Hydrodynamics of classical deterministic circuits
Friedrich Hübner (KCL)
Juan P. Garrahan (Nottingham)
Benjamin Doyon (KCL)
Sun Woo P. Kim, King’s College London
25-07-09, Budapest Integrability Online Seminar
Slides at sunwoo-kim.github.io/en/slides/gateshydro
Hydrodynamics
- Large-scale, coarse-grained description of locally-interacting many-body systems with extensive conserved quantities (CQs) with local densities away from equilibrium
- Thought to be universal in that many microscopic models/systems give rise to the same family of hydrodynamic equations
Local conservation laws
- Extensive CQs, ex. $\vec Q = (N, \bm{P}, E)$
- Local densities $$\vec Q = \int dx \vec q(x, t)$$
- Assuming periodic boundary condition $\frac{d}{dt} \vec Q = 0$ if $$ \partial_t \vec q(x, t) = \partial_x \vec j \left(\vec q(x,t), \partial_x \vec q(x,t), \dots \right)$$
Euler-scale hydrodynamics
- Taylor expand the current wrt gradients $$\partial_t \vec q(x,t) = \partial_x \Big( f(\vec q) + g(\vec q) \partial_x \vec q + \cdots \Big)$$
- Rescale space and time $t = t^\prime L$, $x = x^\prime L$ (i.e. look in terms of $\mathrm{km}$ and $\mathrm{days}$ instead of $\mathrm{cm}$ and $\mathrm{sec}$) $$\partial_{t^\prime} \vec q(x,t) = \partial_{x^\prime} \Bigg( f(\vec q) + \frac{g(\vec q)}{L} \partial_{x^\prime} \vec q + \mathcal{O}(L^{-2}) \Bigg)$$
- As $L \rightarrow \infty$, we can just consider the effects of the first term; this is known as Euler-scale hydrodynamics.
Two approaches
Top-down:
- (Sensible) guess of hydrodynamic equation from physical principles ex. The Navier-Stokes, with $\vec q = (\rho, \rho \mathbf{u})$:
- Does this equation make sense? i.e. Navier–Stokes existence and smoothness (Millenium prize problem)
- Efficient simulations? ex. GenCast by Google, Tensor network methods (Gourianov ‘25)
Two approaches
Bottom-up:
- Microscopic laws $\Rightarrow$ hydrodynamic equations
- Assume thermal relaxation in coarse-grained fluid cells $$ p(\bm{a}) \propto \exp\left[- \sum_x \vec \beta(x,t) \cdot \vec {\mathsf{q}}_x(a_{x:x+l})\right]$$
- Then obtain Euler-scale hydrodynamics with $$\vec q(x, t) = \langle \vec {\mathsf{q}}_{x} \rangle_{\vec \beta(x, t)}, \quad \vec{j}(x,t) = \langle \vec {\mathsf{j}}_{x} \rangle_{\vec \beta(\vec q(x,t))}$$
- Generally hard to evaluate, as $p(\bm{a})$ is not separable ex. Hamiltonian systems. Exceptions: 1D systems (transfer matrix), integrable systems (GHD)
Two approaches
Bottom-up:
- Very hard to prove that relaxation occurs in general, or that coarse-grained dynamics reduces to hydrodynamic description (Hilbert’s 6th problem).
