Viscous Gubser flow with conserved charges to benchmark fluid simulations

This paper presents semi-analytical solutions for viscous Gubser flow with conserved charges to benchmark numerical fluid simulations, demonstrating excellent agreement between these solutions and the newly developed CCAKE Smoothed Particle Hydrodynamics code.

The Gist

Imagine you are trying to predict how a giant, super-hot balloon of invisible gas expands and cools down after a massive explosion. In the world of physics, this "balloon" is the Quark-Gluon Plasma (QGP)—a state of matter so hot and dense that atoms can't even exist, only their tiny building blocks (quarks and gluons) swimming around like a super-fluid.

Scientists use complex computer programs (simulations) to model this explosion. But just like you wouldn't trust a weather app that hasn't been tested against real storms, physicists need to make sure their computer codes are accurate before they use them to understand the universe.

This paper is essentially a new "stress test" or "calibration tool" for those computer programs. Here is the breakdown using everyday analogies:

1. The Problem: The "Perfect" vs. The "Real"

For a long time, scientists had a perfect, mathematical recipe (an exact solution) to describe how this hot gas expands. It was like a perfect, frictionless slide. However, real life is messy. The gas has viscosity (it's sticky, like honey) and it carries charges (like electric charge or "baryon number," which is a type of heavy particle count).

Previous tests for computer codes ignored the "stickiness" and the "charges." They were like testing a car on a perfectly flat, empty track. This paper says: "Let's test the car on a muddy, hilly road with heavy cargo."

2. The Solution: The "Gubser Flow" Recipe

The authors created a new, semi-analytical recipe (a mix of math and numbers) called Viscous Gubser Flow with Conserved Charges (VGCC).

• The Analogy: Imagine you are baking a cake.
  • Old Recipe: You just mix flour and water. It's easy to predict how it rises.
  • New Recipe (VGCC): You add sticky honey (viscosity) and heavy chocolate chips (conserved charges).
  • The Result: The cake rises differently. It might get weird bumps or "shoulders" on the side. The authors figured out exactly how this "sticky, chocolate-chip cake" should behave mathematically.

3. The "Freeze-Out" Moment

In heavy-ion collisions, the hot gas eventually cools down and turns back into normal particles (like protons and neutrons). This moment is called freeze-out.

• The Analogy: Think of a hot cup of coffee cooling down. At a certain temperature, the steam stops rising, and the liquid stops moving.
• The Innovation: The authors calculated exactly what the "surface" of this cooling coffee looks like. In the past, if you had charges (like the chocolate chips), the surface wasn't smooth; it warped. They mapped out this warped surface so computer codes can check: "Did my simulation freeze out at the right shape?"

4. The Test: The "CCake" Code

The authors took a specific computer code called ccake (which uses a method called Smoothed Particle Hydrodynamics, or SPH).

• The Analogy: Imagine SPH is like a swarm of bees. Instead of a grid, the computer tracks individual "bees" (particles) moving around.
• The Test: They fed the "sticky, chocolate-chip" recipe into the ccake code.
• The Result: The computer simulation matched the mathematical recipe almost perfectly. The only tiny errors happened when the "bees" got too far apart or the fluid got too chaotic (far from equilibrium), which is expected.

5. Why This Matters

Why do we care about a math recipe for a theoretical explosion?

• Finding the "Critical Point": Scientists are hunting for a "Critical Point" in the QCD phase diagram (a map of matter states). It's like finding the exact temperature and pressure where water turns to steam. To find this, we need to simulate matter with high "chemical potential" (lots of heavy particles).
• Trustworthy Tools: Before we can trust a computer to tell us where this Critical Point is, we need to know the computer works when things get sticky and charged. This paper provides the "gold standard" answer to check against.

Summary in a Nutshell

The authors built a mathematical ruler for a very specific, complex type of fluid explosion. They used this ruler to measure a new computer program (ccake) and found that the program is accurate. This gives scientists confidence that when they use this code to study the early universe or neutron stars, the results will be reliable, even when the fluid is sticky and full of heavy charges.

The Takeaway: They didn't just solve a math problem; they built a quality control check to ensure our best tools for understanding the universe are working correctly.

Technical Summary

1. Problem Statement

Relativistic hydrodynamics is the standard tool for modeling the Quark-Gluon Plasma (QGP) created in heavy-ion collisions. While benchmark tests exist for ideal and viscous hydrodynamics (e.g., Bjorken and Gubser flows), most existing benchmarks assume vanishing chemical potentials (zero net baryon number, strangeness, and electric charge).

However, understanding the QCD phase diagram, locating the critical point, and modeling neutron star mergers requires hydrodynamic codes that accurately handle finite chemical potentials and conserved charges (Baryon number $B$, Strangeness $S$, Electric charge $Q$). Current numerical codes (such as those using Smoothed Particle Hydrodynamics or grid-based methods) lack rigorous unit tests that combine:

1. Viscous effects (shear viscosity).
2. Finite chemical potentials.
3. Conserved charge currents.

Without such benchmarks, it is difficult to verify the numerical accuracy of codes attempting to simulate systems with large chemical potentials, particularly regarding the coupling between the equation of state (EoS), temperature evolution, and charge densities.

