Density-Phase Formalism Overview

• Density-phase formalism is a family of approaches that encode phase information either explicitly through a density-phase (Madelung) decomposition or implicitly via amplitude and response-based representations.
• It spans various models including hydrodynamic decompositions, phase-field-crystal formulations, and quantum density reformulations where phase details are embedded in complex bilinears and retarded kernels.
• The framework underpins applications in materials science, quantum theory, and phase-space dynamics by clarifying how state identity, elasticity, and coherence are encoded in density-centered theories.

Searching arXiv for papers relevant to “density-phase formalism” and closely related usages. Density-phase formalism is not a single universally standardized construct across contemporary theoretical physics. In the literature represented here, the term covers at least three distinct usages: first, formalisms in which material phase identity is encoded directly in a density field; second, hydrodynamic or phase-field theories in which density is coupled to an explicit phase variable or solid fraction; and third, broader density-centered reformulations that retain phase information only implicitly through amplitudes, response functions, quasidistributions, or temporal operator structure. Some works are therefore genuine density-phase theories in a strict sense, while others are explicitly presented as adjacent but non-Madelung alternatives (Kocher et al., 2014, Toth et al., 2019, Huang et al., 2010, Sutherland, 2020, Blanchard et al., 2010).

1. Terminological scope and core distinctions

The strict hydrodynamic meaning of a density-phase formalism is the decomposition of a wavefunction into density and phase, typically of the form $\psi=\sqrt{\rho}\,e^{iS/\hbar}$. Several papers discussed here state explicitly that they do not adopt that structure. Sutherland’s density formalism does not introduce a local phase field $S(\mathbf x,t)$ or velocity potential, and it instead defines observable densities bilinearly from an initial and a final wavefunction (Sutherland, 2020). Mosquera’s reformulation of time-dependent density-functional theory likewise does not introduce a density-phase decomposition, but argues that exact density dynamics still requires causal memory, contour ordering, and initial-state dependence (Mosquera, 2013). The spacetime density matrix formalism is again not an amplitude/phase construction; it is an operator-valued object encoding correlations among different Cauchy surfaces (Guo, 28 Aug 2025).

By contrast, some materials theories use “phase” in a different sense: not the phase of a wavefunction, but the identity of material states such as solid, liquid, and vapor. In that setting, phase can be encoded directly in a density field or in complex structural amplitudes derived from density waves. This suggests that the phrase “density-phase formalism” is polysemous: in one branch it denotes a local density-plus-phase decomposition, while in another it denotes a density-based representation of phase behavior or crystalline order (Kocher et al., 2014, Huang et al., 2010).

A separate source of ambiguity is the phrase “random phase approximation.” The range-separated ACFD/RPA formalism of Toulouse and coauthors is explicit that this is not a wavefunction-phase formalism. Its language is instead that of density functionals, one-particle Green functions, self-energies, polarization propagators, response kernels, fluctuation-dissipation relations, and orbital-basis response matrices (Toulouse et al., 2010).

2. Single-density and amplitude-based phase descriptions in materials theory

A strict density-based phase formalism appears in phase-field-crystal work on pure materials. Kocher and Provatas formulate a single-order-parameter theory in which the dimensionless density deviation

$$n(\mathbf r)=\frac{\rho(\mathbf r)-\bar\rho}{\bar\rho}$$

distinguishes all three equilibrium phases: solid corresponds to a periodic $n(\mathbf r)$, liquid to a nearly uniform state with intermediate mean density, and vapor to a nearly uniform state with significantly lower mean density (Kocher et al., 2014). In this construction, phase identity is not assigned through separate labels or nonconserved order parameters. It emerges from the structure and mean level of one density field. The free energy combines a local polynomial term, a nonlocal two-point structural kernel, and effective higher-order long-wavelength correlations $\chi^{(3)}$ and $\chi^{(4)}$, which are introduced to stabilize vapor-liquid separation while retaining crystalline ordering. Coexistence is determined by equal chemical potential and equal pressure, implemented through common-tangent constructions, and the dynamics is conserved and diffusive: $$\frac{\partial n}{\partial t}= \Gamma \nabla^2\left(\frac{\delta\mathcal{F}}{\delta n}\right) + N_a\eta.$$ This yields a one-field model with solid-liquid coexistence, solid-vapor coexistence, vapor-liquid coexistence, and a triple point (Kocher et al., 2014).

