CritPt / Challenge_33_main / Consider the Hamiltonian for three kinds of particles labeled by A, B, C:

# Problem setup:

Consider the Hamiltonian for three kinds of particles labeled by A, B, C:

H=−(1)/(2)∑_(i)(v∇_(A_(i))^(α)+w∇_(B_(i))^(β)+w∇_(C_(i))^(β))+∑_(i<j)(z)/(|r_(A_(i))−r_(A_(j))|^(γ))+∑_(i<j)(z)/(|r_(B_(i))−r_(B_(j))|^(η))+∑_(i<j)(z)/(|r_(C_(i))−r_(C_(j))|^(η))−∑_(i,j)(z²)/(√(|r_(A_(i))−r_(B_(j))|^(ξ)+d²))−∑_(i,j)(z⁴)/(√(|r_(A_(i))−r_(C_(j))|^(2η)+f³))+∑_(i,j)(z)/(√(|r_(C_(i))−r_(B_(j))|^(2η)+g⁴)).

Here, v and w are kinetic coefficients; z is the interaction strength; f, g and d are particle layer separations; α and β are dispersion powers; γ, η and ξ are interaction dimensions; and i and j are particle indices.

Suppose

\begin{cases}
(\alpha + z^5 + \xi^{2.1})\ln (g+\alpha^2-3\eta) + \alpha^z \ln(\beta^3 - \xi^2 + \alpha\eta + 6^{\alpha-2}+1) = 0,\\[8pt]
((\log_{10} v)^2 - 2 \log_{10}z \log_{10} v + 5^2 -81)^{4} + 2^{d}\,g^4 = 0,\\[8pt]
v^2(\ln z)^v (9^{\log_{10} (w/z)} -3^4)^v + \dfrac{\ln(1+f^2)}{f^3} = 0,\\[8pt]
(0.25(\log_{10} z + 1)^3 + 3^2 +\ln e^3)^4+2^8)^8 + 1249e^{-d} = 0,\\[8pt]
g^{3.5} + (\alpha + g + \log_{10}(v/w) - 3)^{10+v} = 0,\\[8pt]
(w\,v - 10)^2\,g^6 + (2\xi - \alpha^{2+g})^2 = 0,\\[8pt]
3^{-\frac{z\,v^2}{w}} \dfrac{g \alpha}{\xi} + (\alpha\beta\eta - 2^{2+\alpha}+3\xi)^4 = 0.\\
\end{cases}


Suppose that a phase transition appears when the average distance between particles is less than
a critical value r_(o).

# Main problem:

Which kinds of particles will form a crystal state when r>r_(o)? Suppose r_(o) scales with v^(a)w^(b)z^(c) and r_(o)≥10^(s). Please compute a + 10b + 100c and s.

# Problem setup:

Consider the Hamiltonian for three kinds of particles labeled by $A$, $B$, $C$:

$
H = -\frac{1}{2} \sum_i \left( v\nabla_{A_i}^{\alpha} + w\nabla_{B_i}^{\beta} + w\nabla_{C_i}^{\beta} \right)
+ \sum_{i<j} \frac{z}{|\mathbf{r}_{A_i} - \mathbf{r}_{A_j}|^{\gamma}}
+ \sum_{i<j} \frac{z}{|\mathbf{r}_{B_i} - \mathbf{r}_{B_j}|^{\eta}}
+ \sum_{i<j} \frac{z}{|\mathbf{r}_{C_i} - \mathbf{r}_{C_j}|^{\eta}}
- \sum_{i,j} \frac{z^2}{\sqrt{|\mathbf{r}_{A_i} - \mathbf{r}_{B_j}|^{\xi} + d^2}}
- \sum_{i,j} \frac{z^4}{\sqrt{|\mathbf{r}_{A_i} - \mathbf{r}_{C_j}|^{2\eta} + f^3}}
+ \sum_{i,j} \frac{z}{\sqrt{|\mathbf{r}_{C_i} - \mathbf{r}_{B_j}|^{2\eta} + g^4}}.
$

Here, $v$ and $w$ are kinetic coefficients; $z$ is the interaction strength; $f$, $g$ and $d$ are particle layer separations; $\alpha$ and $\beta$ are dispersion powers; $\gamma$, $\eta$ and $\xi$ are interaction dimensions; and $i$ and $j$ are particle indices.

Suppose

\begin{cases}
(\alpha + z^5 + \xi^{2.1})\ln (g+\alpha^2-3\eta) + \alpha^z \ln(\beta^3 - \xi^2 + \alpha\eta + 6^{\alpha-2}+1) = 0,\\[8pt]
((\log_{10} v)^2 - 2 \log_{10}z \log_{10} v + 5^2 -81)^{4} + 2^{d}\,g^4 = 0,\\[8pt]
v^2(\ln z)^v (9^{\log_{10} (w/z)} -3^4)^v + \dfrac{\ln(1+f^2)}{f^3} = 0,\\[8pt]
(0.25(\log_{10} z + 1)^3 + 3^2 +\ln e^3)^4+2^8)^8 + 1249e^{-d} = 0,\\[8pt]
g^{3.5} + (\alpha + g + \log_{10}(v/w) - 3)^{10+v} = 0,\\[8pt]
(w\,v - 10)^2\,g^6 + (2\xi - \alpha^{2+g})^2 = 0,\\[8pt]
3^{-\frac{z\,v^2}{w}} \dfrac{g \alpha}{\xi} + (\alpha\beta\eta - 2^{2+\alpha}+3\xi)^4 = 0.\\
\end{cases}


Suppose that a phase transition appears when the average distance between particles is less than
a critical value $r_o$.

# Main problem:

Which kinds of particles will form a crystal state when $r > r_o$? Suppose $r_o$ scales with $v^a w^b z^c$ and $r_o \geq 10^s$. Please compute a + 10b + 100c and s.