Open problems of ballistic macroscopic fluctuation theory: rigorous foundations, shock selection, subleading corrections, full counting statistics, and the non-integrable, quantum, and experimental frontiers

Ballistic macroscopic fluctuation theory (BMFT) extends the macroscopic fluctuation theory of diffusive systems to the Euler hydrodynamic scale of one-dimensional many-body systems with ballistic transport: fluctuations of the conserved fields are local thermodynamic fluctuations transported deterministically by inviscid hydrodynamics. The founding paper by Doyon, Perfetto, Sasamoto and Yoshimura left a consolidated list of open problems forming one research objective - completing BMFT - with seven jointly stated directions: (i) a rigorous proof of the Euler-scale Boltzmann-Gibbs principle for Hamiltonian and quantum systems with many conservation laws; (ii) selection of the correct weak (entropy/viscosity) solution of the BMFT saddle equations at shocks; (iii) incorporation of subleading diffusive and superdiffusive corrections and stochastic noise beyond the saddle; (iv) the full scaled cumulant generating function in genuinely inhomogeneous interacting fluids and the precise relation between BMFT and inhomogeneous ballistic fluctuation theory; (v) extension to non-integrable systems with generic shocks; (vi) genuinely quantum regimes; and (vii) experimental tests of BMFT-predicted cumulants and Euler-scale long-range correlations. Every direction remains open: substantial partial progress exists - exact diffusive hydrodynamics of hard rods, ballistic hydrodynamic-noise theory, point-particle rederivations of BMFT, quantum-quench counting statistics, and first experimental observations of ballistic correlations - but each direction retains a precise nonempty open core.

Background

In one-dimensional many-body systems, behavior on space-time scales much larger than microscopic scales is governed by the locally conserved quantities: particle number, momentum and energy, and, in integrable models, an infinite family of stable quasiparticle charges. For stochastic lattice gases with diffusive transport, macroscopic fluctuation theory (MFT) provides a closed universal description: the conserved fields fluctuate around solutions of diffusive hydrodynamic equations, with Gaussian noise fixed by the local equation of state. For systems with ballistic transport - energy and particles propagating without scattering at leading order, as in the hard-rod gas (a one-dimensional gas of impenetrable rods of fixed length, the paradigmatic integrable fluid) and other integrable fluids - the relevant scaling is instead the Euler scale, where transport is governed by the inviscid (hyperbolic) hydrodynamic equations. Generalized hydrodynamics (GHD) is the exact Euler-scale theory for one-dimensional integrable systems, treating the stable quasiparticles as a continuum of species moving ballistically with state-dependent effective velocities.

Ballistic macroscopic fluctuation theory (BMFT), introduced by Doyon, Perfetto, Sasamoto and Yoshimura, adapts MFT to the Euler scale. Its central assumption is a Boltzmann-Gibbs principle at the Euler scale: on mesoscopic cells, local observables relax to functions of the conserved densities (the local equation of state), and fluctuations of the conserved fields are the thermodynamic equilibrium ones, transported deterministically by the Euler hydrodynamics. (The Boltzmann-Gibbs principle, in its classical form, is the statement that a local observable is replaced, at large scales, by the same function of the fluctuating conserved fields evaluated in local equilibrium; it has been proven for stochastic interacting particle systems.) On this basis BMFT derives universal results: long-range equal-time correlations between fluid cells develop over time from long-wavelength inhomogeneous initial states, challenging the usual assumption that Euler-scale cells are uncorrelated; the Gallavotti-Cohen fluctuation relation (a symmetry of the large-deviation rate function for transport under time reversal) follows from time-reversal invariance of the Euler equations; and exact low-order cumulants, such as the variance and third cumulant of the particle number transferred in partitioning protocols (two halves of the system prepared in different stationary states and joined at time zero, a hydrodynamic Riemann problem), are obtained and confirmed by hard-rod simulations.

