Dynamical Freeze-out in Heavy-Ion Collisions

• Dynamical freeze-out is defined by comparing local hydrodynamic expansion rates to microscopic scattering rates, replacing phenomenological fixed-temperature prescriptions.
• It employs the Knudsen number (K = θ/Γ) as a critical parameter, with decoupling occurring when K is about unity, leading to non-isothermal freeze-out surfaces.
• Enhanced models that include event-by-event fluctuations and viscous corrections show improved flow observable predictions, especially in peripheral heavy-ion collisions.

Searching arXiv for recent and foundational papers on dynamical freeze-out in heavy-ion collisions. Dynamical freeze-out in ultrarelativistic heavy-ion collisions denotes a decoupling prescription in which the hydrodynamically evolving medium ceases to behave as a locally equilibrated fluid when microscopic interaction rates become comparable to macroscopic expansion rates, rather than at a fixed temperature imposed phenomenologically. In the heavy-ion literature this idea is implemented most commonly through a comparison of the scalar expansion rate $\theta=\partial_\mu u^\mu$ with a microscopic scattering rate $\Gamma$, or equivalently through a Knudsen-number criterion $K=\theta/\Gamma\sim 1$ [(0710.4476); (Holopainen et al., 2012); (Ahmad et al., 2016)]. Within this framework, freeze-out is a local, rate-based breakdown condition for hydrodynamics, and the corresponding hypersurface is generally not an isotherm. Later work extended this concept from boost-invariant ideal hydrodynamics in central collisions to event-by-event hydrodynamics, to viscous EKRT simulations of peripheral collisions, and to analytic studies based on exact conformal solutions, while related transport and statistical approaches examined chemical freeze-out and species dependence from complementary perspectives [(Holopainen et al., 2012); (Ahmad et al., 2016); (Hirvonen et al., 2022); (Bradley et al., 2024); (Reichert et al., 2020)].

1. Definition and scope

In hydrodynamic modeling of heavy-ion collisions, one assumes that the matter evolves as a locally thermalized fluid up to some freeze-out surface, beyond which particles free-stream to detectors. Two distinct notions are commonly separated: chemical freeze-out, when inelastic reactions cease and particle yields become fixed, and kinetic or thermal freeze-out, when elastic scatterings become too rare to maintain local thermal equilibrium and momentum distributions decouple. The rate-based “dynamical freeze-out” literature in relativistic heavy-ion collisions primarily addresses kinetic freeze-out, especially for pions [(0710.4476); (Holopainen et al., 2012)].

The standard hydrodynamic practice is to define freeze-out by a constant temperature $T=T_{\rm fo}$ or constant energy density. This is geometrically simple but does not follow from the hydrodynamic equations themselves. Dynamical freeze-out replaces that prescription by a local criterion comparing microscopic and macroscopic scales. In the heavy-ion implementations discussed in the cited work, hydrodynamics is valid when many collisions occur on an expansion time scale, and decoupling occurs when this ceases to hold, typically when $\Gamma^{-1}$ becomes comparable to $\theta^{-1}$ [(0710.4476); (Holopainen et al., 2012); (Ahmad et al., 2016)].

A closely related formulation uses the Knudsen number,

$K=\frac{\theta}{\Gamma},$

with freeze-out at $K=K_{\rm crit}$, where $K_{\rm crit}$ is taken to be of order unity [(Holopainen et al., 2012); (Ahmad et al., 2016)]. This formulation connects directly to the microscopic criterion for the breakdown of local kinetic equilibrium.

2. Rate-based formulations in hydrodynamics

The foundational implementation in ideal hydrodynamics determined decoupling surfaces by comparing the local hydrodynamic expansion rate with a microscopic pion-pion scattering rate (0710.4476). In that work, the expansion scalar is

$\theta \equiv \partial_\mu u^\mu,$

and in boost-invariant, azimuthally symmetric geometry with coordinates $\Gamma$0 and

$\Gamma$1

the explicit form is

$\Gamma$2

For a purely longitudinal Bjorken expansion one has $\Gamma$3 (0710.4476).

The microscopic rate in that analysis was approximated for pions by

$\Gamma$4

with

$\Gamma$5

and a temperature-dependent effective $\Gamma$6 cross section parametrized in the hadron resonance gas phase below $\Gamma$7 MeV as

$\Gamma$8

with

$\Gamma$9

The decoupling condition was then written

$K=\theta/\Gamma\sim 1$0

with $K=\theta/\Gamma\sim 1$1 a dimensionless constant of order unity (0710.4476).