A related but more explicitly amplitude-based density-phase formalism is developed for binary PFC systems by Elder, Huang, and Provatas. Starting from DDFT, they introduce total-density and concentration variables,

$$n=\frac{\rho_A+\rho_B-\rho_l}{\rho_l}, \qquad \psi=\frac{\rho_A-\rho_B}{\rho_A+\rho_B},$$

and then derive a slow-modulation representation of the crystalline density field (Huang et al., 2010). Near weak crystallization, the density takes the form

$$n(\mathbf r,t)=n_0(\mathbf R,T)+\sum_j A_j(\mathbf R,T)e^{i\mathbf q_j^0\cdot \mathbf r}+\text{c.c.},$$

with slowly varying average density $n_0$, concentration $S(\mathbf x,t)$0, and complex amplitudes $S(\mathbf x,t)$1. In this framework, the amplitudes are the phase-carrying variables: $S(\mathbf x,t)$2 measures local crystalline order, while $S(\mathbf x,t)$3 encodes lattice translation and deformation. The paper writes

$S(\mathbf x,t)$4

so that the phase of $S(\mathbf x,t)$5 is directly identified with a displacement field $S(\mathbf x,t)$6 (Huang et al., 2010). This is a genuine density-phase formulation in the sense that density, concentration, amplitude, and elastic phase are all retained as coupled mesoscopic fields.

The significance of these PFC constructions is that they make phase behavior and elasticity accessible within one density-centered variational structure. In the single-field case, periodicity and mean density separate solid from fluid and vapor. In the amplitude case, the crystalline phase becomes a slow field variable whose phase component carries translational and elastic information. A plausible implication is that the materials-science meaning of “density-phase formalism” is often closer to “phase encoded by density morphology and amplitude phases” than to Madelung hydrodynamics.

3. Hydrodynamic density-phase coupling in solidification and thermo-density evolution

A more literal density-phase coupling appears in hydrodynamic phase-field theory for single-component solidification with density change. Tóth, Tegze, and Gránásy define the phase field exactly as the local solid mass fraction,

$S(\mathbf x,t)$7

with total density $S(\mathbf x,t)$8 and velocity $S(\mathbf x,t)$9 (Toth et al., 2019). In the fully compressible theory, $$n(\mathbf r)=\frac{\rho(\mathbf r)-\bar\rho}{\bar\rho}$$0 and $$n(\mathbf r)=\frac{\rho(\mathbf r)-\bar\rho}{\bar\rho}$$1 are independent but coupled through a free energy $$n(\mathbf r)=\frac{\rho(\mathbf r)-\bar\rho}{\bar\rho}$$2. The basic equations are

$$n(\mathbf r)=\frac{\rho(\mathbf r)-\bar\rho}{\bar\rho}$$3

A major structural point is that the momentum equation must contain the phase-coupled force $$n(\mathbf r)=\frac{\rho(\mathbf r)-\bar\rho}{\bar\rho}$$4 so that the reversible force density is the divergence of a reversible stress tensor. This is the thermodynamic consistency condition of the model (Toth et al., 2019).

The quasi-incompressible reduction then constrains density to be an explicit function of phase,

$$n(\mathbf r)=\frac{\rho(\mathbf r)-\bar\rho}{\bar\rho}$$5

which removes sound waves while retaining density change across a moving interface. The resulting equations are

$$n(\mathbf r)=\frac{\rho(\mathbf r)-\bar\rho}{\bar\rho}$$6

Within this framework, the special coupling

$$n(\mathbf r)=\frac{\rho(\mathbf r)-\bar\rho}{\bar\rho}$$7

with $$n(\mathbf r)=\frac{\rho(\mathbf r)-\bar\rho}{\bar\rho}$$8 and $$n(\mathbf r)=\frac{\rho(\mathbf r)-\bar\rho}{\bar\rho}$$9 is singled out because it leaves the equilibrium planar phase-field profile unchanged: $n(\mathbf r)$0 It also yields the propagating-front velocity

$n(\mathbf r)$1

so that shrinkage ($n(\mathbf r)$2) decelerates the front and expansion ($n(\mathbf r)$3) accelerates it (Toth et al., 2019).

