# Local structure and glasses under AQS deformation

###### tags: `glasses` `aqs` `local structures`

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### Owners (the only one with the permission to edit the main test)

Saheli, Susana, Frank, Giuseppe

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## Background

In past [work](https://arxiv.org/abs/1908.00425), we have looked into the role of tetrahedral structure in controlling the dynamics of glassy systems. Here, we will explore how this idea extends to glassy systems under shear.

## Plans

We shall work with Wahnstrom system and need to create a shared folder on the cluster for the analysis.

Before heading to the sheared system, Susana will look for number of tetra hedras in inherent structures at different temperature. I will put one configuration per temperature on the shared folder.

For temperature $T=1.5$, Saheli has sheared configurations via AQS across yielding transition, for shear amplitudes $0.02,0.03,0.04,0.08,0.09$. Yielding is close to 0.06. I will also put those on the shared folder so that Susana will run the tetra hedral code on them.
I will find how to calculate local entropy of the above mentioned systems in voronoi cells.

## To Do
:::success
green is for finished tasks
:::
### IS
:::success
* $n_{tet}$ Vs $T$ for IS and Liquid
* $E_{IS}$ Vs $T$ and $E$ Vs $T$
* $E_{pot}$ (IS and Liq.) Vs $n_{tet}$
* $P$ (IS and Liq.) Vs $n_{tet}$
* Check for ICO (and family) in W.
* $s_2$ Vs $T$
:::
* Check the same as above for KALJ (to be filled in a section at the end in the Appendices)

### AQS

* $D$ Vs $\gamma_{max}$ for the trained sample (to see $\gamma_y$) $T=1.5$
* $D$ Vs $\gamma_{max}$ for the trained sample (to see $\gamma_y$) $T=0.8$ low energy
:::success
* $E_{IS}$ Vs $\gamma_{acc}$ for the case where we have the history
* $E_{IS}$ Vs $\gamma_{max}$ for the available temperatures
* $n_{tet}$ Vs $\gamma_{max}$ for $T=1.5$
* $s_2$ Vs $\gamma_{max}$ for $T=1.5$

* $n_{tet}$ Vs $\gamma_{max}$ for $T=0.7$
* better characterization of $s_2$ across shear band

* Add simulations for values of $\gamma$ closer to the critical $\gamma$, to better observe the transition.
:::

### OPEN QUESTIONS

* Now we know that both $s_2$ and $n_{tet}$ both give the same information as the energy. We need to look the local description that these quantities allow. Can we say something about plastic events and rearrangments (see soft spots for example: [https://arxiv.org/pdf/1012.4822.pdf](https://))?
### AQS During cycles:
* To characterize local plastic rearrangement, we need to calculate a non-affine displacement field. For different shear amplitude, during one cycle, we can look at the average variation of the non-affine displacement field $D^2$, seeking how much a particle moves with respect to its neighbours under applied strain. The reference for the transformation matrix, that minimizes $D^2$ can be found in https://arxiv.org/pdf/cond-mat/9712114.pdf .

## METHODS
### Local entropy

> [name=SMA] In the paper *"Local structure in deeply supercooled liquids exhibits growing lengthscales and dynamical correlations"* they have calculated the excess-entropy. It seems more or less straightforward:
> $s_2=-2\pi \rho k_B \int_0^\infty [g_m^i(r)log(g_m^i(r))-g_m^i(r)+1]r^2dr$,
> where $g_m^i$ is the radial distribution function of particle $i$.
> You can find the article in (https://www.nature.com/articles/s41467-018-05371-6)
> [name=saheli] **Calculation of local entropy:** I did calculations for local entropy. It involves several steps. First, the local radial distribution function $g(r)$ for $i^{th}$ particle is highly fluctuatiing. Therefore, it needs introduction of 'mollified' distribution function $g_m^i(r)$, such that with a choice of control parameter $g_m(r) \sim g(r)$ and yet, the curve is smooth and integrable. Second, for each particle we need to integrate $I_m^i(r)=[g_m^i(r)log(g_m^i(r))-g_m^i(r)+1]r^2dr$ , by choosing a large enough cut off for $r$. This gives us for each particle, $S_2^i$, followed by the equation above as Susana has mentioned. But, this $S_2^i$ is a highly fluctuating quantity. To distinguish between more ordered and less ordered regimes in the system, as the last step $S_2^i$ needs to be averaged using is a switching function of the form $f(r_{ij})=\frac{1-(r_{ij}/r_a)^N}{1-(r_{ij}/r_a)^M}$ with $N=6$, $M=12$. The choice of $r_a$ should be good enugh to preserve the locality of $s_2^i$, yet averajed over $j$ neighbours can give a less fluctuating dependable variable $\bar{s_2^i}$ .
Here are the results from calculation of $s=\bar{s_2}$, I am dropping the $i$, although always present.