Event-by-event hydrodynamics generalized this logic to a hadron-resonance gas scattering rate that sums over all hadrons in the equation of state: $K=\theta/\Gamma\sim 1$2 where the cross sections $K=\theta/\Gamma\sim 1$3 are taken from UrQMD and are dominated by resonance formation (Holopainen et al., 2012). In that setting the dynamical criterion is again

$K=\theta/\Gamma\sim 1$4

or equivalently $K=\theta/\Gamma\sim 1$5, with $K=\theta/\Gamma\sim 1$6 tested and $K=\theta/\Gamma\sim 1$7 used in the main comparison to constant-$K=\theta/\Gamma\sim 1$8 freeze-out (Holopainen et al., 2012).

A later ideal-hydrodynamic study of Au+Au at $K=\theta/\Gamma\sim 1$9 GeV and Pb+Pb at $T=T_{\rm fo}$0 GeV used the same Knudsen-number concept,

$T=T_{\rm fo}$1

with a more complete microscopic rate for pions in a hadron resonance gas that includes all hadronic scatterers and UrQMD-based $T=T_{\rm fo}$2 cross sections (Ahmad et al., 2016). This extended the dynamical freeze-out criterion from pion-pion scattering to a broader hadronic environment.

3. Construction of freeze-out hypersurfaces

In the ideal-hydrodynamic implementation for central collisions, the decoupling hypersurface is defined as the set of spacetime points satisfying the rate condition in the hadron resonance gas phase, or at most on the mixed–HRG boundary (0710.4476). Because $T=T_{\rm fo}$3 depends only on temperature while $T=T_{\rm fo}$4 depends on velocity gradients, contours of constant $T=T_{\rm fo}$5 and constant $T=T_{\rm fo}$6 do not coincide. The resulting dynamical freeze-out surface is therefore not an isotherm: near the outer edge it starts at approximately $T=T_{\rm fo}$7 MeV and decreases toward the center, intersecting constant-temperature surfaces such as $T=T_{\rm fo}$8 MeV and $T=T_{\rm fo}$9 MeV (0710.4476).

The physical geometry is systematic across studies. Near the edges of the system, the expansion rate is large and the scattering rate drops rapidly after hadronization, so freeze-out occurs almost immediately after the system enters the hadron gas. Closer to the center, where transverse flow is weaker and expansion slower, decoupling occurs later and at lower temperatures (0710.4476). In smooth 2+1D ideal hydrodynamics, the same qualitative pattern appears: with dynamical freeze-out, the edges freeze earlier at higher temperature, while the center lives longer and freezes later at slightly lower temperature (Holopainen et al., 2012).

In event-by-event hydrodynamics, the local expansion rate inherits strong fluctuations from the initial density profile. Regions with large density gradients and hot spots can have large $\Gamma^{-1}$0, while some dilute regions can even show temporary compression with $\Gamma^{-1}$1. As a consequence, the dynamical freeze-out surface can develop complex topology with long-lived small-scale structures described as “horns” and “fins,” which can be thinner than the mean free path. For this reason, quantitative comparisons in that work focus on smoother initial conditions with Gaussian smearing $\Gamma^{-1}$2 fm (Holopainen et al., 2012).

Later EKRT work implemented a different but related dynamical condition. There the local criterion was

$\Gamma^{-1}$3

with $\Gamma^{-1}$4, supplemented by a global size–mean-free-path condition

$\Gamma^{-1}$5

with $\Gamma^{-1}$6, and an additional restriction $\Gamma^{-1}$7 MeV to forbid freeze-out in the QGP phase (Hirvonen et al., 2022). In that construction the freeze-out surface is determined by simultaneous satisfaction of a local Knudsen-number condition and a global system-size condition.

Exact conformal hydrodynamics gives an analytic view of the same issue. Using a non-boost-invariant viscous generalization of Gubser flow, constant-temperature surfaces and constant-Knudsen-number surfaces can be constructed and compared explicitly. The Knudsen number there is defined as

$\Gamma^{-1}$8

and the comparison shows that the two classes of surfaces can differ strongly when collective flow gradients are large (Bradley et al., 2024).

4. Particlization and observable consequences

Once the freeze-out surface is specified, hadron spectra are computed using the Cooper–Frye prescription,

$\Gamma^{-1}$9

with local equilibrium distributions in the ideal-hydrodynamic studies and no viscous corrections [(0710.4476); (Holopainen et al., 2012)]. In the pion-focused implementation, the distribution was taken to be Bose–Einstein, usually approximated by Boltzmann,

$\theta^{-1}$0

and resonance decays were added after freeze-out (0710.4476).

