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Perfect Fluids

Jan de Boer, Jelle Hartong, Niels A. Obers, Watse Sybesma, Stefan Vandoren

TL;DR

The paper develops a universal framework for perfect fluids that possess translation and rotation symmetry but lack boost invariance, introducing the kinetic mass density $\rho$ to define a general energy-momentum tensor and to capture velocity-dependent thermodynamics. It shows how to recover relativistic, Carrollian, Galilean, and Bargmann limits, derives a conserved entropy current, and presents a generalized Euler equation that includes corrections from $\rho$. A central result is a universal speed-of-sound expression for boostless fluids, with specialized forms under Lifshitz scaling $z$, and it is applied to an ideal Lifshitz gas to obtain explicit classical (Boltzmann) and quantum (Bose/Fermi) results, including explicit formulas for $v_s^2$ and the relevant thermodynamic quantities. The work also establishes a no-go theorem for perfect Schrödinger fluids with $z\neq 2$, discusses hyperscaling violation and charge anomalous dimension, and provides a hydrostatic geometric formulation; together these findings offer a roadmap for exploring Lifshitz hydrodynamics, holographic realizations, and potential experimental tests of boostless fluid dynamics.

Abstract

We develop a new theory of perfect fluids with translation and rotation symmetry, which is also applicable in the absence of any type of boost symmetry. It involves introducing a new fluid variable, the kinetic mass density, which is needed to define the most general energy-momentum tensor for perfect fluids. Our theory leads to corrections to the Euler equations for perfect fluids that might be observable in hydrodynamic fluid experiments. We also derive new expressions for the speed of sound in perfect fluids. Our theory reduces to the known perfect fluid models when boost symmetry is present. It can also be adapted to (non-relativistic) scale invariant fluids with critical exponent $z$. We show that perfect fluids cannot have Schrödinger symmetry unless $z=2$. For generic values of $z$ there can be fluids with Lifshitz symmetry, and as a concrete example, we work out in detail the thermodynamics and fluid description of an ideal gas of Lifshitz particles and compute the speed of sound for the classical and quantum Lifshitz gasses.

Perfect Fluids

TL;DR

The paper develops a universal framework for perfect fluids that possess translation and rotation symmetry but lack boost invariance, introducing the kinetic mass density to define a general energy-momentum tensor and to capture velocity-dependent thermodynamics. It shows how to recover relativistic, Carrollian, Galilean, and Bargmann limits, derives a conserved entropy current, and presents a generalized Euler equation that includes corrections from . A central result is a universal speed-of-sound expression for boostless fluids, with specialized forms under Lifshitz scaling , and it is applied to an ideal Lifshitz gas to obtain explicit classical (Boltzmann) and quantum (Bose/Fermi) results, including explicit formulas for and the relevant thermodynamic quantities. The work also establishes a no-go theorem for perfect Schrödinger fluids with , discusses hyperscaling violation and charge anomalous dimension, and provides a hydrostatic geometric formulation; together these findings offer a roadmap for exploring Lifshitz hydrodynamics, holographic realizations, and potential experimental tests of boostless fluid dynamics.

Abstract

We develop a new theory of perfect fluids with translation and rotation symmetry, which is also applicable in the absence of any type of boost symmetry. It involves introducing a new fluid variable, the kinetic mass density, which is needed to define the most general energy-momentum tensor for perfect fluids. Our theory leads to corrections to the Euler equations for perfect fluids that might be observable in hydrodynamic fluid experiments. We also derive new expressions for the speed of sound in perfect fluids. Our theory reduces to the known perfect fluid models when boost symmetry is present. It can also be adapted to (non-relativistic) scale invariant fluids with critical exponent . We show that perfect fluids cannot have Schrödinger symmetry unless . For generic values of there can be fluids with Lifshitz symmetry, and as a concrete example, we work out in detail the thermodynamics and fluid description of an ideal gas of Lifshitz particles and compute the speed of sound for the classical and quantum Lifshitz gasses.

Paper Structure

This paper contains 20 sections, 164 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Perfect fluids
  3. Thermodynamics and kinetic mass density
  4. A new energy-momentum tensor
  5. Entropy current and Euler equation
  6. Boost invariant equations of state
  7. Lorentz boost invariance
  8. Carroll boost invariance
  9. Massless Galilei boost invariance
  10. Massive Galilei boost invariance
  11. Scale invariance
  12. No-go theorem for perfect Schrödinger fluids
  13. Hyperscaling violation and charge anomalous dimension
  14. Geometry and Equilibrium Partition Function
  15. Speed of sound
  16. ...and 5 more sections

Figures (2)

  • Figure 1: We give a representation of the speed of sound of the Bose gas for $T\leq T_{c}$. There are three distinct regions: speed of sound as in \ref{['eq:sosbelowtc']}, speed of sound is zero $v_{B}^{2}=0$, and no critical temperature and thus speed of sound as in \ref{['eq:speedofsoundBF']}.
  • Figure 2: We present two different situations for the speed of sound of the Bose gas. The upper line ($d=3$ and $z=2$) corresponds to a finite speed of sound in the condensate phase (for $T>0$), whereas the lower line ($d=3$ and $z=5/2$) corresponds to a speed of sound that is equal to zero anywhere in the condensate phase.