Thermalization effects on the dynamics of growing vacuum bubbles

Hardi Veerm\"ae; Marek Lewicki; Martin Vasar; Mateusz Zych; Tomasz Krajewski; Ville Vaskonen

REVIEW 3 major objections 6 minor 1 cited by

Thermalization effects on the dynamics of growing vacuum bubbles

T0 review · 3 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The terminal speed of an expanding vacuum bubble is set by the degree of thermalization of the surrounding plasma, and stationary solutions that exist in steady-state theory may never be reached in practice.

desk verdict A useful three-regime comparison of bubble-wall thermalization; the qualitative message holds, but the intermediate-regime numbers rest on an admittedly imperfect collision scheme. read the letter →

arxiv 2411.15094 v1 pith:5ZA5H7XR submitted 2024-11-22 hep-ph astro-ph.COhep-th

Tomasz Krajewski , Marek Lewicki , Martin Vasar , Ville Vaskonen , Hardi Veermäe , Mateusz Zych This is my paper · ORCID

classification hep-phastro-ph.COhep-th

keywords bubblewallvelocitycosmologicalfirst-orderphasetransitionslocalthermalequilibriumballisticpressureN-bodysimulationsentropyconservationgravitationalwaveselectroweakbaryogenesis

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper asks what limits the growth speed of vacuum bubbles in cosmological first-order phase transitions when the surrounding plasma does not fully thermalize. It argues that the bubble wall's terminal velocity depends on where the particle mean free path lies relative to the wall thickness: entropy conservation fixes the speed when local thermal equilibrium holds everywhere, while matching ballistic pressure against the vacuum-energy difference fixes a different speed when particles cross the wall without thermalizing. The paper finds that the fully ballistic limit gives significantly different wall velocities from the equilibrium predictions, and that intermediate thermalization times can either speed up or slow down the wall. It also finds, in both hydrodynamic and particle-based simulations, that a stationary terminal velocity that exists in steady-state theory may never be reached because the accelerating wall outruns the formation of its heated plasma shell. If this is right, predictions for gravitational-wave signals and electroweak baryogenesis that assume a single LTE terminal velocity carry a systematic uncertainty.

What carries the argument

The organizing quantity is the particle mean free path compared with the bubble-wall thickness, which selects one of three regimes: LTE everywhere, LTE outside the wall with ballistic transport inside it, or a fully ballistic fluid. The terminal velocity is then fixed by one of two closure conditions: entropy-flux conservation, $s_-\gamma_- v_- = s_+\gamma_+ v_+$, in the LTE case, or pressure balance $\Delta P = \Delta V_0$, with $\Delta P$ computed from particle momentum transfer through the wall, in the ballistic cases. For a fully free-streaming Maxwell–Boltzmann fluid, $\Delta P$ takes the closed form of Eq. (3.8), whose relativistic limit reproduces the standard result. Lattice hydrodynamic simulations implement the LTE closure, while N-body simulations with collision dynamics implement the ballistic and intermediate cases and test whether the stationary state is actually approached.

What would settle it

Run the same ballistic setup with a smaller collision-cell size and with collision cells allowed to hold more than two particles; if the inferred terminal velocity changes by more than the quoted error, or if the $\tau \to 0$ limit does not approach the entropy-conservation terminal velocity as the cell shrinks, then the intermediate-thermalization results are a numerical artefact of the collision scheme.

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Extended reading notes

Core claim

On the paper's own terms, the bubble-wall terminal velocity is determined by how well the plasma thermalizes around the wall. In local thermal equilibrium everywhere, the additional condition fixing the wall velocity is entropy conservation across the wall. When equilibrium holds only away from the wall and particles move ballistically through it, the terminal velocity is instead fixed by matching the ballistic pressure with the potential-energy difference driving the bubble; this gives slightly slower walls, because entropy production inside the wall acts as extra friction. When the fluid is fully ballistic, the terminal velocity is set by a closed-form pressure balance and can differ substantially from the other two cases. Beyond these stationary predictions, the paper shows that even when such a solution exists, the actual dynamics matters: bubbles accelerating strongly can run away before the heated plasma shell characteristic of a steady deflagration or hybrid forms, so the terminal state is not automatically realized.