A central result of the heavy-ion literature on dynamical freeze-out is that large geometric differences between constant-$\theta^{-1}$1 and dynamical surfaces do not automatically imply large differences in soft hadronic observables. In central RHIC Au+Au collisions at $\theta^{-1}$2 GeV, BC initial conditions with a constant-temperature freeze-out at $\theta^{-1}$3 MeV reproduce low-$\theta^{-1}$4 pion spectra, while WN initial conditions require $\theta^{-1}$5 MeV. The rate-based criterion $\theta^{-1}$6 with $\theta^{-1}$7–$\theta^{-1}$8 yields spectra very close to those constant-$\theta^{-1}$9 results and in good agreement with RHIC data (0710.4476). At LHC $K=\frac{\theta}{\Gamma},$0 TeV, using the same $K=\frac{\theta}{\Gamma},$1 extracted from RHIC gives pion spectra effectively described by the same constant temperatures, approximately $K=\frac{\theta}{\Gamma},$2 MeV for BC and $K=\frac{\theta}{\Gamma},$3 MeV for WN initialization (0710.4476).

Event-by-event ideal hydrodynamics reached a closely related conclusion. For Au+Au at $K=\frac{\theta}{\Gamma},$4 GeV and 20–30% centrality, with $K=\frac{\theta}{\Gamma},$5 compared against constant $K=\frac{\theta}{\Gamma},$6 MeV, pion and proton spectra from dynamical and constant-$K=\frac{\theta}{\Gamma},$7 freeze-out lie essentially on top of each other for both smooth and moderately fluctuating initial conditions (Holopainen et al., 2012). Charged-hadron elliptic flow $K=\frac{\theta}{\Gamma},$8, calculated with an event-plane method, also shows no visible difference between the two freeze-out prescriptions within the precision of that study (Holopainen et al., 2012).

The ideal-hydrodynamic Knudsen-freeze-out study at RHIC and LHC extended this result to triangular flow. Once the freeze-out temperature $K=\frac{\theta}{\Gamma},$9 and freeze-out Knudsen number $K=K_{\rm crit}$0 are calibrated to produce similar pion and proton $K=K_{\rm crit}$1 spectra, the elliptic and triangular anisotropies are also similar in both event-by-event and averaged initial-state calculations (Ahmad et al., 2016). This supports the view that, for soft observables in large A+A systems, a tuned constant-$K=K_{\rm crit}$2 prescription can mimic a more microscopic dynamical criterion.

The EKRT analysis of peripheral collisions modifies that assessment. There the inclusion of a dynamical freeze-out and bulk viscosity improves the description of flow coefficients in peripheral collisions and extends the centrality range over which the model can describe data. In particular, dynamical freeze-out significantly reduces $K=K_{\rm crit}$3 in peripheral collisions because small systems satisfy the Knudsen and global size criteria earlier in the hadronic phase, shortening the period available to convert spatial eccentricity into momentum anisotropy (Hirvonen et al., 2022). The same study found noticeable effects on flow–$K=K_{\rm crit}$4 correlations, especially in peripheral 5.02 TeV Pb+Pb, where dynamical freeze-out plus bulk viscosity improves agreement with data (Hirvonen et al., 2022).

Exact conformal solutions reinforce the interpretation that the difference between constant-$K=K_{\rm crit}$5 and constant-Knudsen freeze-out is controlled by strong collective flow. In those analytic solutions, the mismatch between the criteria is modest for $K=K_{\rm crit}$6 but becomes large for $K=K_{\rm crit}$7, particularly where radial velocity is largest. Along a constant-Knudsen contour the temperature can vary substantially with radius, with relative peak temperatures above the central value of about $K=K_{\rm crit}$8, $K=K_{\rm crit}$9, and $K_{\rm crit}$0 for $K_{\rm crit}$1, $K_{\rm crit}$2, and $K_{\rm crit}$3, respectively (Bradley et al., 2024).

5. Species dependence, chemical freeze-out, and transport perspectives

The hydrodynamic rate-based literature typically uses pion scattering rates as a proxy for the microscopic scale and assumes all species decouple on the same surface, because pions are the most abundant hadrons [(Holopainen et al., 2012); (Ahmad et al., 2016)]. This is a practical approximation rather than a species-resolved kinetic theory. The authors of the hydrodynamic studies note explicitly that nucleons are expected to decouple later than pions because of strong $K_{\rm crit}$4-resonance effects in $K_{\rm crit}$5 interactions, and that species-dependent freeze-out would require a more detailed hadronic transport treatment (0710.4476).

