Screened Kinetic Condensation Formula

• Screened kinetic condensation formulas are relations that define critical thresholds and delays in kinetic processes by incorporating a screening mechanism.
• They elucidate how finite interaction ranges and density-dependent effects modify condensation dynamics in diverse systems like lattice electrons, self-gravitating bosons, and bipolarons.
• Applications span quantum phase transitions, Bose–Einstein condensation, transport, and capillary phenomena, highlighting the role of screening in both enhancing and delaying condensation.

Searching arXiv for the cited papers and closely related work on screened kinetic condensation across different subfields. The expression screened kinetic condensation formula is used in several technically distinct literatures to denote a relation in which condensation is governed not only by an energetic driving term but also by a screening mechanism that regularizes, suppresses, or geometrically constrains the relevant kinetic process. In the literature considered here, the phrase does not designate a single universal equation. Instead, it refers to a family of threshold formulas, transport-time expressions, jump laws, and non-condensation criteria that arise in systems as different as one-dimensional lattice electrons, self-gravitating bosons with Yukawa interactions, intersite bipolarons, Glasma kinetics, traffic-flow models, opinion-dynamics PDEs, open-pore capillarity, and BGK half-space evaporation or condensation (Ostilli et al., 2019, Chen, 22 May 2026).

1. Terminological scope and general structure

Across these works, “screening” enters in at least four structurally different ways. In lattice-electron and Yukawa-gravity problems, it modifies the interaction range and therefore the competition between kinetic and interaction energies or between infrared transport and relaxation. In transport and opinion models, it acts effectively through the creation of competing clusters or through density-dependent suppression of diffusion. In capillarity, finite geometry and pinned menisci play a role analogous to screening by shifting condensation away from the infinite-system Kelvin law. In BGK half-space problems, the disturbance generated at a boundary is screened mode by mode through exponentially damped spectral contributions (Ostilli et al., 2019, Miedema et al., 2014, Malijevský et al., 2017).

A second unifying feature is that the resulting formula is rarely a bare equilibrium condition. The object of interest may instead be a critical threshold, such as a critical Seitz radius or a critical mass; a condensation time, as in kinetic Bose-star formation; or a jump law, as in weak evaporation or condensation over a wall. This suggests that the common content of the term is not a specific algebraic form but a kinetic reparameterization of condensation in the presence of finite-range, finite-size, or constraint-induced regularization (Chen, 22 May 2026, Calzola et al., 2024, Latyshev et al., 2014).

2. One-dimensional lattice electrons and condensation in the space of states

In the one-dimensional lattice-electron setting, the screened kinetic condensation formula is most explicitly developed for spinless fermions on a ring of $N$ lattice sites with periodic boundary conditions $c_{i+N}=c_i$, at fixed filling

$\varrho=\frac{N_p}{N}.$

The Hamiltonian is written in dimensionless form as

$H=K+gV,$

with hopping term

$$K=-\sum_{i=1}^{N}\left(c_i^\dagger c_{i+1}+c_{i+1}^\dagger c_i\right),$$

and screened Coulomb interaction $V$. The interaction parameter is identified with the Seitz-radius variable through

$g=2a/a_B,\qquad r_s=\frac{g}{4\varrho}.$

At filling $\varrho=3/10$, this becomes $r_s=g/1.2$ (Ostilli et al., 2019).

The decisive step is the decomposition of the Hilbert space into a condensed subspace and a normal subspace,

$$\mathbb{F}=\mathbb{F}_{\mathrm{cond}}\oplus \mathbb{F}_{\mathrm{norm}},$$

together with the condensation-in-state-space criterion

$c_{i+N}=c_i$0

A first-order quantum phase transition occurs if there is a unique finite solution of

$c_{i+N}=c_i$1

so that the first derivative of $c_{i+N}=c_i$2 jumps at $c_{i+N}=c_i$3. In physical terms, small $c_{i+N}=c_i$4 favors a ground state close to the free-fermion state, while large $c_{i+N}=c_i$5 favors a Wigner crystal. The transition is therefore not a smooth crossover but a true first-order quantum phase transition when the screened interaction compares with the kinetic energy (Ostilli et al., 2019).