## RESULTS
>[name=Saheli] I am putting my results and Susana's key results together. First I will put liquid and IS related results for both Wahnstrom and Kob Andersen. Then I will put results of sheared configurations of Wahnstorm system for both high and low temperatures. To look into all earlier discussions, you can scroll down.
>
## Whanstrom
### Liquid and IS
* **Energy in liquid and IS as a function of temperature**
>

| Liquid | IS |
| -------- | -------- |
|  |  |

**Discussions:**
>> [name=Saheli] Here is the energy of liquid and IS as a function of T. I am putting them seperately , because otherwise because of the scales the difference in features is not evident.$e_{IS}$ is constant for high temperatures.
>> > [name=Frank] Again a plateau at high T. So if T is high enough, we end up in a similar inherent state (same IS energy, same IS tetrahedrality) independent of the exact value of the temperature. This makes sense to me. The limiting value is basically what you would get if you minimized energy from a completely random configuration. (This might be very obvious... I haven't thought about ISs very much so far.)

>>>>[name=SMA] Yes it makes quite a lot of sense, if I understood well the goal of calculating the IS. At high temperatures I think the energetic lanscape would be quite similar without having really deep minimum and the average value of the energy should be quite similar.

* **Mean entropy and Mean # of tetrahedrons with temperature**

>

| - | $<s>$ | $\sigma_{s}$ | $<n_{tet}>$ |
| ------------ |:-------------------------------------------:|:-------------------------------------------:|:-------------------------------------------:|
| Wahnstrom |  |  |  |
| Kob Andersen |  |  |  |

**Discussions:**
>>[name=Saheli] Both $<s>$ and $<n_{tet}>$ follow same trend of energy. For IS, at higher temperature it becomes constant. Standard deviation in lower temperatures is higher.
>>[name=Saheli]We find that in liquid the mean entropy keeps increasing from low to high temperature, denoting it becomes more disordered with rise in temperature. In IS however, mean entropy is almost constatnt for higher temperature and for low temperature, order grows. Interestingly, even though local order is higher in low temperature IS, the overall system is more heterogeneous, and therefore the standard deviation is high. We had a similar bservation from hyperuniformity calculation. Where high temperature IS was showing hyperuniformity, and in low temperature IS long range correlation was gone.
>>> [name=gfoffi] This looks exactly as I would expect it. In the IS the systems get more glassy and tetrahedra are indeed boosted.

* **Energy vs Mean # of tetrahedrons**
>[name=SMA] I have removed the temperature from the plots. And I have plotted the EIS vs ntet:
>


>>> [name=gfoffi]Very interesting the Eis is proportional to $n_{tet}$.

* **S in Liquid and IS corresponding to different temperatures:**
> The colour code that I have used is,
-7.0 -3.0
A more negative value implies locally more mordered environment.
>Liquids low to high temperature:::::::::::::::::::::::::::::::::::::::::::::::::::::::::
>Inherent Structures (IS) low to high temperature:::::::::::::::::::::::::::::::::

* **Distribution of S in liquids and IS**
>

|-|Liquid| IS |
|-----| ---- | ---- |
| Wahnstorm | | | |
|Kob-Andersen| | |

**Discussion:**
>[name=Saheli] We see for liquids, mean shifts towards left much faster than IS. In IS for high temperature, curves almost coincides.

* **"Solid" like order in liquids with glassy dynamics:**
>[name=Saheli] Although I have presented the distributions for liquids and IS seperately, it is interesting to look into one high and low temperature case, the differnce between liquid and IS distrubutions when plotted together.

>

|T|Low T (from s2)| High T (from s2)|
|-|-|-|
|Wahnstorm|||
|Kob Andersen|||

* **Distribution of $<n_{tet}>$ in liquids and IS**
>[name=Saheli] I am putting similar plots from tetrahedrons by Susana alongside.

| | Low T (from $n_{tet}$)| High T (from $n_{tet}$)|
| -------- | -------- | -------- |
| Wahnstrom |  |  |
| KA | |  |

**Discussion:**
> [name= Saheli] **From P(s):** Overlap of curves are more in low temperature than in High temperature.
>[name=SMA] $n_{tet}^i$ Histograms for a low temperature and a high temperature. Differently from the entropy distributions, at high temperatures the distribution still have a large overlaping, in both the KA and the Whanstrom systems.
> [name= Saheli] Now, I want to quantify the effect of this overlap for different temperatures. For that, if the local order of $i^{th}$ particle in liquid system is $s^i$, I am computing percentage of particles with $s^i$ low enough that it is within the range of $s$ in the corresponding IS configuration.
Therefore, at each temperature i choose a cut-off, def I:$s_{cut}$=$s_{mean}^{IS}+2\sigma_s^{IS}$ and def II:$s_{cut}$=$s_{mean}^{IS}+3\sigma_s^{IS}$

| | |
| -------- | -------- |
|  |  |

> [name=Frank] Interesting! Would it be nice to check how the the value of $s$ in the IS for a given particle ($s_i^{IS}$) relates to its value in the liquid ($s_i^{liq}$)? E.g. plot them against each other in a 2D density plot?
> [name=Saheli]Hi Frank, so I plotted what you asked for both Wahnstorm and Kob Andersen. I chose two temperatures for each of these cases. One high temperature and one low temperature. The results below are interesting.

