Coordinate Condensation

• Coordinate condensation is the controlled localization of condensation phenomena by selecting specific spatial, temporal, or structural coordinates.
• It enhances simulation and modeling by using methods like Schur complement corrections, dimensional reduction, and active matter modulation to boost convergence and tractability.
• Applications range from phase transitions in physics and lattice models in statistical mechanics to astrophysical condensate modeling and structural principles in set theory.

Coordinate condensation refers to diverse phenomena and methodologies across physics, mathematics, and applied sciences in which condensation is controlled, described, or accelerated by focusing on specific coordinates, subspaces, or spatial structures. The terminology appears in contexts ranging from statistical physics models of phase transitions, numerical methods for simulation, and abstraction principles in set theory, to the molecular and planetary scale formation of condensed phases. The unifying theme is the “coordination” or explicit control/selection of active modes, sites, or subspaces where condensation—whether @@@@1@@@@, energy, chemical species, or density—becomes localized or rapidly processed. This article surveys key domains in which coordinate condensation is prominent, explicates representative mathematical formulations, and highlights its practical consequences.

1. Coordinate Condensation in Physics-Based Simulation

Recent work (Trusty, 14 Oct 2025) introduces coordinate condensation as a subspace-accelerated variant of coordinate descent for solving large-scale systems typical in physics-based simulation. In canonical coordinate descent, global minimization proceeds via sequential updates of each degree of freedom, often suffering slow convergence when the system exhibits strong global coupling (e.g., stiff materials with high mesh resolution). Coordinate condensation augments each local coordinate update with a Schur complement-based subspace correction. Specifically, for each coordinate $i$, a perturbation basis $U_i$ is precomputed (using $U_i = -H_{CC}^{-1}H_{Ci}$), and the per-coordinate update is carried out via

$$\begin{bmatrix} H_{ii} & H_{iC}U_i \ U_i^TH_{iC}^T & U_i^TH_{CC}U_i \end{bmatrix} \begin{bmatrix} \delta x_i \ \delta\alpha_i \end{bmatrix} = -\begin{bmatrix} g_i \ U_i^Tg_C \end{bmatrix}$$

Block elimination yields a condensed update:

$$\delta x_i = -\left(H_{ii} - S\right)^{-1}\tilde{g}_i$$

with

$$S = H_{iC} U_i (U_i^T H_{CC} U_i)^{-1} U_i^T H_{iC}^T, \qquad \tilde{g}_i = g_i - H_{iC} U_i (U_i^T H_{CC} U_i)^{-1} U_i^T g_C$$

This “deflation” removes global coupling from the local stiffness, allowing near-Newton convergence rates while maintaining parallelism. Extensive benchmarks demonstrate robust scaling in both mesh refinement and material stiffness, with convergence far outpacing traditional coordinate descent and earlier subspace-coupling approaches such as JGS2.

2. Real-Space Coordination in Mass Condensation Models

In non-equilibrium statistical mechanics, coordinate condensation denotes the spatial localization of conserved quantities in lattice models such as the zero-range and misanthrope processes (Evans et al., 2015, Seo, 2018). Here, “coordinate” refers to a lattice site or region:

• Zero-range process (ZRP): For hop rates $u(m)$, stationary states factorize as $$P(\{m_i\}) \propto \prod_i f(m_i) \delta(M, \sum_j m_j)$$. When $f(m)$ decays as $m^{-\gamma}$ ($\gamma > 2$), a macroscopic fraction condenses onto one site as the density $\rho$ exceeds a critical value $\rho_c$.
• Misanthrope process: Generalizes ZRP via hop rates $u(m,n)$ depending on both departure and destination. Under constraints, the steady state factorizes, or, in discrete-time parallel dynamics, becomes pair-factorized: $$P(\{m_i\}) \propto \prod_i g(m_i, m_{i+1}) \delta(M, \sum_i m_i)$$.
• Metastability and Markov Chain Reduction: For non-reversible dynamics (Seo, 2018), transitions between “condensed coordinates” (occupied sites) can be described as an effective Markov chain whose jump rates are proportional to the capacity between valleys, established via a novel variational inequality that lifts the divergence-free constraint typical in potential theory.

Applications span from traffic flow and wealth distribution to granular clustering, exploiting the analytic tractability of condensation through factorized and pair-factorized steady states.

3. Dimensional Analysis and Surface Tension Effects in Tube Condensation

In the paper of phase transition in thermodynamic systems, coordinate condensation is characterized by the rigorous reduction of governing equations via generalized dimensional analysis (Dziubek, 2011). For slender tubes, mass, momentum, and energy balances are written in cylindrical coordinates and reduced:

• Generalized dimensional analysis assigns different scales to radial and axial coordinates, leading to nondimensional form:

$$\frac{1}{\hat{r}}\frac{\partial}{\partial \hat{r}}\left(\hat{r}\hat{v}_{r}\right) + \frac{\partial\hat{v}_z}{\partial\hat{z}} = 0$$

$$\frac{\partial \hat{p}}{\partial \hat{z}} = -\frac{1}{\varepsilon\mathrm{Fr}}$$