Load-bearing premise

The load-bearing premise is that the particle-collision algorithm used in the N-body simulations, with at most two particles per collision cell, density- and time-dependent collision probabilities, and finite-size cells, represents physical thermalization accurately enough to determine both the terminal velocity and whether a stationary state is reached; the paper itself reports that this scheme violates causality for wall speeds above about $0.7c$, making those fast-wall results less reliable.

Editorial extensions

If this is right

In the LTE regime, the hydrodynamic simulations and the entropy-conservation matching agree on the wall velocity whenever a steady state is actually reached, so the mapped velocity contours can be used for that regime.
The intermediate scenario, with ballistic transport inside the wall and LTE outside it, predicts slightly lower terminal velocities than full LTE, meaning non-equilibrium effects inside the wall provide extra friction.
The fully ballistic limit predicts substantially different wall velocities, and for strong transitions the wall is typically faster when particles barely interact.
Whether a stationary state is reached can depend on wall tension and on thermalization time, even though the terminal velocity itself does not depend on tension in the large-radius limit; quickly accelerating walls can run away before the heated plasma shell forms.
Because thermalization time affects both the value of the terminal velocity and the existence of a terminal state, time-dependent non-equilibrium simulation is needed to predict the final fate of the bubble.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If these results carry over to realistic models, gravitational-wave spectra computed from a single LTE wall velocity should be treated as a band rather than a line, and electroweak-baryogenesis calculations that assume steady subsonic walls may miss a population of runaways.
The finding that existing stationary states can be unreachable suggests the useful observable is the distribution of wall velocities at bubble collision, not a single terminal velocity from a steady-state analysis.
Extending the three-limit comparison to models with nontrivial quantum reflection coefficients and soft-particle emission would test whether the paper's step-function reflection and neglected emission bracket the error or underestimate it.
Because the paper's collision scheme violates causality for wall speeds above about $0.7c$ by its own account, the fast-wall branch of the thermalization scan is the least trustworthy; a causal kinetic or particle-in-cell treatment would provide a sharper check of whether the short-mean-free-path limit converges to the LTE result.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

This paper deserves a serious read. It compares bubble-wall dynamics across three thermalization regimes—LTE everywhere, LTE outside the wall with ballistic crossing, and fully ballistic—and shows that the thermalization time changes the terminal wall velocity and can determine whether a stationary state is reached at all. That last point is the substantive one: finding a stationary solution is not the same as the bubble reaching it.

What's genuinely new is the unified three-regime map and the finite-thermalization-time scan in Fig. 6, plus the surface-tension effect on reaching a stationary state in the ballistic runs (Fig. 7). The authors are transparent that parts of the non-reachability story already appeared in their earlier work; the novel contribution is the systematic comparison and the explicit time-dependence. The entropy-conservation derivation is clean, and the LTE hydro simulations match the stationary-state matching wherever a steady state is reached. The fully ballistic limit reproduces the analytic formula, which checks the pipeline.

The main soft spot is the intermediate regime. The MPC collision scheme used in the N-body runs cannot reproduce LTE even as τ→0, violates causality for vw ≳ 0.7, and has poor shock resolution. The authors say this in Appendix B, which I respect, but it means the quantitative interpolation in Fig. 6 and the existence statements based on self-interacting runs are the least secure parts. The τ→0 points lying below the analytic LTE limit could be physical nonequilibrium physics, but with a collision scheme whose τ→0 limit isn't LTE, the attribution is not established. I'd phrase those results as indicative, not quantitative. Missing code and error bars make this harder to check. That said, the qualitative picture is robust: the three regimes give clearly different terminal velocities, and the LTE and fully ballistic limits are well anchored.