* **2D density plot of $s_i^{IS}$ versus $s_i^{Liq}$**

|T|Low T|High T|
|-|-|-|
|Wahnstorm|  ||
|Kob Andersen|||

* **2D density plot of $n_{tet}^{IS}$ versus $n_{tet}^{Liq}$**
> [name=Saheli] I am putting similar plots from tetrahedrals by Susana now.

>[name=SMA]Density plots $n_{tet}^{IS}$ vs $n_{tet}^{LIQ}$
>

| | Low T | High T |
| -------- | -------- | -------- |
| Wahnstrom |  |  |
| KA | |  |

**Discussions:**
>[name=Saheli]I think for low temperature , the trend is very evident. Particles that have high order in liquid, maintained that in IS. So, while energy minimization, particles with strong ordered didn't move much.
* **Spearman's rank order correlation between ($s_i^{Liq}$ and $s_i^{IS}$) and ($n_i^{Liq}$ and $n_i^{IS}$)**

>[name=Saheli] I have calculated for both Wahnstrom and Kob-Andersen the Spereman's rank order correlation for configurations at different temperature. At lower temperature indeed we see high correlation. For both the systems the correlation at the lowest temperature is close to 65%.

| Wahnstorm | Kob Andersen |
| -------- | -------- |
|  |  |

>[name=SMA] I've calculated the Spearman Correlation between the $n_{tet}^i$ in the liquid and in the inherent structue. As we were expecting, at low temperatures there is a clear correlation, going upto 0.6 in the Wahnstrom. While going to higher temperatures the system looses correlation which indicates how the system can sample different configurations.
>

* **Correlation Entropy and Tetrahedrality**

>[name=Saheli & SMA] We have now calculated the correlation between the two observables we are focused on.


>**Discussion**
>As we were expecting, both observables are highly correlated in the Wahnstrom mixture, being more evident at lower temperatures where the correlation goes to values upto 0.6. While in the Kob Andersen the correlation is not that strong and it does not change with the temperature, as we have seen before the tetrahedra is not a good descriptor in the KA.

## **Sheared system under AQS**
### WAHNSTROM
* For now, results from Wahnstorm only:
* **Steady state energy , $<s>$ and $n_{tet}$ as a function of shear amplitude $\gamma_{max}$**
>

| Entropy $<s>$ | $\sigma_s$ |
|:------------------------------------:|:------------------------------------:|
|  |  |
| **Energy** | **$<n_{tet}>$** |
|  |  |

**Discussions**
> [name=Saheli]Mean entropy $<s>$ follows the exact trend of energy $U$, as we shear the system. At low temperature system has higher fluctuations ($\sigma_s$) and for high temperature fluctuations increase close to yielding.
> [name=SMA] I have calculated the average number of tetrahedra of the system at a $k_BT=1.5$ and $k_BT=0.7$ for different $\gamma_{max}$ amplitudes.

> With the same range of colour coding , First we see if the local order changes under shearing in absorbing states, below yielding. We are wotking on $IS$ of $\mathbf{T=1.5}$.
> [name=gfoffi] Interesting, the $n_{tet}$ increases with gamma... Let's see what happen for a more annealed configuration, i.e. the lowest T.

> [name=Frank] What happens to the energy as a function of $\gamma$? From Giuseppe's comment in Hangouts, I am guessing that the shear pushes things to lower energy, but is the behavior similar?
>
> [name=gfoffi] Yes, It is in the todo.
>
> [name=Frank] Oops, I see.
>
>[name=SMA] In line with yesterday's discussion, we learn that at lower temperatures (well annealed glass) there is a discontinuity in the energy and in the maximum stress. At $\gamma_{max}$ higher than $\gamma_{Critical}$ it doesn't matter the initial temperature all the systems reach the same energy and the same maximum stress.
>If we see the behavior of the $n_{tet}$ it seems they will follow the same trends, at lower temperature the $n_{tet}$ seems to have a discontinuity. After reaching $\gamma_{critical}$ both systems reach the same state with the same value of $n_{tet}$. On the other hand at high temperature the $n_{tet}$ grows continously, it reaches a maximum before $\gamma_{critical}$ and then "smoothly" goes to lower values of $n_{tet}$.
* **Visualization of local order in sheared system**
>[name=Saheli] With same range of colour coding $s^i$ is presented for sheared systems.
> **Below yielding**::::::::::::::::::::::::::::::::::::
Left to right: IS ($\gamma_{max}=0.0$), $\gamma_{max}=0.02,0.03,0.04,0.06$.
> **Above yielding** :::::::::::::::::::::::::::::::::
[name=Saheli] I wanted to compare this entropy colouring with the shear band in the system. Shear band is the band of particles with higher displacements from one cycle to another. Therefore, in the left pannel I am showing colour map for $s$ and in right pannel colour map for displacements (range 0 to 1) from one cycle to the next in steady states of $\gamma_{max}=0.08$ and $\gamma_{max}=0.09.$

