Phase Transition Curves

• Phase Transition Curves are loci in multidimensional parameter space where systems exhibit non-analytic shifts, indicating phase coexistence and critical points.
• They are derived using methods like Maxwell's construction and the Clapeyron equation to distinguish first-order and continuous transitions.
• These curves apply to a range of systems—from classical fluids and quantum many-body models to gravitational phenomena and complex networks—highlighting universal critical behavior.

A phase transition curve is a locus in the multidimensional space of control parameters (temperature, pressure, density, chemical potential, or other order-parameter couplings) at which a physical system displays a non-analytic change in a macroscopic observable—typically marking the coexistence of distinct thermodynamic phases or a qualitative regime switch in a nonequilibrium or statistical model. The geometric, statistical, and dynamical underpinnings of phase transition curves unify a spectrum of phenomena from classical fluids, quantum many-body systems, random structures, and gravity, with the curves' analytic structure controlled by the underlying free energy landscape, symmetry structure, and critical singularities.

1. Mathematical Definitions and Thermodynamic Foundations

For classical systems governed by an equation of state $P=P(V,T)$, a phase transition curve is most prominently manifest as the binodal line in the $(P,T)$- or $(T,V)$-plane, separating regions of distinct phases (e.g., liquid/gas, solid/liquid). It is determined by global minimization of the thermodynamic potential (e.g., the Gibbs free energy) and characterized by singularities or nonanalyticities in thermodynamic derivatives such as heat capacity, compressibility, or susceptibility.

The curve is constructed via:

• Maxwell's construction: For systems like the van der Waals fluid, the coexistence curve in the $P$-$T$ (or $n$-$T$) plane is determined by enforcing equality of chemical potential (or Gibbs free energy) and mechanical pressure across phases, with the Maxwell equal-area rule replacing the unphysical loop of the mean-field isotherm:

$$\int_{V_1}^{V_2}P(V,T)\,dV = P_\mathrm{coex}\,(V_2-V_1)$$

• Rankine–Hugoniot/Clapeyron equation: The slope of a first-order transition curve, such as in the $(P,T)$ plane, is universally given by:

$\frac{dP}{dT} = \frac{L}{T\Delta V}$

where $L$ is the latent heat, and $\Delta V$ is the jump in specific volume or another order parameter. This equivalence between thermodynamic and conservation-law jump conditions is explicit in modern shock-dynamics analyses of phase diagrams (Moro, 2013).

2. Microscopic and Mean-Field Realizations

• Statistical Mechanics and Model Systems: The phase transition curve arises in diverse lattice and continuum models, such as the Blume-Capel model's first-order/second-order boundary and the Fermi Hubbard model's magnetic transition lines (Cone et al., 2010). In mean-field treatments, first-order boundaries exhibit entropy jumps obeying Clausius–Clapeyron relations, while second-order boundaries display continuous (but slope-changing) isentropes.
• Random Graphs and Networks: In exponential random graph models, the phase transition curve is the set of parameter pairs $(\theta_1,\theta_2)$ where two distinct densities maximize the free-energy functional, i.e., the “coexistence curve” $F(\theta_1,\theta_2)=0$ in terms of a latent-order-parameter equality, ending at a critical point with divergent second derivatives (Radin et al., 2011).
• Social and Percolation Systems: In models such as Axelrod cultural dissemination, the phase transition curve is characterized by the probability $p_\mathrm{int}$ that two agents can interact, with the critical value determined by the emergence of a giant component in an associated Erdős–Rényi random graph. The transition curve collapses onto a universal master curve in both complete- and sparse-network settings (Pinto et al., 2019).

3. Universal Features, Topology, and Critical Endpoints

A generic phase transition curve may terminate at a critical endpoint (second-order critical point), as in the van der Waals liquid–gas system, or branch into more complex structures such as a triple point (where three binodals, or “shocks,” coalesce) (Moro, 2013). Near the endpoint, the curve is marked by diverging susceptibility, compressibility, or correlation length, with universal critical exponents.

In complex models (e.g., QCD), the hadronization boundary in the $(n,T)$ plane is a parabola derived from an effective van der Waals EOS, terminating at the critical point where thermodynamic derivatives vanish, and the system transitions from a first-order phase separation to a crossover (Apostol, 2013).

