First-Order Phase Transition

• First-order phase transitions are defined by a discontinuous jump in order parameters, latent heat release, and phase coexistence.
• They involve nucleation and growth dynamics, where critical droplet formation and hysteresis underpin the transition mechanism.
• Scaling theory and renormalization group analysis integrate FOPT behavior with universal techniques, impacting condensed matter, astrophysics, and cosmology.

A first-order phase transition (FOPT) is a fundamental class of phase transitions characterized by a discontinuity in the first derivative of the free energy with respect to a control parameter (temperature, magnetic field, pressure, or chemical potential). Unlike continuous (second-order) transitions, FOPTs exhibit latent heat, phase coexistence, hysteresis, and nucleation-and-growth dynamics. They play a central role in condensed matter, statistical physics, cosmology, quantum chromodynamics, and astrophysical systems, and have been the subject of recent advances in scaling theory and multi-messenger experimental constraints.

1. Thermodynamic and Field-Theoretic Definition

First-order phase transitions occur when the system's free energy or potential possesses two or more local minima separated by a barrier as a function of an order parameter (e.g., magnetization, density, condensate expectation value). At the equilibrium transition point (critical temperature $T_c$, field $H_c$, or chemical potential $\mu_c$), these minima become degenerate, giving rise to a discontinuous “jump” in the order parameter and a finite latent heat or energy-density jump:

• Latent heat: $L = T_c \Delta S = \Delta E$ at fixed $T_c$.
• Discontinuity in density or other conjugate variables: e.g., $\Delta M$ in magnetic transitions or $\Delta n$ in density-driven transitions (Komoltsev, 8 Apr 2024, Ji et al., 8 Feb 2025).

The canonical Landau-Ginzburg expansion for a FOPT reads

$$F[\phi] = \int d^d x \left[ a(T-T_0)\phi^2 + b\phi^4 + c(\nabla\phi)^2 - h\phi \right]$$

where $T_0$ sets the mean-field spinodal point and $a, b, c > 0$ in the FOPT regime. The transition point is identified where two minima exchange stability (Zhang et al., 31 May 2025).

In dense matter, such as in neutron stars, FOPTs are formally encoded using the Maxwell construction with a pressure plateau at the critical chemical potential $\mu_c$,

$$P(\mu) = \begin{cases} P_H(\mu), & \mu<\mu_c\ P_Q(\mu), & \mu>\mu_c \end{cases}$$

and a finite energy-density jump $$\Delta\epsilon = \epsilon_Q(\mu_c) - \epsilon_H(\mu_c)$$ (Ji et al., 8 Feb 2025). The Gibbs construction can also allow for a mixed phase over a range of densities (Komoltsev, 8 Apr 2024).

2. Microscopic Dynamics: Nucleation, Growth, and Remnants

FOPTs proceed via nucleation of the stable phase within a metastable background, followed by growth of supercritical “droplets” or “bubbles”:

• Nucleation rate for a droplet (field-theoretic/Arrhenius form):

$\Gamma(T) = A(T) \exp[-S_3(T)/T]$

where $S_3(T)$ is the three-dimensional Euclidean bounce action (Lu et al., 2022, Zhong, 2018).

• Critical droplet/barrier: For a radius $r$, the free energy change is

$$\Delta G(r) = -\frac{4}{3}\pi r^3 \Delta g + 4\pi r^2 \gamma$$

yielding a critical radius $r_* = 2\gamma/\Delta g$ and barrier $\Delta G_* = (16\pi/3) \gamma^3 / (\Delta g)^2$ (Choi et al., 2014).

• Avrami law for domain growth and magnetization dynamics,

$$M(t) = M_{\mathrm{eq}} - [M_{\mathrm{eq}} + M_{\mathrm{eq}}^-] \exp\left[ -(t/t_0)^{d+1} \right]$$

where $t_0$ is the nucleation-growth timescale (Zhong, 2018).

• Percolation and false-vacuum remnants: The old-phase region fragments into disconnected remnants at $f_{\mathrm{fv}} \approx 0.29$ volume fraction, with their number and size spectrum derivable from nucleation kinetics and reverse-time formalism (Lu et al., 2022).
• Remnant distribution: Analytical expressions for the remnant spectrum are now available and essential for predicting cosmological relics (e.g., primordial black holes) (Marfatia et al., 22 Jul 2024).

