Non-Equilibrium Trace Anomaly And Bulk Viscosity in Heavy Ion Collisions From Kinetic Theory

This paper investigates the far-from-equilibrium dynamics of a relativistic massive gas with various quantum statistics undergoing Bjorken expansion using the method of moments, revealing that the trace anomaly and bulk viscosity exhibit non-monotonic time evolution, depend sensitively on particle statistics and initial chemical potential, and converge to a universal late-time attractor regardless of initial non-equilibrium configurations.

The Gist

The Big Picture: A Cosmic Balloon Pop

Imagine a heavy ion collision (like smashing two gold atoms together at nearly the speed of light) as a tiny, super-hot fireball being created in a lab. This fireball is made of a "soup" of particles (quarks and gluons) that behaves like a fluid.

The scientists in this paper wanted to understand how this fireball behaves immediately after it is created, before it settles down into a calm, stable state. They were particularly interested in two things:

1. The "Trace Anomaly": A measure of how much the particles are interacting and breaking the rules of perfect symmetry.
2. Bulk Viscosity: Think of this as the "internal friction" or "stickiness" of the fluid when it is being squeezed or stretched.

The Setup: The Stretching Hose

The researchers modeled the fireball using a concept called Bjorken expansion.

• The Analogy: Imagine a long, thin hose filled with hot water. If you stretch the hose very quickly lengthwise, the water inside gets thinner and cooler.
• The Reality: In the collision, the fireball expands incredibly fast in one direction (lengthwise). This rapid stretching pushes the system far away from "equilibrium" (a state of calm balance).

To study this, the team used Kinetic Theory, which is like tracking every single billiard ball in a game rather than just looking at the pool table as a whole. They looked at three different types of "balls" (particles) based on how they behave in nature:

1. Maxwell-Boltzmann: Like standard, predictable marbles.
2. Fermi-Dirac: Like particles that hate being in the same spot (like people in a crowded elevator).
3. Bose-Einstein: Like particles that love to clump together (like a crowd rushing toward a stage).

The Method: The "Relaxation" Game

The team used a mathematical tool called the Relaxation-Time Approximation (RTA).

• The Analogy: Imagine a room full of people running in random directions (chaos). Suddenly, a bell rings, and everyone tries to calm down and stand in a neat line (order). The "Relaxation Time" is how long it takes for the chaos to turn into order.
• The Study: They solved complex equations to see how the "messiness" of the fireball changes over time as it expands and as collisions between particles try to fix the mess.

Key Findings: What They Discovered

1. The "Bumpy" Ride of the Trace Anomaly
The "Trace Anomaly" (a measure of interaction strength) didn't just go up or down smoothly.

• The Behavior: It shot up quickly at the very beginning, then dipped down around the time the "relaxation" started to kick in, and then slowly rose again.
• The Analogy: It's like driving a car over a hill. You go up fast, dip into a valley, and then climb the next slope. This "bump and dip" happens because the fireball is expanding so fast that it fights against the particles trying to settle down.

2. The "Stickiness" Depends on the Crowd
The "Bulk Viscosity" (the stickiness/friction) behaved differently depending on which type of particle statistics were used.

• The Result: The "clumping" particles (Bose-Einstein) showed the strongest friction effects, while the "hating-to-be-close" particles (Fermi-Dirac) showed the least.
• The Takeaway: The rules of the crowd matter. How the particles interact with each other changes how much the fluid resists being stretched.

3. More "Chemical Potential" = More Chaos
They tested what happens if they start with a higher "chemical potential" (which basically means a higher density of particles).

• The Result: The more crowded the fireball started, the harder it was for it to calm down. The "friction" (bulk pressure) got much stronger, and it took longer for the system to return to a stable state.
• The Analogy: If you try to calm down a room with 10 people, it's easy. If you try to calm down a room with 1,000 people, it takes much longer, and the chaos is much more intense.

4. The "Attractor" Phenomenon
This is one of the most interesting parts. They started the simulation with completely random, messy initial conditions (some particles moving fast, some slow, in random directions).

• The Result: Even though they started with different messes, as time went on, all the different scenarios started to look the same. The "stickiness" and the "pressure differences" eventually converged to a single, predictable path.
• The Analogy: Imagine dropping a drop of red ink, a drop of blue ink, and a drop of green ink into a swirling river. At first, they are all in different places. But as the river flows, they all get stretched and mixed until they follow the exact same path downstream. The system "forgets" its messy start and finds a common rhythm.

The Conclusion

The paper concludes that while the fireball eventually settles down into a predictable state (the "attractor"), the journey there is complex.

• The bulk pressure (friction) and pressure differences eventually calm down and look the same regardless of how messy the start was.
• However, the Trace Anomaly (the interaction measure) remembers the messy start for a longer time. It is more sensitive to the history of the explosion.

In short, the universe has a way of smoothing out the chaos of a particle collision, but the "memory" of that initial chaos lingers in specific ways that scientists need to account for to understand the physics of the early universe and heavy ion collisions.

Technical Summary

Technical Summary: Non-Equilibrium Trace Anomaly And Bulk Viscosity in Heavy Ion Collisions From Kinetic Theory

Problem Statement
Heavy ion collisions at facilities like RHIC and LHC create a quark-gluon plasma (QGP) that undergoes rapid expansion, driving the system far from local thermal equilibrium. While lattice QCD provides rigorous results for equilibrium properties, such as the trace anomaly ($\Theta^\mu_\mu = \varepsilon - 3P$) and bulk viscosity, these equilibrium calculations do not capture the dynamical evolution of the fireball. The trace anomaly is a crucial observable for understanding conformal symmetry breaking and is intimately linked to bulk viscosity, which governs the system's response to expansion and compression. However, in the rapidly evolving, non-equilibrium environment of a heavy ion collision, the trace of the energy-momentum tensor receives contributions from both the equilibrium breaking of conformal symmetry and non-equilibrium bulk viscous dynamics. The specific time evolution of these quantities, particularly for massive particles obeying different quantum statistics (Maxwell-Boltzmann, Bose-Einstein, and Fermi-Dirac), requires a kinetic theory approach that goes beyond truncated hydrodynamic approximations.

