{"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":"7e3c544c-7278-53eb-ab9c-d810c34d63a2","task_key":"train--Challenge~5f47~5fmain","task_revision_id":"1","upstream_id":"Challenge_47_main","short_description":"Let $\\vec m(x,t)$ be a classical spin field defined for $x\\in\\mathbb R$ and…","config":"","split":"train","body":"{\"code_template\":\"def answer():\\n    r\\\"\\\"\\\"\\n    Return the value of $\\\\mathrm{Tr}(L^4)$ under the given spin configuration.\\n\\n    Inputs\\n    ----------\\n    None\\n\\n    Outputs\\n    ----------\\n    Tr_L4: float, value of $\\\\mathrm{Tr}(L^4)$ for the spin configuration\\n        $$\\\\vec{m}(x) = ( \\\\sin \\\\theta \\\\cos \\\\phi,\\\\, \\\\sin \\\\theta \\\\sin \\\\phi,\\\\, \\\\cos \\\\theta )$$\\n    \\\"\\\"\\\"\\n\\n    # ------------------ FILL IN YOUR RESULTS BELOW ------------------\\n    Tr_L4 = ...\\n    # ---------------------------------------------------------------\\n\\n    return Tr_L4\",\"problem_description\":\"# Problem setup:\\nLet $\\\\vec m(x,t)$ be a classical spin field defined for $x\\\\in\\\\mathbb R$ and $t\\\\in[0,\\\\infty)$. The field satisfies the unit-length constraint\\n\\\\begin{equation}\\n\\\\vec m\\\\cdot\\\\vec m=1.\\n\\\\end{equation}\\nLet $\\\\sigma_\\\\alpha$ ($\\\\alpha=1,2,3$) denote the Pauli matrices, and define the $2\\\\times2$ matrix-valued field\\n\\\\begin{equation}\\nm=\\\\vec m\\\\cdot\\\\vec\\\\sigma= \\\\sum_{\\\\alpha = 1}^3 m^\\\\alpha\\\\sigma_\\\\alpha.\\n\\\\end{equation}\\nThe Lax operator is given by\\n\\\\begin{equation}\\nL=[\\\\mathcal H,m].\\n\\\\end{equation}\\nIts action should be understood analogously to that of a quantum operator.  Specifically, when acting on any $2\\\\times2$ matrix field $n(x)$, we have\\n\\\\begin{equation}\\nL(n)=\\\\mathcal H(mn)-m\\\\,\\\\mathcal H(n),\\n\\\\end{equation}\\nwhere $\\\\mathcal H$ is the Hilbert transform. For a scalar function $f(x)$, the Hilbert transform is defined as\\n\\\\begin{equation}\\n\\\\mathcal H[f(x)]=\\\\frac{\\\\mathrm P}{\\\\pi}\\\\int_{-\\\\infty}^{\\\\infty}\\\\frac{f(y)}{x-y}\\\\,dy=\\\\frac1{\\\\pi x}\\\\!*f(x).\\n\\\\end{equation}\\nWe may write either $\\\\mathcal H(f)$ or $f_{\\\\mathcal H}$ for this quantity. When $\\\\mathcal H$ acts on a matrix field, it is understood component-wise. We assume the boundary condition\\n\\\\begin{equation}\\n\\\\vec m(x\\\\to\\\\pm\\\\infty)=\\\\text{constant},\\n\\\\end{equation}\\ni.e., the spin field converges to a fixed vector as $x\\\\to\\\\pm\\\\infty$.\\n\\nThe trace $\\\\mathrm{Tr}(\\\\cdot)$ includes both the $2\\\\times2$ matrix indices and the spatial integral:\\n\\\\begin{equation}\\n\\\\mathrm{Tr}(\\\\cdot)=\\\\int_{-\\\\infty}^{\\\\infty}dx\\\\,\\\\mathrm{tr}(\\\\cdot).\\n\\\\end{equation}\\n\\n# Main problem:\\n\\nCompute\\n\\\\begin{equation}\\n\\\\mathrm{Tr}(L^4),\\n\\\\end{equation}\\nfor the following spin configuration:\\n\\\\begin{equation}\\n\\\\vec{m}(x) = ( \\\\sin \\\\theta \\\\cos \\\\phi,\\\\, \\\\sin \\\\theta \\\\sin \\\\phi,\\\\, \\\\cos \\\\theta ),\\n\\\\end{equation}\\nwhich is a wave packet parameterized by\\n\\\\begin{equation}\\n\\\\theta(x) = x, \\\\qquad \\\\phi(x) = \\\\frac{2\\\\pi}{3}e^{-x^2}.\\n\\\\end{equation}\\n\\nEvaluate this quantity to at least six decimal places.\"}","display_format":"text","language":"","answer_status":"published","assets":[],"source_url":"https://critpt.com/","history":"initial import","indexing_mode":"noindex","subproblems":[],"grids":[]}