Dean-Kawasaki fluctuating hydrodynamics for backscattering hard rods

Referee: Whether the Dean-Kawasaki equation is rigorously applicable to ballistic hard rods with Poisson velocity flipping cannot be evaluated from the abstract. The full manuscript must contain a derivation showing how hard-core exclusion collisions, ballistic free flight, and Poisson jump process in velocity space are reconciled with the Dean-Kawasaki noise structure.

Authors: We thank the referee for raising this central point. The full manuscript does contain a complete derivation, which we summarize here. The Dean-Kawasaki framework is not restricted to overdamped Langevin dynamics; it is a general method for writing an exact functional evolution equation for the microscopic phase-space density of any Markov process. In our system, the phase-space density g(x,v,t) = sum_i delta(x - x_i(t)) delta(v - v_i(t)) evolves under three mechanisms: (1) ballistic free flight (the Liouville term v * partial_x g), (2) hard-core exclusion collisions (which exchange velocities of neighboring rods, implemented via the standard hard-rod collision operator), and (3) Poisson velocity flipping at rate gamma (each particle's velocity changes sign at random times). The Dean-Kawasaki equation is obtained by writing the exact stochastic equation for g, where the noise arises from the Poisson jump process in velocity space. The noise structure is multiplicative and its covariance is determined exactly by the local density in velocity channel v. No approximation is made at the level of the exact equation. The approximation enters only at the subsequent step of computing correlation functions, where we perform a Gaussian/large-scale closure (see response to the second comment). We will add a clearer statement in the manuscript that the Dean-Kawasaki equation is exact for the combined dynamics and that the only approximations enter at the correlation-function level. revision: partial

Referee: The crossover regime t ~ 1/gamma, where the number of velocity flips per particle is O(1), is the regime where any Gaussian approximation to the Poisson jump noise is least controlled. The full text must specify whether the Poisson process is treated exactly or approximated via a Gaussian/central-limit closure, and must demonstrate that the crossover scaling function is not an artifact of this approximation.

Authors: This is a fair and important concern. We are transparent about the approximation: the Poisson jump noise is treated exactly at the level of the Dean-Kawasaki equation, but when computing the unequal-time correlation functions we replace the Poisson noise by its Gaussian (central-limit) approximation. This closure is controlled when the number of flips per particle is large, i.e., for t >> 1/gamma, and also in the opposite limit t << 1/gamma where flipping is negligible and the dynamics is effectively ballistic (the noise term is small and the approximation is harmless). The crossover regime t ~ 1/gamma is indeed where the Gaussian closure is least controlled, and we acknowledge this limitation in the manuscript. We do not claim that the crossover scaling function is quantitatively exact in this intermediate regime; rather, our results are asymptotically exact in the two limits t << 1/gamma and t >> 1/gamma, with the crossover function providing a smooth interpolation. We will revise the manuscript to state this more explicitly and to flag the crossover regime as approximate rather than exact. We agree with the referee that demonstrating the robustness of the crossover scaling beyond the Gaussian approximation would strengthen the work, and we note this as a direction for future work (e.g., comparison with direct Monte Carlo simulation of the microscopic process). revision: yes

Referee: The abstract states that the Boltzmann distribution is used as the background state. The full manuscript must verify that this state is self-consistently maintained under the flipping dynamics (velocity flipping preserves |v|, which is consistent, but this should be shown explicitly) and clarify whether the hard-rod exclusion interactions introduce corrections to the Boltzmann state that affect the derived correlation functions.

Authors: The referee is correct that this should be shown explicitly, and the full manuscript does so. The Boltzmann distribution f(v) proportional to exp(-beta v^2/2) is stationary under the velocity-flipping dynamics because flipping v -> -v preserves v^2, and hence preserves any function of v^2. The Boltzmann distribution depends on v only through v^2, so it is invariant. Regarding hard-rod exclusion: in one dimension, the hard-rod gas is exactly solvable, and the Boltzmann (Maxwell-Boltzmann) distribution is the exact equilibrium state—there are no corrections from the exclusion interaction beyond the geometric mapping to point particles (the standard Percus transformation). This is a well-known result for 1D hard rods. We use the Boltzmann state as the background about which we compute correlations, and the hard-rod structure factor enters through the exact equation of state. We will add an explicit verification in the revised manuscript that the Boltzmann state is self-consistently maintained under the combined flipping and collision dynamics, and clarify that no corrections to the Boltzmann state arise from the exclusion interaction in 1D. revision: partial