Several pillars of the framework rest on unproven assumptions or are restricted to special cases. First, the Boltzmann-Gibbs principle is proven only for stochastic dynamics; Hamiltonian chains and quantum systems with many conservation laws - precisely the models BMFT targets - lack such a proof, and only adjacent rigorous results exist (derivation of Euler equations for disordered harmonic chains, hydrodynamic projection formulas in quantum spin chains). Second, the Euler equations are hyperbolic conservation laws whose solutions generically develop shocks (travelling discontinuities); uniqueness of weak solutions (solutions in distributional sense) then fails unless an entropy condition or vanishing-viscosity limit selects one. The BMFT saddle equations inherit this ambiguity: which weak solution the microscopic large-deviation functional selects is unknown, and recent work indicates that the standard Jensen-Varadhan rate function for shocks is not universal outside small Mach numbers. Integrable hydrodynamics is special because it is linearly degenerate (the quasiparticle characteristic speeds vary in a way that forbids shock formation), which is exactly why the founding paper's checks avoid this issue. Third, the theory is leading-order (saddle-level); subleading diffusive and superdiffusive corrections - of Kardar-Parisi-Zhang (KPZ) type, the universality class governing superdiffusive scaling in one dimension - loop terms and determinant prefactors are not yet incorporated, and they are needed to explain the finite-cell-size approach, diffusive tails, and the observed breakdown of large-deviation principles in gapped and isotropic XXZ spin chains (the XXZ chain being the standard one-dimensional spin-1/2 model with anisotropic nearest-neighbour exchange, the canonical integrable lattice system). Fourth, the scaled cumulant generating function $F(\lambda,T)$ - the exponential growth rate of the generating function of a time-integrated counting observable, whose derivatives give its cumulants $c_n$ - is known only at low orders in homogeneous and partitioning states; the genuinely inhomogeneous case, and the precise relation between BMFT and the companion inhomogeneous ballistic fluctuation theory (BFT, a mode-based fluctuation theory which, unlike BMFT, does not assume equal-time long-range correlations), remain open. Fifth, non-integrable fluids such as anharmonic chains (one-dimensional chains of nonlinearly coupled oscillators) and the totally asymmetric simple exclusion process (TASEP, a stochastic lattice model of particles hopping in one direction with exclusion) develop shocks generically, quantum regimes involve coherent mesoscopic fluctuations beyond the current theory, and no experiment has yet observed the BMFT-specific predictions.

In the Conclusion and Outlook of the founding paper the authors state these limitations explicitly as future work. This problem collects them: the directions jointly constitute completing BMFT - turning a successful leading-order, integrable-biased, saddle-level framework into a rigorous, general, and experimentally tested theory of Euler-scale fluctuations.

Problem Statement

Complete the ballistic macroscopic fluctuation theory: resolve the seven open directions stated jointly by Doyon, Perfetto, Sasamoto and Yoshimura in the founding paper, which together would turn BMFT from a leading-order framework verified on integrable, shock-free systems into a rigorous, general, and experimentally tested theory of Euler-scale fluctuations. The directions, with their common objective of completing the single framework, are:

Each of the seven directions is an independently standing open problem, and completing BMFT is their union. Resolving a direction means settling the question that direction poses for the systems it names; a solution must state which direction(s) it addresses, what it establishes for each, and which directions - and which sub-goals within a direction - remain open, so that partial work is reported as partial rather than as completion of a direction. Within (iii) the loop corrections and determinant prefactors and the control of the finite-cell-size approach, the diffusive tails, the relaxation laws toward Euler-scale predictions, and the XXZ large-deviation anomalies are separate sub-goals; within (iv) the full scaled cumulant generating function $F(\lambda,T)$ and the BMFT-BFT relation are separate sub-goals. Each sub-goal is resolvable in its own right, with the other sub-goals of its direction remaining open until they are settled.

(i) Rigorous foundations. Prove the Boltzmann-Gibbs principle at the Euler scale - the local relaxation of fluctuations on which BMFT rests - for Hamiltonian systems and for quantum models with many (in integrable cases infinitely many) conservation laws, where it is currently established only for stochastic dynamics.

(ii) Weak-solution selection. Determine the correct weak (entropy- or viscosity-type) solution of the BMFT saddle equations when hydrodynamic shocks or entropy-nonconserving discontinuities appear, as in non-integrable hyperbolic systems, the totally asymmetric simple exclusion process, and anharmonic chains: identify the criterion by which the microscopic large-deviation functional selects one weak solution.

(iii) Subleading corrections. Incorporate diffusive and superdiffusive (Kardar-Parisi-Zhang-type) corrections and stochastic noise beyond the BMFT saddle - the loop corrections and determinant prefactors of the measure - so as to control the approach at finite mesoscopic-cell size, the diffusive tails, the observed relaxation laws toward Euler-scale predictions, and the large-deviation anomalies of the XXZ spin chain.

(iv) General scaled cumulant generating function. Compute the full scaled cumulant generating function $F(\lambda,T)$ and the higher cumulants $c_{n\ge 4}$ in genuinely inhomogeneous interacting fluids beyond the known low-order cumulants, and clarify the precise relation between BMFT and the inhomogeneous ballistic fluctuation theory, including the regimes where Euler-scale long-range correlations contribute nontrivially.

(v) Non-integrable systems. Apply BMFT to non-integrable interacting systems, in which shocks and weak solutions are generic, and establish whether the predicted long-range correlations and the modifications of intra-cell fluctuations persist there.

(vi) Quantum regimes. Extend BMFT to regimes where quantum coherence or genuinely quantum fluctuations affect mesoscopic variables, combining quantum generalized hydrodynamics with the BMFT path-integral formalism.

(vii) Experimental tests. Design a concrete experiment - for example in cold-atom setups with quantum gas microscopes (instruments that image individual atoms in optical lattices) - that would observe BMFT-predicted cumulants, including the exact partitioning-protocol cumulant derived in the source, and the Euler-scale long-range correlations grown from inhomogeneous initial states, specifying the setup, the observable and the predicted signal; a concrete proposed protocol resolves this direction, and an executed measurement is not required.

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