# NEW: dissipative EDMD
###### tags: `Dissipative self-assembly`

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

EF, AP, FS, GF

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### Pressure in a mixture (langevin not delta):

$$P = \rho T \left(1 + (1+\alpha)\phi \sum_{i,j} x_i x_j g_{ij}^+ \right)$$

where $x_i=N_i/N$.

For a binary mixture:

$$P = \rho T \left(1 + (1+\alpha)\phi \dfrac{N_s^2g_{ss}^+ + N_b^2g_{bb}^+ +2N_bN_sg_{bs}^+}{N^2} \right)$$

At equilibrium, in the solid $\rho$ is larger than in the liquid, which gives the idea that the pair correlation function in the solid should be smaller.

With $S1$ configuration:

$$P = \rho T \left(1 + \dfrac{(1+\alpha)\phi}{4} (g_{ss}^+ + g_{bb}^+ +2g_{bs}^+) \right)$$

with unit radius $\rho=\phi/\pi$ + equilibrium (the sum of pair contact divided by 2 is called $G$):

$$\pi P = \phi T \left(1 + \phi G \right)$$

The pressure and temperature are the same in the solid and in the liquid at equilibrium:

$$\phi_s \left(1 + \phi_s G_s \right) = \phi_l \left(1 + \phi_l G_l \right)$$

Again, meaning that $\phi_s>\phi_l$ implies $G_s<G_l$ at equilibrium.

## [05/09/24]

### Monodisperse hexagonal/fluid coexistence

For a monodisperse gas, assuming gaussian velocity distribution:

$$T_s\phi_s \left(1 + (1+\alpha)\phi_s g_s^+ \right) = T_l\phi_l \left(1 + (1+\alpha)\phi_l g_l^+ \right)$$

The temperature of a system driven by a langevin thermostat at temperature $T_b$, is given by:

$$2\gamma (T_b - T) = \frac{\omega(\phi, T)}{2}T(1-\alpha^2)$$

with $\omega(\phi, T)=8\phi g^+\sqrt{T/\sigma^2\pi m}$ which gives:

$$2\gamma (T_b - T) = 4\phi g^+(T^{3/2}(1-\alpha^2))/(\sigma\sqrt{\pi m})$$

This is not the same as Trizac formula (but somehow, I never managed to get exactly the formula he gave, I always had a factor off somewhere... (and my computations agree with what I measure in my system)).

We define:

$$\Gamma = \frac{2\gamma\sigma\sqrt{\pi m}}{4(1-\alpha^2)}\to \Gamma (T_b - T)= \phi g^+ T^{3/2}$$

There are two awful solutions to this equation. More simply, we eliminate $g^+$ in the equation above obtained from the equality of pressure:

$$T_s\phi_s \left(1 + (1+\alpha)\Gamma (T_b - T_s)T_s^{-3/2} \right) = T_l\phi_l \left(1 + (1+\alpha)\Gamma (T_b - T_l)T_l^{-3/2} \right)$$

$\Lambda = \Gamma (1+\alpha)$ and $\phi_s>\phi_l$ lead to:

$$T_s \left(1 + (1+\alpha)\Gamma (T_b - T_s)T_s^{-3/2} \right) < T_l \left(1 + (1+\alpha)\Gamma (T_b - T_l)T_l^{-3/2} \right)$$

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#### Phenomenology

We define:

$\tilde T_{l,s} =T_{l,s}/T_b$:

$$\tilde T_s\left(1 + \tilde \Lambda (1 - \tilde T_s)\tilde T_s^{-3/2}\right)<\tilde T_l\left(1 + \tilde \Lambda (1 - \tilde T_l)\tilde T_l^{-3/2}\right)$$

With:
$$\tilde \Lambda = \frac{\gamma(1+\alpha)\sigma\sqrt{\pi m/T_b}}{2(1-\alpha^2)}$$

$\tilde \Lambda$ is a ratio of timescale, one associated to the one of the bath $\gamma$ and one associated to dissipative collisions.