I would send this to a serious referee. The topic matters for gravitational-wave predictions and electroweak baryogenesis, and the paper is a clear step toward understanding when equilibrium approximations break down. I'd recommend conditional acceptance: improve or validate the collision model for the intermediate regime, release the code and data, and mark Fig. 6 as proof-of-principle until then. I would cite it for the three-regime comparison even with those caveats.

Referee Report

3 major / 6 minor

Summary. The paper studies the terminal velocity of vacuum bubble walls in a first-order cosmological phase transition, comparing three regimes: local thermal equilibrium (LTE) everywhere, LTE away from the wall with ballistic motion inside the wall, and a fully ballistic fluid. The authors derive terminal velocities from entropy conservation in the LTE case and from a pressure-balance condition in the ballistic cases, and they support the analysis with hydrodynamic lattice simulations and N-body simulations using multi-particle collision dynamics (MPC). The main claims are that the wall velocity depends on the degree of thermalization, that the fully ballistic scenario gives significantly different velocities, and that in both hydrodynamic and ballistic simulations a stationary solution, even when it exists, may not actually be reached.

Significance. If correct, the paper's central message is important: LTE-based predictions of bubble-wall velocities can be unreliable in non-equilibrium regimes, and time-dependent simulations, rather than stationary-state analyses alone, may be needed to determine the final fate of a bubble. The strengths of the paper are the internally consistent analytic derivations (entropy conservation in Sec. 3.1 and pressure balance in Secs. 3.2-3.3), the good quantitative agreement between hydrodynamic simulations and the stationary-state method in the LTE regime (Fig. 2 and Fig. 4, top panel), and the reproduction of the fully ballistic analytic limit in the large-mean-free-path N-body simulations (Fig. 6, tau to infinity limit). The significance is tempered, however, by the reliance of the intermediate-thermalization results on an MPC collision scheme whose limitations are acknowledged in Appendix B.

major comments (3)

[Sec. 4.2 / Fig. 6 and Appendix B] The small-tau N-body points in Fig. 6 are used to argue that the system asymptotically approaches the LTE limit and that the thermalization time changes the stationary wall velocity. Appendix B states, however, that 'even if tau = 0, the N-body simulation algorithm does not result in LTE in front of the bubble, as collisions in the algorithm happen once per timestep.' Because the tau to 0 limit of the algorithm is not LTE, the observed offset between the N-body points and the analytic Delta-P = Delta-V0 limit cannot be unambiguously attributed to physical nonequilibrium effects, as is done in Sec. 4.2. The authors should either demonstrate convergence of the MPC scheme to a known LTE result in a controlled benchmark (for example, a planar wall with decreasing Delta-t and R_cell) or soften the quantitative interpolation claim in Fig. 6 and the associated conclusion that thermalization time changes the stationary wall velocity.
[Sec. 4.3 / Fig. 7 and Appendix B] The conclusion that, in the ballistic regime, the existence of a stationary solution depends on sigma and on whether thermalization is fast enough rests on the same MPC implementation. Appendix B states that the algorithm 'violates causality due to the finite size of the collision size' and that results for v_w >~ 0.7 'become less reliable', and the right panel of Fig. 7 shows wall velocities extending close to v_w = 1. The authors should quantify how much of the parameter region in Fig. 7 lies in the unreliable regime and check whether the 'no stationary solution' region persists when the causality violation is removed or when a different collision scheme is used. As it stands, the abstract's statement that both simulations show a stationary solution may not be reached is stronger than the ballistic evidence supports.
[Appendix B and Fig. 6] The paper does not calibrate the MPC collision probability to a physical transport coefficient, so the mapping between the parameter tau and the physical mean free path or thermalization time is not established. The collision probability contains a factor exp(Delta-t/tau), a density-dependent factor, and a cap of two particles per cell, but no comparison with known transport coefficients (for example, viscosity or mean free path in a Maxwell-Boltzmann gas) is provided. Without this calibration, the horizontal axis of Fig. 6 is only a simulation parameter, and the quantitative statement that thermalization time has an important effect on the stationary wall velocity is not yet tied to a physically meaningful scale.

