Einstein's Diffusion and Probability-Wave Equations of Random ...
Einstein's Diffusion and Probability-Wave Equations of Random ...
Einstein's Diffusion and Probability-Wave Equations of Random ...
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Gauge Institute Journal,<br />
H. Vic Dannon<br />
Pro<strong>of</strong>: We’ll denote<br />
2<br />
t<br />
2<br />
x<br />
( )<br />
x<br />
∂ f = − C∂ f + D ∂ f<br />
1 1<br />
( x − dx ≤ B ζ t ≤ x + dx) = ( B ζ t x)<br />
Pr ( , ) Pr ( , )<br />
2 2<br />
Then, by Bayes’ Theorem,<br />
( B ζ t dt x)<br />
Pr ( , + ) =<br />
<br />
f ( xt , + dtdx )<br />
( ) (<br />
= Pr B(, ζ t + dt) x / B(,) ζ t x −dx Pr B(,)<br />
ζ t x −dx<br />
+<br />
<br />
p f( x−dx, t)<br />
dx<br />
( ) (<br />
+ Pr B(, ζ t + dt) x / B(,) ζ t x + dx Pr B(,)<br />
ζ t x + dx<br />
<br />
That is,<br />
Substituting<br />
we obtain<br />
q<br />
f( x+<br />
dx, t)<br />
dx<br />
f (, x t + dt) = pf ( x − dx,) t + qf ( x + dx,)<br />
t .<br />
f (, x t + dt) ≈ f (,) x t + ( ∂ f (,)) x t dt ,<br />
t<br />
1<br />
2<br />
2 2<br />
x<br />
f ( x − dxt ,) ≈ fxt (,) − ( ∂ fxt (,)) dx + ( ∂ fxt (,))( dx)<br />
,<br />
x<br />
1 2 2<br />
x 2 x<br />
)<br />
f ( x+ dxt , ) ≈ fxt ( , ) + ( ∂ fxt ( , )) dx+ ( ∂ fxt ( , ))( dx ,<br />
1 2 2<br />
x<br />
2 x<br />
<br />
−2 Cdt<br />
( Ddt )<br />
( ∂t<br />
f ( xt , )) dt ≈ ( q − pdx ) ( ∂ fxt ( , )) + ( dx ) ( ∂ fxt ( , ))<br />
<br />
,<br />
)<br />
)<br />
22
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