The Gibbs rotational ensemble represents the possible states of a mechanical system in thermal and rotational equilibrium at temperature T {\displaystyle T} and angular velocity ω {\displaystyle {\boldsymbol {\omega }}} . The Jaynes procedure can be used to obtain this ensemble. An ensemble is the set of microstates corresponding to a given macrostate. The Gibbs rotational ensemble assigns a probability p i {\displaystyle p_{i}} to a given microstate characterized by energy E i {\displaystyle E_{i}} and angular momentum J i {\displaystyle \mathbf {J} _{i}} for a given temperature T {\displaystyle T} and rotational velocity ω {\displaystyle {\boldsymbol {\omega }}} .
p i = 1 Z e − β ( E i − ω ⋅ J i ) {\displaystyle p_{i}={\frac {1}{Z}}e^{-\beta (E_{i}-{\boldsymbol {\omega }}\cdot \mathbf {J} _{i})}}
where Z {\displaystyle Z} is the partition function
Z = ∑ i e − β ( E i − ω ⋅ J i ) {\displaystyle Z=\sum _{i}e^{-\beta (E_{i}-{\boldsymbol {\omega }}\cdot \mathbf {J} _{i})}}
Derivation The Gibbs rotational ensemble can be derived using the same general method as to derive any ensemble, as given by E.T. Jaynes in his 1956 paper Information Theory and Statistical Mechanics. Let f ( x ) {\displaystyle f(x)} be a function with expectation value
⟨ f ( x ) ⟩ = ∑ i p i f ( x i ) {\displaystyle \langle f(x)\rangle =\sum _{i}p_{i}f(x_{i})}
where p i {\displaystyle p_{i}} is the probability of x i {\displaystyle x_{i}} , which is not known a priori. The probabilities p i {\displaystyle p_{i}} obey normalization
∑ i p i = 1 {\displaystyle \sum _{i}p_{i}=1}
To find p i {\displaystyle p_{i}} , the Shannon entropy H {\displaystyle H} is maximized, where the Shannon entropy goes as
H ∼ ∑ i p i ln ( p i ) {\displaystyle H\sim \sum _{i}p_{i}\ln(p_{i})}
The method of Lagrange multipliers is used to maximize H {\displaystyle H} under the constraints ⟨ f ( x ) ⟩ {\displaystyle \langle f(x)\rangle } and the normalization condition, using Lagrange multipliers λ {\displaystyle \lambda } and μ {\displaystyle \mu } to find
p i = e − λ − μ f ( x i ) {\displaystyle p_{i}=e^{-\lambda -\mu f(x_{i})}}
λ {\displaystyle \lambda } is found via normalization:
λ = ln ( ∑ i e − μ f ( x i ) ) = ln ( Z ( μ ) ) {\displaystyle \lambda =\ln \left(\sum _{i}e^{-\mu f(x_{i})}\right)=\ln(Z(\mu ))}
and ⟨ f ( x ) ⟩ {\displaystyle \langle f(x)\rangle } can be written as
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