Transport and microscopic rate-equation studies provide a complementary picture. A coarse-grained UrQMD analysis reconstructed a chemical freeze-out hypersurface microscopically from pion genealogies and found that the chemical freeze-out at all energies coincides with

$K_{\rm crit}$6

while criteria such as $K_{\rm crit}$7 and $K_{\rm crit}$8 are limited to higher collision energies (Reichert et al., 2020). This is a dynamical, transport-based reconstruction of chemical freeze-out rather than the kinetic decoupling criterion used in hydrodynamics.

A microscopic Boltzmann treatment of strange hadrons in a homogeneous, isotropically expanding hadron gas also framed chemical freeze-out dynamically through the competition of reaction rates and cooling rates. There the relevant condition is $K_{\rm crit}$9, or equivalently $\theta \equiv \partial_\mu u^\mu,$0, for each strange species $\theta \equiv \partial_\mu u^\mu,$1. In that study, $\theta \equiv \partial_\mu u^\mu,$2 freeze out sequentially near $\theta \equiv \partial_\mu u^\mu,$3, but in a rapidly expanding system the differences are compressed into a narrow temperature window, so the process appears sudden and nearly simultaneous at the level of final yields (Singh et al., 2020).

At lower SIS18 energies, BUU transport calculations of $\theta \equiv \partial_\mu u^\mu,$4 and $\theta \equiv \partial_\mu u^\mu,$5 mesons argued for a continuous dynamical kinetic freeze-out rather than a sharp decoupling hypersurface. There the fitted transverse-momentum slope parameters become time-independent after about $\theta \equiv \partial_\mu u^\mu,$6 fm/$\theta \equiv \partial_\mu u^\mu,$7, while interaction rates persist until about $\theta \equiv \partial_\mu u^\mu,$8 fm/$\theta \equiv \partial_\mu u^\mu,$9, indicating that spectra can freeze while collisions continue at a diminished dynamical importance (Rabe et al., 2019). This suggests that “freeze-out” in transport is often a broad distribution of last interactions rather than a sharply defined surface.

6. Interpretation, limitations, and broader significance

Across the hydrodynamic literature, the most consistent interpretation is that dynamical freeze-out captures the physical breakdown of hydrodynamics through a competition of local scattering and expansion rates. In the pion-based RHIC and LHC study, comparison with RHIC data indicates that the system indeed decouples when the expansion rate becomes comparable to the pion scattering rate, with $\Gamma$00–$\Gamma$01 (0710.4476). Event-by-event ideal hydrodynamics similarly found that $\Gamma$02 reproduces measured particle spectra reasonably well (Holopainen et al., 2012). This supports the kinetic-theory expectation that hydrodynamics fails when the local mean free time is comparable to the expansion time.

At the same time, several limitations are common. The foundational studies use ideal hydrodynamics with no shear or bulk viscosity and no viscous corrections in the distribution function [(0710.4476); (Holopainen et al., 2012); (Ahmad et al., 2016)]. Most implementations assume boost invariance and are restricted to central or mid-central collisions, although later work extended the framework to peripheral EKRT simulations and exact non-boost-invariant conformal solutions (Hirvonen et al., 2022, Bradley et al., 2024). The microscopic rate is usually taken for pions and then applied to all species, and no hadronic afterburner is used in the hydrodynamic-only setups [(0710.4476); (Holopainen et al., 2012); (Ahmad et al., 2016)]. Event-by-event dynamical surfaces can develop unphysical small-scale structures when the initial fluctuations are too sharp (Holopainen et al., 2012).

The phenomenological impact is therefore nuanced. For large A+A collisions and soft hadronic spectra and flow observables, a tuned constant-temperature freeze-out often approximates a dynamical criterion very well [(0710.4476); (Holopainen et al., 2012); (Ahmad et al., 2016)]. This suggests that the success of constant-$\Gamma$03 freeze-out in such observables is not evidence against a microscopic rate-based picture; rather, the tuned isotherm may effectively mimic the integrated consequences of dynamical decoupling. However, peripheral collisions, flow–$\Gamma$04 correlations, exact-solution studies with strong gradients, and transport analyses all indicate that the distinction becomes more important when system sizes are small, gradients are large, or observables are more directly sensitive to the space–time structure of decoupling (Hirvonen et al., 2022, Bradley et al., 2024, Rabe et al., 2019).

A plausible implication is that dynamical freeze-out is most consequential in regimes where the hydrodynamic description is marginal: peripheral A+A, forward rapidity, and small systems. This interpretation is directly supported by the exact conformal solution study, which highlights the importance of accurately describing freeze-out in collisions with large flow gradients, particularly in small systems (Bradley et al., 2024). Within the heavy-ion context, dynamical freeze-out therefore functions both as a conceptual correction to ad hoc isothermal particlization and as a practical framework for connecting hydrodynamic evolution to microscopic kinetics through local breakdown conditions.