The asymptotic screened kinetic condensation formula appears as rigorous large-$c_{i+N}=c_i$6 bounds on the critical Seitz radius: $c_{i+N}=c_i$7 This formula is obtained from finite-size crossing estimates and energy inequalities involving $c_{i+N}=c_i$8, $c_{i+N}=c_i$9, $\varrho=\frac{N_p}{N}.$0, and the free-fermion ground-state energy

$\varrho=\frac{N_p}{N}.$1

For the representative case $\varrho=\frac{N_p}{N}.$2 and $\varrho=\frac{N_p}{N}.$3, Monte Carlo extrapolation gives

$\varrho=\frac{N_p}{N}.$4

while the rigorous bounds are

$\varrho=\frac{N_p}{N}.$5

A central conceptual point is that screening is essential for the existence of a nontrivial finite critical point. If screening is removed after the thermodynamic limit has been taken, then $\varrho=\frac{N_p}{N}.$6; by contrast, with a bare unscreened Coulomb potential, Wigner crystallization occurs as a smooth crossover rather than as a quantum phase transition (Ostilli et al., 2019).

3. Yukawa screening and condensation times in self-gravitating bosons

A second major usage concerns gravitational Bose-Einstein condensation in the kinetic regime. For an unscreened nonrelativistic self-gravitating Bose gas, condensation in virialized halos or miniclusters is driven by weak gravitational scattering and characterized by a Coulomb logarithm

$\varrho=\frac{N_p}{N}.$7

with kinetic-regime conditions

$\varrho=\frac{N_p}{N}.$8

The long-range interaction is therefore not Debye-screened, but the kinetic integral is logarithmically regulated by the de Broglie scale and the finite halo size (Levkov et al., 2018).

The Yukawa-screened extension replaces this ordinary gravitational Coulomb logarithm by a finite Yukawa transport logarithm

$\varrho=\frac{N_p}{N}.$9

The screened condensation time is then obtained by the replacement

$H=K+gV,$0

or equivalently

$H=K+gV,$1

Here $H=K+gV,$2 is the interaction range. In the unscreened limit $H=K+gV,$3, one has $H=K+gV,$4 and $H=K+gV,$5 (Chen, 22 May 2026).

The physical interpretation is explicit: finite interaction range suppresses the low-$H=K+gV,$6 part of scattering, reduces infrared momentum transport, and therefore delays condensation. The same screening also broadens the equilibrium Bose-star structure relative to the ordinary Newtonian soliton. Fully dynamical pseudospectral simulations with homogeneous and isotropic initial conditions show that increasing $H=K+gV,$7 systematically delays condensation, in good agreement with the screened kinetic prediction after fitting a single normalization parameter, with

$H=K+gV,$8

This is close to the previously known gravitational value $H=K+gV,$9 (Chen, 22 May 2026).

4. Screened interactions, condensation temperatures, and infrared onset

In the extended Holstein–Hubbard setting for intersite bipolarons, screening enters through a Yukawa-type electron–phonon interaction. The condensation temperature of an ideal three-dimensional gas of intersite bipolarons is written as

$$K=-\sum_{i=1}^{N}\left(c_i^\dagger c_{i+1}+c_{i+1}^\dagger c_i\right),$$0

with $$K=-\sum_{i=1}^{N}\left(c_i^\dagger c_{i+1}+c_{i+1}^\dagger c_i\right),$$1. The central dependence is therefore through the polaron mass $$K=-\sum_{i=1}^{N}\left(c_i^\dagger c_{i+1}+c_{i+1}^\dagger c_i\right),$$2, which is determined by the screened electron–phonon force. The paper reports that $$K=-\sum_{i=1}^{N}\left(c_i^\dagger c_{i+1}+c_{i+1}^\dagger c_i\right),$$3 decreases with increasing screening radius $$K=-\sum_{i=1}^{N}\left(c_i^\dagger c_{i+1}+c_{i+1}^\dagger c_i\right),$$4, in both adiabatic and non-adiabatic regimes. For example, in the non-adiabatic regime decreasing $$K=-\sum_{i=1}^{N}\left(c_i^\dagger c_{i+1}+c_{i+1}^\dagger c_i\right),$$5 from $$K=-\sum_{i=1}^{N}\left(c_i^\dagger c_{i+1}+c_{i+1}^\dagger c_i\right),$$6 to $$K=-\sum_{i=1}^{N}\left(c_i^\dagger c_{i+1}+c_{i+1}^\dagger c_i\right),$$7 increases $$K=-\sum_{i=1}^{N}\left(c_i^\dagger c_{i+1}+c_{i+1}^\dagger c_i\right),$$8 from about $$K=-\sum_{i=1}^{N}\left(c_i^\dagger c_{i+1}+c_{i+1}^\dagger c_i\right),$$9 K to $V$0 K; for stronger screening, $V$1 can raise $V$2 by about $V$3 K, and $V$4 can raise $V$5 by about $V$6 K (Yavidov et al., 2013).