Thermo-density coupling in PFC adds another layer to density-phase modeling. Majaniemi, Nonomura, and Provatas derive a framework with conserved internal energy and conserved density, beginning from

$n(\mathbf r)$4

and extending it to a functional entropy variation (Kocher et al., 2018). In the PFC specialization, temperature is promoted from a control parameter to a dynamical field, and the paper derives the minimal thermal equation

$n(\mathbf r)$5

with an optional external extraction term $n(\mathbf r)$6 in simulations. The term involving $n(\mathbf r)$7 acts as a latent-heat source or sink generated by local density rearrangements. This produces a coupled density/temperature formalism rather than a density-plus-wavefunction-phase formalism, but it remains squarely within the family of density-centered phase theories (Kocher et al., 2018).

4. Quantum density-based reformulations without an explicit local phase field

Sutherland’s density formalism for quantum theory is density-based, but not density-phase in the Madelung sense. The central object is a bilinear observable density constructed from an initial wavefunction $n(\mathbf r)$8 and an independent final wavefunction $n(\mathbf r)$9: $\chi^{(3)}$0 The theory defines mass density, charge density, momentum density, energy density, current density, and energy-momentum density in this form, and it extends to many-particle, Dirac, propagator, and QFT settings (Sutherland, 2020). The paper is explicit that this is not a local $\chi^{(3)}$1 decomposition: there is no Hamilton-Jacobi equation, no quantum potential, no velocity field $\chi^{(3)}$2, and no explicit phase variable $\chi^{(3)}$3. Instead, phase information remains encoded implicitly in complex bilinears such as $\chi^{(3)}$4.

Mosquera’s action formalism for TDDFT reaches a related conclusion from a different direction. The density remains the basic variable, but the original Runge-Gross action must be reformulated on the Keldysh contour to avoid the causality paradox (Mosquera, 2013). The resulting theory shows that exact density-based dynamics is not determined by instantaneous density alone. The exchange-correlation potential satisfies a causal variational equation,

$\chi^{(3)}$5

so the missing dynamical information enters through retarded kernels, initial-state dependence, and contour-time ordering rather than through an explicit phase field (Mosquera, 2013). This suggests that some of what hydrodynamic language would treat as “phase information” can reappear as functional memory.

The spacetime density matrix extends density-matrix ideas across time rather than across amplitude and phase. It is defined as an operator $\chi^{(3)}$6 on $\chi^{(3)}$7 such that

$\chi^{(3)}$8

and the formalism generalizes to multiple Cauchy surfaces, admits a Schwinger-Keldysh path-integral representation, and obeys a Liouville-von Neumann-type equation of motion (Guo, 28 Aug 2025). Reduced spacetime density matrices become non-Hermitian for causally connected subsystems. This is not a density-phase decomposition, but it is a density-centered reorganization of temporal quantum information.

5. Phase-space, response, and matrix-valued extensions

Phase-space density-functional theory replaces configuration-space density by the one-body reduced Wigner quasidensity $\chi^{(3)}$9. The basic marginals are

$\chi^{(4)}$0

so the formalism unifies spatial density and momentum density in a single object (Blanchard et al., 2010). The theory establishes exact Hohenberg-Kohn-Levy-type results on phase space, formulates the universal interaction functional by constrained search, and introduces natural Wigner orbitals through the Moyal-eigenvalue equations

$\chi^{(4)}$1

This is not a local density-plus-phase theory, but it retains momentum and coherence information that a pure density theory loses. In that sense it supplies a density-momentum-coherence alternative to hydrodynamic phase variables (Blanchard et al., 2010).

