
In this section almost periodicity appears in a different way, particles no longer move on a discrete grid as before but as a gas in in the Vlasov limit (see [5, 13]). We prove here an existence result in the almost periodic context. Vlasov dynamics are used especially in stellar dynamics and plasma physics.
A finite dimensional system of particles moving on a manifold under a pairwise interaction given by a potential V evolve according to the Newton equations which are the Hamilton equations with Hamiltonian . We assume that the potential V is smooth and that the solutions exist for all times. If the force is rescaled in the limit so that it stays finite, then the dynamics can be extended from point particles to a "particle gas" with an arbitrary density m in the phase space . One evolves then a map , where gives the position and momentum of the particle with initial condition . The corresponding meanfield characteristic equations
are the Hamilton equation for the Hamiltonian
This is an ordinary differential equation (ODE) in an affine linear space of all continuous functions with supremum norm for . By the CauchyPiccard existence theorem for ODEs in Banach spaces and a Gronwall estimate, there is a unique global solution if the gradient is smooth and bounded. The density defined by satisfies then the Vlasov equation
This method of characteristics [1]
is a convenient way to prove the existence and uniqueness of the
solution of this integro PDE.
We extend now this setup. How general can the initial density be?
It can be a finite measure or a signed measure representing charged particles
with different charges. One restriction is that
should be finite.
Only if decays sufficiently fast at infinity can one allow
spatial infinite measures like the product of the Lebesgue measure on
with a compactly supported measure. For periodic V and when the
position space N is the torus , one deals with
particles in a box with periodic boundary conditions. Integration
over gives a finite force. We generalize this now to
the almost periodic case, where we evolve a gas on for which all
physical quantities are almost periodic in the position.
We assume that the initial density on the phase space has the property that for any continuous function h on , the function is a Bohr almost periodic function on N and that there exists a constant r such that m(x,y)=0 for y;SPMgt;r. To define a Vlasov evolution for such measures, we proceed in a similar way as before. Define (f,g)(x,y) = (x + F(x,y),y + G(x,y)), where are continuous with the property that they are Bohr almost periodic functions in x when y is fixed. Such functions form a closed subspace of C(S,S) on which a finite mean
is defined. Especially, when m is 1periodic in x, one has
.
An evolution can now be defined with the Hamilton equations
with initial conditions . The Hamiltonian is
.
Because is both almost periodic in x
and x', we know that is
almost periodic in x. The map is differentiable so that by
the CauchyPiccard existence theorem, there is a solution in the Banach algebra
of almost periodic functions for small times.
A Gronwall estimate assures global existence
of the solution if the gradient satisfies a global Lipshitz estimate.
The corresponding Vlasov equation defines the evolution of spatial almost
periodic measures , where is the initial
measure. For any , the function
on N is almost periodic.
Examples are the physically relevant moments
.
The Fourier transform of the almost periodic function
is a discrete measure on . One writes
.
The measure is supported on the frequency module
of the initial measure . This generalizes the fact that if
is periodic in x, then is also periodic with the same period.
While the frequency module does not
change under the evolution, the weights on the spectrum
change and are expected to shift towards higher and higher
frequencies.
As in the case of almost periodic CA, there are macroscopic
quantities which are invariant under the Vlasov flow. Examples are the
energy H[f,g], the momentum M[g], or the angular momentum .
We can also find almost periodic BernsteinGreenKruskal modes [10],
which are spatial almost periodic equilibrium measures for the actual Vlasov PDE.
These well known solutions are obtained with the separation ansatz
, where
. One gets an equilibrium measure if
Q solves the integral equation
. With , one can find
an almost periodic potential V satisfying
such that P(x,y)=S(y) Q(x) is an equilibrium solution.
We define the Lyapunov exponent
Because
this is the Lyapunov exponent of the finite dimensional cocycle A(f,g) over
the flow .
If is globally bounded, then .
One can readily check that if the measure m is a finite Dirac measure
representing n particles, then this Lyapunov exponent is the classic
Lyapunov exponent of a test particle forming together with the n particles
a restricted (n+1)body problem.
The dynamical entropy
is defined because is spatial almost
periodic for any t. The number
is a measure for the growth rate of complexity of the almost periodic fluid. If
, the almost periodic Dirichlet integral grows
exponentially and the characteristic flow is not recurrent
event hough the dynamics is reversible. If for almost
all ,
one expects that can converge in a weak sense to an equilibrium
solution of the Vlasov equations.
Acknowledgments.
E. Reed acknowledges a SURF research fellowship at
Caltech during the summer of 1995.
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Oliver Knill