In weak-coupling Glasma kinetics, the relevant screened structure is different. The elastic small-angle kernel is regulated by the Debye mass

$V$7

and the inelastic $V$8 sector is reduced, under Gunion–Bertsch and soft/collinear approximations, to effective screened elastic-like and $V$9 contributions. The key small-$g=2a/a_B,\qquad r_s=\frac{g}{4\varrho}.$0 result is that the effective inelastic kernel has a positive leading infrared term because $g=2a/a_B,\qquad r_s=\frac{g}{4\varrho}.$1. Hence inelastic processes globally reduce particle number but locally fill the infrared extremely quickly, driving the low-momentum sector toward a local Bose–Einstein form with $g=2a/a_B,\qquad r_s=\frac{g}{4\varrho}.$2 and catalyzing the onset of dynamical Bose–Einstein condensation earlier than in the purely elastic case (Huang et al., 2013).

These two examples show that screening does not have a universal sign. In the bipolaron problem, stronger screening raises $g=2a/a_B,\qquad r_s=\frac{g}{4\varrho}.$3 by making the relevant quasiparticles lighter. In Yukawa-screened self-gravity, stronger screening delays condensation by suppressing infrared transport. A plausible implication is that the effect of screening depends on whether condensation is controlled primarily by mass renormalization, by long-range momentum transfer, or by particle-number relaxation.

5. Kinetic analogues in transport and opinion dynamics

In one-dimensional transport models with a kinetic constraint, the condensation problem is recast in terms of inactive clusters embedded in active clusters. The necessary and sufficient criterion for condensation is twofold: $g=2a/a_B,\qquad r_s=\frac{g}{4\varrho}.$4 The first condition ensures that inactive clusters do not split; the second ensures that the growth of existing inactive clusters dominates the creation of new inactive clusters upstream. The paper interprets this latter effect as a screening mechanism: upstream jams reduce the inflow to downstream clusters and therefore screen condensate growth. Applied to the Nagel–Schreckenberg model, the criterion implies that true condensates occur only when acceleration out of jammed traffic happens in a single time step, specifically for $g=2a/a_B,\qquad r_s=\frac{g}{4\varrho}.$5 and $g=2a/a_B,\qquad r_s=\frac{g}{4\varrho}.$6; there is no condensate for finite $g=2a/a_B,\qquad r_s=\frac{g}{4\varrho}.$7, for $g=2a/a_B,\qquad r_s=\frac{g}{4\varrho}.$8, or for $g=2a/a_B,\qquad r_s=\frac{g}{4\varrho}.$9 (Miedema et al., 2014).

A formally different but structurally related kinetic condensation mechanism appears in opinion dynamics. There the microscopic interaction frequency is density weighted,

$\varrho=3/10$0

and the diffusion amplitude is suppressed by local density through

$\varrho=3/10$1

In the quasi-invariant limit, the model yields the nonlinear Fokker–Planck equation

$\varrho=3/10$2

The central threshold is

$\varrho=3/10$3

which is precisely the condition for the existence of a finite critical mass. For $\varrho=3/10$4, stationary states are smooth; for $\varrho=3/10$5, a singular concentration forms at the mean opinion $\varrho=3/10$6. Here the screened kinetic condensation formula is not a transport logarithm but a density-enhanced drift coupled to a degenerate diffusion whose effective strength is reduced where $\varrho=3/10$7 is already large (Calzola et al., 2024).

