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Phase Transitions

Field: Statistical Mechanics

The transformation of a thermodynamic system from one phase or state of matter to another.

Sequence of Expressions

Let H\mathcal{H} be the Hamiltonian of the system, and β=1/(kBT)\beta = 1/(k_B T) be the inverse temperature. Define the Helmholtz free energy density f(T,order parameter)f(T, \text{order parameter}) via the partition function Z=Tr(e−βH)Z = \text{Tr}(e^{-\beta \mathcal{H}}). For a continuous (second-order) phase transition, the critical temperature TcT_c is defined by the condition where the coefficient of the quadratic term in the Landau expansion of the free energy density, A(T)A(T), vanishes, while the coefficient of the quartic term, BB, remains positive. Specifically, let the free energy density expansion be f(T,η)=12A(T)η2+14Bη4+…f(T, \eta) = \frac{1}{2} A(T) \eta^2 + \frac{1}{4} B \eta^4 + \dots. The critical temperature TcT_c is determined by the root of A(T)=0A(T) = 0, provided that the susceptibility χ=∂2f/∂η2\chi = \partial^2 f / \partial \eta^2 diverges at this point, satisfying: A(Tc)=0and∂2f∂η2∣T=Tc→∞A(T_c) = 0 \quad \text{and} \quad \left. \frac{\partial^2 f}{\partial \eta^2} \right|_{T=T_c} \to \infty
Definition