A different extension is the classical density matrix formulation of dynamical systems. Here the unnormalized density matrix is built from tangent-space perturbations,

$\chi^{(4)}$2

and it evolves according to

$\chi^{(4)}$3

where $\chi^{(4)}$4 and $\chi^{(4)}$5 are the symmetric and antisymmetric parts of the stability matrix (Das et al., 2021). The determinant sector produces a generalized Liouville equation for $\chi^{(4)}$6. This is again not a scalar density-phase formalism, but it replaces scalar density transport by matrix-valued geometric transport, separating stretching/compression from rotation-like dynamics.

In electronic-structure theory, the range-separated ACFD/RPA framework is best classified as a density-response formalism rather than a density-phase theory. The exact energy is partitioned into an RSH reference plus a long-range correlation term, the long-range interaction is defined by

$\chi^{(4)}$7

and the exact formal development proceeds through Green functions, Dyson equations, polarization propagators, response kernels, and fluctuation-dissipation identities (Toulouse et al., 2010). The central response quantity is the four-point polarization propagator $\chi^{(4)}$8, and practical RPA or RPAx schemes arise by approximating the long-range self-energy. The paper states explicitly that this is not a wavefunction-phase formalism. “Random phase” is historical nomenclature for a response approximation, not a decomposition into density and phase (Toulouse et al., 2010).

6. Spectral and finite-size notions of phase

The coherent-state treatment of the Mermin central-spin model offers yet another meaning of density-phase formalism. In the thermodynamic limit of a monochromatic, symmetrically coupled spin bath, the bath is represented by a spin coherent state $\chi^{(4)}$9, wavefunctions become Majorana polynomials in $$\frac{\partial n}{\partial t}= \Gamma \nabla^2\left(\frac{\delta\mathcal{F}}{\delta n}\right) + N_a\eta.$$0, and the logarithmic derivative

$$\frac{\partial n}{\partial t}= \Gamma \nabla^2\left(\frac{\delta\mathcal{F}}{\delta n}\right) + N_a\eta.$$1

encodes the distribution of roots (Garmon et al., 2010). In the large-$$\frac{\partial n}{\partial t}= \Gamma \nabla^2\left(\frac{\delta\mathcal{F}}{\delta n}\right) + N_a\eta.$$2 limit, the problem reduces to a Riccati-like equation for $$\frac{\partial n}{\partial t}= \Gamma \nabla^2\left(\frac{\delta\mathcal{F}}{\delta n}\right) + N_a\eta.$$3, and the integrated density of states is obtained by contour integration of the branch-cut discontinuity: $$\frac{\partial n}{\partial t}= \Gamma \nabla^2\left(\frac{\delta\mathcal{F}}{\delta n}\right) + N_a\eta.$$4 The phase diagram splits into four regions, distinguished by symmetric versus broken-symmetry behavior and by overlapping versus non-overlapping energy surfaces. Here “phase” refers to symmetry and spectral topology, while “density” enters through the density of states and the condensed distribution of Majorana roots (Garmon et al., 2010).

A different finite-size phase formalism is developed through the pole structure of the isobaric partition in the generalized statistical multifragmentation model. After Laplace-Fourier transformation, the finite-volume grand canonical partition becomes a sum over simple poles $$\frac{\partial n}{\partial t}= \Gamma \nabla^2\left(\frac{\delta\mathcal{F}}{\delta n}\right) + N_a\eta.$$5, each satisfying

$$\frac{\partial n}{\partial t}= \Gamma \nabla^2\left(\frac{\delta\mathcal{F}}{\delta n}\right) + N_a\eta.$$6

The real part of a pole defines the free energy of the corresponding state, and the imaginary part defines the inverse decay or formation time (Bugaev et al., 2011). In this framework, a finite-volume gaseous phase corresponds to a single real pole, a mixed phase to the real pole plus one or more pairs of complex conjugate poles, and a liquid phase to an infinite family of poles accumulating near the liquid singularity as the size cutoff grows. This is a rigorous formalization of phase in finite systems, but not a density-plus-phase-variable theory.

