4.2 Ambiguities using Crocker and Grier’s method in dense systems
4.2.1 Crocker and Grier’s algorithm
The algorithm introduced by Crocker and Grier [35] addresses the tracking problem with the following procedure:
• Start with the spots in the first snapshot𝑡now= 0.
• For each spot: Find all possible successor spots in the next snapshot𝑡next =𝑡now+ 1with displace-ments smaller than𝛿𝑟max.
• Connect unambiguous links, where a spot has exactly one successor and this successor has exactly one predecessor.
• For each “network” of spots with more than one successor and/or more than one predecessor (see the example in Fig.4.1) do the following: Find the most probable selection of links by minimizing the sum of distances squared (Equ.4.3) counting unconnected spots as𝛿𝑟2𝑚𝑎𝑥.
• Label connected spots with the same particle ID to save the connections.
• Save all unconnected spots from snapshot𝑡nowto memory.
• Delete those single spots from memory that are coming from snapshots more than𝛿𝑡maxearlier, namely those with𝑡 <=𝑡now−𝛿𝑡max
• Increase𝑡nowby one and start over again (at point 2) using the spots of snapshot𝑡now, together with all spots in memory.
• ...
Clear advantages are the simplicity and the speed of this algorithm especially in uncomplicated cases, which is due to the direct fixation of the unambiguous links between two snapshots. Time and memory consuming optimization steps only have to be done for networks of spots with ambiguous successors and/or predecessors. Since the calculations are done from one snapshot to the next snapshot these net-works are usually small so that the optimal connections can be found with low computational cost.
4.2.2 Violation of time-reversal symmetry
A physical system in equilibrium obeys the principle of detailed balance, meaning that the probability for a reaction or a particle movement is the same for the forward and for the backward direction in time.
Since we aim to find the most probable selection of links minimizing the PDF in Equation4.2, which has the same value for𝛿𝑟⃗ and−𝛿𝑟⃗, an ideal tracking method should reproduce this law by giving the same resulting trajectories if the timeline is reverted. Crocker and Grier’s algorithm violates time-reversal symmetry because it goes through the snapshots one-by-one starting with the first. As long as all spots are connected from one snapshot to the next one, there is no problem. But as soon as we have to keep unconnected spots in memory the time reversal symmetry is violated: In the next loop of the algorithm the search for successors is done for the spots of the next snapshot together with the spots in memory, so that effectively there are more spots in the left snapshot (𝑡now) than in the right snapshot (𝑡next).
(a) (b)
Figure 4.2: Violation of the time-reversal symmetry with Crocker and Grier’s algorithm: (a) shows the resulting trajectories for the forward-tracking and (b) for the backward-tracking of the same set of spots. Grey arrows and colored areas visualize direction and radius for the search of successors. Red and green colors of the spots indicate the most probable trajectories minimizing the squared displacements (Equ.4.3) for all connections at once, like it is done in the tracking algorithm of Karrenbauer et al. [78].
In a situation where two particles pass by each other at a short distance that coincides with one particle being invisible and the other one doing quite large moves, this can indeed lead to different trajectories.
This is illustrated in Figure4.2, where CG’s algorithm is applied once in forward direction (left panel) and once in backward direction (right panel). While there is only a small difference in the connections at 𝑡= 6, the more severe problems occur in the connections from𝑡= 1to𝑡= 3.
Let us first look at the forward direction (Fig.4.2(a)): From𝑡= 1to𝑡= 2the green particle moves quite far and is therefore quite near to the red particle. But the red particle was not detected, which results in a wrong continuation of the red track and the green particle being saved in memory. From𝑡= 2to𝑡= 3 trouble continues: The red particle at𝑡 = 3is too far away from the green particle in𝑡 = 2, it is even out of the search radius. Therefore the red trajectory is not continued with the next red particle. The virtual green particle, in memory at𝑡= 2, is farther away from the green spot in𝑡= 3than the real green particle, which was already connected to the red trajectory before. Consequently, the red trajectory is again continued with the green particle and a new (blue) trajectory starts with the red spot. Finally, from 𝑡 = 3to𝑡 = 4the wrongly connected green spot at𝑡 = 3is closer to the green spot at𝑡 = 4than the virtual green spot in memory. Therefore, the red trajectory continues wrongly again and the green spot in memory is deleted, because there was no successor for it in three consecutive snapshots. The code leaves us with one long but wrong trajectory and two shorter but correct ones.
