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Entropy

Field: Classical Thermodynamics

A thermodynamic quantity representing the unavailability of a system's thermal energy for conversion into mechanical work.

Sequence of Expressions

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}
Let H\mathcal{H} be the Hamiltonian operator governing the system's dynamics in a phase space Γ⊂R6N\Gamma \subset \mathbb{R}^{6N}. Consider an isolated system constrained to an energy shell ΩE={x∈Γ∣E−ϵ≤H(x)≤E+ϵ}\Omega_E = \{ \mathbf{x} \in \Gamma \mid E - \epsilon \le H(\mathbf{x}) \le E + \epsilon \}. The number of accessible microstates, WW, is defined by the volume of this shell in phase space, normalized by the fundamental volume ℏ3N\hbar^{3N}. Formally, W≈Vol(ΩE)h3NW \approx \frac{\text{Vol}(\Omega_E)}{h^{3N}}. The entropy SS is then defined via the Boltzmann relation:\n\nS=kBln⁡(W)\mathcal{S} = k_B \ln(W)
Let GG be the Gibbs Free Energy, a state function defined for a system in equilibrium at constant pressure PP and temperature TT. The differential change in GG is given by the fundamental thermodynamic relation: dG=VdP−TdSdG = V dP - T dS where VV is the volume, PP is the pressure, and SS is the entropy. Furthermore, GG can be expressed in terms of the enthalpy HH and entropy SS as: G(T,P,N)=H(T,P,N)−TS(T,P,N)G(T, P, N) = H(T, P, N) - T S(T, P, N) The partial derivatives of GG with respect to its natural variables (T,P,N)(T, P, N) define the system's intensive properties: (∂G∂T)P,N=−S\left(\frac{\partial G}{\partial T}\right)_{P, N} = -S (∂G∂P)T,N=V\left(\frac{\partial G}{\partial P}\right)_{T, N} = V (∂G∂N)T,P=μˉ\left(\frac{\partial G}{\partial N}\right)_{T, P} = \bar{\mu} where μˉ\bar{\mu} is the chemical potential.
Let the thermodynamic state of a closed system be defined by the set of extensive variables X=(S,V)\mathbf{X} = (S, V). The internal energy UU is a state function, U=U(S,V)U = U(S, V). The fundamental thermodynamic relation is given by dU=TdS−PdVdU = T dS - P dV. The Helmholtz Free Energy FF is defined as the Legendre transform of UU with respect to S/TS/T, or equivalently, as a function of temperature TT and volume VV: F(T,V)=U(S,V)−TSF(T, V) = U(S, V) - T S. The differential change in FF is then derived by total differentiation: dF=d(U−TS)=dU−TdS−SdTdF = d(U - TS) = dU - T dS - S dT Substituting the fundamental relation dU=TdS−PdVdU = T dS - P dV: dF=(TdS−PdV)−TdS−SdTdF = (T dS - P dV) - T dS - S dT Simplifying yields the rigorous differential form: dF=−SdT−PdVdF = -S dT - P dV Furthermore, the partial derivatives define the system's response functions: \left(\frac{\partial F}{\partial T}\right)_{V} = -S \quad \text{and} \nn\left(\frac{\partial F}{\partial V}\right)_{T} = -P
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
Intermediate
The change in internal energy of a system is equal to the heat added to the system minus the work done by the system: ΔU=Q−W\Delta U = Q - W.
Let S\mathcal{S} be an isolated thermodynamic system, and let dτd\tau be an infinitesimal time interval during a process. Define the entropy SS as a state function such that the differential change dSdS is given by the Clausius inequality: dS≥dQTdS \ge \frac{dQ}{T}, where TT is the absolute temperature and dQdQ is the infinitesimal heat exchanged with the surroundings. For any irreversible process, the entropy production rate S˙irr\dot{S}_{irr} must be non-negative. Formally, the Second Law dictates that the total change in entropy ΔS\Delta S over any process P\mathcal{P} must satisfy:\nΔS≥0\Delta S \ge 0 \nFurthermore, for a reversible process (Prev\mathcal{P}_{rev}), the equality holds, and the change in entropy is defined by the integral of the heat transfer dQrevdQ_{rev} over the process path Γ\Gamma: \nΔS=∫ΓdQrevT\Delta S = \int_{\Gamma} \frac{dQ_{rev}}{T}