Methodology
The authors investigate a relativistic massive gas undergoing boost-invariant Bjorken expansion. The microscopic dynamics are described by the relativistic Boltzmann equation within the Relaxation Time Approximation (RTA). The authors employ the method of moments, a framework pioneered by Grad and adapted for relativistic systems by Denicol, Yan, and Blaizot.

Key methodological steps include:

1. Moment Construction: The single-particle distribution function $f_p$ is reconstructed via an infinite hierarchy of moments $\rho_{n,l}$, defined as integrals over momentum space involving powers of energy ($p^0$) and angular anisotropy ($p_z/p^0$).
2. Evolution Equations: By taking moments of the Boltzmann equation, the authors derive a coupled hierarchy of differential equations for $\rho_{n,l}$. This hierarchy is truncated appropriately to solve for the evolution of thermodynamic quantities.
3. Matching Conditions: The Landau matching conditions are used to define the effective temperature $T(\tau)$ and chemical potential $\mu(\tau)$ by equating the non-equilibrium energy density and particle number density to their equilibrium counterparts.
4. Statistics: The study compares three equilibrium distribution functions: Maxwell-Boltzmann (MB), Fermi-Dirac (FD), and Bose-Einstein (BE).
5. Initial Conditions: Two scenarios are explored:
   • Equilibrium Initialization: The system starts in local equilibrium ($\rho_{n,l} = \rho^{eq}_{n,l}$) and evolves under expansion.
   • Non-Equilibrium Initialization: The system starts with random perturbations to the moments ($\rho_{n,l} \neq \rho^{eq}_{n,l}$), while strictly maintaining the Landau matching constraints for energy and particle number.

Key Results

1. Time Evolution of Temperature and Chemical Potential: Both $T$ and $\mu$ decrease over time, following a power-law cooling characteristic of Bjorken flow, with a subtle modification in the cooling rate around the relaxation time scale $\tau \sim \tau_R$ where collisional effects become comparable to expansion. The evolution rates are sensitive to particle statistics.
2. Non-Monotonic Trace Anomaly: The normalized trace anomaly $\Theta^\mu_\mu/T^4$ exhibits a distinct non-monotonic behavior. It shows a local maximum at very early times, followed by a dip around $\tau \sim \tau_R$, and subsequently increases monotonically. This behavior arises from the interplay between the decreasing temperature and the evolving moments.
3. Bulk Viscous Pressure ($\Pi$):
   • Starting from equilibrium, $\Pi$ is zero initially but develops non-zero values due to rapid longitudinal expansion. It reaches a maximum magnitude around $\tau \sim \tau_R$ and relaxes back to zero at late times.
   • Statistical Dependence: The magnitude of the scaled bulk pressure $|\Pi/P_0|$ depends significantly on statistics, following the ordering $|\Pi_{BE}/P_0| > |\Pi_{MB}/P_0| > |\Pi_{FD}/P_0|$.
   • Chemical Potential Dependence: A larger initial chemical potential ($\alpha_0 = \mu_0/T_0$) significantly enhances the magnitude of both $\Pi$ and $\Theta^\mu_\mu/T^4$. Systems with higher initial chemical potentials take longer to thermalize, evidenced by broader peaks in $\Pi$ and a delayed return to equilibrium.
4. Attractor Behavior: When initialized with random non-equilibrium configurations, the scaled bulk pressure $\Pi/P_0$ and the pressure anisotropy ratio $P_L/P_T$ converge to a common late-time solution, regardless of the initial perturbations. This demonstrates the existence of hydrodynamic attractors in the RTA kinetic theory for non-conformal systems.
5. Trace Anomaly Sensitivity: Unlike the bulk pressure and pressure anisotropy, the normalized trace $\Theta^\mu_\mu/T^4$ retains a noticeable spread among different trajectories even at late times. This suggests that the trace anomaly is more sensitive to the detailed structure of the momentum distribution and retains memory of early-time dynamics for a longer duration.

Significance and Claims
The paper claims to provide a self-consistent, microscopic determination of the trace anomaly and bulk viscous pressure in a far-from-equilibrium setting without relying on the approximations inherent in truncated hydrodynamic theories. By directly computing time-dependent moments, the authors demonstrate that:

• The interplay between equilibrium conformal symmetry breaking and non-equilibrium dynamics creates a complex, non-monotonic evolution of the trace anomaly that differs from static lattice QCD predictions.
• The system exhibits attractor-like behavior for bulk pressure and pressure anisotropy, validating the concept of hydrodynamic attractors in massive, non-conformal systems.
• The trace anomaly, however, does not fully converge to a universal attractor within the studied time window, indicating that non-equilibrium corrections to the equation of state require careful treatment as they retain information about the system's evolution history.
• The results highlight the critical role of particle statistics and initial chemical potential in determining the magnitude and relaxation time of dissipative effects in the QGP.

The study concludes that these kinetic theory results are essential for accurately modeling the transport properties of the evolving QGP, particularly for understanding how bulk viscosity influences the expansion and flow harmonics in heavy ion collisions.