Collision times and partners

For two disks to interact, their centers must be exactly 2R apart

so the time of collision is

Now because disks are non-overlapping, the numerator is always non-positive; for the interaction time to be positive, the denominator must be negative.

To decide when the next collision is going to occur, one could proceed something like the following:

       do j=1,N
         coltime(j)=bigtime
       enddo
       do j=1,N-1
         delx=x(j+1)-x(j)
         delv=v(j+1)-v(j)
         dist=2R-(x_{j+1}-x_j})
         if (dist .lt. 0.0) then
           if (delv .lt. 0.0) then
             coltime(j)=dist/delv
             colpartner(j)=j+1
           endif
         endif
       enddo

Now sort the collision times and see which particles will collide, and move molecules

       timestep=bigtime
       do j=1,N-1
         if (coltime(j) .lt. timestep) then
           timestep=coltime(j)
           k=colpartner(j)
         endif
       enddo

       do j=1,N
         coltime(j)=coltime(j)-timestep
         x(j)=x(j)+v(j)*timestep
       enddo