4. Geometric, Dynamical, and Nonperturbative Perspectives

• Thermodynamic Geometry: The phase coexistence curve can be obtained from the equality of Ruppeiner scalar curvature $|R|$ in coexisting phases, reflecting matched correlation volumes. This “R-crossing” generalizes the Maxwell construction and enables precise definition of Widom lines above the critical point, signaled by the maximum in correlation length or curvature (Ruppeiner et al., 2011).
• Nonlocal Entropies/Nonlinear PDEs: The entire coexistence arc may be derived as a shock solution to nonlinear conservation laws for order-parameter fields, with the Rankine–Hugoniot condition equating to the Clausius–Clapeyron slope. Triple points correspond to shock interactions and confluences (Moro, 2013).

5. Phase Transition Curves in Gravitational and Quantum Models

• Black Hole Thermodynamics: In anti–de Sitter black hole families (e.g., Reissner–Nordström–AdS, Kerr–AdS, Horava–Lifshitz, and regular black holes), phase transition curves are defined in the extended thermodynamic phase space as coexisting small/large black hole branches (analogous to liquid–gas). These curves are constructed using the Maxwell equal-area law and are parameterized in $P$–$T$ or $T$–$V$ space. The slope of the coexistence line is given by a generalized Clapeyron equation, and the critical points (or curves, in the case of HL black holes) are determined by inflection conditions in the equation of state (Zhao et al., 2014, Du et al., 2019, Hegde et al., 2020, Javed et al., 2018).
• Glassy Transitions: In particular regular BH models (e.g., Ayón–Beato–García–Bronnikov), phase boundary curves can display “glassy” (thermodynamically non-equilibrium) transitions, evidenced by a finite Prigogine–Defay ratio $2\leq\Pi\leq5$, reflecting discontinuous second derivatives of the Gibbs free energy but continuous first derivatives (Javed et al., 2018).

6. Empirical and Experimental Determination

• Magnetic and Structural Transitions: Empirical phase transition curves are extracted from experimental observables such as magnetization, resistance, or neutron scattering. For example, a structural vortex-lattice transition in BaFe$_2$(As,P)$_2$ is mapped by the locus of minima in relaxation rate and magnetization, and fit to a theoretical rhombic–square phase boundary with experimentally extracted critical exponents and amplitudes (Jr. et al., 2015).
• Nuclear and Mesoscale Matter: In the context of hot, finite nuclear systems, phase transition curves are operationally probed via constrained caloric curves. The appearance of a backbending (negative-slope) caloric curve at fixed pressure, but not at fixed volume, is a definitive signature of a first-order phase transition, consistent with theoretical expectations for finite systems (Borderie et al., 2012).
• First-Order Reversal Curves (FORC): In complex hysteretic transitions (e.g., magnetostructural FeTe), phase transition curves are revealed as ridges or bands in FORC diagrams. These curves encode the distribution of local switching events, activation barriers, and phase nucleation dynamics, making visible otherwise hidden multi-step transitions (Frampton et al., 2017).

7. Phase Transition Curves in Dark Sector and Exotic Systems

Phase transition curves can encode transitions between fundamentally different particle-regimes (e.g., hot to cold dark matter). In the Bound Dark Matter (BDM) paradigm, the phase boundary is specified by the condition $\rho = E_c^4$, with the transition energy $E_c$ extracted from fits to galactic rotation curves. The (core size, transition energy) pairs across galaxies populate a “phase transition curve” in parameter space, pointing to a universal dark-sector scale possibly connected to neutrino masses (Mastache et al., 2013, Macorra et al., 2011).

8. Interpretations, Universality, and Theoretical Implications

Phase transition curves synthesize information about symmetry breaking, latent heat, criticality, and universality in systems ranging from fluids and magnets to quantum black holes and cosmological structures. Their analytic structure is governed by the nonanalyticities of the free energy (or an equivalent variational principle) and by the universality class of the transition. Phase transition curves serve as boundaries between macroscopic regimes in parameter space, marking loci of coexistence, critical singularities, or dynamical thresholds (e.g., percolation and synchronization), and are thus foundational both for theoretical analysis and experimental characterization of complex systems.