Exact LaTeX body
\begin{proof}[H-Theorem via Symbolic Integration by Parts]
\label{proof:bk1_sketch_direct_evaluation}
\leavevmode
Write the symbolic Fokker--Planck equation in gradient-flow form.
Cf.~Thm.~\ref{theorem:bk1_fundamental_relation_fokker_plank_equation}.
Symbolic drift points along $-\nabla H$, decreasing the Hamiltonian.
Cf.~Def.~\ref{definition:bk1_symbolic_hamiltonian} and
Thm.~\ref{theorem:bk1_the_fokker_planck_equation_theorem}:
\[
\begin{aligned}
\partial_s \rho
&= \nabla\cdot\!\bigl(-\rho D\bigr) + \beta^{-1}\nabla^2\rho \\
&= \beta^{-1}\nabla\cdot\!\bigl(\rho\,\nabla(\log\rho + \beta H)\bigr).
\end{aligned}
\]
\textbf{Step 1: Functional chain rule.}
Since $F[\rho] = \int_M \rho H\,d\mu_g + \beta^{-1}\int_M\rho\log\rho\,d\mu_g$
(Def.~\ref{definition:bk1_symbolic_entropy}, Def.~\ref{definition:bk1_symbolic_hamiltonian}),
\[
\frac{dF}{ds}
= \int_M \frac{\delta F}{\delta\rho}\,\partial_s\rho\,d\mu_g
= \int_M \bigl(H + \beta^{-1}(1+\log\rho)\bigr)\,\partial_s\rho\,d\mu_g.
\]
Since $\int_M\partial_s\rho\,d\mu_g = 0$ (normalization preserved), the constant
$\beta^{-1}$ drops out:
\[
\frac{dF}{ds}
= \int_M (H + \beta^{-1}\log\rho)\,
\beta^{-1}\nabla\cdot\!\bigl(\rho\,\nabla(\log\rho + \beta H)\bigr)\,d\mu_g.
\]
\textbf{Step 2: Integration by parts.}
On the complete Riemannian manifold $(M,g)$
(Lemma~\ref{lemma:bk1_completeness_of_symbolic_distance}) with $\rho$ decaying at infinity,
boundary terms vanish and the divergence theorem gives:
\[
\int_M f\,\nabla\cdot(\rho\,\mathbf{v})\,d\mu_g
= -\int_M \rho\,\langle\nabla f,\mathbf{v}\rangle_g\,d\mu_g.
\]
With $f = H + \beta^{-1}\log\rho$ and $\mathbf{v} = \nabla(\log\rho + \beta H)$:
\[
\nabla f
= \nabla H + \beta^{-1}\nabla\log\rho
= \beta^{-1}(\nabla\log\rho + \beta\nabla H)
= \beta^{-1}\,\mathbf{v}.
\]
Therefore:
\begin{align*}
\frac{dF}{ds}
&= -\beta^{-1}\int_M \rho\,\langle\nabla f, \nabla(\log\rho+\beta H)\rangle_g\,d\mu_g \\
&= -\beta^{-1}\int_M \rho\,\langle\beta^{-1}\mathbf{v},\mathbf{v}\rangle_g\,d\mu_g \\
&= -\beta^{-2}\int_M \rho\,\norm{\nabla\log\rho + \beta\nabla H}_g^2\,d\mu_g \;\leq\; 0.
\end{align*}
\textbf{Step 3: Equality condition.}
$dF/ds = 0$ iff $\nabla\log\rho + \beta\nabla H = 0$ a.e., i.e., $\rho \propto e^{-\beta H}$,
which by normalization is exactly $\rho_{\text{eq}} = Z^{-1}e^{-\beta H}$
(Cor.~\ref{corollary:bk1_equilibrium_distribution}).
\end{proof}
Complete structured record
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"context": "drift points along $-\\nabla H$, decreasing the Hamiltonian. Cf.~Def.~\\ref{definition:bk1_symbolic_hamiltonian} and Thm.~\\ref{theorem:bk1_the_fokker_planck_equation_theorem}: \\[ \\begin{aligned} \\partial_s \\rho &= \\nabla\\cdot\\!\\bigl(-\\rho D\\bigr) + \\beta^{-1}\\nabla^2\\rho \\\\ &= \\beta^{-1}\\nabla",
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"latex_body": "\\begin{proof}[H-Theorem via Symbolic Integration by Parts]\n\\label{proof:bk1_sketch_direct_evaluation}\n\\leavevmode\n\nWrite the symbolic Fokker--Planck equation in gradient-flow form.\nCf.~Thm.~\\ref{theorem:bk1_fundamental_relation_fokker_plank_equation}.\nSymbolic drift points along $-\\nabla H$, decreasing the Hamiltonian.\nCf.~Def.~\\ref{definition:bk1_symbolic_hamiltonian} and\nThm.~\\ref{theorem:bk1_the_fokker_planck_equation_theorem}:\n\\[\n\\begin{aligned}\n\\partial_s \\rho\n&= \\nabla\\cdot\\!\\bigl(-\\rho D\\bigr) + \\beta^{-1}\\nabla^2\\rho \\\\\n&= \\beta^{-1}\\nabla\\cdot\\!\\bigl(\\rho\\,\\nabla(\\log\\rho + \\beta H)\\bigr).\n\\end{aligned}\n\\]\n\n\\textbf{Step 1: Functional chain rule.}\nSince $F[\\rho] = \\int_M \\rho H\\,d\\mu_g + \\beta^{-1}\\int_M\\rho\\log\\rho\\,d\\mu_g$\n(Def.