Entropy-Driven Phase Transition

• Entropy-driven phase transition is a phenomenon where a system undergoes a macroscopic state change primarily due to an increase in entropy rather than energy minimization.
• It manifests in diverse settings such as antiferromagnetic Potts models, colloidal suspensions, polymer networks, and quantum systems, highlighting nontraditional order parameters like configurational and information entropy.
• The transitions are identified by clear entropy-based signatures—including discontinuities, minima, and singularities in entropy observables—offering new insights into self-organization and critical phenomena.

An entropy-driven phase transition is a qualitative change in the macroscopic state of a system triggered primarily by entropy maximization rather than energy minimization. In these scenarios, the system organizes or reorganizes due to the increase in accessible microstates—i.e., configurational, combinatorial, or information entropy—leading to transitions that are not energetically preferred at zero temperature. Entropy-driven transitions are observed in a wide range of systems: statistical mechanics models (e.g., antiferromagnetic Potts models, hidden-state Potts models), soft matter (colloids, polymer networks), out-of-equilibrium networks, and even cosmological and quantum information contexts. Their defining signature is that entropy, broadly construed, is the dominant order parameter or driving force, either by shaping the free energy landscape or by directly identifying the transition point via singularities or extrema in entropy-related observables.

1. Fundamental Principles and Theoretical Framework

Entropy-driven phase transitions contrast sharply with conventional energy-driven transitions. In classic thermodynamics, a first- or second-order transition is characterized by singularities in derivatives of the free energy with respect to an ordering field or temperature, typically reflecting an underlying energy minimization: examples include ferromagnetic ordering, Bose-Einstein condensation, and crystallization. However, when entropy differences between candidate phases are substantial, the entropic term $-TS$ in the free energy $F=U-TS$ (or $G=H-TS$) may become dominant and favor a macroscopic state that is energetically less favored but has vastly more microstates.

Explicitly, in a system with possible phases $\alpha$ and $\beta$, the equilibrium phase boundary for temperature-driven transitions is determined by

$$\Delta G_{\alpha\to\beta}(T) = \Delta U - T\Delta S = 0,$$

where $\Delta U = U_\beta - U_\alpha$ and $\Delta S = S_\beta - S_\alpha$. At low $T$, energy dominates; at higher $T$, entropy can drive $\Delta G$ negative even if $\Delta U > 0$ (Chen et al., 2019, Kollwitz et al., 18 Aug 2025). In model systems, this is observed, for example, when mixed or disordered phases with high combinatorial entropy become more stable than ordered, low-entropy ground states.

In non-equilibrium and information-theoretic settings, entropy-driven transitions are characterized by changes in information content or complexity, rather than only thermodynamic entropy. For example, in the Ginzburg–Landau field theory, the configurational entropy (CE), defined from the fluctuation spectrum’s Shannon entropy, sharply minimizes at the critical point, indicating maximal information storage and spatial self-organization (Sowinski et al., 2016).

2. Distinct Classes and Manifestations

Entropy-driven transitions manifest in diverse physical settings, with mechanisms and observable signatures determined by system-specific entropy sources. Table 1 summarizes archetypal classes:

System/Model	Entropic Mechanism	Observable Transition Signature
Antiferromagnetic Potts models	Order-by-disorder (degeneracy lifting)	Spontaneous magnetization, long-range order at $T>0$
Colloids/hard particles	Excluded-volume, depletion entropy	Crystal/plastic ordering, micellization at high densities
Polymer networks	Combinatorial bond entropy	First-order network/gelling transition, phase coexistence
Multi-phase magnetic systems	Soft-mode entropy	Crossing of free energies, entropy jump at $T_c$
Information-theoretic field models	Configurational/information entropy	Minimum CE at criticality, multi-scale scaling
Coevolutionary networks	Group-size diversity entropy	Maximum entropy, abrupt fragmentation/consensus transitions
Quantum entanglement spectra	Spectral entropy, eigenvalue statistics	Continuous/condensation transitions in spectrum distribution
Nonequilibrium stochastic systems	Trajectory/entropy production large deviations	Dynamical phase coexistence, first-order–like singularities
Cosmology (entropic gravity)	Horizon/internal entropy dynamics	Divergence of specific heat, kinematic phase transition curve

The mechanisms, mathematical frameworks, and numerical/experimental observables differ across these classes. In configurationally or combinatorially rich models (colloids, polymers, Potts with hidden states), entropy is computable from state counting or information-theoretic functionals. In dynamical or information-driven settings, entropy may quantify mode complexity or trajectory statistics rather than thermodynamic microstates.

