Growth Equilibrium in Complex Systems

• Growth Equilibrium is a concept where growth, accumulation, or expansion is balanced by threshold phenomena, optimality principles, or equilibrium conditions.
• It spans diverse fields like statistical mechanics, materials science, and economics, often characterized by control parameters such as inverse temperature or mass density.
• Analytical approaches include spectral analysis, eigenfunction scaling, and dynamic game theory to understand transitions between equilibrium absorption and non-equilibrium growth.

Growth equilibrium is a recurrent but domain-specific concept for systems in which growth, accumulation, coarsening, or expansion is constrained by an equilibrium condition, an optimality principle, or a threshold phenomenon. In the cited literature it does not denote a single universal formalism. Instead, it appears as KMS equilibrium in quasi-lattice ordered monoids, stationary or critical profiles in growth-fragmentation and exchange-driven growth, direct equilibrium growth windows in multicomponent self-assembly, equilibrium-like steady states in economic growth-and-exchange models, fixed-point equilibria in dynamic growth games and mean-field growth models, and self-regulated saturated states in planetary accretion. A common structure is the presence of a control parameter—such as inverse temperature, mass density, chemical potential, growth bias, or externality intensity—that separates equilibration from condensation, kinetic trapping, or continued nonequilibrium growth (Bruce et al., 2017, Balagué et al., 2012, Schlichting, 2018, Liu et al., 2021, Lavigne et al., 21 Jan 2025, Steinmeyer et al., 2024).

1. Conceptual scope

A recurring misconception is that equilibrium in growth problems must imply the cessation of growth. The literature instead uses equilibrium in at least three distinct senses. First, it can mean a genuine stationary state of the reduced dynamics, as in KMS states or equilibria of exchange-driven growth. Second, it can mean a stationary profile after factoring out global amplification, as in

$n(t,x)=e^{\lambda t}g(x),$

where the equilibrium object is the principal eigenfunction $G$ rather than the unrescaled solution itself. Third, it can mean a self-consistent fixed point in a dynamic optimization or market-clearing problem, even when state variables such as capital stocks or population continue to evolve (Bruce et al., 2017, Balagué et al., 2012, Alcala, 2023, Choi et al., 25 Apr 2025).

Domain	Equilibrium object	Governing condition
Quasi-lattice ordered monoids	unique KMS$_\beta$ state	$\beta \ge \beta_c$
Growth-fragmentation	principal eigenfunction $G$	spectral gap and entropy dissipation
Exchange-driven growth	equilibrium $c^e$ or $\omega^\varrho$	subcritical mass/density
Capital games	growth equilibrium	best response in time-average growth
Mean-field growth with externality	strong mean-field game equilibrium	fixed point of the externality path
Rocky-planet accretion	vapor equilibrium envelope	saturation with the magma ocean

Taken together, these formulations suggest that growth equilibrium is best understood as an equilibrium notion adapted to irreversible accumulation, rather than as a synonym for thermodynamic rest.

2. Criticality, partition functions, and relaxation to equilibrium

In the operator-algebraic setting of quasi-lattice ordered monoids, a multiplicative homomorphism

$N:P\to (0,\infty)$

induces the quasi-periodic dynamics

$\alpha_t^N(U_p)=N(p)^{it}U_p.$

The corresponding Hamiltonian satisfies $He_p=(\log N(p))e_p$, and the partition function is

$G$0

The critical inverse temperature is

$G$1

For every $G$2 there is a unique KMS$G$3 state, given by the generalized Gibbs formula

$G$4

and at $G$5 there is also a unique KMS$G$6 state when $G$7. The same Dirichlet series governs both equilibrium and combinatorial growth: $G$8 is the smallest pole of the growth series and the smallest positive real root of the clique polynomial. Subcritical KMS states can only occur at roots of the clique polynomial in $G$9, and no examples are known in which such roots exist. The inversion formula connects the growth series directly to the clique polynomial and skew-growth series, in the sense developed by Albenque–Nadeau, McMullen, and Saito (Bruce et al., 2017).

The approach to equilibrium need not be monotone. In a non-normal approximation of Fokker–Planck dynamics, the Optimal Growth Mode is defined from the singular value decomposition

$_\beta$0

as the right singular vector at the time $_\beta$1 maximizing $_\beta$2. This distinguishes the strongest transient amplification from the slowest asymptotic decay. In the Lorenz 63 example, the relaxation curve $_\beta$3 exhibits a pronounced peak around $_\beta$4, showing that finite-time anti-mixing can dominate the observable approach to statistical equilibrium. This suggests that, in some growth-equilibrium problems, the relevant object is not only the terminal equilibrium state but also the transient mode that organizes the path toward it (Gutiérrez et al., 2024).