Let's define $F(X)=X(1+\tilde \Lambda (1-X)/X^{3/2})$. $F(X)>0$ for $X<1$. From the above inequality, we obtain that:

$$F(\tilde T_s)<F(\tilde T_l)$$

The variations of $F$ (increasing, decreasing, etc). Will gives us a relation for $T_s$ and $T_l$:

$$F'(X) = 1-\dfrac{\tilde \Lambda}{2} \dfrac{1+X}{X^{3/2}}$$

For example: If $F'(X)>0$, $F$ is increasing thus, $\tilde T_s<\tilde T_l$, if $F'(X)<0$: $\tilde T_s > \tilde T_l$.

________________

We recall:

$$\boxed{\tilde T_s\left(1 + \tilde \Lambda (1 - \tilde T_s)\tilde T_s^{-3/2}\right)<\tilde T_l\left(1 + \tilde \Lambda (1 - \tilde T_l)\tilde T_l^{-3/2}\right)\equiv F(\tilde T_s)<F(\tilde T_l)} $$

$$\boxed{\tilde \Lambda = \frac{\gamma(1+\alpha)\sigma\sqrt{\pi m/T_b}}{2(1-\alpha^2)}>0}$$

$$\boxed{F(\tilde T<1)>0}$$

$$\boxed{F'(\tilde T) = 1-\dfrac{\tilde \Lambda}{2} \dfrac{1+\tilde T}{\tilde T^{3/2}}}$$

$$\boxed{\dfrac{\tilde\Lambda}{1+\alpha} (1 - \tilde T)= \phi g^+ \tilde T^{3/2}}$$

We also have for $\tilde\Lambda < 1$: $$F'(T^*)=0\Rightarrow T^*= \frac{1}{6} \left( \sqrt[3]{\tilde{\Lambda}^3 + 6 \sqrt{3} \sqrt{\tilde{\Lambda}^2 \left(\tilde{\Lambda}^2 + 27\right)} + 54 \tilde{\Lambda}} + \frac{\tilde{\Lambda}^2}{\sqrt[3]{\tilde{\Lambda}^3 + 6 \sqrt{3} \sqrt{\tilde{\Lambda}^2 \left(\tilde{\Lambda}^2 + 27\right)} + 54 \tilde{\Lambda}}} + \tilde{\Lambda} \right)
$$

Which notably leads to ($\tilde\Lambda\to 1^-$):

$$T^*=\tilde\Lambda - \dfrac{(\tilde\Lambda-1)^2}{8} + \mathcal{O}\left((\tilde\Lambda-1)^3\right)$$

and ($\tilde\Lambda\to 0$):

$$T^*=\dfrac{\tilde\Lambda^{2/3}}{2^{2/3}}+\dfrac{\tilde\Lambda^{4/3}}{3\times 2^{1/3}}+\mathcal{O}(\tilde\Lambda^{2})$$

In any case, for a large part it is a good assumption to set $T^*\simeq \tilde\Lambda$.

| System limit | Monotocity of $F$ | Conclusion |
| -------- | -------- | -------- |
| Equilibrium at the gaussian level $T_{\{s, l\}}\to 1\Rightarrow\tilde \Lambda \to \infty$ : <br>- $\gamma \to \infty$ and $\alpha \neq 0$<br>- $\alpha\to 1$ and $\gamma\neq 0$<br>- $T_b\to 0$ and $\gamma\neq 0$ | $F'(0<\tilde T<1)<0\Rightarrow F$ is decreasing for $\tilde T<1$ | $$F(\tilde T_s)<F(\tilde T_l)\Rightarrow \tilde T_s> \tilde T_l$$ Unconditionnaly |
| Midly equilibrium case $\tilde \Lambda >1$. | $F'(0<\tilde T<1)<0\Rightarrow F$ is decreasing for $\tilde T<1$ | $$F(\tilde T_s)<F(\tilde T_l)\Rightarrow \tilde T_s> \tilde T_l$$ Unconditionnaly |
|First strong non-equilibrium $T_{\{s, l\}}\to cst\Rightarrow\tilde\Lambda\to cst<1$:<br>- $\alpha\to 0$ and $\gamma\neq \infty$|$F'(0<\tilde T<T^*)<0\Rightarrow F$ is decreasing for $\tilde T<T^*$ and increasing for $\tilde T>T^*$|Don't know!|
|Second strong non-equilibrium $T_{\{s, l\}}\to 0\Rightarrow\tilde\Lambda\to 0$:<br>- $\gamma \to 0$ and $\alpha \neq 1$<br>- $T_b\to \infty$ and $\gamma\neq \infty$|Same as above|Probably $\tilde T_s>\tilde T_l$ (see below)|

