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

Gauge Institute Journal,
H. Vic Dannon
Proof: Since the
P i
are independent,
EP [ ( ζ, t)] = EP [
1 ] + ... + EP [ ]
N
= λNdt
λdt
λdt
Var[ P( ζ, t)] = Var[ P1
] + ... + Var[ PN
] ≈ λNdt
≈λdt
≈λdt
t
t
6.3 The Probability-Wave Equation of the Process
Let ( )
Pr P( ζ , t) = k = p( k, t)
Then The Probability-Wave Equation of X(,)
ζ t for pkt (,) is
the first order differential-difference wave equation
Proof: By Bayes’ Theorem,
( P ζ t + dt = k)
Pr ( , )
∂ pkt (,) = −λΔ
pkt (,)
t
k−1
)+
pkt (, + dt)
= Pr P(, ζ t + dt) = k / P(,) ζ t = k − 1Pr P(,) ζ t = k −1
=
( ) (
p= λdt p( k−1, t)
( )
( )
+ Pr P(, ζ t + dt) = k / P(,) ζ t = k Pr P(,)
ζ t = k
q=−
1 λdt p(
k, t)
That is,
pkt (, + dt) = pk ( − 1,) tλdt+ pkt (,)(1 − λdt)
,
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