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Random packing of lines in a lattice cube
1james.burridge@gmail.com
Summary
Random sequential packing of lines in cubes shows two phases: initial blocking and exponential filling. In large systems, blocking diminishes, leading to a rapid packing rate transition.
Area of Science:
- Physics
- Materials Science
- Statistical Mechanics
Background:
- Understanding random sequential packing is crucial for materials science and statistical physics.
- Previous studies often focused on simpler shapes or lower dimensions.
Purpose of the Study:
- To investigate the random sequential packing of complete lines in integer lattice cubes.
- To analyze the packing fraction and its dependence on system size and time.
- To explain the observed packing dynamics and their relation to object aspect ratio.
Main Methods:
- Exact computation of packing fractions for small cube sizes (N<=15).
- Modeling line occupation as a spatial Poisson process.
- Analysis of packing phases (blocking and non-blocking) and their durations.
- Extension of the analysis to d-dimensional cubes.
Main Results:
- The packing process exhibits two distinct phases: an initial blocking phase and a subsequent exponential filling phase.
- The ratio of blocking to non-blocking phase durations tends to zero as system size (N) approaches infinity.
- In the limit of large N, the packing fraction follows theta(t) = 3/4(1-e^-t).
- A rapid phase transition, appearing as a discontinuity in large systems, is linked to the high aspect ratio of lines.
Conclusions:
- The study provides a physical explanation for packing rate changes and diverging coefficients in packing fraction expansions.
- A conjecture is proposed for the packing fraction in d-dimensional cubes in the large N limit: theta = d/(2d-1).
- The findings offer insights into the fundamental mechanisms governing random sequential packing processes.
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