- Recently (maybe) proven for hard spheres (Deng ‘25)
Brickwork circuits
- Time-discrete many-body system
- Arrange states in a 1D chain (of length $L$ with periodic boundary conditions) then apply a layer of two-site gates at each timestep in a brickwork fashion
- Can be classical (block cellular automata with Margolius neighbourhoods) or quantum
- Can be deterministic or stochastic
Brickwork circuits
- Analytically tractable examples
- Quantum: dual unitaries (Gopalakrishnan ‘19, Bertini ‘19), Haar-random unitaries (von Keyserlingk ‘18)
- Classical: east gate (Klobas ‘24), stochastic charged automata (Klobas ‘18, Krajnik ‘22)
- Can implement them on a quantum computer ($$$)
- Can generalise to $D$ spatial dimensions $\rightarrow$ each gate acts on $2D$ sites
Classical deterministic gates
- Local states $\mathcal{D}=\{0, 1, \dots, d-1\}$
- Two-site determinisitic gates are maps $\sigma: \mathcal{D} \times \mathcal{D} \rightarrow \mathcal{D} \times \mathcal{D}$
- Reversible $\Rightarrow$ $\sigma$ is a bijection, i.e. a permutation on $\{(0,0), (0,1), \dots, (d-1, d-1)\}$
- There are $(d^2)!$ reversible deterministic gates
- Also known as block cellular automata with Margolius neighbourhoods
$(d, \sigma) = (3, 996)$
Classical deterministic gates
- Promote configurations to basis vectors $a \rightarrow | a \rangle$
- Then can represent distributions as
$\ket{p} = \sum_{\bm{a}} p(\bm{a}) \ket{\bm{a}}$ - Gate $\mathsf{u}$ can be written as $$\mathsf{u}=\sum_{a, b}|\sigma_1(a, b), \sigma_2(a, b)\rangle\langle a, b|$$
- Equivalently in tensor network notation
$(d, \sigma) = (3, 996)$
$$\tiny{\begin{aligned}
& |00\rangle \rightarrow|00\rangle \\
& |01\rangle \rightarrow|01\rangle \\
& |02\rangle \rightarrow|10\rangle \\
& |10\rangle \rightarrow|12\rangle \\
& |11\rangle \rightarrow|11\rangle \\
& |12\rangle \rightarrow|21\rangle \\
& |20\rangle \rightarrow|02\rangle \\
& |21\rangle \rightarrow|20\rangle \\
& |22\rangle \rightarrow|22\rangle
\end{aligned}}$$
This work
- Develop machinery to
- Determine the CQs with local densities of (1D classical deterministic) brickwork circuits up to desired locality
- Confirm the number of CQs by confirming relaxation onto (generalised) Gibbs states
- Derive the Euler-scale hydrodynamic equations
- Go out there and search through possible deterministic reversible gates, look for gates with finite number of CQs
- Compare theory with numerics
- Compare Euler-scale hydrodynamics
- Develop non-linear fluctuating hydrodynamics and quantitatively predict KPZ superdiffusion
Why circuits?
- No continuous temporal or spatial translational invariance
$\Rightarrow$ no pesky conserved energy, momentum or particle number to deal with
$\Rightarrow$ can have models with minimum ingredients for hydrodynamic phenomenon - If CQs are 1-local, then Gibbs states are factorisable $$p(\bm{a}) = \exp\left[-\sum_x \vec \beta(x, t) \cdot \vec{\mathsf{q}}_x(a_x) \right] = \prod_x p(a_x)$$
- Classical circuits are very simulable ($L \approx 10^6$)
Finding conserved quantities
- Conserved quantity can be represented as $\bra{F}$ and evaluated as $\langle F | p \rangle$
- Two layer of time evolution (at even timestep) is
- If conserved after (say) $n$ timesteps, then $\bra{F} \mathsf{U} = \bra{F} \lambda^2$ with $\lambda^n = 1$ (such that $\bra{F}{\mathsf{U}}^{n/2}\ket{p} = \bra{F}{p}\rangle \forall \ket{p}$)
- Assume that CQ is translationally invariant under $m$ translations: $\bra{F} \mathsf{T}^2 = \bra{F} \mu^2$, with $\mu^m = 1$
Finding conserved quantities
- Construct ansatz of local densities that may differ on even and odd sites
- ex. for 1-local conserved quantities, the condition for the CQ is
- Can be solved to find exact expressions for CQs or rule out CQs up to $\sim 10$-local densities
Finding conserved quantities
- For local dimension $d=2$, find that all reversible gates are either fully chaotic (i.e. no CQs at all) or (super) integrable (i.e. a tower of CQs)
- Go through all $d=3$ gates ($(3^2)!=362880$ gates)
- Find $\approx 2000$ gates with only 1-local CQs
- ex. For $(d, \sigma) = (3, 996)$, $\lambda=\mu=1$, $\bra{f_\mathrm{e}} = \bra{1}$,
$\bra{f_\mathrm{o}} = -\bra{0}$ and no more! - For $(3, 229117)$, there are two 1-local CQs, one with $\lambda_1=\mu_1=1$ and another with $\mu_2=\lambda_2=e^{i \pi/2}$
Confirming the number of conserved quantities
- To construct the hydrodynamics, require all CQs. How to rule out further CQs ex. 41293-local CQ?
