The nonlinear projection formula of Doyon (J. Stat. Phys. 193, 106 (2026)) expresses the Euler-scale nn-point connected correlation functions (cumulants) of arbitrary local observables in generic many-body systems as a sum over minimal connected covers, whose terms are products of correlation functions of conserved densities (Euler amplitudes) and derivatives of microcanonical averages. Its derivation is physics-level, and the paper itself leaves three linked gaps: (i) a model-independent proof that the products of Euler amplitudes on the right-hand side are well-defined distributions and that the ϵ→0\epsilon\to0 smoothing limit used in their construction commutes with the combinatorial expansion; (ii) a characterization of the space-time set - shocks and other fluid singularities, and contact singularities - where the formula fails (the paper adds that on shock trajectories the formula is not expected to hold, and that it would be interesting to work out the consequences for explicit expressions of correlation functions); (iii) sharp conditions on the microscopic dynamics and initial states under which the derivation's assumptions hold. Solving these turns the proposed universal law into a mathematically established statement with a delimited validity domain.

Background

In a many-body system, the only dynamical quantities that survive at large scales of space and time are the densities qiq_i of conserved quantities (energy, momentum, particle number, and, in integrable models, infinitely many further charges). The Mori-Zwanzig projection principle asserts that under coarse graining any local observable becomes a function of the local values of the conserved densities; combined with the conservation laws this gives Euler hydrodynamics, the deterministic large-scale evolution of the conserved densities. The ballistic macroscopic fluctuation theory (BMFT) of Doyon, Perfetto, Sasamoto and Yoshimura (SciPost Phys. 15, 136 (2023)) upgraded this to fluctuations, via the principle of local relaxation of fluctuations: under coarse graining the joint fluctuations of local observables are exactly those of functions of the fluctuating conserved densities. This is a nonlinear version of the Boltzmann-Gibbs principle (the statement, underlying linear response, that the large-scale fluctuation of a local observable is a linear combination of conserved-density fluctuations).

Consider nn local observables o1,…,ono_1,\ldots,o_n at macroscopic space-time points zk=(xk,tk)z_k=(\mathbf{x}_k,t_k) in a long-wavelength state with variation scale ℓ\ell. Under the ballistic scaling x→ℓx\mathbf{x}\to\ell\mathbf{x}, t→ℓtt\to\ell t, the nn-point connected correlation function (the cumulant, i.e. the part of the expectation of the product not reducible to products of lower-order cumulants) scales as ℓ(1−n)d\ell^{(1-n)d} in dd spatial dimensions; the scaled limit defines the Euler amplitude So1,…,on\mathsf S_{o_1,\ldots,o_n}, which is generically a distribution in the macroscopic coordinates rather than a function. For n=2n=2 the projection formula of linear response reads So1,o2(z1,z2)=∑ij∂o1∂qi∂o2∂qj Sqi,qj(z1,z2),\mathsf S_{o_1,o_2}(z_1,z_2)=\sum_{ij}\frac{\partial\mathsf o_1}{\partial\mathsf q_i}\frac{\partial\mathsf o_2}{\partial\mathsf q_j}\,\mathsf S_{q_i,q_j}(z_1,z_2), where o(q)\mathsf o(\mathsf q) is the microcanonical average of oo (its average in the equilibrium state with conserved-density values qi\mathsf q_i) and Sqi,qj\mathsf S_{q_i,q_j} are Euler amplitudes of the conserved densities. This two-point formula has been established with mathematical rigour for one-dimensional isolated systems (Doyon, Commun. Math. Phys. 391, 293 (2022)) and, at all frequencies and wavelengths, for quantum spin lattices (Ampelogiannis and Doyon, Ann. Henri Poincare 25, 65 (2024)).

For n≥3n\geq 3 the projection is genuinely nonlinear: one must retain the full fluctuations of the conserved densities, and derivatives of o(q)\mathsf o(\mathsf q) up to order n−1n-1. Using Malyshev's formula - an extension of the classical moment-cumulant expansion to partial cumulants, originally developed for cluster expansions in lattice statistical mechanics (Malyshev, Russ. Math. Surveys 35 (2), 1 (1980)) - Doyon (J. Stat. Phys. 193, 106 (2026)) derived the general formula. Its right-hand side is organised as a sum over minimal connected covers of the point set {z1,…,zn}\{z_1,\ldots,z_n\}: a minimal connected cover is a collection of patches (subsets of two or more points) built so that each newly adjoined patch meets the union of the previous ones in exactly one point; to each patch one attaches a connected correlation function of conserved densities at the points of the patch, and to each point covered by mm patches the mm-th mixed derivative of the corresponding microcanonical average with respect to the conserved densities, with all free conserved-charge indices summed (unit coefficients). The formula corrects earlier higher-point principles for integrable systems (Doyon, SciPost Phys. 5, 054 (2018); Fava et al., PNAS 118, e2106945118 (2021)), whose assumption of local entropy maximisation at higher orders fails because of the hydrodynamic long-range correlations of Doyon et al. (Phys. Rev. Lett. 131, 027101 (2023)), and yields explicit two- and three-point stationary-state formulas in terms of purely thermodynamic data (the covariance matrix of conserved densities and the flux Jacobian of Euler's equation).

