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stability) in the vertical direction (i.e. gravity, electric and/or magnetic forces); for completeness,
it might also include the confinement potential ensuring horizontal stability in experiments
[Samsonov, 2002].
2.1 2d equation of motion
Considering the motion of the n−th dust grain in both the longitudinal (horizontal, ∼ ˆx) and
the transverse (vertical, off-plane, ∼ ẑ) directions (i.e. suppressing the transverse in-plane –
shear – component, ∼ ˆx; see Fig. 1), so that r n = (x n ,z n ), we have the 2d equation of motion
M
( d 2 r n
dt 2
+ ν dr )
n
= − ∑ dt
n
∂U nm (r nm )
∂r n
+ F ext (r n ) ≡ q E(r n ) + F ext (r n ), (1)
where E j (x) = −∂φ(r)/∂x j is the (interaction) electrostatic field and F ext,j = −∂Φ ext (r)/∂x j
accounts for all external forces in the j− direction (j = 1/2 for x j = x/z); the usual ad hoc
damping term was introduced in the left-hand-side of Eq. (1), involving the damping rate ν due
to dust–neutral collisions.
2.2 Anharmonic vertical substrate potential
Assuming a smooth, continuous variation of the levitation (electric E ext and/or magnetic B ext )
field(s) and of the grain charge q (which varies due to charging processes) near the equilibrium
position z 0 , the electric and magnetic force(s) may be combined into an overall vertical force
F ext (z) = F el/m (z) − Mg ≡ −∂Φ ext (z)/∂z ≈ −M[ω 2 gδz n + α(δz n ) 2 + β (δz n ) 3 ] + O[(δz n ) 4 ]
(δz n = z n − z 0 ) where the phenomenological substrate potential Φ ext (z) is of the form
[ 1
Φ(z) ≈ Φ(z 0 ) + M
2 ω2 g δz2 n + α 3 (δz n) 3 + β ]
4 (δz n) 4 + O[(δz n ) 5 ]. (2)
Recall that F e/m (z 0 ) = Mg at equilibrium. The gap frequency ω g and the phenomenological
anharmonicity coefficients α and β are defined via (derivatives of) E ext and B ext (see in [Kourakis
& Shukla, 2004c; d] for the exact definitions).
The anharmonic potential Φ ext (z) is depicted in Fig. 2, as it results from ab initio calculations
[Sorasio, 2002].
2.3 Discrete equations of motion
Let (δx n , δz n ) = (x n − x (0)
n , z n − z n (0) ) denote the displacement of the n−th grain from the
equilibrium position (x (0)
n ,z n (0) ) = (nr 0 , 0). Assuming small displacements from equilibrium,
one may Taylor expand the interaction potential energy U(r) around the equilibrium inter-grain
distance lr 0 = |n − m|r 0 (between l−th order neighbors, l = 1,2,...), i.e. around δx n ≈ 0 and
δz n ≈ 0, viz.
U(r nm ) =
∞∑
l ′ =0
∣
1 d l′ U(r) ∣∣∣r=l
l ′ (x n − x m ) l′ ,
! dr l′ |n−m|r0
where l ′ = 2 denotes the potential parabolicity, l ′ = 3 denotes cubic nonlinearity in the interaction,
and so forth. Notice that the inter-grain distance r = [(x n −x m ) 2 +(z n −z m ) 2 ] 1/2 also needs
to be expanded near |x n − x m | = lr 0 and z n − z m = 0, viz. ∂U(r)/∂x j = [∂U(r)/∂r](∂r/∂x j ) ≈
3