Critical Point

Let F(T,V)F(T, V) be the Helmholtz free energy density of a system, where TT is temperature and VV is volume. The equation of state is defined by the pressure P=−(∂F∂V)TP = -\left(\frac{\partial F}{\partial V}\right)_{T}. The critical point (Tc,Pc,Vc)(T_c, P_c, V_c) is mathematically characterized by the simultaneous vanishing of the second and third derivatives of the free energy with respect to volume (or density ρ=1/V\rho = 1/V) at the point of maximum compressibility, such that the isothermal compressibility κT\kappa_T and the heat capacity CVC_V exhibit critical exponents. Specifically, the following conditions must hold at (Tc,Vc)(T_c, V_c): \begin{enumerate} \item ∂2P∂V2∣Tc=0\left.\frac{\partial^2 P}{\partial V^2}\right|_{T_c} = 0 \item ∂3P∂V3∣Tc=0\left.\frac{\partial^3 P}{\partial V^3}\right|_{T_c} = 0 \end{enumerate} Furthermore, the critical point is defined by the divergence of the generalized susceptibility χ\chi: \begin{equation} \chi^{-1} = \frac{\partial P}{\partial V} \bigg|_{(T, V)} \end{equation} \text{such that at } (T_c, V_c), \lim_{|(T, V) - (T_c, V_c)| \to 0} \chi^{-1} = 0.Thisimpliesthecriticalexponent. This implies the critical exponent \gamma = 0forthesusceptibility,orequivalently,thevanishingofthecoefficientofthequadratictermintheLandauexpansionofthefreeenergydensity for the susceptibility, or equivalently, the vanishing of the coefficient of the quadratic term in the Landau expansion of the free energy density f(\phi)$: \begin{equation} f(\phi) \approx \frac{1}{2} a(T-T_c) \phi^2 + \frac{1}{4} b \phi^4 + \frac{1}{6} c \phi^6 \end{equation} \text{where } a(T_c)=0 \text{ and } b>0 \text{ for the second-order transition.}
Let H\mathcal{H} be a system described by a Hamiltonian H(q,p,N)\mathcal{H}(\mathbf{q}, \mathbf{p}, \mathbf{N}) and subject to a volume VV and temperature TT. In the grand canonical ensemble, the system is characterized by the grand partition function Z(μ,V,T)\mathcal{Z}(\mu, V, T), where μ\mu is the chemical potential. The Gibbs Free Energy GG is defined via the relation:\n\nG(μ,V,T)=−kBTln⁡Z(μ,V,T)G(\mu, V, T) = -k_B T \ln \mathcal{Z}(\mu, V, T)\n\nAlternatively, considering the fundamental thermodynamic potential, the differential form is given by:\n\ndG=−SdT+VdP+μdNdG = -S dT + V dP + \mu dN\n\nWhere SS is the entropy, PP is the pressure, and NN is the number of particles. The state variables (μ,V,T)(\mu, V, T) define the equilibrium manifold where GG is minimized for fixed TT and PP.
Let S\mathcal{S} be a thermodynamic system undergoing a phase transition from an initial phase α\alpha to a final phase β\beta at constant pressure PP. Define the enthalpy HH as the thermodynamic potential H=U+PVH = U + PV, where UU is the internal energy. The enthalpy change of transition, ΔHα→β\Delta H_{\alpha \to \beta}, is defined by the path integral along the equilibrium coexistence curve Cα→β\mathcal{C}_{\alpha \to \beta} in the (T,P)(T, P) plane:\n\nΔHα→β=∫αβdH=∫T1T2Cp,α→β(dTT)⋅P⋅dTTor, more simply, using the definition of heat capacity at constant pressure:\Delta H_{\alpha \to \beta} = \int_{\alpha}^{\beta} dH = \int_{T_1}^{T_2} C_{p, \alpha \to \beta} \left( \frac{dT}{T} \right) \cdot P \cdot \frac{dT}{T} \quad \text{or, more simply, using the definition of heat capacity at constant pressure:} \n\nΔHα→β=∫TinitialTfinalCp(T,P)dTT⋅Pwhere Cp(T,P) is the specific heat capacity at constant pressure, and the integral is taken over the temperature range of the transition.\Delta H_{\alpha \to \beta} = \int_{T_{initial}}^{T_{final}} C_{p}(T, P) \frac{dT}{T} \cdot P \quad \text{where } C_{p}(T, P) \text{ is the specific heat capacity at constant pressure, and the integral is taken over the temperature range of the transition.} \n\nAlternatively, using the Clausius-Clapeyron relation for the coexistence curve P(T)P(T): \ndPdT=ΔSΔV=ΔHTΔV\frac{dP}{dT} = \frac{\Delta S}{\Delta V} = \frac{\Delta H}{T \Delta V} \nThis yields the rigorous statement for the latent heat (enthalpy change): \nΔH=∫P1P2TΔVVfinal−VinitialdP\Delta H = \int_{P_1}^{P_2} \frac{T \Delta V}{V_{final} - V_{initial}} dP