Three gaps remain, explicitly acknowledged in the source paper. First, the right-hand side contains products of Euler amplitudes, and the product of distributions is in general undefined (Hörmander's wave-front-set criterion, a condition on the directions in which the Fourier transform of each factor fails to decay, decides when a product is defined); the paper's argument that the products here are harmless - singular parts should occur only when ballistically transported positions collide, and two patches of a minimal connected cover share at most one point - is heuristic, and the formula is established only under the caveat that the right-hand side makes sense as a distribution, with the ϵ→0\epsilon\to0 smoothing used to realise the amplitudes required to commute with the expansion. Second, the formula is stated only away from fluid singularities: shocks (spontaneously forming discontinuities in solutions of nonlinear systems of conservation laws) and contact singularities at coinciding points; the paper takes the position that the result holds almost everywhere and leaves a precise analysis of the failure conditions open. Integrable one-dimensional systems are the safe case, since their hydrodynamics (generalized hydrodynamics, the Euler theory of integrable models in terms of a Bethe-ansatz root density with infinitely many charges) is linearly degenerate - its characteristic speeds are constant along the modes, so shocks do not form, and GHD solutions exist and are unique (Hübner and Doyon, Phys. Rev. B 112, L241101 (2025), arXiv:2406.18322; arXiv:2411.04922); generic systems enjoy no such protection. Third, the derivation rests on assumptions (existence of the scaled limits, local relaxation of fluctuations, differentiability of Euler solutions under source perturbations) whose necessary and sufficient conditions on the microscopic dynamics and initial states are uncharacterised. The paper additionally lists open extension programs - sub-Euler-scale corrections, controlled inclusion of quantum fluctuations at low entropy, and generalisation to higher-dimensional integrable or soliton-gas systems - which are separate objectives outside this problem's completion boundary.

Problem Statement

Establish the nonlinear projection formula for Euler-scale correlation functions (Doyon, J. Stat. Phys. 193, 106 (2026)) as a mathematically valid statement with a sharply delimited domain of validity. The formula expresses the Euler amplitude So1,…,on\mathsf S_{o_1,\ldots,o_n} - the ℓ(1−n)d\ell^{(1-n)d}-scaled ballistic limit of the nn-point connected correlation function of local observables at distinct macroscopic space-time points, in d≥1d\geq1 dimensions, in and out of equilibrium - as a sum over minimal connected covers of the point set, whose terms are products of connected correlation functions of conserved densities (Euler amplitudes) and mixed derivatives of microcanonical averages, summed over free conserved-charge indices. Solve the following three linked problems that the source paper leaves unresolved:

(1) Well-definedness. Prove, model-independently (for classical or quantum microscopic dynamics, Hamiltonian, circuit, or stochastic), that the products of Euler amplitudes appearing on the right-hand side are well-defined distributions on the macroscopic space-time coordinates, and that the ϵ→0\epsilon\to0 smoothing used to construct the amplitudes commutes with the minimal-connected-cover expansion - the argument must cover general nn and the multiplicities with which covers occur - or exhibit an explicit model and observables for which the products are ill-defined (the regularisation diverges or is non-unique).

(2) Failure set. Characterise the set of macroscopic space-time configurations and classes of initial states on which the projection formula fails - intrinsic fluid singularities (shocks) of the Euler solution, extrinsic singularities induced by the observables themselves, and contact singularities at colliding points.

(3) Hypotheses. Identify sharp conditions, necessary and/or sufficient, on the microscopic dynamics and on the long-wavelength initial states under which the derivation's three assumptions hold: existence of the scaled ballistic limits, local relaxation of fluctuations, and differentiability of the Euler solution under macroscopic source perturbations.

All three parts are required and jointly decide when and in what sense the formula is a valid universal law; a refutation under (1) does not on its own resolve (2) and (3), which must then be given for the exhibited model.

References

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