{"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":"aa640307-facc-58b5-b802-2e0182b96568","task_key":"train--Challenge~5f49~5fmain","task_revision_id":"1","upstream_id":"Challenge_49_main","short_description":"In a long-range dispersal model, a cluster of species can grow from a single…","config":"","split":"train","body":"{\"code_template\":\"import sympy as sp\\n\\nz, K = sp.symbols('z K')\\n\\ndef answer(z, K):\\n    r\\\"\\\"\\\"\\n    Return the expression of $\\\\varphi$ in Sympy format.\\n\\n    Inputs\\n    ----------\\n    z: sympy.Symbol, defined as $\\\\log_2 t$\\n    K: sympy.Symbol, constant from the self-consistent equation\\n            $$\\\\frac{1}{\\\\ell^{\\\\mu}(t)} \\\\int_0^t \\\\ell(\\\\tau) \\\\ell(t - \\\\tau) d\\\\tau = K$$\\n\\n    Outputs\\n    ----------\\n    varphi: sympy.Expr, expansion of $\\\\varphi$ in terms of $z$ for large $t$\\n        and retain terms up to constant order in $z$\\n    \\\"\\\"\\\"\\n\\n    # ------------------ FILL IN YOUR RESULTS BELOW ------------------\\n    varphi = ...  # SymPy expression in terms of z (and possibly K),\\n                  # retaining z, log_2(z), log_2(z)**2 terms; omit constants\\n    # ---------------------------------------------------------------\\n\\n    return varphi\",\"problem_description\":\"\\n\\n# Problem setup:\\nIn a long-range dispersal model, a cluster of species can grow from a single seed. The cluster expands into space with long-range power-law interactions. In one dimension, the asymptotic size of the cluster,  denoted by $\\\\ell(t)$, satisfies a self-consistent equation:\\n\\\\begin{equation}\\n    \\\\frac{1}{\\\\ell^{\\\\mu}(t)} \\\\int_0^t \\\\ell(\\\\tau) \\\\ell(t - \\\\tau) d\\\\tau = K.\\n\\\\end{equation}\\nHere, $\\\\mu$ is the power-law index of the long-range interaction, and $K$ is a constant independent of time.\\n\\n# Main problem:\\n\\nConsider the specific scenario where $\\\\mu = 2$, and let us define $\\\\varphi = \\\\log_2 \\\\ell$ and $z = \\\\log_2 t$. Expand $\\\\varphi$ in terms of $z$  for large $t$ and retain terms up to constant order in $z$. That is, you should fix any $\\\\text{polylog}(z)$ corrections such as $\\\\log_2(z)$ and $\\\\log^2_2(z)$ if they are present; constant corrections can be ignored.\"}","display_format":"text","language":"","answer_status":"published","assets":[],"source_url":"https://critpt.com/","history":"initial import","indexing_mode":"noindex","subproblems":[],"grids":[]}