Let F(x,T)\mathcal{F}(\boldsymbol{x}, T) be the Helmholtz free energy density of a multi-component system, where x=(x1,x2,…,xN)\boldsymbol{x} = (x_1, x_2, \dots, x_N) are the mole fractions, subject to ∑i=1Nxi=1\sum_{i=1}^N x_i = 1. Assume the free energy density can be expanded in a polynomial form incorporating interaction parameters LijL_{ij}: F(x,T)=∑i=1NxiFi(T)+12∑i=1N∑j=1NxixjΩij(T)+∑i=1N∑j=i+1N∑k=1MLij(k)(xi−xj)k⋅ψk(x)+…\mathcal{F}(\boldsymbol{x}, T) = \sum_{i=1}^N x_i \mathcal{F}_i(T) + \frac{1}{2} \sum_{i=1}^N \sum_{j=1}^N x_i x_j \Omega_{ij}(T) + \sum_{i=1}^N \sum_{j=i+1}^N \sum_{k=1}^M L_{ij}^{(k)} (x_i - x_j)^k \cdot \psi_k(\boldsymbol{x}) + \dots. The Redlich-Kister Criterion dictates the stability of the phase transition by analyzing the coefficients Lij(k)L_{ij}^{(k)}. For a system to exhibit a stable phase transition at composition x∗\boldsymbol{x}^*, the coefficients must satisfy the condition that the Hessian matrix of the free energy density, H={∂2F∂xi∂xj}i,j=1N\mathbf{H} = \left\{ \frac{\partial^2 \mathcal{F}}{\partial x_i \partial x_j} \right\}_{i,j=1}^N, must be positive semi-definite (or negative definite, depending on the definition of F\mathcal{F}) in the vicinity of the equilibrium state x∗\boldsymbol{x}^*. Specifically, the stability requires that the coefficients Lij(k)L_{ij}^{(k)} must be constrained such that the second-order mixing term Ωij(T)\Omega_{ij}(T) dominates the higher-order terms for the transition to be continuous (second-order), or that the coefficients lead to a minimum in the Gibbs free energy, ΔG=min⁡x(F(x,T)−∑xiμi0)\Delta G = \min_{\boldsymbol{x}} \left( \mathcal{F}(\boldsymbol{x}, T) - \sum x_i \mu_i^0 \right), ensuring ∂2ΔG∂xi∂xj≥0\frac{\partial^2 \Delta G}{\partial x_i \partial x_j} \ge 0 for all i,ji, j.
Let H\mathcal{H} be the Hamiltonian operator governing the system's time evolution, and let A^\hat{A} be an observable operator in the Hilbert space H\mathcal{H}. The time evolution of the expectation value ⟨A^⟩\langle \hat{A} \rangle is governed by the equation:\n\nddt⟨A^⟩=1iℏ⟨[A^,H]⟩+∂⟨A^⟩∂t\frac{d}{dt} \langle \hat{A} \rangle = \frac{1}{i\hbar} \langle [\hat{A}, \mathcal{H}] \rangle + \frac{\partial \langle \hat{A} \rangle}{\partial t}\n\nFor an operator A^\hat{A} that depends on position x^\hat{x} and momentum p^\hat{p}, the commutator [A^,H][\hat{A}, \mathcal{H}] can be expanded using the canonical commutation relations [x^,p^]=iℏ[\hat{x}, \hat{p}] = i\hbar. This leads to the generalized form:\n\nddt⟨A^⟩=1ℏ⟨{A^,H}P⟩+∂⟨A^⟩∂t\frac{d}{dt} \langle \hat{A} \rangle = \frac{1}{\hbar} \langle \{ \hat{A}, \mathcal{H} \}_{P} \rangle + \frac{\partial \langle \hat{A} \rangle}{\partial t}\n\nwhere 1ℏ⟨{A^,H}P⟩\frac{1}{\hbar} \langle \{ \hat{A}, \mathcal{H} \}_{P} \rangle represents the expectation value of the Poisson bracket structure, linking the quantum expectation value to the classical dynamics derived from the Hamiltonian H\mathcal{H}.
Let H\mathcal{H} be the Hamiltonian operator of a system with a discrete set of energy eigenstates {Ei}i=1N\{E_i\}_{i=1}^{N}. The canonical partition function ZZ is defined as the sum over all accessible states: Z(β,H)=∑i=1Ne−βEiZ(\beta, \mathcal{H}) = \sum_{i=1}^{N} e^{-\beta E_i}, where β=1kBT\beta = \frac{1}{k_B T} is the inverse temperature parameter. The probability PiP_i that the system occupies the state ii with energy EiE_i is given by the Boltzmann distribution:\nPi=e−βEiZ(β,H)P_i = \frac{e^{-\beta E_i}}{Z(\beta, \mathcal{H})}
Let ρ(t)\rho(t) be the time-evolving density matrix of an isolated quantum system, governed by the Hamiltonian H^\hat{H}. The evolution is described by the von Neumann equation: dρdt=−iℏ[H^,ρ]\frac{d\rho}{dt} = -\frac{i}{\hbar}[\hat{H}, \rho]. Define the von Neumann entropy functional S(ρ)S(\rho) as: S(ρ)=−kBTr(ρln⁡ρ)S(\rho) = -k_B\text{Tr}(\rho \ln \rho). The Second Law of Thermodynamics mandates that the rate of change of entropy must be non-negative: dS(ρ)dt≥0\frac{d S(\rho)}{dt} \ge 0. Specifically, substituting the evolution equation yields: dS(ρ)dt=−kBTr(dρdtln⁡ρ+ρd(ln⁡ρ)dt)≥0\frac{d S(\rho)}{dt} = -k_B\text{Tr}\left(\frac{d\rho}{dt} \ln \rho + \rho \frac{d(\ln \rho)}{dt}\right) \ge 0. This inequality holds for all physical Hamiltonians H^\hat{H} and represents the irreversible flow towards equilibrium.
Let H\mathcal{H} be the Hamiltonian of a system defined on a lattice Λ\Lambda with local interactions JijJ_{ij}. Consider the partition function Z=Tr(e−βH)Z = \text{Tr}\left(e^{-\beta \mathcal{H}}\right), where β=1/kBT\beta = 1/k_B T. Define the order parameter Φ\Phi as the expectation value of a symmetry-breaking field operator ϕ(r)\phi(\mathbf{r}): Φ=⟨ϕ(r)⟩=1ZTr(ϕ(r)e−βH)\Phi = \langle \phi(\mathbf{r}) \rangle = \frac{1}{Z} \text{Tr}\left(\phi(\mathbf{r}) e^{-\beta \mathcal{H}}\right). In the context of mean-field theory (e.g., Ising model), Φ\Phi minimizes the Landau free energy functional F(Φ)F(\Phi): F(Φ)=a(T−Tc)Φ2+bΦ4+cΦ6+12∑r(∇Φ(r))2F(\Phi) = a(T - T_c)\Phi^2 + b\Phi^4 + c\Phi^6 + \frac{1}{2} \sum_{\mathbf{r}} \left( \nabla \Phi(\mathbf{r}) \right)^2 The equilibrium value Φ0\Phi_0 is determined by the minimization condition ∂F∂Φ∣Φ=Φ0=0\frac{\partial F}{\partial \Phi} \bigg|_{\Phi=\Phi_0} = 0. For a second-order transition, Φ0∝Tc−T\Phi_0 \propto \sqrt{T_c - T} as T→Tc+T \to T_c^+.
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.