Longitudinal Hydrodynamic Evolution

Updated 25 January 2026

Longitudinal hydrodynamic evolution is the study of the time and spatial dynamics of matter expanding along the beam direction, emphasizing rapidity profiles and velocity gradients.
It incorporates Landau and Bjorken paradigms to model initial conditions, viscous corrections, and event-by-event fluctuations that affect observable particle correlations.
Modern 3D hydrodynamic simulations leverage these models to accurately extract QGP transport coefficients and interpret rapidity-dependent experimental data.

Longitudinal hydrodynamic evolution refers to the temporally and spatially resolved dynamics of matter expanding along the beam (longitudinal) direction in high-energy collisions. Unlike transverse evolution, longitudinal hydrodynamics directly controls the mapping from initial rapidity distributions and momentum asymmetries into final-state observables such as rapidity profiles, anisotropic flows, and particle correlations. Heavy-ion collision phenomenology distinguishes between two limiting paradigms: Landau's strongly stopped, locally thermalized “pancake” initial condition featuring rapid initial compression and subsequent 3D expansion, and Bjorken's boost-invariant scaling flow, with matter produced in a rapidity plateau and expansion dominated by longitudinal velocity gradients. Recent advances supplement these with event-by-event fluctuations, nontrivial correlations, viscous corrections, and coupling to electromagnetic fields and baryon currents.

1. Initial Conditions and Paradigms of Longitudinal Expansion

Longitudinal hydrodynamics is initialized with either Landau or Bjorken conditions. Landau initial states generate a Gaussian energy density profile in space-time rapidity $\eta_s$ , $H(\eta_s;\Delta)\equiv\exp(-\eta_s^2/(2\Delta^2))$ , often with transverse inhomogeneity from Glauber Monte Carlo methods (Sen et al., 2014). The flow velocity is parameterized as $u^\mu = \gamma(1, v_x, v_y, v_z)$ , with $v_z(x,y) = \tanh y_f(x,y)$ for each transverse coordinate. The degree of baryon stopping and local asymmetry between projectile and target participants controls the local rapidity shift and the mapping of orbital angular momentum into the fluid (Ryu et al., 2021).

Bjorken initial conditions assume full transparency (no net baryon stopping), yielding a uniform plateau in rapidity, $dN/dy \approx$ constant, and initial flow velocity $u^\mu=(\cosh\eta_s,0,0,\sinh\eta_s)$ . Fluctuating and string-driven models (e.g., AMPT-based) introduce eventwise variation in the distribution of color flux tubes or strings along $\eta_s$ , producing essential initial-state fluctuations (Pang et al., 2015).

2. Hydrodynamic Equations and Viscous Corrections

Longitudinal evolution is governed by energy–momentum conservation $\partial_\mu T^{\mu\nu}=0$ , with $T^{\mu\nu}=(e+p)u^\mu u^\nu + p g^{\mu\nu}$ for ideal fluids. Viscous dynamics introduces shear and bulk stress: $T^{\mu\nu}=e\,u^\mu u^\nu-(P{+}\Pi)\Delta^{\mu\nu}+\pi^{\mu\nu}$ , obeying relaxation-type second-order equations (Israel–Stewart, DNMR) (Monnai et al., 2011, Du, 2021): $\tau_\pi D\pi^{\langle\mu\nu\rangle} + \pi^{\mu\nu} = 2\eta \sigma^{\mu\nu} + ... + \xi^{\mu\nu}$ with fluctuating stochastic noise $\xi^{\mu\nu}$ and constraint conditions arising from the underlying equation of state, bulk viscosity, and baryon diffusion.

Longitudinal expansion in Milne coordinates $(\tau,\eta_s)$ couples the rapidity-dependent pressure gradients to the build-up of longitudinal flow—nontrivial acceleration when $\partial_{\eta_s}P \neq 0$ —and entropy flux from midrapidity outward. For non-boost invariant evolution, derivatives in $\tau$ and $\eta_s$ mix in the comoving derivative and expansion scalar.

3. Transition to Boost Invariance and Dynamical Correlations

In realistic 3D hydrodynamic scenarios, the system approaches, but does not fully reach, boost invariance even at late times. The flow rapidity $y_f(\tau,\eta_s)$ evolves slowly toward space-time rapidity $y_s$ , with local deviations $\delta(\tau,\eta_s) = y_f/y_s - 1$ persisting at freezeout ( $\sim$ 0.3–0.4 at midrapidity) (Sen et al., 2014). These violations are strongly correlated to both the magnitude of the transverse flow $\langle v_T\rangle$ and elliptic flow $v_2$ , so more energetic or anisotropic regions display greater non-boost invariant behavior.

This incomplete development of longitudinal flow invalidates the assumption that collective expansion is strictly Bjorken-like near midrapidity, which underpins traditional methods for extracting quark-gluon plasma transport coefficients such as $\eta/s$ .

