01.

## “Dark particle” or mini black hole.

Special circumstances arise when the mass of the black hole is the reduced Planck mass

\begin{align}m_p = \sqrt{\frac{\hbar c}{8\pi G}}\notag\end{align}

\begin{align}\delta x_p &= \frac{\hbar}{2 m_p c} = 2 \pi R_S \notag\ = \frac{4 \pi G m_p}{c^2} \notag\end{align}

\begin{align}k_B T_{\text{Hawking}} = \frac{\hbar c^3}{8 \pi G m_p} = m_p c^2\notag\end{align}

For a Dark Particle, the event horizon (solid line) is the smaller than quantum boundary (dotted line) of dark particles and thus the surface temperature is defined by the variance (quantum boundary) and is at arbitrary temperature T.

\begin{align}R_{QB} = \frac{\Delta x_0}{2 \pi} = \frac{\hbar}{4 \pi \sqrt{m_p k_B T}}\notag\end{align}

\begin{align}\kappa (r) = \frac{-1}{2} \frac{\partial}{\partial r} \left( -1 + \frac{2 G m_p}{c^2 r} \right ) = \frac{G m_p}{c^2 r^2}\notag\end{align}

\begin{align}T_{DP} = \frac{\kappa (R_{QB})\hbar}{2 \pi k_B c} = \frac{8 \pi G m^2 T}{\hbar c} = T\notag\end{align}

02.

## Dual solutions to quantum diffusion & general relativity

Quantum Mechanics Length Scale

Quantum variance between positive and negative particles

\begin{align}\ell^2 \equiv 2 \cdot (\Delta x)^2\notag\end{align}

Where,

\begin{align}m(\ddot{x}) = -\frac{m}{\tau}(\dot{x}) - \frac{m}{\tau^2} x\notag\end{align}

With,

\begin{align}\overline{x(\dot{x})}(t=0) = \frac{\hbar}{2m} = \frac{D}{2}\notag\end{align}

A modified Langevin equation is the solution to both realms

\begin{align}\ell^2 = 2\overline{x^2} = 4 D \tau \left( 1- e^{-t/\tau}\right)\notag\end{align}

General Relativity Length Scale

Maximum distance between positive and negative particles in relaxation time "tau"

\begin{align}L \equiv 2 \tau c = \frac{\hbar c}{k_B T}\notag\end{align}

Where,

\begin{align}\left (\frac{(\dot{L})}{L}\right)^2 = \frac{8\pi G}{3} \rho_{DP}{L}\notag\end{align}

With,

\begin{align}\rho_{\text{DarkParticle}} = \frac{3}{8 \pi G} \left(\frac{\hbar}{mL^2} - \frac{k_B T_0}{\hbar} \right)^2\notag\end{align}

A modified Langevin equation is the solution to both realms

\begin{align}L_{DP}^2 = 4 D \tau_0 \left( 1- e^{-t/\tau_0}\right)\notag\end{align}

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