* **From local order $s_2$ and shear band**

| High T (colour range of s -3 to -7)| Low T (colour range of s -4 to -7)|
| -------- | -------- |
|  |  |

* **Slab wise change along shear direction of s :**
>
**Discussion**
>[name=saheli] I have plotted in first rwo the displacements along shear direction between two consecutive cycles. In second row is the change in local entropy $s$. Very expectedly, it showed the change across the band. Higher displaced bands with lower order and less displaced particles with higher order. Anyways, from the MSD, it feels we are toward high shear amplitude, meaning, the and is quite big. Probably its good to run for little lower amplitudes.

* **From tetrahedral calculations the above yielding senario**
>
| $\gamma_{max}=0.08$ | $\gamma_{max}=0.09$ |
| -------- | -------- |
| |

| High tet |  |Low tet |

* **Along shear direction from $n_{tet}$:**
>

| $T=1.5$ $\gamma_{max}=0.08$ | $T=1.5$ $\gamma_{max}=0.09$ |
|:------------------------------------:|:------------------------------------:|
|  |  |
| $T=0.7$ $\gamma_{max}=0.08$ | |
|  | |

**Discussion:**
>[name=SMA] Profile graphs of the $<n_{tet}>$ and $<\delta r>$ corresponding to $T=1.5$ paralel to the shear band. The cyan color corresponds to $<n_{tet}>$ and the green to $<\delta r>$. There is a nice and clear behavior along the shear band the displacements are greater and the average number of tetrahedra is smaller, which is in agreement withigh tetrahedrality the system seems to be more arrested.

* **Number of tetrahedra as a function of cycles**

>[name=SMA] In line with today's discussion, we want to see how the structure evolves through cyclic shearing. To do so, I've calculated the $<n_{tet}>$ as a function of $\gamma_{acc}$. The biggest rearrangements occur at the first cycles, which could be that the shearing band is forming. After some cycles the structures is stabilized and fluctuates around one value.
>
| Low Temperature | High T |
| -------- | -------- |
| | |

* **Energy as a function of deformation cycles**
>[name=Saheli] Here is the energy as a function of deformation cycles. I have put results of two temperatures together.
>

* **$S_2$ as a function of cycles**
> [name=Saheli] For high and low temperatures, $<S_2>$ is plotted as a function of cycle of deformation for different shear amplitudes $0.02-0.09$.

| T=1.5 | T=0.7 |
| -------- | -------- |
|  |  |

## During deformation cycles:
>[name=Saheli]
>For Whanstrom system, here I report results for two temperatures, $T=1.5$ and $T=0.7$. I choose two cases.
> I. What happens during the very first cycle of deformation?
> II. Then second, what happens during a cycle in the steady state?
> So, here I am plotting evolution of energy and average non affine squared displacement $D^2$ (averaged over N=64k particles for single configuration).
> For computation of $D^2$ for $i^{th}$ particle, I used,
> $D^2_{min}=\frac{1}{n} \sum_n [(r_j(t)-r_i(t))-\Gamma(r_j(0)-r_i(0))]^2$
>Here, t denotes AQS steps during deformation cycle, $t=0$ starting point of that deformation cycle. $n$ is number of neighbours at $t=0$.
> The matrix $\Gamma$ is such that it minimizes $D^2$.
> For detail of the form of matrix, check:
> [M. L. Falk and J. S. Langer, Phys. Rev. E57, 7192 (1998)](https://arxiv.org/pdf/cond-mat/9712114.pdf)
> I have used Lees-Edwards boundary conditions for sheared system in the calculation of $d(t)=(r_j(t)-r_i(t))$ and $d'(0)=\Gamma(r_j(0)-r_i(0))$.
>
### Results
>[name=Saheli]
> Here in the plots I have used steps/T in x axis, but this T is basically time period: To scale all the changes in one cycle from 0 to 1.

| T=1.5| Energy $U$| $D^2_{min}$ |
| -------- | -------- | -------- |
| 1st cycle |  ||
| steady state|||

>

| T=0.7| Energy $U$| $D^2_{min}$ |
| -------- | -------- | -------- |
|1st cycle| | |
|steady state| ||

>[name=SMA] The same as previous for the number of tetrahedra.
>
| T=0.7 | $n_{tet}$ ||
|:---------:|:------------------------------------:|:---:|
| 1st cycle | |
|
| Steady |  |
|

| T=1.5 | $n_{tet}$ | |
|:---------:|:------------------------------------:|:------------------------------------:|
| 1st cycle | | |
| Steady | |  |