4. Fluctuations, Rapidity-Decorrelation, and Collective Observables

Initial longitudinal fluctuations and hydrodynamic (thermal) noise both introduce decorrelation of anisotropic flows as a function of pseudo-rapidity. The event-plane angle $\Psi_n(\eta)$ and flow amplitude $v_n(\eta)$ acquire a twist and random variation along $\eta$ , quantifiable via the factorization ratio $r_n(\eta_a,\eta_b) = \langle V_n(-\eta_a) V_n^*(\eta_b)\rangle/\langle V_n(\eta_a) V_n^*(\eta_b)\rangle$ (Pang et al., 2015, 1901.10120, Sakai et al., 2021). Hydrodynamic evolution preserves but further propagates these initial twists. Hydrodynamic simulations matching experimental data require both sources of rapidity fluctuation (Sakai et al., 2021).

Long-range rapidity correlations in multi-particle production—measured via factorial moments over well-separated rapidity bins—are predicted by hydrodynamics to be fully determined by total multiplicity fluctuations, with characteristic “sum-rule” constraints (Bialas et al., 2011). Any violation signals additional, non-hydrodynamic sources of correlation (e.g. strings, independent clusters).

5. Viscosity, Baryon Diffusion, Magnetohydrodynamics, and Critical Dynamics

Viscous corrections alter the longitudinal expansion via entropy production and entropy flux. Shear and bulk viscosities are implemented using kinetic-theory estimates and relaxation times, with bulk viscosity in the EOS crossover region producing observable modification in freezeout lifetimes and emission radii (Efaaf et al., 2010, Monnai et al., 2011, Monnai et al., 2011, Du, 2021). Baryon diffusion, modeled in DNMR theory, broadens and fills mid-rapidity baryon distributions, though with minimal impact on total energy-momentum conservation (Du, 2021).

Resistive relativistic magnetohydrodynamics (RRMHD) provides analytic solutions for the impact of electromagnetic fields and finite conductivity, introducing longitudinal acceleration and faster cooling relative to ideal Bjorken flow (Moghaddam et al., 2020). In this context, the local acceleration parameter $\lambda(\tau,\eta)=Y(\tau,\eta)/\eta$ captures deviations from ideal scaling.

6. Spin Polarization and Angular Momentum Injection

Spin polarization observables—especially global polarization of $\Lambda$ and $\bar{\Lambda}$ hyperons—probe the coupling of orbital angular momentum (OAM) and vorticity to the longitudinal hydrodynamic flow (Ryu et al., 2021). Mapping the collision geometry into macroscopic hydrodynamic fields requires local energy-momentum conservation and a parameter $f$ for the fraction of initial longitudinal momentum transferred into fluid velocity. Simultaneous agreement with the slope of pion directed flow and $\Lambda$ polarization uniquely constrains $f$ and the midrapidity OAM (typically $100$– $200\,\hbar$ at RHIC energies for central collisions).

Distinct gradient contributions—thermal vorticity, $\mu_B/T$ gradients (baryon-induced polarization), shear-induced polarization—can invert the hierarchy between baryons and anti-baryons, matching experimental trends.

7. Experimental Probes and Phenomenological Implications

Dedicated measurements of rapidity-dependent observables discriminate between Landau and Bjorken evolution. Transverse HBT radii $R_{side}(y)$ vary steeply with $y$ for Landau-type initial states, remaining flat under Bjorken scaling (Sen et al., 2014, Efaaf et al., 2010). Jet quenching observables $R_{AA}(y)$ and rapidity-dependent momentum correlators directly reflect the evolving longitudinal geometry. Flatness of long-range multiparticle rapidity correlations indicates hydrodynamic origin, whereas observable decorrelation signals non-equilibrium or string-driven sources (Bialas et al., 2011).

Extraction of QGP transport coefficients requires full (3+1)D event-by-event hydrodynamic simulation, with explicit inclusion of longitudinal flow variations, fluctuations, and non-boost invariant effects. Ignoring these contributions systematically biases the interpretation of experimental data, underestimating $\eta/s$ and mis-attributing features of rapidity-dependent observables.

In summary, longitudinal hydrodynamic evolution incorporates initial-state rapidity structure, dynamically generated correlations, viscosity and baryon transport, and electromagnetic field coupling, producing complex, non-boost invariant dynamical patterns. Contemporary phenomenology demands event-resolved, fully three-dimensional fluctuating hydrodynamic frameworks for precision modeling and robust comparison to experimental data on rapidity-dependent, correlation-based, and polarization observables (Sen et al., 2014, Pang et al., 2015, Sakai et al., 2021, Ryu et al., 2021, Efaaf et al., 2010, Du, 2021, Moghaddam et al., 2020, Bialas et al., 2011, 1901.10120, Monnai et al., 2011, Monnai et al., 2011, Sakai et al., 2018, Yu et al., 2017).