- Dimension of linear equation for $l$-local CQ $\sim \exp(l)$
- Previous method cannot capture quasi-local CQs, i.e. those with exp. decaying tails (Sharipov ‘25)
Confirming the number of conserved quantities
- Use / test assumptions of hydrodynamics: relaxation onto (generalised) Gibbs states
- Degree of freedom for Gibbs states is $N_\mathrm{cq}$
$\Rightarrow$ local expectation values of local observables should lie on a $N_\mathrm{cq}$-dimensional manifold and agree with microscopic predictions
Constructing the Euler-scale hydrodynamic equations
- Once all CQs have been found, can construct Euler-scale hydrodynamics
- For 1-local CQs with $\lambda=\mu=1$, $$\braket{\mathsf{q}}_\beta = \braket{f_\mathrm{o}}_\beta + \braket{f_\mathrm{e}}_\beta \quad \braket{\mathsf{j}}_\beta = \braket{f_\mathrm{o}}_\beta - \braket{f_\mathrm{e}}_\beta$$ where the current is given by
Constructing the Euler-scale hydrodynamic equations
- For $(d, \sigma) = (3, 996)$,
$$ \braket{\mathsf{q}}_{\beta} = \frac{1}{2 e^{\beta}+1}+\frac{2}{e^{\beta}+2}-1 $$ $$ \braket{\mathsf{j}}_{\beta} = \frac{1}{-2 e^{\beta }-1}+\frac{2}{e^{\beta}+2}-1 $$
- So $j(q) = \frac{1}{3} \left(2 - \sqrt{9 q^2+16}\right)$ and $$ \partial_t q(x) = -v(q(x)) \partial_x q(x) $$ with $v(q) = {3q}/{\sqrt{16+9q^2}}$
Constructing the Euler-scale hydrodynamic equations
- With change of variables $\rho = v(q)$, obtain Burger’s equation
- First example of deterministic and closed microscopic model with one giving rise to Burger’s equation
- Impossible to get with Hamiltonian systems with one CQ
Non-linear fluctuating hydrodynamics
- The hydrodynamic equations are for averages, i.e. one-point functions
- Now consider fluctuations around the mean, $\delta q(x,t) = q(x,t) - q_0$ where $q_0 = \langle \mathsf{q} \rangle_\beta$
- Expanding the Euler-scale equation (and considering higher order terms), $$\partial_t \delta q(x,t) + j^\prime(q_0) \partial_x \delta q + \frac{1}{2} j^{\prime \prime}(q_0) \partial_x \delta q^2 + D \partial_x^2 \delta q + (\mathrm{noise}) = 0$$
Non-linear fluctuating hydrodynamics
- Can be mapped onto KPZ equation ($\partial_x h = \delta q/C$) from which we can predict the fluctuations (Spohn ‘14) $$\langle\delta q(t, x) \delta q(0,0)\rangle \sim \frac{1}{(\lambda t)^{2 / 3}} f_{\mathrm{KPZ}}\left(\frac{x-j^{\prime}(q_0) t}{(\lambda t)^{2 / 3}}\right)$$ where superdiffusion constant $\lambda$ is $$\lambda = \sqrt{2 \langle \delta q^2\rangle} j^{\prime \prime}(q_0)$$
Other phenomenology
Outlook
- Machinery is straight-forward to extend to quantum and higher $D$ and other topologies
- Currently: identifying interesting quantum gates and gates in higher $D$
- We find non-integrable models with exact hydrodynamic predictions!
- Study quasi-local charges
- Combine with other circuit techniques? “Solvability” or generalisation of dual-unitarity