Circuits as a simple platform for the emergence of hydrodynamics in deterministic chaotic many-body systems

TL;DR

This work demonstrates that classical deterministic brickwork circuits with local conserved quantities naturally realize hydrodynamic behavior at large scales, including Euler-scale flows, diffusion, and KPZ superdiffusion. By systematically extracting conserved quantities and constructing the exact Gibbs manifold, the authors derive explicit Euler-scale equations and verify both diffusion and KPZ scaling through large-scale simulations, even in regimes with shocks and entropy production. A striking result is that a simple circuit with a single CQ yields Burgers-type dynamics, making deterministic, closed microscopic models capable of generating rich hydrodynamic phenomenology without stochastic noise or Hamiltonian structure. The approach is readily extendable to higher dimensions and quantum circuits, offering a versatile platform to probe turbulence, non-equilibrium thermodynamics, and the emergence of hydrodynamics from microscopic deterministic chaos.

Abstract

The emergence of hydrodynamics is one of the deepest phenomena in many-body systems. Arguably, the hydrodynamic equations are also the most important tools for predicting large-scale behaviour. Understanding how such equations emerge from microscopic deterministic dynamics is a century-old problem, despite recent progress in fine-tuned integrable systems. Due to the universality of hydrodynamics, the specific microscopic implementation should not matter. Here, we show that classical deterministic circuits provide a minimal, exact, and efficient platform that admits non-trivial hydrodynamic behaviour for deterministic but chaotic systems. By developing new techniques and focusing on 1D circuits as a proof of concept, we obtain the characteristic dynamics, including relaxation to Gibbs states, exact Euler equations, shocks, diffusion, and exact KPZ super-diffusion. Our methods can be easily generalised to higher dimensions or quantum circuits.

Figures (6)

• Figure 1: Hydrodynamics of deterministic circuits, illustrated with gate $(d, \sigma)=(3,996)$. (a) Local transition rules. (b) Brickwork structure of the space-time circuit. (c) Example microscopic trajectory. (d) Coarse-grained evolution of the conserved quantity, displaying formation of shocks followed by relaxation to thermal equilibrium. The results from large-scale microscopic numerics, with system size $L=3\times 10^6$ and timesteps $T=8L$ (blue shading) is accurately captured by the hydrodynamic predictions (red dashed). Here, the numerics is from a single realisation, with fluid-cell averaged over $3 \times 10^4$ sites.
• Figure 2: Expectation values of observables embedded in low-dimensional manifolds. Randomly generated observables of random initial states overlaid on theoretical thermal expectation values, for $\beta \in [-5, 5]$ for $L=20000$, $T=1000$, averaged over the last $10$ even timesteps. (a) For gate $(d, \sigma)=(3, 996)$, we expect only a single CQ. For both kinds of initial states with spatial periodicities $m=2,4$, we see good evidence that the observables lie on a one-dimensional manifold, and that their values agree with the thermal expectations. (b) For gate $(d, \sigma) = (3, 229117)$, we expect two conserved quantities, one with $m_1 = 2$ and another "staggered" quantity with $m_2=4$. For initial states with $m=2$, the data lies on a one-dimensional curve corresponding to thermal expectation on only the first CQ, as it does not couple to the staggered CQ. Using instead initial states with $m=4$, we find that data lies on a two-dimensional surface of thermal expectation on both CQs.
• Figure 3: Rapid thermalisation from formation of shocks in deterministic circuits, illustrated with gate $(d, \sigma)=(3, 996)$. (a) Formation of shocks. Microscopic numerics for $L=10^5$ averaged over $10^5$ samples (solid lines, linewidth corresponds to standard deviation) verifies weak hydrodynamic solutions (dashed lines) given by Rankine-Hugoniot conditions. (b) Entropy production from shocks. $S(t)$ is constant until shock formation time $t_{\mathrm{shock}} \approx 0.95L$, which facilitates entropy production until saturating to a theoretical maximum value $S_\rm{theory}(\infty) \approx 1.758L$ at $t_{\rm{eq}} \approx 16 L$. Here, $L=8\times 10^4$ and the result is averaged over $10^3$ samples.
• Figure 4: KPZ super-diffusion in deterministic circuits with one conserved quantity, illustrated with gate $(d, \sigma)=(3, 996)$. (a) Super-diffusive scaling of the full-width half maximum (FWHM) of the correlation function $\langle q(x,t) q(0,0) \rangle^\rm{c}_{\beta=0}$ obtained from microscopic numerics (red crosses, $L=2^{14}$ and averaged over $10^6$ samples), consistent with $t^{2/3}$ scaling (purple line). (b) Data collapse of $\langle q(x,t) q(0,0) \rangle^\rm{c}_{\beta=0}$, which also converges to the universal KPZ function $f_\rm{KPZ}$ (red line). (c) Theoretical prediction of the super-diffusive constant $\lambda_\rm{B}$ (blue line) compared to the value extracted from the numerics (orange markers, $L=10^3$ and averaged over $10^6$ samples).
• Figure 5: Coexistence of diffusion and KPZ superdiffusion in deterministic circuits, illustrated with gate $(d,\sigma)=(3,1092)$. At special point $q^*=\sqrt{3}\sin(\pi/9)$, (a) the correlations show diffusive $t^{1/2}$ scaling, and (b) converge to a standard normal $\mathcal{N}(0,1)$ when suitably scaled with the diffusion, $D=0.91690(1)$, and height, $C'=0.17840(1)$, constants. (c) Away from $q^*$, the correlations show superdiffusion.