The two-particle exchange
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0.0 0.2 0.4 0.6 0.8 1.0
kxi−xi,0k minimization s0s0+4000s0+8000
spring
s0= 0 bisection 3
s0= 4400 TS bisection 3
s0= 4400 spring s0= 0
Figure 5.2: Time series of the edge tracking process. In the top row, from left to right are different initial conditions, and each column shows snapshots of the corresponding minimization. The outermost columns show the two closest trajectories generated with the initial spring displacements. On the left, the two central particles rotate counterclockwise, back to their original position. On the right, they rotate clockwise and eventually swap their positions. Correspondingly, the inner columns show the initial conditions and their minimization after three consecutive applications of the edge tracking routine. The central column finally shows the transition state itself, determined by Newton’s algorithm. Colors indicate displacements with respect to the reference minimum in the lower left panel. The dashed lines visualize the distortion of the lattice. The corresponding time lines of the procedure are shown in figure 5.3.
many particles in the surroundings have to be pushed away to free the space. Since the system approaches a force equilibrium, this process slows down. The escape towards the minima on the other hand happens on a faster timescale, at least as soon as the system is slightly off the separating manifold. More importantly, as soon as the two central particles leave the intermediate position, the two particles which should be pushed outwards will quickly relax towards their original minimum positions, reversing the approach towards the transition state. Therefore, we use heuristics to improve convergence of the algorithm by reducing both the maximum separation of neighboring trajectories devolve and the bi-section threshold εbisect. This can also be seen in the time line in figure 5.3: The initial bisection takes a rather large separation and bisection threshold so that a rough estimate of the transition state is quickly obtained. After the bisection has finished, the two min-ima of the final initial conditions are compared with the two minmin-ima of the starting initial conditions. In principle, at this point a third, intermediate minimum might have been
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5.3 Finding the transition state: An example
0 2000 4000 6000 8000 10000
stepss
0.0 0.2 0.4 0.6 0.8 1.0
d=max(kxi−xi,0k) spring
bisection 1 bisection 2 bisection 3 transition state
Figure 5.3: Time line of the edge tracking process. Each line represents a different tra-jectory and its distance to the reference minimum. At s = 0, we start with initial conditions generated using the spring algorithm. Minimization yields either the orig-inal minimum (d = 0) or the swapped configuration (d= 1). Snapshots of the two innermost trajectories are shown in figure 5.2 in the outermost columns. Ats= 3000, 3500, and 4400 the bisection is restarted due to neighboring trajectories separating beyond the threshold value. Over time, this approximates the transition state, in this particular case it was found after the third bisection by a final Newton-search.
encountered, which would correspond to a chain of several transitions leading to the in-vestigated exchange of particles. However, the minima remained the same, and thus the next bisection will investigate the same transition. Therefore, the maximum separation is reduced so that the separating manifold is kept closely bracketed, giving particles far away from the center of displacement time for relative equilibration. After the second bisection, only one positive eigenvalue remains, indicating that the algorithm almost reached the transition state. Thus, the bisection threshold is decreased to pin the transition state.
The final initial conditions are then refined using Newton’s algorithm: the transition state is found.
It is shown in figure 5.4, along with the displacement field. In the left part of the figure, we see the resulting transition with the two central particles exchanging their positions. In the middle panel, the transition state itself is depicted. Color-coded are the individual displacements with respect to the starting minimum on a logarithmic scale.
Similarly, arrows indicate the displacement from the starting configuration in blue and towards the final minimum in red. We observe that the two particles exchanging their positions align with two more particles, significantly displacing them. On the other hand, particles along the perpendicular axis are pulled towards the center of deformation, filling the emerged void. Consequently, the transition state is twofold mirror symmetric. The corresponding displacement field hence exhibits a quadrupolar structure, which is shown in the rightmost panel of the figure. We see that displacements quickly drop below 10%
of the lattice spacing r0. Moreover, the mirror symmetry observed close to the center is broken by the boundary conditions, so that only a twofold rotational symmetry persists.
Additionally shown as solid lines are points of vanishing radial displacement. They spiral out of the center such that they connect across the cell boundaries. Correspondingly, at
The two-particle exchange
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Figure 5.4: The two-particle exchange (left, indicated by arrows), the corresponding tran-sition state (middle) and displacement field (right). Color coded is the logarithm of displacements with respect to the starting configuration. The arrows in the mid-dle panel represent the displacement vectors from the starting configuration (blue) and towards the final minimum (red). The solid black lines in the right panel indi-cate vanishing radial displacements, whereas the dashed lines mark d= 0.025. The displacement field shows a quadrupolar structure (orange-red) with particles pushed outwards towards the upper right and lower left, and pulled inwards in the perpen-dicular direction.
the boundary the total displacements diminish as well. Although the localization of the transition state is recognizable from the displacement field qualitatively, we still have to quantify it, e.g. by computing a participation number as is done in the next section.
5.3.1 Localization of the transition state: The participation number
When investigating a transition, it is of interest to know which particles, or on a more basic level, how many particles participate in it. For a global transition, we would expect almost all particles to take part. On the other hand, for a local rearrangement we would expect that only few particles participate. Following the example of Bell and Dean (1970), we define the participation number as
PNk=2= P
i|di|22
P
i|di|4 , (5.4)
where di are particle displacements with respect to a reference configuration. It gives a measure of the number of particles involved in an event. For only one particle participating, di =δij, it yields PN2 = 1, whereas for all particles being displaced equally,di = 1/N, the result is PN2 =N, the number of particles.
A similar definition is given by Swayamjyoti et al. (2014) fork= 4. We generalize this by using normalized weights of order k,
ωi(k)= |di|k P
j|dj|k. (5.5)
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