Many-Body Chaos in Thermalized Fluids

Linking thermodynamic variables like temperature $T$ and the measure of chaos, the Lyapunov exponents $\lambda$, is a question of fundamental importance in many-body systems. By using nonlinear fluid equations in one and three dimensions, we show that in thermalized flows $\lambda \propto \sqrt{T}$, in agreement with results from frustrated spin systems. This suggests an underlying universality and provides evidence for recent conjectures on the thermal scaling of $\lambda$. We also reconcile seemingly disparate effects—equilibration on one hand and pushing systems out of equilibrium on the other—of many-body chaos by relating $\lambda$ to $T$ through the dynamical structures of the flow.

Figure 1

Probability distribution functions of the $x$ component of the thermalized velocity field from Galerkin-truncated 3D Euler and (inset) 1D Burgers simulations for different energies; dashed lines denote the corresponding Gibbs distribution.

Figure 2

Representative plots of the difference field $|\mathit{\delta }\mathbf{v}(\mathbf{x},t){|}^2$, along the $z=0$ plane of a 3D thermalized fluid (with energy $E=2.0$ and a perturbation amplitude $\epsilon ={10}^{-6}$ at (a) early ($t=0.4$) and (b) later (0.7) times. The inset of panel (a) shows the same early time data, with a magnified scale, to reveal a somewhat self-similar spatial structure that arises from nonlocal interactions (see main text and movie in Supplemental Material [32]).

Figure 3

Semilog plots of $\varphi (r,t)$ ($E=1.0$) and (lower inset) $|{\widehat{\mathrm{\Delta }}}_k{|}^2$ ($E=2.0$) showing an initial exponential growth and eventual saturation. The dashed lines, from linearized theory, are in excellent agreement with DNSs at early times. Upper inset: semilog (left $y$ axis) plot of $\mathrm{\Phi }(t)$ (3D fluid) along with results from our linearized theory (dashed line). $\lambda$, extracted from $\mathrm{\Phi }(t)$, shown as dash-dotted horizontal line, agrees well with ${\lambda }_S$ (linearized theory, right $y$ axis).

Figure 4

Log-log plot of $\lambda /N_G$ versus $T$ for the 3D (axes in red) and 1D (axes in blue) thermalized fluids corresponding to different values of $\epsilon$, $N_G$, and, for the 1D fluid, $k_p$. The dashed line $\propto \sqrt{T}$ confirms our theoretical prediction.

Figure 5

A plot of $C(t)$ for a (a) nearly and (b) completely thermalized 3D fluid along with the theoretical Gaussian prediction. Lower inset: a magnified view shows that for a fluid which is not completely thermalized, $C(t)$ falls off to zero much more slowly. Upper inset: representative plots of $\tau$ and the average (negative) eigenvalue (compressional direction) versus $\lambda$.