Monitored Fluctuating Hydrodynamics

We introduce a hydrodynamic framework for describing monitored classical stochastic processes. We study the conditional ensembles for these monitored processes – i.e., we compute spacetime correlation functions conditioned on a fixed, typical measurement record. In the presence of global symmetries we show that these conditional ensembles can undergo measurement-induced “sharpening” phase transitions as a function of the monitoring rate; moreover, even weak monitoring can give rise to novel critical phases, derived entirely from a classical perspective. We give a simple hydrodynamic derivation of the known “charge-fuzzy phase” for weakly monitored diffusive many-body quantum systems. We show that although the unmonitored symmetric and asymmetric exclusion processes are in different universality classes of transport, the fluctuations in their conditional ensembles flow to the same fixed point with emergent relativistic invariance under monitoring. On the other hand, weakly monitored systems with non-Abelian symmetries enter a novel strongly coupled fixed point with non-trivial dynamical exponent, which we characterize. Our formalism naturally accounts for monitoring general observables, such as currents or density gradients, and allows for a direct calculation of information-theoretic diagnostics of sharpening transitions, including the Shannon entropy of the measurement record.

💡 Research Summary

The authors introduce a hydrodynamic framework for describing classical stochastic processes that are continuously monitored, which they call “monitored fluctuating hydrodynamics.” The central idea is to study the ensemble of trajectories conditioned on a typical measurement record, rather than the usual unconditioned ensemble. By employing a replicated Martin‑Siggia‑Rose (MSR) path‑integral formalism, they are able to treat the non‑linear observables that arise when one averages over the measurement‑conditioned ensemble (e.g., variances of conserved quantities).

The paper first clarifies the notion of strong versus weak symmetries in a classical Markov chain. A strong symmetry guarantees that a configuration’s conserved charge is exactly preserved along each trajectory; a weak symmetry only preserves the symmetry on average. When a strong symmetry is present, the problem of “charge sharpening” – the reduction of uncertainty about the global charge due to measurements – is well defined. The authors define trajectory ensembles, show how Bayes’ theorem incorporates measurement outcomes, and explain why linear observables are unaffected by conditioning while non‑linear ones (such as the variance of the total particle number) are dramatically altered.

Using the replicated MSR action, they derive a field theory for a single diffusive scalar charge ρ(x,t) obeying ∂ₜρ = D∇²ρ + ∇·ξ, with Gaussian noise ξ. Weak monitoring of either the charge density itself or its spatial derivative (the current) leads to a measurement‑induced sharpening transition. Below a critical monitoring rate γc the variance of the total charge grows diffusively; above γc it saturates to a finite value, reproducing the previously known “charge‑fuzzy” and “charge‑sharp” phases. Monitoring the current instead of the charge produces a distinct fixed point with its own scaling exponents.

The authors then apply the formalism to concrete lattice‑gas models. For the symmetric simple exclusion process (SSEP), they consider random measurements of local occupation. The measurement outcomes act as “impurities” that constrain possible particle trajectories, but because the outcomes are drawn from the evolving state, one‑replica quantities such as the average current remain unchanged. In contrast, two‑replica quantities like the variance of the total particle number are strongly suppressed once the monitoring rate exceeds γc, signalling a sharpening transition.

Next they study the asymmetric simple exclusion process (ASEP), which in the unmonitored limit belongs to the Kardar‑Parisi‑Zhang (KPZ) universality class with dynamical exponent z≈3/2. Remarkably, any non‑zero monitoring eliminates the non‑linear replica couplings, causing both SSEP and ASEP to flow to the same relativistic fixed point with z=1. Thus transport properties (e.g., average current) are unchanged, but the conditional trajectory ensemble is dramatically reorganized. This demonstrates a “monitoring‑induced universality” that overrides the underlying non‑equilibrium universality class.

The paper finally addresses systems with non‑Abelian symmetries (e.g., SU(2) spin conservation). Monitoring a component of the spin density is a relevant perturbation that destabilizes the Gaussian fixed point of the unmonitored theory. Numerical renormalization‑group calculations and a mean‑field scaling analysis reveal a new strongly‑coupled fixed point with a dynamical exponent 1<z<2. Correlation functions of the monitored spin current exhibit anomalous power‑law decay distinct from both diffusive and KPZ behavior, indicating a genuinely new universality class induced by measurement.

In addition to dynamical properties, the authors compute information‑theoretic diagnostics directly from the replicated action. The Shannon entropy of the measurement record, S = −∑ₘP(m)logP(m), shows a sharp change across the sharpening transition, quantifying how efficiently the measurement stream extracts the global charge information.

Overall, the work provides a unified hydrodynamic description of measurement‑induced phenomena in classical stochastic systems, reproduces known quantum measurement‑induced phase transitions in a purely classical setting, uncovers emergent relativistic invariance under monitoring, and discovers a novel strong‑coupling fixed point for non‑Abelian symmetries. By linking statistical physics, non‑equilibrium dynamics, and information theory, it opens new avenues for studying learnability and inference in many‑body systems subject to continuous observation.