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Almost periodic particle dynamics in the Vlasov limit

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 tex2html_wrap_inline492 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 tex2html_wrap_inline494 under a pairwise interaction given by a potential V evolve according to the Newton equations tex2html_wrap_inline498 which are the Hamilton equations with Hamiltonian tex2html_wrap_inline500 . We assume that the potential V is smooth and that the solutions exist for all times. If the force is rescaled in the limit tex2html_wrap_inline504 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 tex2html_wrap_inline508 . One evolves then a map tex2html_wrap_inline510 , where tex2html_wrap_inline512 gives the position and momentum of the particle with initial condition tex2html_wrap_inline514 . The corresponding mean-field characteristic equations

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are the Hamilton equation for the Hamiltonian

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This is an ordinary differential equation (ODE) in an affine linear space of all continuous functions tex2html_wrap_inline520 with supremum norm for tex2html_wrap_inline522 . By the Cauchy-Piccard existence theorem for ODEs in Banach spaces and a Gronwall estimate, there is a unique global solution if the gradient tex2html_wrap_inline524 is smooth and bounded. The density tex2html_wrap_inline526 defined by tex2html_wrap_inline528 satisfies then the Vlasov equation

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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 tex2html_wrap_inline532 be? It can be a finite measure or a signed measure representing charged particles with different charges. One restriction is that tex2html_wrap_inline534 should be finite. Only if tex2html_wrap_inline524 decays sufficiently fast at infinity can one allow spatial infinite measures like the product of the Lebesgue measure on tex2html_wrap_inline494 with a compactly supported measure. For periodic V and when the position space N is the torus tex2html_wrap_inline544 , one deals with particles in a box with periodic boundary conditions. Integration over tex2html_wrap_inline544 gives a finite force. We generalize this now to the almost periodic case, where we evolve a gas on tex2html_wrap_inline492 for which all physical quantities are almost periodic in the position.

We assume that the initial density tex2html_wrap_inline550 on the phase space tex2html_wrap_inline552 has the property that for any continuous function h on tex2html_wrap_inline492 , the function tex2html_wrap_inline558 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 tex2html_wrap_inline570 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

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is defined. Especially, when m is 1-periodic in x, one has tex2html_wrap_inline586 .

An evolution can now be defined with the Hamilton equations

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with initial conditions tex2html_wrap_inline590 . The Hamiltonian is tex2html_wrap_inline592 . Because tex2html_wrap_inline594 is both almost periodic in x and x', we know that tex2html_wrap_inline600 is almost periodic in x. The map tex2html_wrap_inline604 is differentiable so that by the Cauchy-Piccard 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 tex2html_wrap_inline524 satisfies a global Lipshitz estimate. The corresponding Vlasov equation defines the evolution of spatial almost periodic measures tex2html_wrap_inline608 , where tex2html_wrap_inline610 is the initial measure. For any tex2html_wrap_inline612 , the function tex2html_wrap_inline614 on N is almost periodic. Examples are the physically relevant moments tex2html_wrap_inline618 .

The Fourier transform of the almost periodic function tex2html_wrap_inline620 is a discrete measure tex2html_wrap_inline622 on tex2html_wrap_inline494 . One writes tex2html_wrap_inline626 . The measure tex2html_wrap_inline628 is supported on the frequency module of the initial measure tex2html_wrap_inline532 . This generalizes the fact that if tex2html_wrap_inline632 is periodic in x, then tex2html_wrap_inline636 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 tex2html_wrap_inline642 .

We can also find almost periodic Bernstein-Green-Kruskal modes [10], which are spatial almost periodic equilibrium measures for the actual Vlasov PDE. These well known solutions are obtained with the separation ansatz tex2html_wrap_inline644 , where tex2html_wrap_inline646 . One gets an equilibrium measure if Q solves the integral equation tex2html_wrap_inline650 . With tex2html_wrap_inline652 , one can find an almost periodic potential V satisfying tex2html_wrap_inline656 such that P(x,y)=S(y) Q(x) is an equilibrium solution.

We define the Lyapunov exponent

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Because

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this is the Lyapunov exponent of the finite dimensional cocycle A(f,g) over the flow tex2html_wrap_inline666 . If tex2html_wrap_inline668 is globally bounded, then tex2html_wrap_inline670 . 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 tex2html_wrap_inline680 is defined because tex2html_wrap_inline682 is spatial almost periodic for any t. The number tex2html_wrap_inline460 is a measure for the growth rate of complexity of the almost periodic fluid. If tex2html_wrap_inline486 , the almost periodic Dirichlet integral tex2html_wrap_inline690 grows exponentially and the characteristic flow tex2html_wrap_inline692 is not recurrent event hough the dynamics is reversible. If tex2html_wrap_inline694 for almost all tex2html_wrap_inline696 , one expects that tex2html_wrap_inline698 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.

Next: References Up: Complexity Growth in Almost Previous: Almost periodic lattice gas

Oliver Knill
Mon Jun 29 14:18:53 CDT 1998