Ballistic macroscopic fluctuation theory (BMFT) is the Euler-scale large-deviation theory for one-dimensional integrable systems, in which all late-time fluctuations arise from initial-state noise convected by deterministic generalized hydrodynamics. It has been worked out concretely and microscopically verified only for the force-free hard-rod gas, where it yields explicit long-range correlations of the conserved phase-space density from inhomogeneous initial states. The source work (Kundu, J. Stat. Mech. 2025) posed a set of extensions: counting statistics beyond the homogeneous state, joint statistics of two tracers, a general fluctuating hydrodynamic description for inhomogeneous initial states, application to other integrable systems such as the Toda lattice, and slowly varying external potentials. The objective of this task is to establish the following three extensions: (1) BMFT for hard rods, or the equivalent point-particle representation, in slowly varying external potentials, including the modified saddle-point equations and correlation formulas, in a setting where confinement is known to break integrability; (2) a mesoscopic fluctuating hydrodynamics for inhomogeneous initial states with a systematically derived coarse-grained noise term and a precise connection to BMFT, where current proposals disagree on gradient and dissipative terms; and (3) BMFT for the classical Toda lattice, where generalized hydrodynamics and equilibrium correlations are known but quasiparticle fluctuations are not yet characterized. The common objective is to test whether the initial-noise-plus-deterministic-convection mechanism underlying BMFT survives beyond its verified setting.

Background

The hard-rod gas is the one-dimensional classical gas of equal line segments that move freely and collide elastically, exchanging velocities. It is the simplest interacting integrable system and is exactly solvable because a collision only swaps velocities and shifts the coordinates by the rod length, which makes the gas equivalent to free point particles supplemented by a two-body scattering shift (the net displacement of the two trajectories at a collision). Its exact large-scale dynamics is described by generalized hydrodynamics (GHD): the state is the quasiparticle phase-space density ρ(x,θ)\rho(x,\theta), the density of quasiparticles carrying velocity label θ\theta at position xx, obeying a continuity equation with an effective velocity veff(x,θ)v^{\mathrm{eff}}(x,\theta) fixed self-consistently by the scattering shifts and the local density. The relevant hydrodynamic scaling is the Euler scale, on which x/tx/t is of order one and transport is purely ballistic.

For diffusive systems, macroscopic fluctuation theory (MFT) describes the probability of large-scale fluctuations through a large-deviation functional, with a stochastic noise added to the hydrodynamic current to represent coarse-grained non-conserved degrees of freedom. For ballistic systems such as integrable models, the analogous theory is ballistic macroscopic fluctuation theory (BMFT), introduced by Doyon, Perfetto, Sasamoto and Yoshimura (SciPost Phys. 15, 136 (2023)): on the Euler scale the only source of fluctuations is the initial state, and the theory is a saddle-point principle for the phase-space density with a nonlocal entropy functional. A central prediction is long-range correlations (LRC): when the initial state varies inhomogeneously on a length scale ℓ\ell, the conserved densities in two regions of size ℓ\ell separated by a distance of order ℓ\ell acquire correlations of order ℓ\ell, a coherent remnant of the initial fluctuations carried by the ballistic evolution, which is absent in diffusive nonequilibrium steady states.

Kundu (J. Stat. Mech. 2025 103203, the source of this problem) made BMFT concrete for hard rods: starting from the microscopic model with a slowly varying initial profile of the phase-space density, he derived the hard-rod fluctuation action, solved the saddle-point equations to linear order in a counting field λ\lambda conjugate to the phase-space density (the generating field whose derivatives yield cumulants of additive observables), and obtained an explicit two-point, two-time correlation formula that is straightforward to evaluate numerically and reproduces known equilibrium results. Full counting statistics (FCS) denotes the complete distribution of a time-integrated counting observable, beyond its variance. A tracer is a tagged particle; in one-dimensional no-crossing (single-file) systems, the joint statistics of several tracers encode the collective rearrangements of all particles between them.

The source's conclusion names the extensions addressed here, two of which connect to active neighboring programs. First, fluctuating hydrodynamics of the Dean-Kawasaki type, meaning stochastic conservation laws with white-in-time, spatially correlated conserved noise, was derived for generalized hard-rod dynamics in equilibrium by Ferrari and Olla; connecting such equations to BMFT and generalizing them to inhomogeneous initial states at mesoscopic (Euler-to-diffusive) scales is open, and current proposals disagree on gradient and dissipative terms. Second, the Toda chain, the classical lattice with exponential nearest-neighbor interactions that is integrable through its Lax pair (a matrix whose spectrum encodes the conserved quantities), has established GHD and known equilibrium spacetime correlations, but its quasiparticle fluctuations are only beginning to be characterized. A further constraint shapes the external-potential direction: external potentials generally destroy hard-rod integrability, since harmonically confined rods are known to possess only two conserved quantities, so extending BMFT to forced rods is not a routine substitution.

Problem Statement

Extend ballistic macroscopic fluctuation theory (BMFT), established and microscopically verified for the force-free hard-rod gas, to the following three regimes. Each of the three components below is a required, independently reportable deliverable, and resolving any single one of them does not complete the task. The components share one objective, to determine whether the BMFT mechanism (all late-time large-scale fluctuations are initial-state fluctuations convected by deterministic Euler-scale hydrodynamics) survives beyond its currently verified setting.

(1) External potentials. Determine whether a BMFT fluctuation description exists for hard rods, or the equivalent point-particle representation, subject to a slowly varying external potential V(x)V(x); if it does, derive the Euler-scale evolution with the force term, the corresponding fluctuation action, the modified saddle-point equations, and explicit correlation formulas, controlling the fact that confinement breaks the integrability on which the quasiparticle description rests (harmonically confined rods admit only two conserved quantities). A negative result—that no consistent BMFT fluctuation description exists for forced rods—resolves this component.

(2) Mesoscopic fluctuating hydrodynamics from inhomogeneous initial states. Derive, rather than postulate, a fluctuating hydrodynamic equation for the hard-rod phase-space density valid at mesoscopic (Euler-to-diffusive) scales and for inhomogeneous initial states, with a systematically established coarse-grained noise term; make precise its relation both to the BMFT long-range-correlation predictions and to the existing homogeneous-state Dean-Kawasaki-type equations, resolving the discrepancies on gradient and dissipative terms identified in current proposals, and incorporating the diffusive-scale corrections induced by initial long-range correlations.

(3) BMFT for the Toda chain. Develop the BMFT fluctuation fields, long-range correlations, and counting statistics for the classical Toda lattice evolving from inhomogeneous initial states, in a system where generalized hydrodynamics and equilibrium spacetime correlations are known but quasiparticle fluctuations are not yet fully characterized.

References

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