minor comments (6)

[Footnote 1] The sentence 'will not make this distinction in terminology' is missing a subject; it should read 'we will not make this distinction'.
[Appendix B] The phrase 'violates causality due to the finite size of the collision size' should presumably read 'collision cell size'; please correct this and clarify which characteristic scale is meant.
[Sec. 2.1] The text says that the effect of the statistics is 'quantified explicitly in Sec. 3.1', but the quantitative comparison of Maxwell-Boltzmann, Bose-Einstein, and Fermi-Dirac distributions appears in Sec. 4.2 (Fig. 5).
[Sec. 3.2, Eq. (3.5)] The notation T_j and R_j in Eq. (3.5) is not defined; the text defines a reflection coefficient R and a transmission coefficient T = 1 - R. Please clarify whether the subscript j labels the side of the wall and how T_j and R_j relate to R.
[Fig. 6] The horizontal axis label 'tau / Delta t infinity' is ambiguous; please indicate explicitly that the right boundary corresponds to the free-streaming limit tau to infinity.
[Sec. 2.3, Eq. (2.17)] The definition 'Delta V0 = V(0,T) - V(phi0,T)' appears inconsistent with the earlier definition of Delta V0 as the bare potential difference in Eq. (2.14) and with the later statement in Sec. 3.2 that Delta V0 omits thermal corrections. Please clarify whether V here denotes the bare potential V0 or the thermal potential V.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: terminal velocities are obtained from conservation and matching conditions and from N-body simulations, not from inserting the desired result; self-cited formulas are independent kinetic-theory inputs.

full rationale

The paper's three limiting wall velocities are derived by solving equations rather than by assuming the answer. The LTE velocity follows from entropy-current conservation (Sec. 3.1, Eqs. 3.1-3.3), combined with the stationary fluid equations (2.16)-(2.17). The intermediate scenario solves the ballistic-pressure condition ΔP = ΔV0 using the pressure integral (3.5), and the fully ballistic scenario solves Eq. (3.7) with the analytic pressure (3.8). In every case v_w is the output of a self-consistency equation, not a fitted parameter. The analytic pressure formulas cited from the authors' prior papers [74,75] are parameter-free kinetic-theory results with stated assumptions (Maxwell-Boltzmann distributions, step-function reflection, no self-interactions), and they are benchmarked in this paper against independent simulations: the large-τ N-body results agree with Eq. (3.8), and the LTE hydrodynamics agree with the entropy-conservation matching method. The intermediate-τ results are generated by N-body simulation with τ as an input and v_w as an output, so they are not 'predictions' obtained by construction. The paper itself flags the important numerical caveat in Appendix B that the MPC algorithm does not reach LTE even in the τ→0 limit and is unreliable for v_w ≳ 0.7; this is a real limitation and a correctness risk for the quantitative interpolation in Fig. 6 and existence claims in Fig. 7, but it is not a circularity. No step in the derivation reduces to its own inputs by definition, and no load-bearing conclusion rests solely on a self-citation. The self-citations that do appear supply independent analytic ingredients or prior observations that the present simulations re-check, so they do not raise the circularity score.

