.\cfrac { \partial \, T }{ \partial \, x }\right|_{a_{\psi\,\pi}}$

$\left.\cfrac { dq }{ dx }\right|_{ a_{ \psi \,c } } =3\left.\cfrac{\partial V}{\partial x}\right|_{a_{\psi\,\pi}}-1.49824\left.\cfrac{\partial T}{\partial x}\right|_{a_{\psi\,\pi}}$

where $\left.\cfrac{\partial\psi}{\partial x}\right|_{a_{\psi\,\pi}}=\left.\cfrac{\partial V}{\partial x}\right|_{a_{\psi\,\pi}}+\left.\cfrac{\partial T}{\partial x}\right|_{a_{\psi\,\pi}}=k$

since,

$\cfrac{\partial\psi}{\partial x}=\cfrac{\partial V}{\partial x}+\cfrac{\partial T}{\partial x}=k$

is true for values of $x\le a_{\psi\,\pi}$.  So,

$\left.\cfrac { dq }{ dx }\right|_{ a_{ \psi \,c } }=\left[3 \cfrac { \partial V\,  }{ \partial \, x } -1.49824\cfrac { \partial \, T }{ \partial \, x } \right]_{x=a_{\psi\,\pi}}$

but,

$\left.\cfrac { dq }{ dx }\right|_{ a_{ \psi \,\pi } }=\left[3 \cfrac { \partial V\,  }{ \partial \, x } +\cfrac { \partial \, T }{ \partial \, x } \right]_{x=a_{\psi\,\pi}}$

$\cfrac { dq }{ dx }$ is also a force; from the two expressions, the resultant interaction of $a_{\psi\,c}$ and $a_{\psi\,\pi}$ is attractive, as the common component in the two sums,

$3 \cfrac { \partial V\,  }{ \partial \, x }$

cancels and the leading coefficients of the term

$\cfrac { \partial \, T }{ \partial \, x }$

sums to a negative value ($-0.49824$) and results in a decrease in $T$ (kinetic energy $KE$) along $x$.

In the case when $a_{\psi\,\pi}$ interacts with $a_{\psi\,\pi}$ both particle experience positive change in kinetic energy in the direction of $x$ and they move further apart.  The potential component still cancels, as both particles change $PE$ in the opposite directions.  The interaction of two $a_{\psi\,c}$ particles however, is still attractive due to the negative coefficient to the $KE$ term,

$\cfrac { \partial \, T }{ \partial \, x }$

So, $a_{\psi\,c}$ particles will coalesce to size $n=n_e$, where

$\cfrac{a_{\psi\,\pi}}{a_{\psi\,ne}}=ratio_e$

until the substitution into the expression,

$\left.\cfrac { dq }{ dx }\right|_{ a_{ \psi \,c } } =3k-2\left.\cfrac { \partial \, T }{ \partial \, x }\right|_{a_{\psi\,c}}$

does not result in a negative coefficient to the term,

$\cfrac { \partial \, T }{ \partial \, x }$

ie,

$2*ratio_e\lt 3$

$ratio_e\lt 1.5$

or

$\cfrac{a_{\psi\,\pi}}{a_{\psi\,ne}}\lt 1.5$

Only after attaining the size $n_e$, $a_{\psi\,ne}$, do the particles repel each other.  There is not just one opposite charge particle but $a_{\psi\,c}$ till $a_{\psi\,ne}$ are all opposite charge to $a_{\psi\,\pi}$.

If $a_{\psi\,c}$ is the complement negative particle, from the post  "Sizing Them Up" dated 3 Dec 2014,

$a_{\psi_c}$ (nm)	$f_c$ (GHz)	$\lambda_c$(nm)	$a_{\psi\,\pi}$(nm)	$f_{\pi}$(GHz)	$\lambda_{\pi}$(nm)
19.34	2466067.5	121.57	43.50	1096415.0	273.43
16.32	2922728.6	102.57	36.71	1299446.7	230.71
15.48	3082568.8	97.25	34.82	1370511.8	218.75
14.77	3230699.3	92.79	33.22	1436370.7	208.70

where the spectra lines are due to basic particles $a_{\psi\,c}$, $n=1$.  And the size of big particles ${a_{\psi\,\pi}}$ are given by,

$\cfrac{a_{\psi\,\pi}}{a_{\psi\,c}}=2.24921$

and

$2\pi a_{\psi}=\lambda$

It was expected that $a_{\psi\,c}$ are negative particles.  But the value of $2466067\,Hz$ suggests integer reduced micro wave frequency that agitates $T^{+}$ particles.  Based on this, $a_{\psi}=19.34\,nm$ is a $T^{+}$ particle.  What gives?

Simple; an input of energy at reduce resonance frequency (by an integer divisor) imparts energy onto the particle;  $a_{\psi\,c}$, a negative temperature particle, grows into $a_{\psi\,\pi}$, a positive temperature particle;  the matter heats up.

Goodnight...