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Potential Dynamics
The analysis and understanding of the amplitude equation are vastly simplified by a remarkablefeature of the equation, namely the existence of a potential or (loosely) a Lyapunov function. Thisquantity is an integral functional of the amplitude and its low order derivatives over the domainthat for periodic, and a few other boundary conditions, has properties analogous to the mechanicalenergy (potential and kinetic) of a frictionally (over)damped ball in a potential, namely that thefunctional monotonically decreases in the dynamics.Just as for the damped ball, for which we know that the motion will eventually cease with the ballat a minimum of the mechanical energy (zero kinetic energy and potential energy at a minimum,not necessarily the lowest), the potential for the amplitude equation tells us that the amplitude willapproach a configuration giving a minimum of the potential, and that there the dynamics willcease. Not surprisingly, the minima (as well as the maxima) of the potential correspond to thesolutions of the amplitude equations. In fact, the relation between the two is the same as thatbetween the Lagrangian and the Newton’s equations.This is a very restrictive result that imposes strong constraints on the dynamics. The existence of apotential often provides a powerful tool for understanding the system. We refer to the existence ofthe potential as remarkable, because it is not a property that we would generally expect for asystem far from equilibrium. Indeed, more careful analysis shows that the existence of this type ofpotential is an artifact of the lowest order truncations in the expansions leading to the amplitudeequation, since higher order amplitude equations are generally no longer potential.
Let us show that the dynamics predicted by the lowest order amplitude equation are potential forcertain kinds of boundaries. The dynamics of the amplitude predicted by the equationtogether with particular but common boundary conditions, must be consistent with the continualdecrease of the potential
V^
given by
(&)
This quantity evolves according to
(&&)
We can easily verify this by computing the time derivative of (&)In order to remove the spatial derivatives from
we need to integrate by parts in the last term.
This operation generates terms that are evaluated at the boundaries of the domain. For
V^
to be a
potential, all of these boundary terms must vanish. This is the case for periodic boundaryconditions as well as for the homogeneous boundary conditions for suppressing boundaries in arectangular domain (see previous lecture).
2 0 2 2
(^20)
0
A
A
g A
i q
A
A^
y c x
t^
^
]
[
2 2
(^20) 4 0 2
∫∫^
^
=^
A
i q
A
g A
dxdy
A V^
y c x
2
0
∫∫^
A
dxdy
dVdt
t
(^
)^
c.c.
*^
2
2
(^20)
2 0
∫∫^
^
^
^
^
=^
A
i q
A
i q
A
A
A
g A
dxdy
dVdt
t y c x y c x t
* A
∂ t
Several caveats should be stated. First of all, we can only describe nonlinear competition betweendifferent patterns composed of stripes in terms of a potential, if the amplitude equations
for
superimposed stripes
are potential. Luckily, it turns out that they are (at leading order in
ε).
Second, the result applies
only
in the context of an experiment in which the competition between
bulk saturated regions,
in contact via a domain wall
, occurs. Other experimental conditions, such
as the growth from small initial conditions, may give
different
results. This is because in general
there will be a potential barrier between the two ideal states (see figure below) and only in specialphysical circumstances will there be a dynamical path between the two that flows monotonicallydown the potential. Finally, the motion of the domain wall may be impeded, for example, by pinning of the wall to thestripes themselves, in which case there may be no motion even if the potentials are different. Thiswould then give a finite range of parameters for coexistence.
stripesdomi-nate
hexa-gonsdomi-nate
A second application of the potential is to the question of wave number selection, i.e., determiningthe precise value of the wave number in an
ideal
stripe state (or unit cell size in the
ideal
lattice
states). Again we can argue that any local dynamics that mediates between two ideal statesoccupying large portions of the system, which will then dominate the integral that forms thepotential, will favor the state with a lower value of the potential
V (
q ). (Here
V (
q ) is the potential
evaluated for the ideal periodic state with wave number
q .) In the case of potential dynamics,
different dynamical mechanisms that allow the wave number to change, such as defect motion,boundary relaxation etc., will all tend to yield the
same
wave number, namely the one that
minimizes
V (
q ). This has been confirmed by numerical experiments. A similar test could be used
to decide whether a particular experimental system is potential or not.
Amplitude Equation for Stripes in Anisotropic Systems
The amplitude equation takes on a different form for the uniaxial systems (systems with reflectionbut no rotational symmetry). This form can again be derived phenomenologically by exploitingthe symmetry properties of the linear instability. If we orient the
x^ direction along the preferred
axis, the amplitude equation for stripes with the critical wave vector
takes the form
The constants
ξ x^
and
ξ y^
define the spatial scales for modulation of the amplitude in the
x^ and
y
directions and can be related to the expansion of the growth rate about the critical wave vector
ˆ x qc. 2 0 2 2 2 2 0 A A g A A A
A^
y y x x
t^
[^
].
(^
2 2 2
2 (^10)
y y c x x^
q
q q
ξ
ξ ε τ σ^
=^
− q
