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Entropy Definition

A measure of the disorder or randomness within a system, often quantified as the unavailable energy for doing work.
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The statement of the theorem

Let H\mathcal{H} be a physical system described by a Hamiltonian H^\hat{H}. Let ρ\rho be the probability density operator (or the normalized density matrix) in the Hilbert space H\mathcal{H}. The entropy SS is defined by the von Neumann entropy formula (for quantum systems) or the Gibbs entropy formula (for classical ensembles):\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. For a classical system with a phase space Γ\Gamma and a probability distribution ρ(q,p)\rho(\mathbf{q}, \mathbf{p}), the differential entropy is given by:\n\ndS=−kB∑ipiln⁡piordS=−kB∫Γρ(q,p)ln⁡ρ(q,p)dNqdNphNdS = -k_B \sum_{i} p_i \ln p_i \quad \text{or} \quad dS = -k_B \int_{\Gamma} \rho(\mathbf{q}, \mathbf{p}) \ln \rho(\mathbf{q}, \mathbf{p}) \frac{d^N q d^N p}{h^N}