Manuscript En01-B@2 · En01-B-v2.tex · section · f226d1eb

Different mass and packet prescriptions

Declared regularized model and its dynamics › Different mass and packet prescriptions · lines 1090–1146

En01-B@2#eq:spectral

For two displaced intrinsic profiles,

E_{G} (d) = \frac{G}{2} \int \int \frac{δ f ( r ) δ f ( r ^{'} )}{∣ r - r ^{'} ∣} d r d r^{'}, δ f (r) = f_{r_{c}} (r) - f_{r_{c}} (r - d e),

and Fourier substitution gives $E_{G} (d) = ℏ Γ_{r_{c}} (d) \geq 0$ .

A continuous homogeneous sphere has

μ (k) = M F (∣ k ∣ R), F (u) = 3 \frac{sin u - u cos u}{u ^{3}}, F (0) = 1,

Γ_{r_{c}} (d) = \frac{2 G M ^{2}}{π ℏ} \int_{0}^{\infty} e^{- r_{c}^{2} k^{2}} F (k R)^{2} [1 - sinc (k d)] d k .

A microscopic constituent specification instead has

μ = j \sum m_{j} δ_{a_{j}}, f_{r_{c}} (k) = e^{- r_{c}^{2} ∣ k ∣^{2} /2} j \sum m_{j} e^{- ik \cdot a_{j}} .

Its cross terms cannot be discarded without an approximation argument. Writing

W_{r_{c}} (r) = \frac{erf ( r / ( 2 r _{c} ))}{r}, W_{r_{c}} (0) = \frac{1}{π r _{c}},

its rate is

ℏ Γ_{r_{c}} (d) = G j, l \sum m_{j} m_{l} [W_{r_{c}} (∣ a_{j} - a_{l} ∣) - W_{r_{c}} (∣ a_{j} - a_{l} + d e ∣)] .

This follows by convolving two Gaussian mass profiles and using their Newtonian potential.

Finite-width centre-of-mass packets are states of $x$ , not intrinsic mass profiles. Their coherence kernel experiences $Γ_{r_{c}} (x - x^{'})$ at every position pair. Packet width is not interchangeable with $r_{c}$ .

Displayed formulas in this passage

En01-B@2#eq:spectral · line 1109
$Γ_{r_{c}} (d) = \frac{2 G M ^{2}}{π ℏ} \int_{0}^{\infty} e^{- r_{c}^{2} k^{2}} F (k R)^{2} [1 - sinc (k d)] d k .$
unlabeled · line 1093
$E_{G} (d) = \frac{G}{2} \int \int \frac{δ f ( r ) δ f ( r ^{'} )}{∣ r - r ^{'} ∣} d r d r^{'}, δ f (r) = f_{r_{c}} (r) - f_{r_{c}} (r - d e),$
unlabeled · line 1102
$μ (k) = M F (∣ k ∣ R), F (u) = 3 \frac{sin u - u cos u}{u ^{3}}, F (0) = 1,$
unlabeled · line 1117
$μ = j \sum m_{j} δ_{a_{j}}, f_{r_{c}} (k) = e^{- r_{c}^{2} ∣ k ∣^{2} /2} j \sum m_{j} e^{- ik \cdot a_{j}} .$
unlabeled · line 1125
$W_{r_{c}} (r) = \frac{erf ( r / ( 2 r _{c} ))}{r}, W_{r_{c}} (0) = \frac{1}{π r _{c}},$
unlabeled · line 1130
$ℏ Γ_{r_{c}} (d) = G j, l \sum m_{j} m_{l} [W_{r_{c}} (∣ a_{j} - a_{l} ∣) - W_{r_{c}} (∣ a_{j} - a_{l} + d e ∣)] .$

Nearby in this section

proposition: Proposition: Quadratic coefficient and remainder