2. Methodology

The authors developed a semi-analytical solution for Viscous Gubser Flow with Conserved Charges (VGCC). The methodology involves the following steps:

• Symmetry and Coordinates: They utilize the $SO(3)_q \otimes SU(1,1) \otimes Z_2$ symmetry of Gubser flow. The system is described in Milne coordinates $(\tau, r, \phi, \eta)$ and transformed via Weyl rescaling and coordinate mapping into de Sitter space $(\rho, \theta, \phi, \eta)$. This transformation simplifies the fluid velocity to a static profile ($\hat{u}^\mu = (-1, 0, 0, 0)$), converting partial differential equations (PDEs) into ordinary differential equations (ODEs).
• Conservation Laws:
  • Energy-Momentum: Solved using the Israel-Stewart formalism for viscous hydrodynamics, including a relaxation-type equation for the shear stress tensor $\pi^{\mu\nu}$.
  • Conserved Currents: The authors prove that in exact Gubser symmetry, diffusion currents vanish identically (Appendix A). Consequently, the evolution of charge densities ($n_B, n_S, n_Q$) decouples from the energy-momentum tensor and evolves independently via $D n_Y + n_Y \theta = 0$.
• Coupling via Equation of State (EoS): Although charge evolution is decoupled from momentum, the temperature ($T$) and chemical potentials ($\mu_Y$) are coupled through the EoS. The authors utilize two distinct conformal EoS models:
  1. EoS1: A massless quark-gluon plasma model with explicit $\mu$ dependence.
  2. EoS2: A parameterized conformal model used specifically for code benchmarking.
     These EoS models allow the mapping between hydrodynamic variables ($E, n_Y$) and thermodynamic variables ($T, \mu_Y$).
• Freeze-out Hypersurface: The authors derive the freeze-out hypersurface defined by a constant energy density ($E = E_{FO}$). They calculate the normal vectors to this hypersurface analytically, providing a new metric for testing numerical codes.

3. Key Contributions

• First Semi-Analytical VGCC Solution: The paper presents the first semi-analytical solution for viscous Gubser flow that simultaneously evolves temperature, chemical potentials, and shear stress for systems with conserved charges.
• Proof of Vanishing Diffusion: A rigorous derivation showing that Gubser symmetry forbids diffusion currents, simplifying the conservation equations to simple continuity equations for charge densities.
• New Benchmark for Numerical Codes: The authors provide the freeze-out hypersurface profiles and normal vectors as a new "unit test" for hydrodynamic codes. This tests not just the bulk evolution but also the code's ability to reconstruct complex hypersurfaces.
• Benchmarking ccake: The solution is used to benchmark ccake, a newly developed (2+1)D relativistic viscous hydrodynamic code based on Smoothed Particle Hydrodynamics (SPH) that includes BSQ conserved charges.

4. Results

• Evolution Dynamics:
  • The inclusion of finite chemical potentials significantly pushes the system further from equilibrium compared to the $\mu=0$ case.
  • Temperature Profiles: For large initial ratios of $\mu/T$, the temperature profiles develop "shoulders" away from the core. This is caused by entropy production from large viscous corrections at finite chemical potential, which must compensate for the vanishing $n_Y \mu_Y$ term at the system's edge (via the Gibbs-Duhem relation).
  • Freeze-out: At $\mu \neq 0$, the freeze-out surface is neither isothermal nor isentropic. The entropy density is lower in the core than at the tails, a stark contrast to the $\mu=0$ case.
• Code Benchmarking (ccake vs. Semi-Analytical):
  • Agreement: The numerical solutions from ccake show excellent agreement with the semi-analytical solution for energy density, number density, and fluid velocity.
  • Discrepancies: The largest deviations occur in the off-diagonal shear stress components ($\pi_{xy}$) and at late times ($\tau > 1.6$ fm/c) in the radial region $1-2$ fm. These deviations are attributed to the system becoming far-from-equilibrium, leading to causality violations and negative entropy in SPH particles, causing the simulation to crash.
  • Freeze-out Surface: Despite the eventual crash of the simulation at late times, ccake accurately reproduces the freeze-out hypersurface and normal vectors derived semi-analytically. This confirms that the code is valid within the hydrodynamic regime before non-equilibrium effects dominate.

5. Significance

• Validation of Future Simulations: This work provides a critical "gold standard" for validating hydrodynamic codes intended for the Beam Energy Scan (BES) program at RHIC and future experiments (FAIR, NICA), where finite baryon density is the primary focus.
• Understanding Non-Equilibrium Effects: The results highlight how finite chemical potentials alter the spacetime evolution of the QGP, specifically by inducing non-isentropic freeze-out and modifying temperature profiles in ways not seen in zero-density simulations.
• Tool for Critical Point Search: By enabling accurate simulations of systems with conserved charges, this benchmark supports the search for the QCD critical point, where the interplay between viscosity, chemical potentials, and the EoS is crucial.
• Open Source Availability: The authors provide the Python implementation of the solution and benchmark data, facilitating immediate adoption by the community for testing new hydrodynamic models.

In summary, this paper bridges the gap between exact analytical solutions and complex numerical simulations for viscous fluids with conserved charges, offering a robust framework to verify the accuracy of next-generation heavy-ion collision simulations.