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Fluctuations of Entropy

The inherent statistical fluctuations in entropy contribute to the driving force behind phase transitions, reflecting the tendency towards disorder.
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The statement of the theorem

Let H\mathcal{H} be the Hamiltonian operator governing the system, and let β=1/(kBT)\beta = 1/(k_B T) be the inverse temperature. The canonical partition function is defined as Z(β)=Tr(e−βH)Z(\beta) = \text{Tr}\left(e^{-\beta \mathcal{H}}\right). The entropy SS is related to the free energy FF by S=−(∂F∂T)VS = -\left(\frac{\partial F}{\partial T}\right)_V. The fluctuation of the entropy, ΔS\Delta S, is formally defined by its variance, Var(S)\text{Var}(S), which can be derived from the cumulant generating function of the energy fluctuations. Specifically, the variance of the entropy is given by the second derivative of the free energy with respect to temperature, scaled by the heat capacity CVC_V: Var(S)=1kBT2⟨(S−⟨S⟩)2⟩\text{Var}(S) = \frac{1}{k_B T^2} \left\langle (S - \langle S \rangle)^2 \right\rangle. In the thermodynamic limit, the fluctuation is often related to the susceptibility χ\chi: Var(S)∝1kBT2⟨(S−⟨S⟩)2⟩≈1kBT2(∂2F∂T2)V,N⋅Volume\text{Var}(S) \propto \frac{1}{k_B T^2} \left\langle (S - \langle S \rangle)^2 \right\rangle \approx \frac{1}{k_B T^2} \left( \frac{\partial^2 F}{\partial T^2} \right)_{V, N} \cdot \text{Volume}. For a system undergoing a phase transition, the magnitude of this fluctuation, ΔS\Delta S, is proportional to the correlation length ξ\xi and the generalized susceptibility χS\chi_S: ΔS∝kBξ2χS(β)\Delta S \propto k_B \xi^2 \chi_S(\beta). This relationship quantifies how the system's entropy deviates from its mean value due to critical fluctuations.