For the last case, we can do a perturbation around small $\tilde\Lambda$ because terms are of same order:

$$\underbrace{\frac{\tilde\Lambda}{(1+\alpha)\phi g^+}}_{L} (1 - \tilde T)= \tilde T^{3/2}\Rightarrow \tilde T = L^{2/3}\Rightarrow F'(T_{\{l, s\}}) \simeq 1-\dfrac{\tilde \Lambda}{2}\dfrac{1+\left[\tilde\Lambda/(1+\alpha)\phi_{\{l, s\}} g_{\{l, s\}}^+\right]^{2/3}}{\tilde\Lambda/(1+\alpha)\phi_{\{l, s\}} g_{\{l, s\}}^+}$$

From this we obtain that the liquid and solids are in the zone where $F$ is most likely decreasing ($(1+\alpha)\phi_{\{l, s\}}g^+_{\{l, s\}}>2$):

$$F'(T_{\{l, s\}}) = 1-\dfrac{(1+\alpha)\phi_{\{l, s\}}g^+_{\{l, s\}}}{2}(1 + \mathcal{O}(\tilde \Lambda^{2/3}))\stackrel{probably}{<}0$$

which implies that the solid is hotter than the liquid.

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We can see it differently

$T^*\simeq\left(\dfrac{\tilde\Lambda}{2}\right)^{2/3}$ while here $\tilde T_{\{l, s\}}=\left(\dfrac{\tilde\Lambda}{(1+\alpha)\phi_{\{l, s\}} g_{\{l, s\}}^+}\right)^{2/3}$.

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### Coexistence with $\Delta$ model

For the $\Delta$ model, at the gaussian level, the density does not influence the energy of the system. Thus we would not be able to see anything for the monodisperse system. For the polydisperse system, the equipartition is broken, thus some difference would be visible but it's hard to analyze. Thus, let's proceed with the monodisperse $\Delta+\gamma$ model for which the steady state temperature is density dependent.

#### Coexistence temperature for $\Delta+\gamma$ model:

$$T_s\phi_s \left(1 + (1+\alpha)\phi_s g_s^+ \right) = T_l\phi_l \left(1 + (1+\alpha)\phi_l g_l^+ \right)$$

The temperature of a system driven by a $\Delta+\gamma$ , is given by:

$$2\gamma T = \frac{\omega(\phi, T)}{2}\left(-T(1-\alpha^2) + m\Delta^ 2 + \alpha\Delta \sqrt{\pi m T}\right)$$

with $\omega(\phi, T)=8\phi g^+\sqrt{T/\sigma^2\pi m}$ which gives:

$$\underbrace{\dfrac{\gamma \sigma\sqrt{\pi m}}{2} \dfrac{\sqrt{m\Delta^2}}{m\Delta^2}}_{\tilde \Gamma}\frac{\sqrt {T/(m\Delta^2)}}{-\dfrac{T}{m\Delta^2}(1-\alpha^2) + 1 + \alpha \sqrt{\pi } \sqrt{T/(m\Delta^2)}} = \phi g^+.$$

equivalently:

$$\tilde \Gamma\frac{\sqrt {\tilde T}}{1+\alpha \sqrt{\pi\tilde T} -\tilde T(1-\alpha^2)} = \phi g^+ \text{ and } \tilde\Gamma=\dfrac{\gamma \sigma\sqrt{\pi}}{2\Delta} $$