Assumptions & free parameters 4 free parameters · 8 assumptions · 0 invented entities

The model introduces no new particles or forces. The free parameters are numerical and model inputs that set the thermalization and resolution scales; the axioms are standard approximations inherited from the phase-transition literature plus the paper-specific MPC collision model.

free parameters (4)

thermalization time tau = scanned simulation parameter
Chosen by hand in N-body simulations to interpolate between LTE (tau -> 0) and free-streaming (tau -> infinity); central to Fig. 6 and the claim that thermalization time affects terminal velocity.
collision cell length Rcell = 2 R_b / 91
Numerical resolution parameter of the MPC scheme; chosen under GPU memory constraints and partly responsible for the inability to resolve shock-like behavior.
timestep dt = R_b / 10^4
Temporal resolution of N-body runs; collision probability and wall evolution depend on it, and the authors note causality violations for fast walls.
wall surface tension sigma = sigma / Delta m^3 = 0.01, 0.001
Physical model parameter varied in Fig. 7; affects whether a stationary state is reached and the acceleration rate, though not the terminal velocity.

assumptions (8)

domain assumption Perfect-fluid description with Maxwell-Boltzmann statistics is adequate for the plasma.
Used in both hydrodynamic and stationary-state calculations; Fig. 5 shows FD and BE statistics change vw only mildly, but the quantitative predictions assume this equation of state.
domain assumption The fluid consists of a single species that is massless in the false vacuum and gains mass m_psi^2 = y^2 phi^2 in the true vacuum.
This defines the model and the pressure formulas; no additional Standard Model species are included.
domain assumption The thin-wall approximation and planar-wall geometry apply; bubble curvature and cosmic expansion are neglected.
Used for the wall equation of motion (3.4) and the mode analysis in Appendix A; valid when wall thickness is much smaller than bubble radius.
ad hoc to paper Thermal potential V = V0 - p(phi,T) remains valid inside the wall even when local thermal equilibrium is questionable.
Appendix A admits that non-local modes make local equilibrium inside the wall not always meaningful, yet Eq. (2.5) is used throughout the LTE calculations.
domain assumption Reflection coefficient R_-(u) = theta(Delta m^2 - u^2) with R_+ = 0 captures the ballistic pressure.
Assumed in Eq. (3.6) and in N-body simulations; the authors argue reflections matter only in a narrow momentum shell but do not quantify the effect on vw.
domain assumption Stationary fluid profiles are self-similar with shocks or rarefaction discontinuities and matching conditions (2.17).
Standard fixed-point analysis used to compute stationary states and terminal velocities; assumes the wall has already reached a fixed velocity.
ad hoc to paper The MPC collision algorithm with two-particle cells approximates physical thermalization for the simulated fluid.
The algorithm was built for this study; Appendix B documents that it cannot reach Maxwell-Boltzmann with more particles per cell and violates causality for fast walls.
domain assumption Soft-particle emission and higher-order non-equilibrium friction can be neglected.
Section 3 states these effects are not considered because they matter mainly for ultrarelativistic walls not studied here.

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Pith. "Pith review of Thermalization effects on the dynamics of growing vacuum bubbles." pith.science (2026). https://pith.science/paper/5ZA5H7XR

@misc{pith2026241115094,
  author       = {Pith},
  title        = {Pith review of: Thermalization effects on the dynamics of growing vacuum bubbles},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5ZA5H7XR}},
  note         = {Machine review of arXiv:2411.15094}
}

abstract

We study the evolution of growing vacuum bubbles. The bubble walls interact with the surrounding fluid and may, consequently, reach a terminal velocity. If the mean free path of the particles in the fluid is much shorter than the bubble wall thickness, the fluid is locally in thermal equilibrium and the wall's terminal velocity can be determined by entropy conservation. On the other hand, if local thermal equilibrium inside the wall cannot be maintained, the wall velocity can be estimated from the pressure impacted by ballistic particle dynamics at the wall. We find that the latter case leads to slightly slower bubble walls. Expectedly, we find the largest differences in the terminal velocity when the fluid is entirely ballistic. This observation indicates that the non-equilibrium effects inside walls are relevant. To study bubble evolution, we perform hydrodynamic lattice simulations in the case of local thermal equilibrium and $N$-body simulations in the ballistic case to investigate the dynamical effects during expansion. Both simulations show that even if a stationary solution exists in theory it may not be reached depending on the dynamics of the accelerating bubble walls.

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