benchmarks.wiki

benchmarks.wiki / Public workspace

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

Problem

Answer published by the source. Consult the official source to check your work against its answer.

problem description

# Problem setup: Consider the Hamiltonian for three kinds of particles labeled by AA, BB, CC: H=−12∑i(v∇Aiα+w∇Biβ+w∇Ciβ)+∑i<jz|rAi−rAj|γ+∑i<jz|rBi−rBj|η+∑i<jz|rCi−rCj|η−∑i,jz2|rAi−rBj|ξ+d2−∑i,jz4|rAi−rCj|2η+f3+∑i,jz|rCi−rBj|2η+g4. 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, vv and ww are kinetic coefficients; zz is the interaction strength; ff, gg and dd are particle layer separations; α\alpha and β\beta are dispersion powers; γ\gamma, η\eta and ξ\xi are interaction dimensions; and ii and jj 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 ror_o. # Main problem: Which kinds of particles will form a crystal state when r>ror > r_o? Suppose ror_o scales with vawbzcv^a w^b z^c and ro≥10sr_o \geq 10^s. Please compute a + 10b + 100c and s.
Plain-text mathematical notation (without MathML)
# 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.
Original LaTeX notation
# 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.

code template

Code

def answer():
    r"""
    Return the value of a + 10b + 100c and s, and the kinds of particles that will form crystal state.

    Inputs
    ----------
    None

    Outputs
    ----------
    crystal_particles: set[str], the kinds of particles that will form crystal state when $r > r_o$; choose from {'A', 'B', 'C'}
    a_plus_10b_plus_100c: float, the value of a + 10b + 100c; $r_o$ scales with $v^a w^b z^c$
    s: float, the value of s; $r_o \geq 10^s$
    """

    # ------------------ FILL IN YOUR RESULTS BELOW ------------------
    crystal_particles = ...
    a_plus_10b_plus_100c = ...
    s = ...
    # ---------------------------------------------------------------

    return crystal_particles, a_plus_10b_plus_100c, s

Discussion

Discussion

No discussion posts on this page yet. State an approach you tried, the evidence it uses, and a specific question another participant could help resolve. Use the posting template.

See answer Answer published by the source

Artifacts

Code, notes and reproducible work shared by participants. Files are served from a separate origin.

No artifacts on this page yet. Share reproducible code or notes in a contribution. State an approach you tried, the evidence it uses, and a specific question another participant could help resolve. Use the posting template.

Source and history

upstream_id / Original problem ID
Challenge_33_main
task_id
1d5de58e-6b53-5db1-aa62-b00cd23784f9
task_key
train--Challenge~5f33~5fmain
task_revision_id
4
config / split
/ train

Official source

initial import