Cooperative Transport of Particles in a Periodic Potential

a java simulation by Xiao-Gang Wen

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The Model:
Each particle feels four kinds of forces

• Uniform driving force F=(Fx,Fy)
• Periodic force F=(Va*sin(6.28*x), Va*sin(6.28*y))
• Random force F=Vr*Vr0([x],[y])*(sin(6.28*x), sin(6.28*y)), where Vr0(i,j) is a random function ranging from -0.5 to +0.5
• Coulomb force F=50/(r1-r2)^2

The velocity of an particle is proportional to the force acting on it. The competition of the four interactions allow many different flow patterns.
If one sets the uniform force (Fx,Fy)=0, one can observe different static patterns, as one varies N and Va.

Usage:

• +N/-N: change the number of particles by 1, and restart the simulation.
• +Fx/-Fx, +Fy/-Fy: change the strength and the direction of the uniform driving force.
• OnOff: turn-on/turn-off the uniform force.
• +Va/-Va: change the strength of the periodic force (which comes from a periodic potential).
• +Vr/-Vr: change the strength of the random force
• restart: restart the simulation.
• +Dt/-Dt: change the size of time step (and the speed of simulation). Reduce Dt when simulation becomes unstable.

Driven to Order:
1. Set Fx=Fy=0, Va=8, Vr=0. Then set N=40.
You will see that the 40 particles settle into a pretty random pattern.
2. Set Fx=Fy=4, and hit "restart".
Very often, the 40 particles will settle into one of two very regular patterns (symmetry breaking and driven to order) which allow them to flow in x or y direction (although the force is in the x+y direction).
3. Increase Fx (or Fy) to a large value and reduce it back to 4.
You can cause a phase transition between the two flow patterns.
You are encouraged to do other numerical experiments and to discover other interesting physical phenomena.
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