Let S\mathcal{S} be an isolated thermodynamic system, and let ∂S\partial \mathcal{S} be its boundary. Consider a process where heat δQ\delta Q is transferred across ∂S\partial \mathcal{S} at a local temperature TT. The change in entropy δS\delta S is defined by the Clausius relation δS=δQT\delta S = \frac{\delta Q}{T}. The Second Law dictates that the total entropy change ΔStotal\Delta S_{total} must be non-negative for any spontaneous process: ΔStotal≥0\Delta S_{total} \ge 0. Specifically, for a heat transfer Qcold→hotQ_{cold \to hot} from a body at temperature TCT_C to a body at temperature THT_H (where TC<THT_C < T_H), the spontaneous transfer requires ΔStotal=Qcold→hotTC+−Qcold→hotTH<0\Delta S_{total} = \frac{Q_{cold \to hot}}{T_C} + \frac{-Q_{cold \to hot}}{T_H} < 0, which is forbidden. Therefore, the mathematical constraint for spontaneous processes is:\ΔStotal≥0$.Furthermore,foranycyclicprocess,\Delta S_{total} \ge 0\$. Furthermore, for any cyclic process, \oint \frac{\delta Q}{T} \ge 0$.
Let S\mathcal{S} be a set of physical systems, and let ρA\rho_A and ρB\rho_B be the respective microstates of systems AA and BB. Define the interaction Hamiltonian HAB(ρA,ρB)H_{AB}(\rho_A, \rho_B) such that the system is in thermal equilibrium if and only if the generalized potential Φ(A,B)=−kBTAB(SA+SB)+EAB\Phi(A, B) = -k_B T_{AB} \left(S_A + S_B\right) + E_{AB} is maximized, where SAS_A and SBS_B are the entropies of AA and BB, and EABE_{AB} is the internal energy. Assume the existence of a third system CC with state ρC\rho_C. If ρA\rho_A is in equilibrium with ρC\rho_C, then ∂Φ(A,C)∂t=0\frac{\partial \Phi(A, C)}{\partial t} = 0, implying Φ(A,C)=f(TA,TC)\Phi(A, C) = f(T_A, T_C). Similarly, if ρB\rho_B is in equilibrium with ρC\rho_C, then Φ(B,C)=f(TB,TC)\Phi(B, C) = f(T_B, T_C). The Zeroth Law asserts that the function ff depends only on the absolute temperature parameter TT, such that f(TA,TC)=g(TA,TC)f(T_A, T_C) = g(T_A, T_C) and f(TB,TC)=g(TB,TC)f(T_B, T_C) = g(T_B, T_C). Consequently, the equilibrium condition between AA and BB must also be determined solely by TAT_A and TBT_B, i.e., Φ(A,B)=g(TA,TB)\Phi(A, B) = g(T_A, T_B). This establishes the existence of a universal temperature parameter TT such that Φ(A,B)=g(TA,TB)\Phi(A, B) = g(T_A, T_B).
Let S\mathcal{S} be the system and R\mathcal{R} be the surroundings. Consider a process P\mathcal{P} occurring between initial state X1\mathbf{X}_1 and final state X2\mathbf{X}_2. The total entropy change is defined as ΔStotal=ΔSS+ΔSR\Delta S_{total} = \Delta S_{\mathcal{S}} + \Delta S_{\mathcal{R}}. For any irreversible process P\mathcal{P}, the entropy production S˙gen\dot{S}_{gen} is defined by the rate of change of total entropy: S˙gen=dStotaldt\dot{S}_{gen} = \frac{d S_{total}}{d t}. The Second Law of Thermodynamics dictates that the entropy production must be non-negative, S˙gen≥0\dot{S}_{gen} \ge 0. For a strictly irreversible process, the inequality is strict: ΔStotal=∫t1t2S˙gendt>0\Delta S_{total} = \int_{t_1}^{t_2} \dot{S}_{gen} dt > 0. Specifically, if the process involves irreversible fluxes Ji\mathbf{J}_i driven by generalized forces Xi\mathbf{X}_i (e.g., heat flux Jq\mathbf{J}_q driven by temperature gradient ∇T\nabla T), the entropy production rate is given by the sum of products of fluxes and forces: S˙gen=∑iJi⋅Xi>0\dot{S}_{gen} = \sum_{i} \mathbf{J}_i \cdot \mathbf{X}_i > 0