Entropy transport in closed quantum many-body systems far from equilibrium

TL;DR

Entropy transport in closed quantum many-body systems far from equilibrium demonstrates that the von Neumann entropy $S$ is conserved while entropy migrates between infrared and ultraviolet momentum scales, decreasing at long distances and increasing at short distances. The authors combine spatially resolved ultracold-atom experiments with ab initio relativistic field theory calculations to reveal a dynamical scale separation that produces macroscopic order robust to microscopic disorder. They find a small mutual information between IR and UV sectors and interpret the dynamics as approaching a near Gaussian nonthermal fixed point where the Boltzmann-Einstein entropy $H$ tracks $S$. The results suggest that low-entropy macroscopic descriptions can emerge dynamically in closed quantum systems, with implications for early universe cosmology and ultracold quantum gases.

Abstract

We investigate entropy transport for universal scaling phenomena in closed quantum many-body systems far from equilibrium. From spatially resolved experimental data of a spinor Bose gas, we demonstrate that entropy decreases on long-distance scales while it increases at short distances. A dynamical separation of scales leads to macrophysics with long-range order, which is insensitive to the highly entropic microphysical processes. Since the total von Neumann entropy is conserved on a fundamental level for the quantum system, our analysis reveals a reciprocal connection between the emergence of macroscopic structure and microscopic disorder. To illustrate the scope of this connection, we exemplify the universal phenomenon also in a relativistic quantum field theory calculation from first principles, which is relevant for particle physics and early-universe cosmology.

Figures (10)

• Figure 1: Illustration: a) Experimental scheme for the spinor Bose gas with three hyperfine states. After an initial quench, the complex spin observable $F_\perp=F_x+iF_y$ is analyzed. b) Scale separation during the far from equilibrium evolution, encoded in the dual cascade of the particle distribution function $f(t,p)$. c) Entropy transport from low momenta (IR) to higher momenta (UV) with conserved total entropy $S$. For the systems considered, the total entropy can only be approximately estimated and the mutual entropy $S_{\mathrm{IR}\cap\mathrm{UV}}$ is small, such that $S\simeq S_\mathrm{IR}+S_\mathrm{UV}$. d) Entropy change momentum profile $\dot{\mathcal{S}}(t,p)$ drawn in the scaling regime. The vanishing integral of the curve (hatched areas) visualizes entropy conservation.
• Figure 2: Distribution function$f(t,p)$ for the spinor Bose gas as a function of momentum for different times in the scaling regime.
• Figure 3: a) Boltzmann-Einstein ($H$) vs. Shannon ($I$) entropy estimates from the measured data as a function of time. An initial fast growth is followed by a relatively flat evolution after the time $t_\mathrm{ref}=1\,\mathrm{s}$, where the system enters the scaling regime. b) IR entropies are decreasing in the scaling regime. Both estimates for the IR Boltzmann-Einstein and Shannon entropies agree remarkably well, as expected even in this non-perturbative regime when the dynamics becomes Gaussian at low momenta. c) UV entropies increase correspondingly. Both estimates agree rather well in this perturbative regime. d) Relative mutual information between the low- and high-momentum regimes. The inset displays the same data points on a sub-percentage level, showing that $I_{\mathrm{IR}\cap\mathrm{UV}}$ makes up only a small fraction of the total $I$.
• Figure 4: a) Mode entropy change for the Boltzmann-Einstein entropy, and b) for the Shannon entropy at different times in the scaling regime. Both estimates from the experimental data show great similarity, with the negative contributions occurring below the characteristic scale $Q$.
• Figure 5: Distribution function as a function of momentum at different times for the relativistic quantum field theory. The time evolution exhibits a dynamical separation of scales, with transport of particles towards low momenta and energy transport towards high momenta. The initial distribution with momentum scale $Q$ is given by the gray line.