3. Representative Models and Mathematical Structures

3.1 Lattice Models: Potts, Antiferromagnets, Hidden States

In low-temperature antiferromagnetic Potts models on bipartite lattices, “order-by-disorder” emerges: ordering one sublattice increases the entropy (number of allowed colorings) on the other, stabilizing a symmetry-broken ordered phase not favored by energetics alone. This is rigorously established for the 3-state model on diced and quadrangulated lattices via Peierls contour and random-cluster representations (Kotecký et al., 2012). The key point is that entropy terms effectively induce a ferromagnetic coupling through sublattice degeneracy.

For Potts models with additional “hidden” spin states, only the visible states interact; the hidden states simply increase the entropy. The mean-field Ginzburg–Landau free energy (Kim et al., 16 Jan 2024) demonstrates that increasing the ratio $x=r/q$ of hidden to visible states generically separates the continuous (second-order) and discontinuous (first-order) transition lines, with a tricritical point at which thermodynamic singularities merge. The analytic phase diagram is determined by entropy-induced changes in the free-energy expansion coefficients.

3.2 Entropy-Driven Ordering in Soft Matter

In dense suspensions of hard particles (spheres, rods, superballs), the competition between configurational and excluded-volume entropy produces first-order transitions to crystalline or plastic (rotator) phases. Classical nucleation theory quantitatively describes the critical nucleus size and free-energy barrier solely in terms of entropy and geometry, absent significant cohesive energy (Ni, 2012). Similarly, in associative polymer networks with alternating “stickers,” combinatorial gain from inter-polymer bond formation can drive a sharp first-order gelation transition at high density (Rovigatti et al., 2023).

Depletion-induced micellization of asymmetric colloidal dumbbells, as well as nematic–isotropic transitions in liquid crystals mixed with non-mesogenic impurities, provide further evidence of entropic order-disorder transitions in complex fluids (Jana et al., 2019).

3.3 Information-Theoretic Models

Configurational entropy (CE), defined as the Shannon entropy of the Fourier amplitude distribution of field fluctuations, acts as an “order parameter in information space” (Sowinski et al., 2016). Near criticality, the system’s spatial complexity is minimized, indicating maximal compressibility and predictability. The CE’s wavenumber-resolved density exhibits universal scaling regimes (scale-free, turbulent, critical), and its global minimum at $T_c$ provides a direct informational signature of the phase transition.

Non-equilibrium settings reveal entropy-driven dynamical transitions via the large deviations of entropy production. In driven kinetic networks, the scaled cumulant generating function (SCGF) $\psi(\lambda)$ develops a cusp, corresponding to a linear “tie-line” in the large deviation function $I(\sigma)$, and is physically associated with coexisting “phases” of trajectory classes (e.g., delocalized vs. localized entropy generation) (Vaikuntanathan et al., 2013).

3.4 Entropic Transitions in Quantum and Cosmological Contexts

Purely entropic mechanisms appear in quantum systems: the distribution of the entanglement spectrum under constraints of fixed von Neumann entropy displays two continuous phase transitions, corresponding to changes in the small- and large-eigenvalue behaviors and controlled entirely by entropy, not energy (Facchi et al., 2013). In cosmology, augmenting gravitational dynamics with entropic source terms or considering entropy on evolving horizons yields novel formulation of phase transitions, where, e.g., the divergence of specific heat at constant volume is entirely a function of the deceleration and jerk kinematics, with no energy-based singularity required (Chakrabarti, 19 Mar 2024).