Starting the tracking backwards from right to left (Fig.4.2(b)) the situation is completely different: From 𝑡 = 3 to𝑡 = 2there is no possible successor for the red particle, while the green particle is correctly connected. From𝑡 = 2to𝑡 = 1the virtual red particle in memory finds a new partner, as well as the green particle finds its successor somewhat farther away but still in the search radius. There is only one error at𝑡= 6: Despite the fact that on average the red particle is closer the two other red particles at𝑡= 7 and𝑡 = 5it joins the green trajectory. This is because the distance to the green spot at𝑡 = 7is slightly closer than the distance to the red one.
It is obvious that finding the most probable connections strictly minimizing all the squared distances would solve the discrepancy and retain time reversal symmetry. For the red spot at 𝑡 = 6the sum of squared distances to the other two red dots is smaller than to the two green dots, it would therefore join the red trajectory. And the wrong connection at𝑡= 2can be prevented by punishing unconnected spots (green at 𝑡 = 1and red at 𝑡 = 3in Fig. 4.2(a)) with additional “costs” compared to connected spots (Fig. 4.2(b)). This would favour the somewhat longer connection of the two green spots at𝑡 = 1 and 𝑡= 2, avoiding costs for any unconnected spots. However, in order to realize a strict minimization of the
4.2 Ambiguities using Crocker and Grier’s method in dense systems squared distances favouring connected spots, it is not possible to go from one snapshot to the next as in Crocker and Grier’s method. One needs to do the optimization for all possible links at once! This is the approach of Karrenbauer and Wöll [78].
4.2.3 Tracking the small particles of a binary mixture
Crocker and Grier never meant their code to be used in such ambiguous situations like the one presented in the example above. The important requirement, that the distance moved by a particle between two snapshots has to be sufficiently small compared to the average particle distances, is not fulfilled here.
However, in three-dimensional measurements of small (and therefore fast) particles such a situation is not so unlikely. The time needed to record a 3D snapshot is in the range of several seconds. A freely diffusing small particle with a diameter of 1µm on average moves a distance of𝛿𝑟 ∼ 2µm within 4 seconds. At higher volume fractions this becomes comparable to the interparticle distances. Larger particles are somewhat easier to follow due to their slower diffusion. For instance a2µm particle will only move1.5µm within 4 seconds, which is already less than its own diameter.
With a typical binary system of2µm and 1µm particles (see section3.3.3 for more details), fast 3D measurements were carried out with time intervals of4.2seconds between two snapshots. As expected, the big particles are easy to follow but for the small particles this is not the case at all. In this dense binary system at ∼ 30 % volume fraction the average distance between small particles is only about 2µm. However, the particles cannot diffuse freely, so that the average distance a particle moves within 4seconds is only about0.4µm. Nevertheless, observed maximal displacements are rather in the range of1µm (see Figure3.28for the van-Hove function of small particles). Additionally, the small particles are not consistently detected in each snapshot, compared to the big particles their luminosity is lower.
Sometimes a particle stays invisible for more than10consecutive snapshots.
Figure4.3shows a subset of particles of such a binary mixture, where the tracking problems are easy to see. Similar to Figure4.2, where the violation of the time-reversal symmetry was illustrated, at certain points a particle moves somewhat faster than usually, while another nearby particle is not continuously detected. This leads to the establishment of wrong links. In Figure 4.3 those wrong trajectories are
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time t [s]
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Figure 4.3:Problems of Crocker and Grier’s code tracking the small particles in a typical binary mixture used in the experiments. Tracking parameters are𝛿𝑟max = 0.9µm and𝛿𝑡max = 20snapshots. Only a small subset of the trajectories is shown, which is representative for the problems. Particle diameters are1.0µm, snapshots are recorded every4.2seconds. In the coordinate-over-time plots one can see that similar problems occur as shown in Figure4.2, which illustrated the violation of time-reversal symmetry.
particles that are suddenly moving, within a few steps, directly to the place of another particle. At the same time that other particle disappears or has not moved at all (Most obvious examples are the light green trajectory and the brown trajectory). It is clear that by reducing the search radius𝛿𝑟maxand the time𝛿𝑡maxthat determines how long a spot is saved in memory one could avoid such large moves. Then one would be left with very many short tracks that are correct. But this would render it impossible to compute physical quantities like correlation functions at large time intervals, which is a necessity for the characterization of glassy dynamics. Therefore the standard Crocker and Grier algorithm is not adaptable to the here investigated samples.