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

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Gauge Institute Journal, H. Vic Dannon Δ=∞ ∫ Δ=−∞ ϕ( Δ) dΔ = 1, ϕ( Δ) ≠ 0, only for very small Δ , ϕ( Δ ) = ϕ( −Δ). 7) f (,) xt is the particles’ density at x , at time t We first note that ϕ( Δ ) is the Delta Function that was established already in 1882 by Kirchhoff. [Temple, p.158], and was similarly presented without mentioning Kirchhoff by Dirac. Recently, we established the Delta Function as a hyper-real function in infinitesimal Calculus. [Dan4]. Then, δ( x) ⎧ 1 dx dx , x ∈ ⎡− , ⎤ = ⎪ dx ⎢ 2 2 ⎥ ⎨ ⎣ ⎦ ⎪⎪ 0, otherwise ⎩ Consequently, according to assumption 6 dn = nδ( Δ) dΔ ⎧ 1 dΔ dΔ dΔ, Δ ∈ ⎡− , ⎤ = n ⎪ dΔ ⎢ 2 2 ⎥ ⎨ ⎣ ⎦ ⎪⎪ 0, otherwise ⎩ ⎧ dΔ dΔ n, Δ∈⎡− , ⎤ = ⎪ ⎢ 2 2 ⎥ ⎨ ⎣ ⎦ ⎪⎪ 0, otherwise ⎩ 4
Gauge Institute Journal, H. Vic Dannon That is, all the particles are in an infinitesimal interval at the origin… This bizarre conclusion leads to no contradiction since neither n , nor dn , appear in the following Einstein’s derivation… 0.2 Einstein’s Derivation of the Diffusion Equation Einstein starts with equation (17), p.131, claiming that Δ=∞ ∫ f (, xt+ τ) = fx ( + Δ,)( tδ Δ) dΔ Δ=−∞ In fact, the sifting by δ( Δ ) gives f (,) xt , and no equality. Einstein’s next claim mandates that τ f fxt (, + τ) = fxt (,) + τ ∂ , ∂ t must be an infinitesimal. This makes assumptions 2, and 3, meaningless, but leads to no contradiction, since the assumptions are not used in the derivation that follows… Then, Einstein’s expands his integrand ∂fxt (,) 1 ∂ fxt (,) 2 fx ( +Δ , t) = fxt ( , ) + Δ+ Δ + ... ∂x 2! 2 ∂x 2 5
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