#### $<S_2>$ during deformation cycles:
>

>
| T=0.7 | $S_2$ | $S_2$|
|:---------:|:------------------------------------:|:---:|
| 1st cycle | ||
| Steady | ||

>
| T=1.5 | $S_2$ |$S_2$|
|:---------:|:------------------------------------:|:---:|
| 1st cycle | ||
| Steady | ||

### Spearman's correlation non affine displacement $D^2$ and $n_{tet}$
>[name=SMA] From the results of the squared non-affine displacement $D^2$, I calculate the Spearman correlation between ntet and $D^2$. The number of tetrahedra is calculated from the configuration at step=0.
>
| Correlation ntet and $D^2$ | T=0.7 |
| -------------------------- |:------------------------------------:|
| 1st cycle ||
| steady state ||

>[name=SMA]CHANGE

| Correlation ntet and $D^2$ | T=1.5 |
| -------------------------- |:------------------------------------:|
| 1st cycle | |
| steady state |  |

### Spearman's correlation non affine displacement $D^2$ and $\bar{n}_{tet}$
>[name=SMA] In order to be consistent with the definition of the non-affine displacement and to improve the correlation. We average the number of tetrahedra over a cut off radius $r_c=1.4$. Later, we correlate $\bar{n}_{tet}$ and $D^2$
>
| Correlation $\bar n_{tet}$ and $D^2$ | T=0.7 |
| -------------------------- |:------------------------------------:|
| 1st cycle |  |
| steady state |  |

>[name=SMA] CHANGE

| Correlation $\bar n_{tet}$ and $D^2$ | T=1.5 |
| -------------------------- |:------------------------------------:|
| 1st cycle | |
| steady state ||

### Spearman's correlation non affine displacement $D^2$ and $\bar{S}_{2}$
>[name=SMA] To have another structural quantity, we calculated the correlation between the non affine displacemente and the $\bar{S}_2$
>
| Correlation $\bar S_{2}$ and $D^2$ | T=0.7 |
| -------------------------- |:------------------------------------:|
| 1st cycle | |
| steady state | |

>[name=SMA] CHANGE

| Correlation $\bar S_{2}$ and $D^2$ | T=1.5 |
| -------------------------- |:------------------------------------:|
| 1st cycle | |
| steady state |  |

>[name=SMA]**TO DO:**
>::: success
>* We were thinking we could improve the correlation by calculating the ntet averaged over a cutoff radius
>* Calculate the Spearman correlation with $S_2$
>:::
>* Correlation with ntet from the sheared configurations?

>**Conclusion**
>[name=Saheli]
> So we see that with the averaged <$n_{tet}$> and $<s_2>$ within a cut-off radius $r_{cut}=1.4$, the maximum correlation is now close to $0.4$ .
> * Also, we have this maximum correlation when the simulation box is maximally titlted for any amplitude below yielding.
> * Above yielding the correation becomes constant
> * Low temperature system ($T=0.7$) is better correlated compared to high temperature system ($T=1.5$).
>

### Density Plots

>[name=SMA] In order to check whether the $n_{tet}$ or the $S_2$ captures the changes on non-affine displacements, I have calculated the density plots of all possible combintations. In this image you will see all the data for the different $\gamma_{max}$ at the STEADY STATE. The data is the following:
>* First column: The **histogram of the number of tetrahedra at $t=0$** in the Steady state. As we can see, the distribution do not have extreme changes at different $\gamma_{max}$.
>* Second column: **Density plot of $\bar{S}_2$ and $\bar{n}_{tet}$**. As we saw previously these two quantities are highly correlated, in order to check whether we are seeing similar information with both. I calculated the density plot of these two quantities calculated at $t=0$ in the steady state. We can see in almost all $\gamma$ that they follow a clear linear relation giving us more hints that both might be capturing similar features of the systems.
>* Third column: **Histogram of the non-affine displacement**. In order to have a feeling of the dynamical behavior of the system, I checked the distribution of non-affine displacement at the maximum shearing point i.e. when $\gamma_{inst}=\gamma_{max}$ and $\gamma_{inst}=-\gamma_{max}$. Seems that only a few particles have larger $D^2$ and most of them are around a value of 0.
>* 4th column: **Density plot of $D^2$ and $n_{tet}$** at $\gamma_{inst}=\gamma_{max}$
>* 5th column: **Density plot of $D^2$ and $\bar{n}_{tet}$** at $\gamma_{inst}=\gamma_{max}$
>* 6th column: **Density plot of $D^2$ and $\bar{S}_{2}$** at $\gamma_{inst}=\gamma_{max}$

**Note that all these calculations are with $D^2_{min}$**
>In the last density plots the colors are in logaritmic scale, as we can see the data is concentrated in values close to 0 and the $n_{tet}$ and $S_2$ extend over all values.