Jump conditions (e.g., Young–Laplace for interface momentum):

$$\hat{p} - \hat{p}_v = -\frac{1}{\mathrm{We}}\frac{1}{\hat{h}}$$

Yield a reduced ODE for condensate film thickness:

$$\frac{d\hat{h}}{d\hat{z}} = \frac{\mathrm{We}}{\varepsilon \mathrm{Fr}}\frac{\hat{h}^2}{2} \pm \sqrt{\left(\frac{\mathrm{We}}{\varepsilon \mathrm{Fr}}\right)^2 \frac{\hat{h}^4}{4} - \cdots}$$

This generalized framework matches classical models (Nußelt, Chen), but enables direct extension to varying tube geometries and incorporation of surface tension effects, improving predictions for heat exchanger design, particularly for fluids such as R134a.

4. Dynamic and Statistical Coordination: Active Matter and Atmospheric Advection

Coordinate condensation extends to dynamical control in active matter systems (Berx, 2023), where the “temporal coordinate” (time under a protocol) directly tunes phase transitions:

• In models with a density-dependent diffusivity and confining potential $U(r,t)$, condensation occurs when peak density reaches a threshold $\rho_c$:

$\phi_c(t) = 1 - [f(t) / f(t, T_c)]^{d/2}$

$$k_BT_c = \frac{\sigma^2}{2} \frac{(\bar{\rho} / \rho_c)^{2/d} e^{4H(t_c,0)} -1}{\mathcal{G}(t_c)}$$

Temporal modulation yields steady-state, growing, or reentrant transitions, the latter causing oscillatory condensate fractions due to periodic driving.

Atmospheric moisture condensation is coordinated by interplay of large-scale advection (coherent stirring), stochastic mixing, and instantaneous local removal (Tsang et al., 2017). Moisture parcels obey stochastic differential equations—advective drying aligns humidity with streamlines, while subsequent stochastic excursions produce gradual further drying, elucidating Hadley-cell-like structures.

5. Condensation Coordination in Chemistry and Astrophysics

The notion extends to planetary science and astrophysical chemistry (Ebel, 2023, Kitzmann et al., 2023):

• Equilibrium Calculations: “Coordinate condensation” refers to the systematic determination of conditions and sequences for element and phase segregation as vapor cools, controlled via coupled nonlinear equations that enforce element conservation, phase activity, and selection by phase rule (Gibbs):

$n_i = K_i(T) \prod_j n_j^{\nu_{ij}}$

$$\ln(a_c) = \ln K_c(T) + \sum_j \nu_{cj} \ln n_j = 0 \;\mathrm{(stable)}$$

$$0 = \ln K_c(T) + \lambda_c + \sum_j \nu_{cj} \ln n_j \;\mathrm{(all)}$$

Astrophysical codes, such as FastChem Cond, iteratively solve for all possible condensates, automatically selecting the stable subset according to the phase rule constraints. Rainout approximations address removal of condensates by gravitational settling, updating gas-phase abundances in stratified planetary atmospheres.

These coordinated computations are essential for modeling planet formation, disk chemistry, and stellar atmospheres, linking micrometer-scale mineral formation to macro-scale astrophysical evolution.

6. Mathematical Coordination: Condensation Principles in Set Theory

In abstract set theory and logic, “coordinate condensation” acquires a structural interpretation. The local club condensation principle (Fernandes, 2021) generalizes condensation properties of the constructible universe $L$ by defining filtrations $(M_\alpha : \alpha < \kappa)$ satisfying continuity, closure, and coherence (e.g., Mostowski collapses holding on a club of ordinals). The equivalence between failure of local club condensation and subcompact cardinals in extender models formally restricts the principle to intervals without certain large cardinals. Forcing notions (Holy–Welch–Wu) allow the preservation of GCH, cardinals, and cofinalities while producing models in which local club condensation, and associated combinatorial principles ($\square_\kappa$, $\diamond(S)$, $\Dl_{S}^{*}(\Pi^{1}_{2})$), hold on prescribed intervals.

7. Structural Duality: Cluster Condensation in Nuclei

Coordinate condensation also manifests in nuclear structure via mathematical equivalence between the generator coordinate Brink cluster model and nonlocalized THSR wave functions (Ohkubo, 2020). Integration over generator coordinates transforms spatially localized (“crystalline”) α-cluster states into coherent, delocalized condensates, establishing a duality central to the observed supersolid properties of nuclei. The Pauli principle induces both geometric localization and wave-like coherence; the latter results in a non-zero condensate fraction and Nambu–Goldstone modes associated with spontaneous symmetry breaking.

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In summary, coordinate condensation encompasses a suite of rigorous, model-dependent mechanisms that leverage spatial, temporal, structural, or algebraic coordination to control condensation phenomena. In physics-based simulation, mathematics, statistical physics, planetary science, and logic, the principles underlying coordinate condensation facilitate both analytic tractability and practical execution of complex condensation and localization processes. The flexibility and breadth of methodologies—from Schur complement correction and equilibrium Gibbs minimization to structural abstraction and combinatorial forcing—underscore its centrality in contemporary research.