6. Geometric and half-space formulations

In capillary condensation, screening is implemented by finite geometry rather than by a modified pair potential. For an infinite slit of width $\varrho=3/10$8, the Kelvin shift is

$\varrho=3/10$9

For an open slit of finite height $r_s=g/1.2$0, the liquid-like state must support two menisci pinned near the open ends, and the condensation pressure is shifted closer to bulk saturation according to

$r_s=g/1.2$1

The edge contact angle $r_s=g/1.2$2 is determined by

$r_s=g/1.2$3

so that $r_s=g/1.2$4 for finite $r_s=g/1.2$5 and $r_s=g/1.2$6 as $r_s=g/1.2$7. For complete wetting, $r_s=g/1.2$8, one has

$r_s=g/1.2$9

for large $$\mathbb{F}=\mathbb{F}_{\mathrm{cond}}\oplus \mathbb{F}_{\mathrm{norm}},$$0. In this context, the modified Kelvin equation is the screened condensation law: finite height screens the infinite-slit condensation shift through the interfacial cost of the pinned menisci (Malijevský et al., 2017).

A kinetic half-space analogue appears in one-dimensional BGK problems of moderately strong evaporation or condensation with constant collision frequency. After linearization around the absolute Maxwellian far from the wall, the solution is represented spectrally as

$$\mathbb{F}=\mathbb{F}_{\mathrm{cond}}\oplus \mathbb{F}_{\mathrm{norm}},$$1

For $$\mathbb{F}=\mathbb{F}_{\mathrm{cond}}\oplus \mathbb{F}_{\mathrm{norm}},$$2, the jump coefficients are given in closed form by

$$\mathbb{F}=\mathbb{F}_{\mathrm{cond}}\oplus \mathbb{F}_{\mathrm{norm}},$$3

The mode-by-mode factor $$\mathbb{F}=\mathbb{F}_{\mathrm{cond}}\oplus \mathbb{F}_{\mathrm{norm}},$$4 acts as a kinetic screening factor for the wall disturbance. In the weak evaporation or condensation formulation of the constant-frequency BGK model, the jump laws become explicitly

$$\mathbb{F}=\mathbb{F}_{\mathrm{cond}}\oplus \mathbb{F}_{\mathrm{norm}},$$5

Here the screened kinetic formula is a linear response relation between the macroscopic jumps and the driving fields, with coefficients determined by spectral factorization of the half-space collision operator (Latyshev et al., 2014, Bugrimov et al., 2014).

7. Comparative interpretation and limiting behavior

Taken together, these formulations show that the phrase screened kinetic condensation formula refers to a class of relations in which condensation is controlled by a screened competition. In the lattice-electron problem, screening creates a finite nontrivial critical point and permits a first-order quantum phase transition into a Wigner crystal. In Yukawa-screened self-gravity, screening replaces the Coulomb logarithm by a finite transport logarithm and delays Bose-star formation. In the bipolaron problem, stronger screening raises $$\mathbb{F}=\mathbb{F}_{\mathrm{cond}}\oplus \mathbb{F}_{\mathrm{norm}},$$6 by reducing the effective mass. In Glasma kinetics, Debye-screened inelastic processes reduce the total particle number but nonetheless accelerate the transient approach to condensation onset by rapidly filling the infrared sector. In transport models, the relevant screening is the formation of competing upstream jams. In opinion dynamics, it is the density-dependent suppression of self-thinking. In capillarity and BGK half-space theory, it is imposed by finite geometry or by exponentially damped spectral propagation (Ostilli et al., 2019, Chen, 22 May 2026, Yavidov et al., 2013, Huang et al., 2013, Miedema et al., 2014, Calzola et al., 2024, Malijevský et al., 2017).

Several misconceptions are directly excluded by these results. Screening does not uniformly suppress condensation: it can delay a kinetic process, raise a condensation temperature, or convert a crossover into a genuine phase transition, depending on which quantity it renormalizes. Conversely, unscreened interactions do not necessarily produce a sharper transition. In the one-dimensional lattice-electron problem, removing screening after the thermodynamic limit forces $$\mathbb{F}=\mathbb{F}_{\mathrm{cond}}\oplus \mathbb{F}_{\mathrm{norm}},$$7, and the bare unscreened Coulomb case yields a smooth crossover rather than a quantum phase transition (Ostilli et al., 2019). The most faithful general definition is therefore operational: a screened kinetic condensation formula is any condensation law in which the relevant threshold, time scale, or jump coefficient is determined by a kinetic process after the singular long-range, boundary, or constraint structure has been regularized by screening.