~\\ref{definition:bk1_symbolic_entropy}, Def.~\\ref{definition:bk1_symbolic_hamiltonian}),\n\\[\n\\frac{dF}{ds}\n= \\int_M \\frac{\\delta F}{\\delta\\rho}\\,\\partial_s\\rho\\,d\\mu_g\n= \\int_M \\bigl(H + \\beta^{-1}(1+\\log\\rho)\\bigr)\\,\\partial_s\\rho\\,d\\mu_g.\n\\]\nSince $\\int_M\\partial_s\\rho\\,d\\mu_g = 0$ (normalization preserved), the constant\n$\\beta^{-1}$ drops out:\n\\[\n\\frac{dF}{ds}\n= \\int_M (H + \\beta^{-1}\\log\\rho)\\,\n \\beta^{-1}\\nabla\\cdot\\!\\bigl(\\rho\\,\\nabla(\\log\\rho + \\beta H)\\bigr)\\,d\\mu_g.\n\\]\n\n\\textbf{Step 2: Integration by parts.}\nOn the complete Riemannian manifold $(M,g)$\n(Lemma~\\ref{lemma:bk1_completeness_of_symbolic_distance}) with $\\rho$ decaying at infinity,\nboundary terms vanish and the divergence theorem gives:\n\\[\n\\int_M f\\,\\nabla\\cdot(\\rho\\,\\mathbf{v})\\,d\\mu_g\n= -\\int_M \\rho\\,\\langle\\nabla f,\\mathbf{v}\\rangle_g\\,d\\mu_g.\n\\]\nWith $f = H + \\beta^{-1}\\log\\rho$ and $\\mathbf{v} = \\nabla(\\log\\rho + \\beta H)$:\n\\[\n\\nabla f\n= \\nabla H + \\beta^{-1}\\nabla\\log\\rho\n= \\beta^{-1}(\\nabla\\log\\rho + \\beta\\nabla H)\n= \\beta^{-1}\\,\\mathbf{v}.\n\\]\nTherefore:\n\\begin{align*}\n\\frac{dF}{ds}\n&= -\\beta^{-1}\\int_M \\rho\\,\\langle\\nabla f, \\nabla(\\log\\rho+\\beta H)\\rangle_g\\,d\\mu_g \\\\\n&= -\\beta^{-1}\\int_M \\rho\\,\\langle\\beta^{-1}\\mathbf{v},\\mathbf{v}\\rangle_g\\,d\\mu_g \\\\\n&= -\\beta^{-2}\\int_M \\rho\\,\\norm{\\nabla\\log\\rho + \\beta\\nabla H}_g^2\\,d\\mu_g \\;\\leq\\; 0.\n\\end{align*}\n\n\\textbf{Step 3: Equality condition.}\n$dF/ds = 0$ iff $\\nabla\\log\\rho + \\beta\\nabla H = 0$ a.e., i.e., $\\rho \\propto e^{-\\beta H}$,\nwhich by normalization is exactly $\\rho_{\\text{eq}} = Z^{-1}e^{-\\beta H}$\n(Cor.~\\ref{corollary:bk1_equilibrium_distribution}).\n\\end{proof}",
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"context": ".e., i.e., $\\rho \\propto e^{-\\beta H}$, which by normalization is exactly $\\rho_{\\text{eq}} = Z^{-1}e^{-\\beta H}$ (Cor.~\\ref{corollary:bk1_equilibrium_distribution}). \\end{proof}",
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"context": "bf{Step 1: Functional chain rule.} Since $F[\\rho] = \\int_M \\rho H\\,d\\mu_g + \\beta^{-1}\\int_M\\rho\\log\\rho\\,d\\mu_g$ (Def.~\\ref{definition:bk1_symbolic_entropy}, Def.~\\ref{definition:bk1_symbolic_hamiltonian}), \\[ \\frac{dF}{ds} = \\int_M \\frac{\\delta F}{\\delta\\rho}\\,\\partial_s\\rho",
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"context": "damental_relation_fokker_plank_equation}. Symbolic drift points along $-\\nabla H$, decreasing the Hamiltonian. Cf.~Def.~\\ref{definition:bk1_symbolic_hamiltonian} and Thm.~\\ref{theorem:bk1_the_fokker_planck_equation_theorem}: \\[ \\begin{aligned} \\partial_s \\rho &= \\nabla\\cdot\\!\\bigl",
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"context": "+ \\beta H)\\bigr)\\,d\\mu_g. \\] \\textbf{Step 2: Integration by parts.} On the complete Riemannian manifold $(M,g)$ (Lemma~\\ref{lemma:bk1_completeness_of_symbolic_distance}) with $\\rho$ decaying at infinity, boundary terms vanish and the divergence theorem gives: \\[ \\int_M f\\,\\nabla\\cdot(\\rh",
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"context": "f:bk1_sketch_direct_evaluation} \\leavevmode Write the symbolic Fokker--Planck equation in gradient-flow form. Cf.~Thm.~\\ref{theorem:bk1_fundamental_relation_fokker_plank_equation}. Symbolic drift points along $-\\nabla H$, decreasing the Hamiltonian. Cf.~Def.~\\ref{definition:bk1_symbolic_hamiltonian",
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