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Statistical mechanics formulated on classical phase space, using probability densities or measures over positions and momenta together with classical microscopic dynamics.
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The ergodic hypothesis supplies one route from long-time microscopic averages to invariant ensemble averages, but modern equilibrium statistical mechanics does not require strict ergodicity of every model.
The source supplies a substantive formal ingredient, dynamical sector, law, field content, or mechanism of the target theoretical framework.
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Hamiltonian flows supply the microscopic phase-space evolution used in classical statistical mechanics, including invariant Liouville measure; ensemble reasoning adds probabilistic and macroscopic structure.
A mathematical concept supplies part of the formal language, state space, representation, or analytic machinery used by a scientific or mathematical-physics concept. The source is the mathematical predecessor; this does not claim that physical content follows from mathematics alone.
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Hamiltonian mechanics supplies the microscopic equations of motion and invariant phase-space structure used by classical statistical mechanics; probability measures, ensembles, and macroscopic observables are additional statistical ingredients.
The source supplies a substantive formal ingredient, dynamical sector, law, field content, or mechanism of the target theoretical framework.
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Classical statistical mechanics represents each microscopic state as a phase-space point and ensembles as measures or densities over phase space, subject to the selected Hamiltonian and constraints.
The source supplies a substantive formal ingredient, dynamical sector, law, field content, or mechanism of the target theoretical framework.
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Classical statistical mechanics is recovered for appropriate observables when quantum level spacing and exchange effects are negligible, for example in a dilute high-temperature regime; this is not a universal replacement of quantum dynamics.
The target is recovered from the source either in a stated mathematical or asymptotic limit, or through a quantitatively controlled approximation with identified small parameters or omitted terms. The edge label and detail must state which case applies and its regime of validity.
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Classical statistical mechanics specializes the microscopic state space and evolution to classical phase space and classical dynamics.
A more specific theory, model, entity class, or regime is obtained by restricting or extending the scope of a broader framework.
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