Einstein's Diffusion and Probability-Wave Equations of Random ...

Gauge Institute Journal,
H. Vic Dannon
or to the right, with probability
p ,
q
= 1 − p.
At time t , after
N infinitesimal time intervals dt ,
N =
t
dt
, is an infinite hyper-real,
the particle is at the point
x .
At the i th step we define the Bernoulli Random Variable,
Bi (right step)
Bi (left step)
where i = 1,2,..., N .
= dx , ζ
1
= right step .
=−dx, ζ
2
= left step .
Pr( B = dx)
= p,
i
Pr( B =− dx)
= q,
i
E[ B ] = dx ⋅ p + ( −dx ) ⋅ q = ( p − q)
dx ,
i
2 2 2
i
EB [ ] = ( dx) ⋅ p+ ( −dx) ⋅ q = ( dx)
2 2
i i
2
( dx)
( p−q)
dx
Var[ Bi] = E[ B ] −( E[ B ])
= (1
+ p −q)(1 − p + q)( dx) = 4 pq( dx)
2p
2 2
q
2
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