Our Derivations
01.
Newton’s law derivations
From Newton’s laws
[Newton’s 2nd law]
\begin{align}F_{\text{entropic}} + F_{\text{conservative}} = m a_{\text{particle}} \notag\end{align}
[Newton’s 3rd law]
\begin{align}m a_{\text{particle}} = F_{\text{particle}} = - F_{\text{vacuum}} = - ma\notag\end{align}
[Newton’s 1st law]
\begin{align}F_{\text{entropic}} + F_{\text{conservative}} + F_{\text{vacuum}} = 0 \notag\end{align}
Other formulas
[Euler - Lagrange]
\begin{align}F_{\text{vacuum}} = ma = \frac{d}{dt} (m v) = \nabla \left (\frac{mv^2}{2} \right ) \notag\end{align}
[Potential Energy]
\begin{align}F_{\text{conservative}} = - \nabla V(r) \notag\end{align}
[Entropic Force]
\begin{align}F_{\text{entropic}} = T\nabla S = k_B T \nabla \log(\Omega) \notag\end{align}
Combining ALL formulas
[All Formulas]
\begin{align}k_B T \nabla \log(\Omega) - \nabla V(r) + \nabla \left (\frac{m v^2}{2} \right ) = 0 \notag\end{align}
\begin{align}k_B T \cdot \log(\Omega) = V - \frac{m v^2}{2} + E_0 = E_0 - \mathcal{L} \notag\end{align}
\begin{align}\tau = \frac{\hbar}{2 k_B T} \notag\end{align}
\begin{align}\log (\Omega) = \left (\frac{2}{\hbar} \right ) (E_0 - \mathcal{L} ) \tau \notag\end{align}
02.
Path integral derivations
FORMULAS USED
[Canonical Distribution]
\begin{align}\log (\Omega ) = S / k_B = - E [\log (p) ] \notag\end{align}
[Concavity of Log]
\begin{align}\log (\Omega ) = - E [\log (p) ] = - \log (p)\notag\end{align}
[Rules of Qm]
\begin{align}p = \psi^* \psi \notag\end{align}
[Path Integral]
\begin{align}\psi = e^{-i\mathcal{H} d t/\hbar} \notag\end{align}
[Slow massive particle]
\begin{align}\mathcal{H} = m_0 c^2 + \frac{m v^2}{2} + V\notag\end{align}
[Minkowski transformation]
\begin{align}it \rightarrow \tau \notag\end{align}
[Langrangian]
\begin{align}\mathcal{L} = \frac{mv^2}{2} - V\notag\end{align}
Resulting formula
[Result]
\begin{align}S/k_B = \left (\frac{2}{\hbar}\right ) \int (mc^2 -\mathcal{L} ) d\tau \notag\end{align}
03.
Capstone derivation - information theory
[Mutual Information]
\begin{align}dC = \frac{1}{2} \log \left ( 1 + \frac{S}{N} \right ) \notag\end{align}
\begin{align}S = (\Delta x)^2_{\text{Vaccum,} K=1} = \left ( 1 - \left (\frac{v}{c} \right )^2 \right ) 2 (\delta x)^2 \notag\end{align}
[Conditional Entropy]
\begin{align}H_2(\beta) = -\beta \log(\beta) - (1-\beta)\log(1-\beta) \notag\end{align}
\begin{align}H_2(\beta) = \log(2) - \frac{1}{2} \left(\frac{v}{c}\right)^2 \notag\end{align}
[Mutual Information]
\begin{align}N = 2 D_\mu \tau = 2 \mu k_B T \tau = \frac{v\hbar}{F} \notag\end{align}
\begin{align}d I_M = 2 d C = \frac{2 \left ( 1- \left (\frac{v}{c} \right )^2 \right ) F c^2 (\delta t)^2}{v \hbar} \notag\end{align}
\begin{align}\frac{dI_M}{dt} = \frac{2Fc^2}{\frac{\hbar v}{dt}} - \frac{2 F v d t}{\hbar} \notag\end{align}
\begin{align}\frac{dI_M}{dt} = \frac{2}{\hbar} (mc^2 + V - V_0) \quad \quad \quad \quad 0 = \frac{2}{\hbar} (mc^2 - V_0) \notag\end{align}
[Conditional Entropy]
\begin{align}H_C (\beta) = H_2 (\beta) + \log\left(\frac{e}{2}\right) = 1-\frac{1}{2}\left(\frac{v}{c}\right)^2 \notag\end{align}
\begin{align}\frac{dH_C}{dt} = \frac{2}{\hbar} \left( mc^2 - \frac{m v^2}{2}\right) = \frac{2}{\hbar}(mc^2 - K) \notag\end{align}
\begin{align}-E[\log(p)] = H_C + I_M = \frac{2}{\hbar} \int (mc^2 - (K - V)) dt \notag\end{align}
[Final Formula]
\begin{align}\frac{S}{k_B} \equiv \log(\Omega) = -E[\log(p)] = \left(\frac{2}{\hbar}\right) \int (mc^2 - \mathcal{L}) dt \notag\end{align}
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