4. Signatures, Observables, and Scaling Behavior

Entropy-driven transitions are characterized by direct entropy-based signatures: discontinuities, extrema, or singularities in entropy or its density, corresponding precisely to the phase boundary or critical point. Key manifestations include:

• Minimum of configurational entropy: As in the Ginzburg–Landau field model, CE exhibits a pronounced minimum at criticality, with three distinct $k$-scaling regimes: scale-free ($s_C(k)\sim k^0$), turbulent ($s_C(k)\sim k^{-5/3}$), and critical ($s_C(k)\sim k^{-7/4}$). The minimum marks maximal spatial information storage and compressibility (Sowinski et al., 2016).
• Peak in group-size entropy: In coevolving opinion networks, the entropy of the cluster-size distribution maximizes precisely at critical plasticity, reflecting maximal diversity of group structures and identifying the consensus-fragmentation transition (Burgos et al., 2014).
• Large-deviation “tie-lines” in entropy production: The appearance of linear segments in $I(\sigma)$, nonanalyticities in $\psi(\lambda)$, and plateaus in the $\lambda$-biased mean entropy production are dynamical analogs of equilibrium first-order transitions, but driven by rare, entropy-fluctuating trajectories (Vaikuntanathan et al., 2013).
• Discontinuities in microcanonical entropy curve: In the $q=3$ Potts model, the detachment and merging of entropy branches correspond to first- and second-order transitions between cluster configurations; the microcanonical approach reveals entropy-driven crossings invisible to canonical ensemble observables (Ferreira et al., 2022).
• Entanglement and Rényi entropies as disorder indicators: In directed percolation and large random pure states, rare configurations (absorbing states or spectral condensation) dominate the entropy and drive nonanalytic changes, such as cusps or jumps, at phase boundaries (Harada et al., 2019, Facchi et al., 2013).

5. Broader Implications, Universality, and Applications

Entropy-driven phase transitions reveal fundamental principles of self-organization, order–disorder phenomena, and collective behavior beyond classical energy-based paradigms. The central insights include:

• Universality of entropic control: Across disparate model systems, entropy maximization alone governs the transition—even in the absence of significant energetic differences or explicit Hamiltonians. This extends to Maxwell relations in quantum and electronic systems where, for instance, entropy per particle acts as a direct fingerprint of Fermi surface topology and band inversions (Galperin et al., 2018).
• Information-theoretic vs. energetic order parameters: Especially in complex networks, adaptive systems, and field-theoretic approaches, entropy (in various guises) quantifies multi-scale organization, predictability, and the ability to encode, compress, and process information. CE and related information entropies offer new order parameters complementary to traditional symmetry-breaking ones (Sowinski et al., 2016).
• Experimental and computational diagnostics: Entropy-driven transitions can be detected by calorimetry (measuring $\Delta S_{orb}$ in correlated oxides), fluctuations (cluster-size distributions in soft matter), and information-theoretic analysis (entropy per particle in 2D crystals, compressibility or complexity in data streams). Monte Carlo and Wang–Landau methods that directly probe the density of states or entropy functionals have emerged as indispensable tools (Ferreira et al., 2022, Desgranges et al., 2020).

Potential applications arise in material science (delta-temperatural transport in nickelates for temperature-sensing devices (Chen et al., 2019)), soft-matter engineering (stabilization of ordered/disordered phases by tuning entropy sources (Ni, 2012, Rovigatti et al., 2023)), network and opinion dynamics (identifying critical fragmentation for consensus modeling (Burgos et al., 2014)), and quantum information theory (structural transitions in entanglement spectra and resource architectures (Facchi et al., 2013)).

6. Open Directions and Universal Phenomenology

Current research addresses unifying connections between entropy-driven transitions across scales and disciplines. Key avenues include:

• Formulating entropy-centered universality classes and scaling theories for nonequilibrium, adaptive, or quantum-entangled systems, distinct from energy-centric frameworks.
• Generalizing information-theoretic order parameters to capture transitions in high-dimensional or complex systems where no conventional macroscopically ordered state exists.
• Investigating the detailed interplay between entropy-driven and energy-driven transitions, such as hybrid or successive transitions in Potts and percolation models with added entropy sources (hidden states) (Kim et al., 16 Jan 2024).
• Exploring the fundamental and practical implications of entropy-driven criticality for self-organizing computation, synthetic materials, and cosmological evolution.

The entropy-driven transition paradigm continues to expose new facets of criticality, organization, and information transfer wherever the balance of microstate accessibility and macroscopic constraints dictates emergent behavior. It provides a coherent framework for phenomena previously interpreted in strictly energetic or order-parameter terms, with rigorous analytic tools and experimental accesses now widely established.