**Temperature $T=0.7$**

**Temperature $T=1.5$**

Are there any changes between the training and the steady?
**Temperature $T=0.7$**

<!-- ### Dividing "fast" and "slow" particles -->
<!-->[name=SMA] From the distribution of the non-affine displacement, we can identify two types of particles. Ones with "slow dynamics", these particles are the majority. And, other particles with "fast dynamics", these particles are outlaiers in the distribution of $D^2$. These particles are pressumably the ones that have plastic rearrangements. We want to see if the two types of particles have different structural environment. To do so, first we identify the particles into slow and fast from the the distribution of $D^2$ at $1/4$ of the cycle. Then we calculate the distriburion of $S_2$ for each type of particles, we expect to see a difference in the variance in the distribution or a more skewed distribution.-->
<!-- > -->
<!-- >[name=SMA]Note that $S_2$ is calculated in the stroboscopic configuration (at step 0). As we can see from the left column, the distribution of $S_2$ is essentially the same for both types of particles. (The plastic rearrangements are totally random? Is there something different in the environment of the particles that will have a plastic rearrangement?)

#### TCC
>[name=SMA] Is there any structural difference between particles involved in plastic rearrangements? To try to answer this question, we used once more TCC to detect all types of structures. We calculate the fraction of particles involved in one of the clusters for each type of partices:
>\begin{equation}
>N_{slow(fast)}^{cluster}=\frac{n_{slow(fast)}^{cluster}}{N_{slow(fast)}}
>\end{equation}


>[name=SMA] It seems that there are no big differences between the first or the second type of particle. (However, we are comparing with dynamics when $\gamma=\gamma_{max}$, at that point many rearrangements have already happened, and a correlation between the structure at the beginning and the dynamics after this big ammount of rearrangements I think is not direct. In any case, if we would like to see how the structure is related to plastic rearrangement I would think that we have to locate the fisrt rearrangement and in there compare the structure of the particles involved in that rearrangement and the ones that haven't undergo plastic rearrangements.) -->

## Connecting local structure to plastic rearrangement

>[name=Saheli] I have recalculated $D^2_{min}$ with Lees Edwards and now things should be okay. Their evolution during cycles is below,

>

| T=0.7 | First cycle | A cycle in steady state |
| -------- | -------- | -------- |
| In the first cycle maximum of $D^2_{min} < 0.1$ |  |  |
|T=1.5 | First cycle | A cycle in steady state|
| In the first cycle maximum of $D^2_{min} < 0.25$ |  |  |

>[name=Saheli]
>We have seen that the effect of local structures $S_2$ and tetrahedral are more pronounced in low temperature system.
> Therefore, I am considering for now the case of $T=0.7$ and $\gamma_{max}=0.08$. The average values $<D^2_{min}>$ against number of steps looks quite smooth in the graphs above, even though they are not. Due to averaging with $N=64000$, the drops in $<D^2_{min}>$ are not clearly visible.
> For this reason, I am plotting $<D^2_{min}>(n)-<D^2_{min}>(n-1)$ with $n$ where $n$ is $n^{th}$ step of deformation in a cycle.

>

| | Events in 1st cycle | zoom in 1st quarter |
| -------- | -------- | -------- |
||  |  |

>[name=Saheli]
>In the above plot, we can detect the events easily. We see that, as we proceed from $0\to \gamma_{max}$, the rearrangements have higher magnitude. Then from $\gamma_{max}\to 0$ we have negative values. From $0 \to -\gamma_{max}$ again the amplitudes increase. Atlast in the section of $-\gamma_{max} \to 0$ we have small values and negative values.
> When we zoom in the first quarter, we can find, that the very first event occurs from step $n=2$ to step $3$. We want to see which particles moved more from steps $2 \to 3$. Then, We will plot the distribution of $S_2$, that is ,$P(S_2)$ in the intial undeformed configuration, (step $n=0$) for these two classes (Higher change in $<D^2{min}>$ and Lower change in $<D^2{min}>$ as we move from $n=2\to3$ using some cut-off).
> Next, we will repeat it for steps $n=0 \to 1600$, that is, what happens at the end of first cycle.
* First rearrangement:
>


* End of first cycle
>
> [name=Saheli]
Now we compare this distribution of $P(S_2)$ at the end of first cycle to $P(n_{tet})$.

>

>[name=Saheli] we can see that at the end of first cycle, there is a shift in the distributions. So, the particles that moved more, has an average value of $S_2$ that is less than other particles. Same is true for tetrahedras.
> I am now putting the same plots, without normalization,
> 

#### TCC
>[name=SMA] Is there any structural difference between particles involved in plastic rearrangements? To try to answer this question, we used once more TCC to detect all types of structures. We calculate the fraction of particles involved in one of the clusters for each type of partices:
>\begin{equation}
>N_{slow(fast)}^{cluster}=\frac{n_{slow(fast)}^{cluster}}{N_{slow(fast)}}
>\end{equation}


>Moreover, we can see reflected the structural changes in the counts of different clusters. In particular, we can see differences related to defective ico (10B) and ico (13A). (And they belong to the family related to the ICO check
[SI of the tetrahedra paper](https://journals.aps.org/prl/supplemental/10.1103/PhysRevLett.124.208005 )=>9B,10B, 11C, 11E, 12B, 12D, 13A...) For the rest of the different types of clusters there is no relevant diference between the slow and the fast population.

