Ideal-Gas Approach to Hydrodynamics

Abstract

Transport is one of the most important physical processes in all energy and length scales. Ideal gases and hydrodynamics are, respectively, two opposite limits of transport. Here, we present an unexpected mathematical connection between these two limits; that is, there exist situations that the solution to a class of interacting hydrodynamic equations with certain initial conditions can be exactly constructed from the dynamics of noninteracting ideal gases. We analytically provide three such examples. The first two examples focus on scale-invariant systems, which generalize fermionization to the hydrodynamics of strongly interacting systems, and determine specific initial conditions for perfect density oscillations in a harmonic trap. The third example recovers the dark soliton solution in a one-dimensional Bose condensate. The results can explain a recent puzzling experimental observation in ultracold atomic gases by the Paris group and make further predictions for future experiments. We envision that extensive examples of such an ideal-gas approach to hydrodynamics can be found by systematical numerical search, which can find broad applications in different problems in various subfields of physics.

Alternate abstract:

Plain Language Summary

Understanding the transport of matter is important at all scales, from quarks to galaxies. For all types of transport, there are two extremes: the hydrodynamic regime, where collisions among particles dominate the dynamics, and the collisionless regime, where interactions among particles do not factor in (such as in ideal gases). It is well accepted that the physics in these two regimes is drastically different. Here, we upend that conventional wisdom and report that the solutions to certain hydrodynamic equations can be directly constructed from solutions to the dynamics of ideal gases, thus providing analytical results in a regime where previously only numerical solutions were possible.

Specifically, we show that, for certain initial conditions, the solution to the one-particle Liouville equation for ideal gases can be used to construct a set of hydrodynamic equations if a generalized local-equilibrium condition can be satisfied. As an application of this result, we explain recent experimental observations in 2D ultracold Bose gases, where a puzzling periodic density oscillation has recently been observed.

We envision that this mapping between the hydrodynamic equation and the free-particle Liouville equation can find more broad applications in different branches of physics.