From Osher Doctorow
I should emphasize that the Croatian Marin-Slobodan Tomas of Rudjer
Boskovic Institute Croatia is undoubtedly unfamiliar with Probable
Causation/Influence (PI), but Tomas' equations show that the force is
proportional to Probable Causation/Influence (PI) when normalized.
The paper is "Enhanced van der Waals interaction at interfaces,"
arXiv: 0710.3503 v1 [quant-ph] 18 Oct 2007, 8 pages.
The key force equation relevant to this is Tomas' (24b) which has the
form:
1) F_Az(rA, rB) proportional to 1 + y - x
where:
2) x = w_A^2 /w_S^2
with w_S the surface-mode frequency and w_A is the transition
frequency of atom A (an excied atom). Here atom B is a ground-state
atom, both embedded in an inhomogeneous magnetoelectric system
described by permittivity e(r, w) and permeability u(r, w). I assume
that x is normalized.
In Probable Causation/Influence (PI), x of (1) and (2) is the Cause
(Causal Probability) and y of (1) is the Effect (Effect Probability),
so (2) tells us that the Cause is the ratio of frequencies squared of
the above types.
The effect y is rather ponderous algebraically but involves rational
functions of z_A and z_B (with magnitudes the respective distances of
atoms A and B from the interface (boundary) or sapphire surface) and
also is directly proportional to 1 - w_A^2 / w_B^2 and also L(w_B,
w_A, gamma_B) where the latter comes from:
3) L(x, y, z) = (definition) x^4/[(x^2 - y^2) + (yz)^2]
where x isn't necessarily the quantity referred to as x in (1) and
(2). In fact, in (3), w_B is x, w_A is y, and gamma_B is z, so that L
increases the closer w_A is to w_B and the bigger w_B is for "all else
constant".
I should point out that the force F_Az in (1) is the perpendicular
component of the force F_A. Because of the presence of atom B, F_A
gets a parallel component and its perpendicular component is
diminished or increased due to the relative positions of surface and
atomic resonances.
w_A < w_S is attractive, w_A > w_S is repulsive.
Osher Doctorow
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