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Boltzmann Constant

A fundamental physical constant (kB) relating temperature to energy, central to kinetic theory's thermodynamic relationships.
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The statement of the theorem

Let H:Phase Space→R\mathcal{H}: \text{Phase Space} \to \mathbb{R} be the Hamiltonian operator for a system of NN particles. Consider the canonical ensemble defined by the partition function Z(β)=Tr(e−βH)Z(\beta) = \text{Tr}\left(e^{-\beta \mathcal{H}}\right), where β=1/(kBT)\beta = 1/(k_B T). The average internal energy ⟨E⟩\langle E \rangle is given by ⟨E⟩=−∂∂βlnZ(β)\langle E \rangle = -\frac{\partial}{\partial \beta} \text{ln} Z(\beta). By definition, the thermodynamic temperature TT is related to the energy and entropy SS via 1T=(∂S∂⟨E⟩)V,N\frac{1}{T} = \left(\frac{\partial S}{\partial \langle E \rangle}\right)_{V, N}. Since S=kBlnZ+β⟨E⟩S = k_B \text{ln} Z + \beta \langle E \rangle, we have 1T=∂∂⟨E⟩(kBlnZ+β⟨E⟩)\frac{1}{T} = \frac{\partial}{\partial \langle E \rangle} \left(k_B \text{ln} Z + \beta \langle E \rangle\right). Equating the two expressions for 1/T1/T yields the definition of the Boltzmann constant kBk_B as:\n\nkB=⟨E⟩T(∂lnZ(β)∂⟨E⟩)β or equivalently, kB=⟨E⟩T(∂β∂⟨E⟩)βk_B = \frac{\langle E \rangle}{T} \left( \frac{\partial \text{ln} Z(\beta)}{\partial \langle E \rangle} \right)_{\beta} \text{ or equivalently, } k_B = \frac{\langle E \rangle}{T} \left( \frac{\partial \beta}{\partial \langle E \rangle} \right)_{\beta}
Source: Wikipedia