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Statistical Interpretation of Entropy

Entropy is proportional to the number of microstates corresponding to a given macrostate, reflecting the probability of different arrangements.
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The statement of the theorem

Let H\mathcal{H} be the Hamiltonian operator governing the system's energy levels, and let ρ\rho be the system's density matrix, defined in the Hilbert space Hsys\mathcal{H}_{sys}. The system's macrostate is characterized by the expectation value of the energy, ⟨E⟩\langle E \rangle. The statistical entropy SS is defined by the von Neumann entropy formula:\n\nS=−kBTr(ρln⁡ρ)S = -k_B \text{Tr}(\rho \ln \rho)\n\nwhere kBk_B is the Boltzmann constant, and Tr(⋅)\text{Tr}(\cdot) denotes the trace operation over the density matrix ρ\rho. For an isolated system in the microcanonical ensemble, the density matrix is given by ρ=1W∑i=1W∣ψi⟩⟨ψi∣\rho = \frac{1}{\text{W}} \sum_{i=1}^{\text{W}} |\psi_i\rangle\langle\psi_i|, where W\text{W} is the number of accessible microstates, leading to the Boltzmann formulation:\n\nS=kBln⁡(W)S = k_B \ln(\text{W})\n\nFurthermore, the probability distribution pip_i over the microstates ∣ψi⟩|\psi_i\rangle must satisfy the normalization condition ∑ipi=1\sum_{i} p_i = 1 and the Gibbs entropy formulation:\n\nS=−kB∑ipiln⁡piS = -k_B \sum_{i} p_i \ln p_i
Source: Wikipedia