3. Stationary profiles, spectral gaps, and critical masses

For the growth-fragmentation equation

$_\beta$5

equilibrium is formulated through the principal eigenfunction $_\beta$6, which solves

$_\beta$7

The dual eigenfunction $_\beta$8 determines the conserved observable

$_\beta$9

The equilibrium profile has fine asymptotics: at large size,

$\beta \ge \beta_c$0

while near zero its behavior is governed by $\beta \ge \beta_c$1. The entropy method yields

$\beta \ge \beta_c$2

and therefore

$\beta \ge \beta_c$3

Equilibrium here is thus a rescaled asymptotic shape, together with exponential relaxation under power-law coefficient assumptions of the type used after Cáceres, Cañizo, and Mischler (Balagué et al., 2012).

Exchange-driven growth exhibits a different threshold structure. In the detailed-balance case, both formulations in the cited EDG papers identify a critical mass or density—denoted $\beta \ge \beta_c$4 or $\beta \ge \beta_c$5—that separates two regimes. Below threshold, there exists a unique equilibrium distribution, and solutions converge strongly to it. Above threshold, no equilibrium exists with the full conserved mass; instead, solutions converge only weakly to the critical equilibrium, while the excess mass migrates to larger and larger clusters. In the weak limit, this excess disappears from every fixed cluster size. The papers are explicit that “mass loss” does not mean destruction of mass in the dynamics, but escape to infinity in cluster size. Under detailed balance, the free energy

$\beta \ge \beta_c$6

acts as a Lyapunov function and also drives a formal gradient-flow structure. For type II kernels without detailed balance, equilibrium is obtained by contraction arguments, with exponential convergence either in the number-of-clusters norm or, under small-mass assumptions, in the total-mass norm (Schlichting, 2018, Esenturk et al., 2019).

4. Equilibrium growth, kinetic trapping, and morphology in materials

In materials science and self-assembly, growth equilibrium commonly refers to the competition between thermodynamic target states and kinetic constraints. In Volmer–Weber growth on a foreign substrate under incomplete wetting, the equilibrium morphology of strained three-dimensional islands is a truncated pyramid rather than a full pyramid. Tensile islands have smaller aspect ratios compared with compressed islands owing to their better adhesion to the substrate. The aspect ratio $\beta \ge \beta_c$7 increases with $\beta \ge \beta_c$8 but saturates at about

$\beta \ge \beta_c$9

The mechanistic reason is rapid strain relaxation with thickness: the top layers become nearly unstrained, so there is no strong energetic incentive to sharpen the pyramid to a point. The paper therefore rejects older continuum or harmonic expectations that sufficiently large misfit should eliminate the upper base (Prieto et al., 2010).

For multicomponent equilibrium self-assembly, direct growth of the target structure requires a separation between designed and undesigned interactions. In the lattice model with $G$0 distinct block types, high-fidelity growth is most probable when designed interactions are drawn from a distribution that is as narrow as possible, and direct equilibrium growth can occur even in the presence of substantial attractive undesigned interactions if the designed interaction scale is chosen appropriately. The same analysis is used to motivate a DNA-brick sequence-selection principle based on narrowing the distribution of hybridization free energies, with SantaLucia thermodynamics used to compare random sequence choice and narrowed complementary sets (Hedges et al., 2014).

A noteworthy counterexample to the common intuition that mild nonequilibrium minimizes defects is provided by layer-by-layer growth in a 3D lattice gas. There the vacancy density obeys the scaling relation

$G$1

so growth equilibrium corresponds to $G$2 and $G$3. Yet, for fixed observation time, the best quality structure is obtained with strong interactions and far-from-equilibrium growth conditions, because stronger bonding lowers the thermodynamic vacancy floor more rapidly than kinetic trapping raises it (Whitelam, 2017).

Other growth problems show that far-from-equilibrium assembly can produce robust but non-equilibrium outcomes. In binary solids with a checkerboard ground state, kinetic trapping generates a rate-insensitive nonequilibrium stoichiometry or “magic number,” which is predicted by a mapping to Flory’s jammed random tiling of dimers. In two dimensions the prediction

$G$4

agrees with the observed plateau $G$5, while in three dimensions the predicted $G$6 agrees with simulations near $G$7 (Mannige et al., 2015). Irreversible magnetic thin-film growth in a temperature gradient exhibits a continuous order-disorder phase transition at

$G$8

distinct from the homogeneous-bath case, showing that growth equilibrium is not required for criticality (Candia et al., 2012). In calamitic liquid crystals, equilibrium phase behavior and nonequilibrium coarsening are likewise decoupled: the nematic phase obeys $G$9, whereas the SmB-H phase exhibits a two-time-scale scenario with an early $c^e$0 regime and a later regime strongly suggesting $c^e$1 (Birdi et al., 2022).

