The two-particle exchange
the orientation of the transition in reference to the computational box changes and thus the transition state might be different. Likewise, a larger domain might change the behav-ior as well: With more particles to compensate the stresses, displacements and ultimately the edge state could be altered.
In order to estimate the quality of our results, we compare systems of sizes N1 = 2500 and N2 = 10000. We use either periodic or fixed boundaries, where particles outside a circle of radiusrfix=Ly/2−4.75 around the origin are fixed at the global minimums’ posi-tions. This definition allows us to use the same cut-off radius as with periodic boundaries.
The difference of the new boundaries is that forces are balanced at the walls, whereas with periodic conditions forces constitute additional pressures across the cell boundaries.
In addition, we consider the transition parallel to thex-axis in the periodic 2500-particle system. It will provide us insight into the dependence on the orientation of the lattice in relation to the computational domain. For each of the five systems we compute the transition state of the two-particle exchange. In figure 5.5 we show the transitions of the 2500-particle systems, both the horizontal transition with periodic boundaries (a) and with fixed boundaries (b). Naturally, in the latter case the orientation of the transition is irrelevant. Visually, close to the center of deformation no qualitative differences between the states can be spotted. They both show the same quadrupolar structure as the original transition state in figure 5.4. However, the displacement fields differ in the region close to the boundaries: With a periodic domain, displacements extend to the next cell and have to be continued across the boundary. When considering the horizontal transition in figure 5.5(a), distortions of the field are minimized since the transition state fits symmet-rically into the box, showing mirror-symmetries about both axes. With fixed boundaries, on the other hand, displacements have to vanish at a set radius. As a result, the shape of the computational domain becomes irrelevant, the accompanying artifacts completely disappear, and the displacement field decays much faster and only exhibits the symmetry of the transition state.
To quantify the similarities, we compute several indicator functions, namely the energies, barrier heights, participation numbers PNk and the spectra, which we characterize by the unstable mode and the effective frequency. The results are summarized in table 5.3 for different boundary conditions in the 2500-particle system and in table 5.4 for the 10000-particle system, and will be discussed in the next two paragraphs.
5.4.1 Dependence on boundary conditions
As would be expected, changing the boundary conditions leads to slightly different results.
When aligning the transition with the x-axis, the energy barrier slightly increases. The source can be found in the symmetry of the displacement field: Since it shows mirror symmetries both along the x- and y-axis, displacements have to drop off to zero on the boundaries. As a consequence, relative displacements have to be larger and thus the energy is increased. Likewise, for the same reason the participation numbers are smaller.
The same arguments apply to the system with fixed boundaries. Here, the effect is even emphasized because particles are fixed on their positions beyond a radius ofrfix=Ly/2− 4.75. In both cases, the higher order participation numbers, PN4 and PN6, are only slightly affected indicating that the transition state itself is quite unaffected by the change
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5.4 Effects of the system size and boundary conditions
−2 0 2
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−20 0 20
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log10(d) a)
−2 0 2
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−2 0 2
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−2 0 2
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Figure 5.5: The two-particle exchange (left, indicated by arrows), the corresponding tran-sition state (middle) and displacement field (right), both for a horizontal trantran-sition with periodic boundary conditions (a), and for particles fixed on their lattice sites beyond rfix=Ly/2−4.75 (b). Comparison with figure 5.4 shows that in either case structures are qualitatively identical, and that the displacement field is mostly affected close to the boundary. Color coded in the figure is the logarithm of displacements with respect to the starting configuration. The arrows in the middle panel represent the displacement vectors from the starting configuration (blue) and towards the final minimum (red). The solid black lines in the right panel indicate vanishing radial displacements, whereas the dashed lines markd= 0.025.
of boundaries. This is confirmed by the corresponding spectra: Both the unstable mode and the effective frequency are in good agreement in all three cases. Therefore, the small difference between the results indicates that the effect of the central deformation diminishes quickly with distance and is negligible for systems of N ≥ 2500 particles. Hence, when investigating further transition states we will not inspect different orientations on the lattice.
5.4.2 Dependence on system size
For both system sizes, the absolute energy barrier is the same within 1%. Consequently, the per-particle barrier is approximately four times smaller for the larger system. When taking weighted averages (eqn. (5.7)), we observe that both systems converge with increasing order in k. This indicates that the transition state is well localized in both systems and thus independent of the number of particles for N ≥ 2500. The larger deviation of h∆EiPN2 can be understood by looking at the displacements as a function of distance to the center, figure 5.6: They exhibit a long tail which drops to zero only close to the
The two-particle exchange
Table 5.3: Comparison of several quantities such as energies, energy barriers, averaged en-ergy barriers, participation numbers PNk, the unstable mode and effective frequency for a system size of 2500 particles and different boundary conditions. With fixed boundary conditions, 1034 free particles remain. Orientation of the transition state has merely an effect on thermodynamically relevant quantities, the same is true for fixed boundary conditions.
periodic periodic deviation fixed deviation
diagonal horizontal
Emin 409.185 409.185 409.185
ETS 409.337 409.337 <10−3% 409.338 <10−3%
∆E 0.152002 0.152444 0.29% 0.153366 0.90%
h∆EiN 6.08009×10−5 6.09776×10−5 0.29% 6.13465×10−5 0.90%
h∆EiPN2 4.44217×10−3 5.25764×10−3 18.36% 6.33172×10−3 42.54%
h∆EiPN4 1.10507×10−2 1.11782×10−2 1.15% 1.13772×10−2 2.95%
h∆EiPN6 1.15770×10−2 1.16340×10−2 0.49% 1.17164×10−2 1.20%
PN2 35.28 25.61 27.32% 18.06 48.72%
PN4 3.62 3.58 1.10% 3.49 3.59%
PN6 2.45 2.45 <10−3% 2.43 0.82%
λ+ 0.282862 0.286159 1.17% 0.292931 3.56%
ωeff/2π 2.849 2.870 0.74% 2.948 3.47%
Table 5.4: The same quantities as in table 5.3 for a system size of 10000 particles, compared to the corresponding 2500-particle systems. With fixed boundary conditions, ≈5390 free particles remain. Additionally, deviations of the corresponding 2500-particle sys-tem subject to identical boundary conditions and between the 10000-particle syssys-tems are given. The thermodynamically relevant quantities coincide nicely for both system sizes.
periodic deviation fixed deviation
diagonal 2500 2500 10000
Emin 1636.74 1636.74
ETS 1636.89 1636.89 <10−3%
∆E 0.151941 0.04% 0.152210 0.76% 0.18%
h∆EiN 1.51941×10−5 300.16% 1.52210×10−5 303.04% 0.18%
h∆EiPN2 3.78302×10−3 17.42% 4.86983×10−3 30.02% 28.73%
h∆EiPN4 1.10210×10−2 0.27% 1.11019×10−2 2.48% 0.73%
h∆EiPN6 1.15622×10−2 0.13 % 1.15920×10−2 1.07% 0.26%
PN2 48.40 27.12% 29.48 38.74% 39.09%
PN4 3.64 0.43% 3.60 3.06% 1.10%
PN6 2.46 0.14% 2.45 0.82% 0.41%
λ+ 0.282396 0.17% 0.284426 2.99% 0.72%
ωeff/2π 2.840 0.32% 2.869 2.8% 1.02%
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