> [name=Saheli]
* A cycle in steady state:
> here the first big rearrangemnt was from step 13 to step 14. And, we compare it with $S_2$ of cycle $400$. We will again see what happens after a full cycle, that is, we will compare from cycle $400$ to cycle $401$. To remember, there are $1600$ steps in between these two cycles.
* * 1st major rearrangement in a cycle in steady state, where shear band has formed.
> 
* * Before and after end of a cycle in steady state
>

>[name=Saheli]
>We can gain see the shift in the distribution of two classes.

### Repeating the analysis of rearrangements for different cases at T=0.7----------------

> Each case has 4 plots:
> 1. scatter plot of $D^2_{min}$ at the end of a cycle, w.r.t at the beginning of a cycle.
> 2. Distribution of the above.
> 3. Distribution of $S_2$ at the beginning of a cycle with a classification of Higher and lower $D^2_{min}$. The cut-off is chosen to be $10\%$ of the maximum of $D^2_{min}$.
> 4. Same as above but with tetrahedra.

#### At the end of first cycle T=0.7:
>

| $\gamma_{max}$ | $S_2$ and tetra dist |
| -------- | -------- |
| 0.02 ||
| 0.03 ||
| 0.04 |  |
| 0.05 | |
| 0.06 | |
| 0.07 | |
| 0.08 | |

### In steady state just below and just above yielding T=0.7

>[name=Saheli] I am choosing to amplitudes 0.05, 0.06 below yielding and 0.07, 0.08 above yielding. It is because, 0.05, 0.06 is still evolving to energy minimum. Below that, for lower amplitudes we cannot make a difference between configuration at consecutive cycles. at 0.07, shear band has formed.

>

| Amplitude | Comparing n and n+1th cycle |
| -------- | -------- |
| 0.05 | |
| 0.06 | |
| 0.07 | |
| 0.08 | |

### Events in steady states

>

| Below | Events | Above | Events|
| -------- | -------- | -------- |---|
| 0.02 |  | 0.07 |  |
| 0.04 |  | 0.08 |  |
| 0.06 |  | 0.09 |  |

### Steady state, maximum displacement in a cycle
>

| Amplitude | Comparing n and n+1th cycle |
| -------- | -------- |
| 0.02 ||
| 0.03 ||
| 0.05 ||
| 0.06 ||
| 0.07 ||
| 0.08 ||
| 0.09 ||

#### Changing the cut off above yielding,
| Amplitude | Comparing n and n+1th cycle |
| -------- | -------- |
|0.09, cutoff=1.16| |
|0.09, cutoff=1.74 ||
|0.09,cutff=2.32 ||

#### changing percentage
>[name=saheli] I ranked the particles according to $D^2_{min} max$ in a cycle. Then for a certain percentage p, I chose top p% and last p% of the particles as Blue (More displaced) and Red(Less displaced).
#### S2
>

| Amplitude | Comparing n and n+1th cycle |
| -------- | -------- |
| 0.02 ||
| 0.03 ||
| 0.05 ||
| 0.06 ||
| 0.07 ||
| 0.08 ||
| 0.09 ||

#### ntet
>

| Amplitude | Comparing n and n+1th cycle |
| -------- | -------- |
| 0.02 ||
| 0.03 ||
| 0.05 ||
| 0.06 ||
| 0.07 ||
| 0.08 ||
| 0.09 ||

>

#### S2 , T=1.5
| Amplitude | Comparing n and n+1th cycle |
| -------- | -------- |
|0.02||
|0.03||
|0.04||
|0.05||
|0.06||
|0.07||
|0.08||
|0.09||

#### ntet, T=1.5

| Amplitude | Comparing n and n+1th cycle |
| -------- | -------- |
|0.02||
|0.03||
|0.04||
|0.05||
|0.06||
|0.07||
|0.08||
|0.09||

#### average:
>

|entity|T=0.7|T=1.5 |
|--|--|--|
|S2|||
|ntet|||

#### standard deviation $\sigma$:
>

|entity|T=0.7|T=1.5 |
|--|--|--|
|S2|||
|ntet|||

#### scatter plot for blue and red
> To check correlation. Yaxis is $D^2_{min}$.
>

| Amplitude | $D^2_{min}$ versus $S_2$ |
| -------- | -------- |
|0.02||
|0.03||
|0.05||
|0.06||
|0.07||
|0.08||
|0.09||

| Amplitude | $D^2_{min}$ versus $n_{tet}$ |
| -------- | -------- |
|0.02||
|0.02||
|0.02||
|0.02||
|0.02||
|0.02||
|0.02||
|0.02||