Together these examples show that “phase” can denote symmetry sectors, spectral topology, or finite-volume collective states rather than a conjugate variable to density. A plausible implication is that the phrase “density-phase formalism” must always be interpreted contextually: the word “phase” may refer to material phase, wavefunction phase, response approximation, or thermodynamic state classification.

7. Conceptual boundaries and recurring misconceptions

The most persistent misconception is to treat all density-centered formalisms as if they were hydrodynamic density-plus-phase theories. The record assembled here does not support that equivalence. In materials models such as single-field PFC and quasi-incompressible solidification, phase usually means solid, liquid, vapor, or solid fraction, and density is the field that carries the distinction (Kocher et al., 2014, Toth et al., 2019). In binary PFC amplitude theory, phase becomes the argument of complex structural amplitudes and therefore a measure of lattice translation and strain, not a quantum-mechanical action field (Huang et al., 2010).

A second misconception is to read “phase” literally into RPA. The range-separated ACFD formalism explicitly rejects such a reading: its central objects are $$\frac{\partial n}{\partial t}= \Gamma \nabla^2\left(\frac{\delta\mathcal{F}}{\delta n}\right) + N_a\eta.$$7, $$\frac{\partial n}{\partial t}= \Gamma \nabla^2\left(\frac{\delta\mathcal{F}}{\delta n}\right) + N_a\eta.$$8, $$\frac{\partial n}{\partial t}= \Gamma \nabla^2\left(\frac{\delta\mathcal{F}}{\delta n}\right) + N_a\eta.$$9, $$n=\frac{\rho_A+\rho_B-\rho_l}{\rho_l}, \qquad \psi=\frac{\rho_A-\rho_B}{\rho_A+\rho_B},$$0, $$n=\frac{\rho_A+\rho_B-\rho_l}{\rho_l}, \qquad \psi=\frac{\rho_A-\rho_B}{\rho_A+\rho_B},$$1, and orbital-basis matrices $$n=\frac{\rho_A+\rho_B-\rho_l}{\rho_l}, \qquad \psi=\frac{\rho_A-\rho_B}{\rho_A+\rho_B},$$2 and $$n=\frac{\rho_A+\rho_B-\rho_l}{\rho_l}, \qquad \psi=\frac{\rho_A-\rho_B}{\rho_A+\rho_B},$$3, and the “random phase approximation” is a response approximation in many-body theory rather than a wavefunction-phase representation (Toulouse et al., 2010).

A third recurring issue concerns quantum density reformulations that do not introduce $$n=\frac{\rho_A+\rho_B-\rho_l}{\rho_l}, \qquad \psi=\frac{\rho_A-\rho_B}{\rho_A+\rho_B},$$4 explicitly. Sutherland’s two-state density formalism, TDDFT action theory on the Keldysh contour, and the spacetime density matrix all preserve density as a primary or generalized variable while relocating phase-sensitive information into complex bilinears, retarded kernels, contour ordering, or non-Hermitian temporal operators (Sutherland, 2020, Mosquera, 2013, Guo, 28 Aug 2025). This suggests that the presence or absence of an explicit phase field is not the only way to classify density-based theories. One must also ask where coherence, momentum, memory, or temporal ordering is stored.

Finally, phase-space and matrix-valued approaches indicate that density-phase questions can be reframed as questions about what additional structure supplements density. In Wigner DFT the supplement is momentum-resolved quasidensity $$n=\frac{\rho_A+\rho_B-\rho_l}{\rho_l}, \qquad \psi=\frac{\rho_A-\rho_B}{\rho_A+\rho_B},$$5; in classical density-matrix transport it is the matrix geometry of tangent-space deformation; in coherent-state spin models it is the branch-cut topology of $$n=\frac{\rho_A+\rho_B-\rho_l}{\rho_l}, \qquad \psi=\frac{\rho_A-\rho_B}{\rho_A+\rho_B},$$6 and the resulting density of states (Blanchard et al., 2010, Das et al., 2021, Garmon et al., 2010). Density-phase formalism is therefore best understood not as one doctrine, but as a family of density-centered representations that differ in how phase, order, deformation, or state identity is encoded.