Again $\Lambda=(1+\alpha)\Gamma$:

$$ \tilde T_s\left(1 + \tilde \Lambda \dfrac{\sqrt {\tilde T_s}}{1+\alpha \sqrt{\pi\tilde T_s} -\tilde T_s(1-\alpha^2)}\right)<\tilde T_l\left(1 + \tilde \Lambda \dfrac{\sqrt {\tilde T_l}}{1+\alpha \sqrt{\pi\tilde T_l} -\tilde T_l(1-\alpha^2)}\right)$$

Here:

$$F(\tilde T) = \tilde T\left(1 + \Lambda \dfrac{\sqrt {\tilde T}}{1+\alpha \sqrt{\pi\tilde T} -\tilde T(1-\alpha^2)}\right)$$

and:

$$F'(T)=\frac{\tilde \Lambda \sqrt{T} \left(\left(\alpha ^2-1\right) T+2 \sqrt{\pi } \alpha \sqrt{T}+3\right)}{2 \left(\left(\alpha ^2-1\right) T+\sqrt{\pi } \alpha \sqrt{T}+1\right)^2}+1$$

which is always larger than 0 $F(\tilde T_s)<F(\tilde T_l)\Rightarrow \tilde T_s<\tilde T_l$.

Still we have to be careful because the function is weird:

But I think the divergence observed is for the temperature of the system without damping, hence, sunce the temperatur in our system is always below it, we are fine. (checked at $\alpha \simeq 1$)

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### Check temperature diff:

#### $\Delta+\gamma$ detailed model analysis :

/home/syrocco/Documents/Coexistence/hexagons/dump delta + gamma diff N/
/home/syrocco/Documents/Coexistence/code/hexagon.py



The coexistence is nice enough to try to obtain a direct measurement of the temperature difference!!!!
(just relaunched a simu)

| Density | Energy |
| -------- | -------- |
|  |  |

WHAT WE EXPECT! nice!

Maybe no hexatic (or very narrow if it exists (as in equilibrium in fact))

| Coexistence 0.71021| Above coexistence 0.712684|
| -------- | -------- |
|  |  |
|||

#### Langevin
fuck langevin



| Probably hexatic? $\phi = 0.72034$|wtf $\phi = 0.7248$ |
| -------- | -------- |
|  |  |
|bad bragg peak |
|

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$\gamma = 0.01$ and $T$ is varying (which might be the issue!).
| $\alpha = 1$ | $\alpha = 0.9999$ | $\alpha=0.999$ | $\alpha = 0.99$ | $\alpha= 0.8$ |
| -------------------------------------------------- | --- | -------------------------------------------------- | ----------------------------------------------------------------------------------------------------------------------------------- | -------------------------------------------------- |
|  |  |  | $\gamma = 0.01$  $\gamma=0.1$  |  |

Take for example $\alpha = 0.9999$.
Seems to be a loop?
In the middle of the pressure drop, here is what we see:

close to beginning
long time
evolution:
evolution larger system (same density)
smaller density:

Compare this with the Delta model:


Maybe the coexistence is metastable somehow, with respect to the second order transition?

Note that the systems seems equilibrated

A check would be to look at the evolution of the Mayer loop (that should not exist), with system size and better statistics.

#### Andrea's idea of mixing models


|| | | | | Don't know.. | coexistence|
| -------- | -------- | --- | --- | --- | --- | -------- |
| ... | ... | ... | ... | ... |  |  |

weird energy behavior for before last (nothing strange on the fluctuation side)

Larger system:


$T= 0.1$

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$T = 0.417$
$\phi = 0.703$

$\phi = 0.709$

### Micro phase separation $\Delta+\gamma+$ attractive well:

(steady state result)

#### Hexatic at coexistence?

If we use a density slightly above the coexistence densit for the $\Delta+\gamma$ model: 
We obtain a quasi long ranged solid which is strange :)