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Thermodynamic stability of superflows in General Relativity and Newtonian gravity

Lorenzo Gavassino

TL;DR

This paper recasts frictionless superfluid motion as a local minimum of the grand potential $\Omega=U-T_\star S-\mu_\star^X N_X$ in a general relativistic setting and derives a comprehensive stability criterion for multicomponent superfluids. By formulating relativistic hydrodynamics with multiple order parameters and entrainment, it identifies stationary-point conditions and constructs a positive-definite information-current-based quadratic form $2\mathcal{E}$ that signals true minima against hydrodynamic fluctuations. The authors then specialize to a single-component fluid and derive explicit matrix positivity conditions, followed by a non-relativistic limit that yields Newtonian analogues and connections to Andreev-Melnikovsky-type criteria. The results yield a set of necessary, physically interpretable inequalities on thermodynamic and kinetic quantities, applicable to systems as diverse as finite-temperature superfluids and neutron-star interiors, where Landau's criterion may fail. While macroscopic in scope, the work provides a framework to bound flow stability and motivates future work on gapless superfluids and astrophysical applications.

Abstract

Landau's criterion for superfluidity is a special case of a broader principle: A moving fluid cannot be stopped by frictional forces if its state of motion is a local minimum of the grand potential. We employ this general thermodynamic criterion to derive a set of inequalities that any superfluid mixture (with an arbitrary number of order parameters) must satisfy for a certain state of motion to be long-lived and unimpeded by friction. These macroscopic constraints complement Landau's original criterion, in that they hold at all temperatures, and remain valid even for gapless superfluids. Unfortunately, they are only necessary conditions for the existence of a frictionless hydrodynamic motion, since they presuppose the validity of a fluid description. However, they do provide sufficient conditions for stability against stochastic hydrodynamic fluctuations. We first formulate our analysis within the framework of General Relativity, and then we take the Newtonian limit.

Thermodynamic stability of superflows in General Relativity and Newtonian gravity

TL;DR

This paper recasts frictionless superfluid motion as a local minimum of the grand potential in a general relativistic setting and derives a comprehensive stability criterion for multicomponent superfluids. By formulating relativistic hydrodynamics with multiple order parameters and entrainment, it identifies stationary-point conditions and constructs a positive-definite information-current-based quadratic form that signals true minima against hydrodynamic fluctuations. The authors then specialize to a single-component fluid and derive explicit matrix positivity conditions, followed by a non-relativistic limit that yields Newtonian analogues and connections to Andreev-Melnikovsky-type criteria. The results yield a set of necessary, physically interpretable inequalities on thermodynamic and kinetic quantities, applicable to systems as diverse as finite-temperature superfluids and neutron-star interiors, where Landau's criterion may fail. While macroscopic in scope, the work provides a framework to bound flow stability and motivates future work on gapless superfluids and astrophysical applications.

Abstract

Landau's criterion for superfluidity is a special case of a broader principle: A moving fluid cannot be stopped by frictional forces if its state of motion is a local minimum of the grand potential. We employ this general thermodynamic criterion to derive a set of inequalities that any superfluid mixture (with an arbitrary number of order parameters) must satisfy for a certain state of motion to be long-lived and unimpeded by friction. These macroscopic constraints complement Landau's original criterion, in that they hold at all temperatures, and remain valid even for gapless superfluids. Unfortunately, they are only necessary conditions for the existence of a frictionless hydrodynamic motion, since they presuppose the validity of a fluid description. However, they do provide sufficient conditions for stability against stochastic hydrodynamic fluctuations. We first formulate our analysis within the framework of General Relativity, and then we take the Newtonian limit.

Paper Structure

This paper contains 22 sections, 2 theorems, 52 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Multicomponent superfluids in general relativity
  3. The macroscopic degrees of freedom of a superfluid mixture
  4. Constitutive relations
  5. Equations of motion
  6. Some convenient notation
  7. Stability criterion in general relativity
  8. Stationary points of the grand-potential
  9. Actual minima
  10. Vector decomposition in the rest frame of the bath
  11. Main Theorem
  12. Application to single-component superfluids
  13. Derivation of the quadratic form
  14. Matrix representation of the quadratic form
  15. Non-relativistic case
  16. ...and 7 more sections

Key Result

Theorem 1

A superfluid state $\{u^\nu (x^\alpha),T(x^\alpha),\mu^A(x^\alpha),\varphi^I(x^\alpha)\}$ that fulfills equillo is a local minimum of the grand potential $\Omega =U-\mu_\star^X N_X$ only if the quadratic form is non-negative at every spacetime point, and for all admissible values of $\{\delta \mu^X,\delta u^\nu,\delta w^I_\nu,V^\nu \}$. If $\mathcal{E}$ is strictly positive definite, then stateme

Figures (1)

  • Figure 1: Thermodynamic explanation for the long lifetime of superflows. Left panel: A simple experimental setup, where a superfluid is confined in a circular ring-shaped pipe (with negligible thickness). Right panel: In this experimental setup, the grand potential \ref{['grandone']} admits a discrete list of local minima $\psi_z$, given by \ref{['minimina']}, corresponding to different rotation rates with quantized angular velocity. In the plot, we graph the grand potential $\Omega(\lambda)$ (with $g=\rho=5$) along a continuous one-parameter family of states $\psi(\lambda)=\sum_z c_z(\lambda)\psi_z$, with $c_z =(1-|\lambda{-}z|)\Theta(1-|\lambda{-}z|)$. This family is constructed so that the state $\lambda=z$ has order parameter $\psi_z$, and it thus matches a local minimum (red dots). Mathematically, the grand potential barriers separating the minima exist because changing the winding number of the phase of $\psi$ requires $|\psi|$ to vanish at some location. This entails lifting the order parameter to the top of the Mexican hat potential.

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2