Infinite hard-sphere system
The time-evolution for the system of infinitely many particles in space interacting by a hard-sphere potential is constructed. Examples abound of configurations of the infinite system having more than one solution to the Newtonian equations of motion. A regularity condition is imposed on the solutions sought, which limits the growth of velocities and of the length of chains of particles close together as absolute value x ..-->.. infinity; it is proven that through any point of the phase space there passes at most one regular solution. Every point in a subset X bar of the phase space X is the initial point of a regular solution which is defined for all time. The subset X bar is of full measure for every Gibbs state and is invariant under the one-parameter group T/sup t/ of shifts along solution trajectories. Moreover, the flow T/sup t/ leaves every Gibbs state invariant. The solutions constructed are limits, as R ..-->.. infinity, of motions in which particles inside the sphere of radius R are elastically reflected from its boundary while those outside remain fixed. For this reason, one also studies the motion of finite systems. For finitely many hard-sphere particles in a region of space with piecewise smooth boundary, the set of points of the phase space through which solutions exist for all time without triple or grazing collisions, are of full Lebesgue measure and are residual in the sense of Baire. Liouville's Theorem holds for the one-parameter group of shift-transformations T/sup t/. Finally, we give examples in which a single billiard moving in the plane is reflected infinitely often from a boundary curve in finite time, and necessary conditions for such singularities to occur are established.
- Research Organization:
- California Univ., Berkeley (USA). Lawrence Berkeley Lab.
- DOE Contract Number:
- W-7405-ENG-48
- OSTI ID:
- 7189579
- Report Number(s):
- LBL-4801
- Resource Relation:
- Other Information: Thesis
- Country of Publication:
- United States
- Language:
- English
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Related Subjects
MANY-BODY PROBLEM
SPHERICAL CONFIGURATION
BOLTZMANN-VLASOV EQUATION
BOUNDARY CONDITIONS
ELASTIC SCATTERING
EQUATIONS OF MOTION
PHASE SPACE
REFLECTION
SINGULARITY
TIME DEPENDENCE
TRAJECTORIES
VELOCITY
CONFIGURATION
DIFFERENTIAL EQUATIONS
EQUATIONS
MATHEMATICAL SPACE
SCATTERING
SPACE
990200* - Mathematics & Computers