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| Here we have the ''N-particle distribution function''
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| (Ref. 1 Eq. 2.2)
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| :<math>\mathcal{G}_{(N)} ({\mathbf X}_{(N)},t)= \frac{\Gamma_{(N)}^{(0)}}{\mathcal{N}} \frac{{\rm d}\mathcal{N}}{{\rm d}\Gamma_{(N)}}</math>
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| where <math>\Gamma_{(N)}^{(0)}</math> is a normalized constant with the dimensions
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| of the [[phase space]] <math>\left. \Gamma_{(N)} \right.</math>.
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| :<math>{\mathbf X}_{(N)} = \{ {\mathbf r}_1 , ..., {\mathbf r}_N ; {\mathbf p}_1 , ..., {\mathbf p}_N \}</math>
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| Normalization condition (Ref. 1 Eq. 2.3):
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| :<math>\frac{1}{\Gamma_{(N)}^{(0)}} \int_{\Gamma_{(N)}} \mathcal{G}_{(N)} {\rm d}\mathcal{N} =1</math>
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| it is convenient to set (Ref. 1 Eq. 2.4)
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| :<math>\Gamma_{(N)}^{(0)} = V^N \mathcal{P}^{3N}</math>
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| where <math>V</math> is the volume of the system and <math>\mathcal{P}</math> is the characteristic momentum
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| of the particles (Ref. 1 Eq. 3.26),
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| :<math>\mathcal{P} = \sqrt{2 \pi m \Theta}</math>
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| Macroscopic mean values are given by (Ref. 1 Eq. 2.5)
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| :<math>\langle \psi ({\mathbf r},t)\rangle= \frac{1}{\Gamma_{(N)}^{(0)}}
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| \int_{\Gamma_{(N)}} \psi ({\mathbf X}_{(N)}) \mathcal{G}_{(N)} ({\mathbf X}_{(N)},t) {\rm d}\Gamma_{(N)}
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| </math>
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| ===[[Ergodic hypothesis |Ergodic theory]]===
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| Ref. 1 Eq. 2.6
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| :<math>\langle \psi \rangle = \overline \psi</math>
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| ===[[Entropy]]===
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| Ref. 1 Eq. 2.70
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| :<math>S_{(N)}= - \frac{k_B}{ V^N \mathcal{P}^{3N}} \int_\Gamma \Omega_1,... _N \mathcal{G}_1,... _N {\rm d}\Gamma_{(N)}</math>
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| where <math>\Omega</math> is the ''N''-particle [[thermal potential]] (Ref. 1 Eq. 2.12)
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| :<math>\Omega_{(N)} ({\mathbf X}_{(N)},t)= \ln \mathcal{G}_{(N)} ({\mathbf X}_{(N)},t)</math>
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| ==References==
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| # G. A. Martynov "Fundamental Theory of Liquids. Method of Distribution Functions", Adam Hilger (out of print)
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| [[category: statistical mechanics]]
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