An Equation of State for Turbulence in the Gross-Pitaevskii model

Abstract

We report the numerical observation of a far-from-equilibrium equation of state (EOS) in the Gross-Pitaevskii model. We first show that the momentum distribution of the turbulent cascade is well described by wave-turbulent kinetic theory in the appropriate limits. Calculating the energy and particle fluxes $Π_\varepsilon(k)$ and $Π_N(k)$, we show that the turbulent state possesses the hallmarks of a direct energy cascade. Building on this, we show that the GP model encodes a universal EOS in the form of a relationship between the turbulent cascade's momentum distribution amplitude $n_0$ and the energy flux $ε$ in the steady state. We find that in our regime of `mixed' turbulence - where both vortices and waves play a significant role - $n_0\propto ε^{0.67(2)}$, a result that is not captured by any existing theory of turbulence but that agrees with a recent experimental measurement for large energy fluxes. Finally, we find that the concept of quasi-static thermodynamic processes between equilibrium states extends to far-from-equilibrium steady states.

Figures (11)

• Figure 1: The direct turbulent cascade in the GP model. (a) Cartoon of the simulation geometry and the driving protocol. We use a cylindrical box trapping potential of length $L$ and radius $R$, and the energy is injected into the system by applying a time-periodic potential gradient $V_\text{drive}(\boldsymbol{r},t)=U_{\textrm{s}} \sin(\omega t)z/L$ (see text for typical parameters). (b) The build-up of the turbulent cascade; the mode occupation number $N_k$ is shown for various shaking times $t$. At long times, the system is in a steady state with $N_k\propto k^{-\gamma}$ (the dashed line corresponds to $\gamma=3.5$). The inset shows the cascade exponent $\gamma(k)\equiv-\textrm{d}\ln[n(k)]/\textrm{d}\ln{[k]}$, calculated from the continuous momentum distribution $n(k)$; the dashed line is $\gamma(k)=3.5$. Here, the simulation parameters are $L=50\,\upmu{\rm m}$, $R=15\,\upmu{\rm m}$, $U_{\textrm{s}}=1.0\zeta$, $\omega=2\pi\times10\,{\rm Hz}$, and $a=100a_0$ (corresponding to $\xi=1.2\,\upmu{\rm m}$ and $\tau=10\,$ms).
• Figure 2: Energetics of the direct turbulent cascade. (a) Energy input and dissipation rates. We show $\tau\epsilon/(n\zeta)$ where $\epsilon$ is either the energy injection rate calculated as $\epsilon_{\textrm{in}}\equiv \langle Fvn\rangle$ (symbols) or the particle dissipation rate $\dot{N}$ multiplied by $U_{\textrm{D}}/V$ (solid lines). Both $\epsilon_{\textrm{in}}/n$ and $\dot{N}U_{\textrm{D}}/N$ are constant at long times (dashed lines; see also Supplementary), but $\epsilon_{\textrm{in}}/n$ is higher by a factor of $\approx1.3$. The vertical solid lines mark the onset time of dissipation $t_{\textrm{D}}$; for an analytical calculation of $t_{\textrm{D}}$, see the End Matter. Inset: the dissipation spectrum $D_\varepsilon$ for $U_{\textrm{s}}=\zeta$ and $a=100a_0$. The average dissipation momentum $\langle k_{\textrm{diss}}\rangle\approx1.15k_{\textrm{D}}$, predicting $\epsilon V/(\dot{N}U_{\textrm{D}})\approx1.32$ (see text). (b) Energy flux $\Pi_\varepsilon$ (left) and particle flux $\Pi_N$ (right) for different dissipation scales $k_{\textrm{D}}$ (vertical dashed lines). Both fluxes are scale independent. The dotted line ($\propto k^{-2}$) shows that $\Pi_N\propto k_{\textrm{D}}^{-2}$, while $\Pi_\varepsilon$ is (nearly) independent of $k_{\textrm{D}}$; horizontal dashed lines are $\epsilon_{\textrm{in}}$ (resp. $\tau\dot{N}/N$) in the left (resp. right) panel (see text).
• Figure 3: A universal equation of state of the GP model. (a) The numerical calculations for different $a$ collapse onto a universal curve when the state variables $\epsilon$ and $n_0$ are expressed in the instantaneous natural scales of the GP model ($n$, $\xi_t$, $\zeta_t$, $\tau_t$). The dashed line is a power-law fit to the data that gives $n_0/n=29(2)(\tau_t\epsilon/[n\zeta_t])^{0.67(2)}$, and the pink band shows the fit uncertainty. (b) The comparison of our numerical EOS (gray symbols) with the experimental data from Dogra:2023 (colored symbols). The color coding of the experimental data is based on the gas parameter $na^3$; note that the experimental fluxes are multiplied by $1.3$ compared to the data of Dogra:2023 to account for $\alpha\neq1$. The vertical error bars of the experimental data represent the uncertainties due to different $\gamma_0$ in simulations and experiments (see Supplementary).
• Figure 4: Quasi-static process far from equilibrium. The instantaneous far-from-equilibrium state variables for different $t$ follow the EOS, in analogy with a quasi-static process in equilibrium thermodynamics. The data correspond to $a=100a_0$ and the blue star here corresponds to the green star in Fig. \ref{['figEoS']}(a). The inset shows the fraction of particles $N/N_0$ left in the system.
• Figure 5: The cascade exponent $\gamma(k)$ for different system parameters. (left) $\gamma(k)$ for different interactions is not universal, and it bends up around $k_{\textrm{D}}/2$ (solid line). (right) The rescaled data for $\gamma$ versus $k\xi$ collapse in the range $2.5\lesssim k\xi\lesssim k_{\textrm{D}}\xi/2$, demonstrating that the effective injection scale $k_0$ of the isotropic $4$-wave cascade is $\propto k_\xi\equiv1/\xi$. The dashed line shows the theoretical prediction $\gamma(k)-3=1/[3\ln(k/k_0)]$Nazarenko:2011Zhu:2023a with $k_0=1.64k_\xi$, and the red shading indicates the region of momentum space where the weak interaction approximation is not valid (see text).