3. Scaling, Universality, and RG Structure

While FOPTs were historically believed not to exhibit scaling laws or universality, recent developments have integrated FOPTs within the renormalization-group (RG) paradigm. The core ideas are:

• Instability fixed point: Expanding the Landau-Ginzburg theory near the spinodal yields an effective cubic ($\phi^3$) field theory, with an imaginary RG fixed point (Yang-Lee class). This nontrivial fixed point yields real, universal “instability exponents” (β, δ, ν, z) despite the absence of criticality in the microscopic sense (Zhong, 2017, Zhong, 2018, Zhang et al., 31 May 2025, Zhong, 30 Mar 2024).
• Finite-time scaling (FTS): Under a linearly varying control parameter (e.g., $H(t) = H_s + R t$), the nonequilibrium response obeys

$$M(H, R; L) - M_s(R) = R^{\beta/(\nu r)}\, f\left( (H - H_s) R^{-\beta\delta/(\nu r)},\, L^{-1} R^{-1/r} \right)$$

with $r = z + \beta\delta/\nu$ and $f$ a universal scaling function (Zhong, 2018, Zhong, 2017, Zhang et al., 31 May 2025, Zhong, 30 Mar 2024).

• Logarithmic corrections: In two dimensions, additional multiplicative logarithmic factors occur due to tunneling, e.g., $(-\ln R)^{-3/2}$ in the scaling variable (Zhong, 2018, Zhong, 2017).
• Universality: The scaling hypothesis, RG exponents, and scaling functions are independent of microscopic parameters (noise amplitude, system size, sweep rate), up to nonuniversal amplitudes. This places FOPTs on the same theoretical footing as continuous transitions (Zhong, 30 Mar 2024, Zhang et al., 31 May 2025).
• Irrelevance of nucleation/growth for scaling: The nucleation growth parameters, though crucial for kinetics, become formally RG-irrelevant at the instability point (Zhong, 2018).

4. Experimental and Numerical Signatures

FOPTs are probed in diverse contexts, from condensed-matter systems to nuclear matter and astrophysics:

• Hysteresis loop scaling: Experiments in magnets and simulations (e.g., in the 2D Ising model) reveal the scaling of the coercive field and magnetization jump as power laws of the sweep rate, in agreement with RG predictions (Zhong, 2018, Zhong, 2017, Zhang et al., 31 May 2025, Zhong, 2017).
• Nano-critical phenomena: In FeRh, a nominally strong FOPT shows “critical-like” growth of domain size and phase fraction over a finite window, governed by interfacial frustration and exhibiting effective exponents (e.g., domain size $L(T) \propto |T - T_t|^{-\nu}$ with $\nu \simeq 1$) (Choi et al., 2014).
• First-order signatures in QCD: In heavy-ion collisions, the construction of the Landau potential from net-baryon-number cumulants allows for discrimination between crossover and FOPT behaviors by analyzing higher-order cumulant ratios and discriminant parameters; precise cumulant measurements near $\sqrt{s_{NN}}=7.7$ GeV may directly reveal the FOPT (Lu et al., 2022).
• Astrophysical consequences: In neutron stars, an FOPT leads to a discontinuous softening of the EoS at high density, producing distinctive signatures in the NS mass-radius relation (twin branches), tidal deformabilities observed by GW detectors, and possible features in post-merger gravitational waves (Ji et al., 8 Feb 2025, Komoltsev, 8 Apr 2024).

5. Cosmological and Quantum Field Applications

First-order phase transitions in the early Universe have profound implications:

• Baryogenesis and Leptogenesis: Strong FOPTs provide the required nonequilibrium conditions for electroweak baryogenesis and for triggering out-of-equilibrium decays of heavy neutrinos (FOPT leptogenesis), where the transition induces a large mass jump for sterile neutrinos, enhancing the efficiency of baryon-number generation (Borah et al., 2023, Huang et al., 2022).
• Gravitational waves: FOPT bubble nucleation and expansion generate a stochastic gravitational-wave background, with the spectrum depending on nucleation rate, wall velocity, and transition strength (α, β/H*) (Borah et al., 2023, Kersten et al., 23 Dec 2024, Gonçalves et al., 11 Jun 2024). Extended bubble size distributions broaden the GW spectrum and shift the peak to lower frequencies compared to the monochromatic case (Marfatia et al., 22 Jul 2024).
• Primordial black holes (PBHs): Supercooled FOPTs can leave false-vacuum remnants that shrink and collapse to form Fermi-balls or PBHs, with the remnant and resulting PBH mass spectra directly calculable from nucleation dynamics and remnant statistics (Lu et al., 2022, Kawana et al., 2021, Marfatia et al., 22 Jul 2024, Gonçalves et al., 11 Jun 2024).
• Bubble spin and cosmological perturbations: Recently, the angular momentum acquired by spherical FV bubbles as a consequence of cosmological perturbations has been computed, leading to a distribution of spins in PBHs formed via FOPTs that depends on the transition rate (β/H), wall velocity (v_w), and sector temperature ratio (Acuña et al., 14 May 2025).