5. Economic, game-theoretic, and network formulations

In economic growth theory, equilibrium is often an optimal or market-clearing path rather than a static allocation. A discrete-time Lucas–Uzawa model with physical and human capital defines equilibrium as an optimal solution to the planner’s dynamic program. Under assumptions $c^e$2–$c^e$3, a solution exists, and the value function satisfies the Bellman equation

$c^e$4

With externalities, existence is retained when $c^e$5, after a homogeneity-restoring transformation of human capital (Alcala, 2023). In a multi-sector mean-field growth model with a common external variable $c^e$6, equilibrium is a pair $c^e$7 such that $c^e$8 is optimal given $c^e$9 and $\omega^\varrho$0 is induced by the aggregate effect of $\omega^\varrho$1. The equilibrium is characterized by a coupled FBSDE system, and existence and uniqueness follow from the contraction condition

$\omega^\varrho$2

In the numerical brown–green transition example, rising pollution lowers brown productivity and shifts investment toward green capital (Lavigne et al., 21 Jan 2025). In a Radner economy with Poisson population growth, equilibrium annuity prices are constructed recursively in the current population size; the existence proof proceeds from truncated systems to the unlimited-growth economy, and numerically increasing the birth rate reduces oscillations in the annuity price (Choi et al., 25 Apr 2025).

A separate line of work redefines equilibrium directly in terms of growth rates. In capital games, payoffs are capital observables rather than assumed von Neumann–Morgenstern utilities, and players maximize the time-average growth rate

$\omega^\varrho$3

A growth equilibrium is a strategy profile in which every player’s strategy is a best response under that criterion. For positive capital games with linearizable dynamics, the growth equilibria correspond exactly to the Nash equilibria of a transformed standard game with

$\omega^\varrho$4

This creates an exact representation theorem linking growth-optimal play and ordinary game-theoretic equilibrium (Abramowitz, 1 Oct 2025).

Macroscopic and agent-based models add threshold phenomena resembling phase transitions. In the generalized Yard-Sale model with economic growth, $\omega^\varrho$5 separates an equilibrium-like phase with mobility and exponentially growing wealth of all agents from a non-stationary phase with wealth condensation and no mobility. For $\omega^\varrho$6, the wealth metric satisfies $\omega^\varrho$7, rank correlations decay to zero, and the energy distribution is consistent with a Boltzmann form; for $\omega^\varrho$8, those properties fail. The same paper stresses that apparent critical exponents depend on the path to the transition unless the Ginzburg parameter is held fixed (Liu et al., 2021). In a temporal-network model, value and cost obey

$\omega^\varrho$9

and growth persists only while $N:P\to (0,\infty)$0; the characteristic stopping time is

$N:P\to (0,\infty)$1

This is a finite-growth equilibrium in which preferential attachment is endogenously inhibited by competition cost and internal structural cost (Hasani et al., 2019). A much broader macroeconomic construction posits the physical equilibrium

$N:P\to (0,\infty)$2

with human capacity and physical capital forming two stabilizing feedback loops whose only analytic solutions are S-functions (Danielmeyer et al., 2012).

6. Self-regulation, stability criteria, and cross-domain themes

Some formulations of growth equilibrium are explicitly energetic. In a vapor-equilibrium model of accreting rocky planets, the envelope is assumed to be in hydrostatic equilibrium and in vapor equilibrium with the underlying magma ocean, with SiO used as the heavy vapor proxy. Pebbles therefore do not undergo sublimation in the envelope and survive until they plunge into the magma ocean. Once $N:P\to (0,\infty)$3, an inner radiative region forms because SiO condensation suppresses convection; for $N:P\to (0,\infty)$4, the temperature and pressure near the surface reach the supercritical point of SiO; and for $N:P\to (0,\infty)$5, all accreted pebble material must contribute to maintaining vapor equilibrium in the envelope, so the non-vapor mass ceases to increase. Growth equilibrium here is a saturated, condensation-stabilized buffer state that still permits direct core growth to much higher masses than earlier envelope-sublimation arguments suggested (Steinmeyer et al., 2024).

In fracture mechanics, safe equilibrium is formulated variationally for a cracked inhomogeneous elastic body. The crack remains in safe equilibrium as long as

$N:P\to (0,\infty)$6

and it starts to grow when

$N:P\to (0,\infty)$7

The growth direction is the argmax of $N:P\to (0,\infty)$8. This energetic criterion implies the interface-deflection and kinking criteria of He and Hutchinson for bimaterial cracks (Le et al., 2021).

Across these literatures, several structural themes recur. Thresholds determine whether growth can be absorbed by an equilibrium family or instead produces condensation, excess-mass escape, kinetic trapping, or self-limiting saturation. Lyapunov functionals, free energies, entropy-dissipation inequalities, or variational principles typically control the approach to equilibrium. Weak and strong convergence often differ sharply, especially when mass or wealth escapes to large scales. Finally, several papers emphasize unresolved points rather than closed classifications: no example is known of sporadic subcritical KMS states generated by roots of the clique polynomial inside $N:P\to (0,\infty)$9 (Bruce et al., 2017), and the late-time $\alpha_t^N(U_p)=N(p)^{it}U_p.$0 smectic growth law is presented as strongly suggested rather than definitively established (Birdi et al., 2022). This suggests that growth equilibrium is less a single theorem than a family of threshold-governed equilibrium concepts adapted to systems whose most salient feature is continued growth.