### Correlation fastest and slowest
>[name=SMA] I have calculated the Spearman correlation of the $n_{tet}$ with the new quantity (accumulated $D^2_{min}$ or $max[D^2_{min}]$) for all the particles, and for a subset of them formed by the $10\%$ slowest and fastest.
>

| All particles | Subset $10\%$ fastest & slowest |
|:------------------------------------:|:------------------------------------:|
|  |
|

### Structural differences fastest and slowest (TCC)

I analyzed all the clusters from TCC of the $5\%$ particles with largest and smallest rearrangements/

| Amplitude | Clusters |
| -------- | -------- |
|0.02||
|0.03||
|0.04||
|0.05||
|0.06||
|0.07||
|0.08||
|0.09||

>[name=SMA] Things to remark: The particles that do not have big rearrangements are mainly complex structures such as the icosahedral cluster and the deffective icosahedral (and in general the family of five-fold symmetry). Meanwhile some clusters seem to be more present in the particles with larger rearrangements, such as the BCC structure-like, which it has been shown that is highly present in dense liquids. And the other one that is more present in the highly mobile particles is the cluster 6A, which corresponds to an octahedron (it is four particles forming a square, and two extra particles one above and one below). I can imagine that the way these particles are arrange is easier for them to slip while shearing.

>[name=SMA] For the clusters with major differences (13A, 10B and 6A) I plot the percentage of particles involved on those clusters as a function of $\gamma_{max}$:
>


The percentage of this structure above and below yielding do not change significantly. Just for the icosahedral case it seems to have a peak below yielding. Note than this plots are made with the 5% more mobile and more static particles.
### How are things distributed in space?
In order to answer this question I'm focusing in the more static particles. In particular, I'm checking how the icosahedral clusters are located in space.
Color Coding:
Orange particles: All particles involved in icosahedral clusters.
Red particles: Particles that I can presume they are in the center of icosahedral clusters, i.e. theu value of $n_{tet}=20$.

| $\gamma_{max}$ | |$\gamma_{max}$ ||
| -------- | -------- |--- |---|
|0.03||0.04||
| 0.05 | |0.06||
|0.07||0.08||

### Cluster analysis temperature $T=1.5$
Same as the analysis for $T=0.7$ but for $T=1.5$.

| Amplitude | Clusters |
| -------- | -------- |
|0.02||
|0.03||
|0.04||
|0.05||
|0.06||
|0.07||
|0.08||
|0.09||


## Appendices
## Clusters in Liquids and Inherent Structures
>[name=SMA] I've quantified all the possible clusters for the Wahnstrom mixture for all the temperatures of both the liquid and the IS. Interestingly, some of the clusters have a plateau at high temperatures. And, in general at the IS all the clusters have higher values than in the liquid state.
In here, I am showing you the behavior of the defective icosahedra (10B) and the icosahedral cluster (13A) in the IS and in the liquid.
>
|  |  |
| ------------------------------------ | --- |
| Defective Icosahedra | Icosahedra |

## Pressure as a function of $<n_{tet}>$
In here we shoe the pressure as a function of the $n_{tet}$ for the liquid and the inherent states of the Wahnstrom mixture:

> [name=saheli] I think it will also be nice to plot as a function of pressure. Since for HS systems.
> [name=saheli] Here is with pressure. For liquid we have power law behaviour. For IS, we have liniear behaviour. Same trend is observed for Energy versus $n_{tet}$.

| Pressure versus $n_{tet}$ |
|  exponent -0.89 |  slope -0.32 |
| ------------------------------------ | --- |
| LIQUID | IS |

---
## Correlation between $\delta r$ and $n_{tet}$ inside and outside shear band
**Correlation displacement $\delta_r$ and $n_{tet}$**
>[name=SMA] I've calculated the Spearman correlation between the $<n_{tet}>$ and the displacement in a sheared system for Wahnstrom at $T=1.5$. The displacement is taken from the last 20 cycles:
>$\delta r^i=|r^i(cycle_{final})-r^i(cycle_{final}-20)$|
>And the $n_{tet}$ is taken from the configuration $cycle_{final}-20$.
> The correlation between these two quantities is rather weak, increasing a little while crossing the yielding transition.

Correlation between displacement and $n_{tet}$ inside and outside the shear band:

* **High Temperature**

| | Outside Shear Band | Inside Shear Band |
|:----------------------------:|:------------------------------------:| ------------------------------------ |
| $\gamma_{max}=0.09$ |  |  |
| Corr$(n_{tet}^i,\delta r^i)$ | -0.0105998 | 0.0454993 |
| $\gamma_{max}=0.08$ |  |  |
| Corr$(n_{tet}^i,\delta r^i)$ | 0.0935454 | 0.0555202
* **Low Temperature**

| | Outside Shear Band | Inside Shear Band |
| ------------------- |:------------------------------------:|:------------------------------------:|
| $\gamma_{max}=0.08$ |  |  |
| Corr$(n_{tet}^i,\delta r^i)$ | 0.205294 | 0.105626 |

---