{"kind":"task","effective_mode":"full","benchmark":{"kind":"benchmark","effective_mode":"full","slug":"critpt","formal_name":"CritPt","introduction":"研究水準の物理問題で、科学的理解と多段階の推論・計算を評価するベンチマークです。公開データには70の課題があり、問題文とコード雛形を組み合わせて解答を構成します。\n\nCritPt evaluates scientific understanding and multi-step reasoning and computation on research-level physics problems. Its public dataset contains 70 challenges with problem descriptions and code templates.","introduction_ja":"","introduction_en":"","category":"Category not supplied","task_count":null,"acquisition_status":"Acquisition status not supplied","official_url":"https://critpt.com/","indexing_mode":"noindex"},"task_id":"9af1cbe6-92c0-51b0-8314-cec0f36289d8","task_key":"train--Challenge~5f36~5fmain","task_revision_id":"1","upstream_id":"Challenge_36_main","short_description":"In the following autocatalytic reaction cycle, each component catalyzes the…","config":"","split":"train","body":"{\"code_template\":\"import sympy as sp\\n\\nk, n = sp.symbols('k n')\\nX_tot = sp.symbols('X_tot')\\n\\ndef answer(k, n, X_tot):\\n    r\\\"\\\"\\\"\\n    Return the expression of $\\\\mathbb E\\\\left[C^2\\\\right]$ in Sympy format,\\n    and the minimal value of $n$ need to be to observe such oscillatory behavior.\\n\\n    Inputs\\n    ----------\\n    k: sympy.Symbol, reaction rate constant $k$\\n    n: sympy.Symbol, number of components in the cycle $n$\\n    X_tot: sympy.Symbol, total population size $X_{tot}$\\n\\n    Outputs\\n    ----------\\n    E_C2: sympy.Expr, expression of $\\\\mathbb E\\\\left[C^2\\\\right]$ in terms of model parameters $k$ and $n$\\n    n_min: sympy.Expr, minimal value of $n$ to observe such oscillatory behavior\\n    \\\"\\\"\\\"\\n\\n    # ------------------ FILL IN YOUR RESULTS BELOW ------------------\\n    E_C2  = ...  # SymPy expression of Inputs\\n    n_min = ...\\n    # ---------------------------------------------------------------\\n\\n    return E_C2, n_min\",\"problem_description\":\"# Problem setup:\\nIn the following autocatalytic reaction cycle, each component catalyzes the production of the next one in a cycle\\n\\\\begin{equation}\\n    X_{i-1} \\\\xrightarrow{\\\\; k\\\\;}X_{i-1}+ X_i,\\n\\\\end{equation}\\nfor $1<i\\\\leq n$ and\\n\\\\begin{equation}\\n    X_n \\\\xrightarrow{\\\\; k\\\\;}X_n+ X_1.\\n\\\\end{equation}\\nAsymptotically, these reactions lead to exponential growth and homeostasis of all the components. The goal of this problem is to understand the stochastic transient dynamics of the approach to this asymptotic state. We start with a single copy of $X_1$ and no $X_i$ for $i>1$ at time zero. For large $t$ and for large enough $n$, the number of $X_j$ molecules approaches\\n\\\\begin{equation}\\n    X_j \\\\to \\\\frac{1}{n}\\\\left(X_{tot}+2\\\\, C\\\\cos(\\\\omega t+\\\\Phi) e^{\\\\lambda t}\\\\right),\\n\\\\end{equation}\\nwhere $X_{tot}=\\\\sum_j X_j$, $\\\\omega$ and $\\\\lambda$ are unknown constants, and $C$ and $\\\\Phi$ are random variables with unknown distributions. Physically, this says that the approach of $X_j$ to its steady exponential growth differs from that of the total population size with a relatively decaying oscillating component with the stochastic amplitude $2C$.\\n\\n# Main problem:\\n\\nFind the mean-squared value of $C$, $\\\\mathbb E\\\\left[C^2\\\\right]$, in terms of the model parameters $k$ and $n$, and determine how large $n$ needs to be to observe such oscillatory behavior.\"}","display_format":"text","language":"","answer_status":"published","assets":[],"source_url":"https://critpt.com/","history":"initial import","indexing_mode":"noindex","subproblems":[],"grids":[]}