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Physics — Statistical Mechanics · maximum-entropy-lagrange-multipliers
Problem Maximize the Shannon entropy S = − k ∑ i P i log P i S = -k\sum_i P_i \log P_i S = − k ∑ i P i log P i ∑ P i = 1 \sum P_i = 1 ∑ P i = 1 ⟨ f ( x ) ⟩ = ∑ P i f ( x i ) \langle f(x)\rangle = \sum P_i f(x_i) ⟨ f ( x )⟩ = ∑ P i f ( x i ) P i = 1 Z ( β ) e − β f ( x i ) , Z ( β ) = ∑ e − β f ( x i ) , ⟨ f ( x ) ⟩ = − d d β ln Z ( β ) . (15.26) \begin{aligned}
P_i &= \frac{1}{Z(\beta)}e^{-\beta f(x_i)}, \tag{15.26} \\
Z(\beta) &= \sum e^{-\beta f(x_i)}, \\
\langle f(x)\rangle &= -\frac{\mathrm{d}}{\mathrm{d}\beta}\ln Z(\beta).
\end{aligned} P i Z ( β ) ⟨ f ( x )⟩ = Z ( β ) 1 e − β f ( x i ) , = ∑ e − β f ( x i ) , = − d β d ln Z ( β ) . ( 15.26 ) Browse the problem library