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Bacterial populations exhibit exponential growth when conditions such as nutrient availability and temperature are favorable. In this phase, cells reproduce through binary fission, where each cell divides into two identical daughter cells. This process causes the population to double at regular intervals, resulting in a growth rate that is directly proportional to the current number of cells. As the population increases, the number of new cells formed during each generation also grows, creating...
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Updated: Jun 8, 2026

Automated Delivery of Microfabricated Targets for Intense Laser Irradiation Experiments
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Published on: January 28, 2021

Exponential energy growth in a Fermi accelerator.

Kushal Shah1, Dmitry Turaev, Vered Rom-Kedar

  • 1Faculty of Mathematics and Computer Science, Weizmann Institute of Science, PO Box 26, Rehovot 76100, Israel. kushal.shah@weizmann.ac.il

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|September 28, 2010
PubMed
Summary

Particles bouncing in oscillating 2D polygons show unbounded energy growth. For a specific rectangle setup, this energy grows exponentially, and the rate is controllable by adjusting geometry and initial conditions.

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Automated Delivery of Microfabricated Targets for Intense Laser Irradiation Experiments
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Published on: January 28, 2021

Area of Science:

  • Physics
  • Dynamical Systems
  • Chaos Theory

Background:

  • Bouncing particle systems (billiards) can exhibit complex dynamics.
  • Static polygons typically lead to predictable, non-chaotic motion (zero Lyapunov exponents).
  • Investigating particle behavior in dynamic, non-static environments is crucial for understanding energy transfer.

Purpose of the Study:

  • To investigate particle energy growth in two-dimensional (2D) oscillating polygonal billiards.
  • To analyze the conditions leading to unbounded and exponential energy growth.
  • To derive and verify a predictive model for the energy growth rate.

Main Methods:

  • Analytical derivation of energy growth rate for a specific oscillating rectangular billiard.
  • Numerical simulations of particle trajectories within the oscillating polygon.
  • Ensemble averaging of initial conditions to study statistical behavior.

Main Results:

  • Observed unbounded energy growth for particles in 2D oscillating polygons.
  • Demonstrated exponential energy growth for a rectangle with a vertically oscillating bar.
  • Derived an analytical expression for the exponential growth rate.
  • Numerical results closely matched the predicted growth rate, confirming controllability.

Conclusions:

  • Particle energy growth in certain 2D oscillating billiards is unbounded and exponential.
  • The rate of energy growth is analytically predictable and controllable via system parameters.
  • This finding has implications for understanding energy dynamics in non-static physical systems.