6. Theoretical and Computational Methodologies

• RG analysis: The scaling structure near the instability point is derived from the $\phi^3$ RG fixed point, with the cubic coupling becoming formally imaginary for $d<6$, but all critical exponents remain real (Zhong, 30 Mar 2024).
• Finite-time and finite-size scaling: Scaling forms are established for both the external sweep rate and system size, enabling comprehensive numerical tests via data collapse in simulations (Zhang et al., 31 May 2025, Zhong, 2017, Zhong, 2018).
• Bayesian inference in astrophysics: Large ensembles of EoS models constrained by Gaussian-process priors and astrophysical likelihoods (X-ray masses/radii, tidal deformability, GW signals) yield probabilistic constraints on FOPT location and strength in NS matter (Komoltsev, 8 Apr 2024, Ji et al., 8 Feb 2025).
• Statistical mechanics of remnants: Reverse-time formalism allows calculation of the full size and number distribution of old-phase remnants post-percolation, which is critical for mapping FOPT physics to observational signatures (Lu et al., 2022, Marfatia et al., 22 Jul 2024).
• Multi-messenger synergy: Cosmological, astrophysical, and laboratory probes are jointly sensitive to FOPT characteristics via complementary observables: GW spectra, PBH mass/spin distributions, X-ray/radio signals, and collider signatures (Gonçalves et al., 11 Jun 2024, Ji et al., 8 Feb 2025).

7. Frontier Topics and Ongoing Challenges

• Universality and crossover: While scaling and universality are now established for FOPTs at the instability point, the extent of universality with respect to microscopic or external-parameter variations, and the precise classification of universality classes beyond the 2D Ising or scalar field cases, are active areas.
• Fluctuation-induced “criticality” in nominally strong FOPT: Nano-scale frustration and interfacial wetting effects can induce apparent critical scaling over finite windows, blurring the textbook dichotomy between discontinuous and continuous transitions (Choi et al., 2014).
• Extended distribution effects: Realistic FOPT-induced relics, such as PBHs, have broad non-monochromatic spectra. These effects directly shape the possibility of discovery or exclusion via microlensing, evaporation gamma-ray spectra, and stochastic GW background measurements (Marfatia et al., 22 Jul 2024, Gonçalves et al., 11 Jun 2024).
• Quantum and driven FOPTs: Dynamics in low-dimensional, strongly quantum, or highly driven systems may involve novel scaling behavior and stochasticity, extending beyond current RG-based frameworks.
• Experimental discrimination in QCD and NS matter: Distinguishing FOPT from rapid crossovers or “destabilizing” transitions in QCD and dense-matter systems requires high-precision fluctuation measurements or multi-modal constraints, with ongoing efforts in heavy-ion and neutron-star observations (Lu et al., 2022, Ji et al., 8 Feb 2025, Komoltsev, 8 Apr 2024).

---

References:

• (Choi et al., 2014) Critical phenomena of nano phase evolution in a first order transition.
• (Zhong, 2017) Compelling evidence for the theory of dynamic scaling in first-order phase transitions.
• (Zhong, 2017) Universal scaling in first-order phase transitions mixed with nucleation and growth.
• (Zhong, 2018) Universal scaling in first-order phase transitions mixed with nucleation and growth.
• (Lu et al., 2022) Old Phase Remnants in First Order Phase Transitions.
• (Lu et al., 2022) Revealing the Signal of QCD Phase Transition in Heavy-Ion Collisions.
• (Borah et al., 2023) LIGO-VIRGO constraints on dark matter and leptogenesis triggered by a first order phase transition at high scale.
• (Zhong, 30 Mar 2024) Complete universal scaling in first-order phase transitions.
• (Komoltsev, 8 Apr 2024) First-order phase transitions in the cores of neutron stars.
• (Gonçalves et al., 11 Jun 2024) Primordial Black Holes from First-Order Phase Transition in the xSM.
• (Marfatia et al., 22 Jul 2024) Phenomenology of bubble size distributions in a first-order phase transition.
• (Ji et al., 8 Feb 2025) Observational and Theoretical Constraints on First-Order Phase Transitions in Neutron Stars.
• (Acuña et al., 14 May 2025) Angular momentum of vacuum bubbles in a first-order phase transition.
• (Zhang et al., 31 May 2025) Complete universal scaling of first-order phase transitions in the two-dimensional Ising model.