A Nonequilibrium Equation of State for a Turbulent 2D Bose Gas

Abstract

Nonequilibrium equations of state can provide an effective thermodynamic-like description of far-from-equilibrium systems. We experimentally construct such an equation for a direct energy cascade in a turbulent two-dimensional Bose gas. Our homogeneous gas is continuously driven on a large length scale and, with matching dissipation on a small length scale, exhibits a nonthermal but stationary power-law momentum distribution. Our equation of state links the cascade amplitude with the underlying scale-invariant energy flux, and can, for different drive strengths, gas densities, and interaction strengths, be recast into a universal power-law form using scalings consistent with the Gross-Pitaevskii model.

Figures (3)

• Figure 1: The equation of state (EOS) for an equilibrium system (left) and a flux-carrying turbulent system in a steady state, with matching injection and dissipation rate $\epsilon$ (right). For an equilibrium system, coupled to a reservoir, state variables such as pressure, $p$, are functions of the reservoir temperature $T$. For a flux-carrying turbulent system, the amplitude, $n_0$, of the emergent power-law spectrum is a function of the underlying flux $\epsilon$.
• Figure 2: Turbulent steady state and the underlying energy flux. (a) An example of a steady-state momentum distribution $n_k(k)$ for our turbulent 2D gas, held in a square box trap of side length $L=30\,{\upmu{\rm m}}$ and driven on a large length scale ($\sim L$) by a time-periodic force. Between the inverse healing length $k_\xi$ and the dissipation scale $k_{\textrm{D}}$, set by our trap depth, $n_k$ exhibits a power-law spectrum, $\propto n_0\,k^{-\gamma}$, with $\gamma\approx 2.6$ (dashed line). Here, the gas has density $n = 25\, \upmu{\rm m}^{-2}$ and dimensionless interaction strength $\tilde{g} = 0.04$, and the force amplitude is $F_0 = 1.6 \,\zeta/L$, where $\zeta = \hbar^2\tilde{g}n/m$. (b) The corresponding energy flux $\epsilon$. The steady-state particle flux $\Pi$ is measured through atom loss for various $k_{\textrm{D}}$. In agreement with the zeroth law of turbulence, we observe $\Pi \propto 1/k_{\textrm{D}}^2$ (solid line), and thus deduce the $k_{\textrm{D}}$-independent $\epsilon = \Pi\,\hbar^2k_{\textrm{D}}^2/(2m)$ (inset). Note that $n_k$, $\Pi$, and $\epsilon$ are normalized per particle, and the spectrum in (a) was measured with the largest $k_{\textrm{D}}$ value in (b).
• Figure 3: Universal EOS. Here we generalize our measurements of the steady-state $n_k$ to various $\tilde{g}$, $n$, and $\epsilon$. (a) The cascade exponent $\gamma$. Our data show no systematic variation and are consistent with $\gamma = 2.7(1)$ (dotted line and shading). (b) The cascade amplitude $n_0$ (extracted with fixed $\gamma=2.7$) plotted versus $\epsilon$. (c) Plotting $n_0$ versus the dimensionless energy flux $\hbar\epsilon/\zeta^2$ collapses all our data onto a universal EOS. The solid line shows a power-law fit, with exponent $b = 0.47(5)$. Inset: we highlight the two data points corresponding to the steady states established before and after a change in the driving-force amplitude (see text).