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Bifurcation diagram for compartmentalized granular gases
D van der Meer1, K van der Weele, D Lohse
1Department of Applied Physics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands.
Summary
Granular gas distribution in connected compartments transitions from uniform to clustered states as driving intensity decreases. This transition exhibits pitchfork or saddle-node bifurcations and hysteresis, becoming more pronounced with more compartments.
Area of Science:
- Physics
- Complex Systems
- Statistical Mechanics
Background:
- Granular materials exhibit unique behaviors distinct from ideal gases.
- Understanding phase transitions and stability in confined granular systems is crucial.
- Vibrofluidization is a key technique for manipulating granular materials.
Purpose of the Study:
- To construct and analyze the bifurcation diagram for a vibrofluidized granular gas in N compartments.
- To identify the critical driving intensities and bifurcation types leading to state transitions.
- To investigate the role of hysteresis and multi-peaked solutions in system dynamics.
Main Methods:
- Construction of the bifurcation diagram for varying numbers of compartments (N).
- Analysis of stability for uniform and clustered states.
- Identification of bifurcation types (pitchfork, saddle-node) and hysteresis effects.
- Stability analysis of transient, multi-peaked solutions.
Main Results:
- At high driving, a stable uniform distribution (equipartitioned gas) exists.
- Decreasing driving intensity leads to instability of the uniform state and emergence of clustered states.
- For N=2, the transition occurs via a pitchfork bifurcation; for N>2, saddle-node bifurcations dominate.
- Hysteresis effects intensify with increasing N.
- Transient multi-peaked solutions are observed in the bifurcation diagram.
Conclusions:
- The system exhibits complex phase transitions driven by changes in driving intensity.
- The number of compartments significantly influences the nature of the bifurcation and the degree of hysteresis.
- Multi-peaked solutions represent transient dynamics, with their physical relevance dependent on stability analysis.