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Monitored Fluctuating Hydrodynamics

Sarang Gopalakrishnan1, Ewan McCulloch2,1, and Romain Vasseur3

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Phys. Rev. X 16, 011024 – Published 12 February, 2026

DOI: https://doi.org/10.1103/295c-lj1w

Abstract

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 nontrivial 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.

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In statistical physics, a fundamental challenge is understanding how much an observer can learn about a many-body system from partial measurements of its evolution. We develop a general hydrodynamic framework based on a replicated Martin-Siggia-Rose formalism to describe monitored classical stochastic processes. By computing correlation functions of fluid densities conditioned on typical measurement records, we show that these conditional ensembles undergo measurement-induced phase transitions as the monitoring rate increases. We find that even weak monitoring can give rise to novel critical phases, such as the “charge-fuzzy” phase, and drive systems with different unmonitored universality classes into the same monitored fixed point. Our results establish that learnability transitions previously identified in quantum dynamics have direct classical analogs, providing a simpler route to studying inference in ubiquitous stochastic processes.

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Phys. Rev. X 16, 011024– Published 12 February, 2026
  • Received 12 June 2025
  • Revised 11 September 2025
  • Accepted 14 January 2026
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