The Transition to the Jamming Cluster

1. The Transition to the Jamming Cluster

2. What is a jamming cluster? • The point at which a percolating network of forces “solidifies”. • The jamming cluster is most likely to occur in a system made of many smaller pieces of matter (like sand). • When a system is “jammed” it can apply a unified force on something else. (Through the multitude of littler forces between each grain) • Demo

3. What do we know about the jamming cluster? • We do not yet have a complete understanding how it works. There are a myriad of forces between each grain of sand (or other particle) in such a system. • It would be ludicrous to try to calculate the forces on each grain of sand or piece of matter in the system. Statistical models of this particular system are also not very accurate. • Modern physics is still pursuing a model to accurately describe and predict how a jamming cluster behaves.

4. Mechanical Stability It has been noted that system with a certain packing fraction, or a certain number of contacts is stable is perturbations. For example, if sand were unpacked and the grains had an average of 2 or less contacts it would easily shift when the bottle was picked up However if the sand had an average of 3 or more contacts it would have forces on it from more directions and would be much more stable to perturbations. It could be said that this system is “jammed”.

5. Research In order to better understand how a jamming cluster functions, I added more code to our percolation code. “The Jam Test”: k-core percolation or k-core constraint model change=1 while change==1: update=0 for i in range (N): xpos=i%L ypos=(i/L) if occupation[i]==1: print "%4d" % (i) sum=0 if occupation[i-1+L*(xpos==0)]==1: sum=sum+1 if occupation[i+1-L*(xpos==(L-1))]==1: sum=sum+1 if occupation[i-L+N*(ypos==0)]==1: sum=sum+1 if occupation[i+L-N*(ypos==(L-1))]==1: sum=sum+1 if sum<K: occupation[i]=0 update=update+1 • if update==0: • change=0 • for i in range (N): • if occupation[i]==1: • ptr[i]=-1 • # print • else: • ptr[i]=EMPTY

6. Data I studied the size of the spanning cluster after the k-core percolation (“jam test”) for differing values of L (the size of the lattice) and p (the percent of percolation, or density of the lattice) for sum<3, <2 and <1. I used L values 1-20 and p values between 0.1 and 1.0. (p=0 would not create a lattice) I also used three different seeds for the random number generator, but kept them the same between sums.

7. Sample Data Table Average Spanning Cluster Size for sum <2

8. Some Interesting RelationshipsSum<3 Every cluster not terminated by the “jam test” is a spanning cluster. This is due to the nature of the jam test code. If the occupied spot does not have at least 3 nearest neighbors, it will be deleted. So, it is necessary for a cluster to span the lattice and loop in order to survive. The spanning cluster= 0 for all values of L=1 and L=2. This can also be expected. A single square has no neighbors and a 2 by 2 lattice will be quickly terminated because it is only possible for it to have two. Spanning clusters in sum <3 also tend to appear as solid shapes (without any squares missing). If there are “holes” they tend to occur in even numbers and in patterns. L=8 p=0.9 sum<3

9. sum<2 A spanning cluster occurs at significantly lower values of p when the k-core percolation (“jam test”) lowers the bar to only 2 nearest neighbors. This, from the nature of the code is to be expected. The magnitude of the spanning cluster as the same values of L as is inversely proportional to k in sum<k. Spanning clusters with “holes” in them occur more often than in sum <3 and appear more random. Spanning clusters occur in p values as low as 0.5 in the higher tested L values (18, 19 and 20), whereas spanning clusters even at p=0.9 occurred only 1/3 of the time in higher p values when sum <3. L=8 P=0.9 Sum <2

10. Sum <1 Work in progress I have not finished collecting data for sum< 1, but I hypothesis it with show the same patterns from sum<2 and sum<3. (Spanning cluster size will be inversely proportional to k in sum<k). I also expect it to be very similar to sum<2, and to represent very closely what the cluster were to look like if there were no jam test at all. (Only completely isolated squares will be terminated).

11. What does this tell us about the jamming cluster? This data gives us a better idea of how the transition to the jamming cluster operates. The fact that it so closely mimics the real lattice when sum<1 tells us that it is closely related to known patterns of percolation.

12. Applications of the transition to the jamming cluster • Granular packing • Mimics percolation transition from liquid to solid • High Rigidity • Same relative displacement between smaller objects when whole object is moved • High viscosity materials like glass • Mimics percolation transition but instead of changing the packing fraction (distance between smaller particles) to make it solid, it is cooled.