Ballistic Macroscopic Fluctuation Theory

• BMFT is a large-deviation framework that defines ballistic transport as evolution driven solely by initial thermodynamic fluctuations propagated by nonlinear Euler equations.
• The approach employs a path-integral formalism and saddle-point approximations to derive explicit expressions for full counting statistics and long-range correlations.
• BMFT challenges conventional diffusive theories by revealing that all macroscopic fluctuations in integrable and hard-rod gas models stem from deterministic convection of initial noise.

Ballistic Macroscopic Fluctuation Theory (BMFT) is a universal large-deviation framework describing the evolution of fluctuations and correlations in many-body systems with strictly ballistic (Euler-scale) transport. BMFT generalizes the core principles of diffusive Macroscopic Fluctuation Theory (MFT), replacing stochastic bulk noise with deterministic transport of initial thermodynamic fluctuations by nonlinear Euler equations. This approach reveals how all late-time fluctuations and long-range correlations in ballistic systems, classical and quantum, stem exclusively from initial condition noise convected by Euler-scale hydrodynamics, a result that fundamentally alters conventional understandings of non-equilibrium fluctuations in integrable and strongly interacting models (Kethepalli et al., 23 May 2025, Doyon et al., 2022, Hübner et al., 2024).

1. Fundamental Structure and Scope

BMFT applies to systems where transport is solely ballistic—meaning the characteristic length scale grows linearly with time ($x\sim t$), in contrast to the diffusive regime ($x\sim \sqrt{t}$). In such systems, hydrodynamics is governed by conservation laws

$\partial_t q_i(x,t) + \partial_x j_i(x,t) = 0$

for a vector of conserved densities $q_i$ and associated currents $j_i$. At the Euler scale, the system is locally constrained to a family of generalized Gibbs ensembles (GGEs), and the evolution of averages is captured by nonlinear Euler equations

$$(\partial_t + A[\underline{\beta}]\,\partial_x)\,\underline{q}(x,t) = 0$$

where $A$ denotes the flux Jacobian and $\underline{\beta}$ are the hydrodynamic Lagrange multipliers. In BMFT, fluctuations of observables at large scales are entirely determined by initial thermodynamic fluctuations, with no noise generated dynamically in the bulk, in stark contrast to standard (diffusive) MFT (Kethepalli et al., 23 May 2025, Doyon et al., 2022).

2. BMFT Formalism and Path Integral Representation

BMFT constructs a path-integral over hydrodynamic fields and Lagrange multipliers, thereby extending the large-deviation principle to ballistic (Euler-scale) trajectories. For a generic system, the BMFT action takes the form: $$S[\underline{q},\underline{H}] = \mathcal{F}[\underline{q}(\cdot,0)] + \int\!dx\,dt\, H^i(\partial_t q_i + \partial_x j_i[\underline{q}]) - O[\underline{q}],$$ where $\mathcal{F}$ is the large-deviation free energy of the initial condition, $H^i$ are conjugate fields enforcing the continuity equations, and $O[\underline{q}]$ encodes observable functionals (e.g., currents or full counting statistics). The path-integral is evaluated at leading order by a saddle-point solution, and at subleading order by Gaussian expansion around this saddle (Kethepalli et al., 23 May 2025, Doyon et al., 2022).

For quantum or classical integrable models, quasiparticle degrees of freedom can be mapped onto a gas of point particles with straight-line trajectories, with interactions mapped via a two-body scattering shift that plays the role of an effective rod length. The phase-space density $\rho_t(x,\theta)$ of quasiparticles obeys a deterministic generalized hydrodynamics (GHD) equation: $$\partial_t n_t(x,\theta) + v_\text{eff}[n_t](x,\theta) \partial_x n_t(x,\theta) = 0$$ with a nontrivial, density-dependent velocity $v_\text{eff}$ (Kethepalli et al., 23 May 2025).

3. Fluctuations, Full Counting Statistics, and Long-Range Correlations

The primary predictions of BMFT are twofold: (i) the large-deviation statistics of time-integrated currents (full counting statistics), and (ii) the explicit form of Euler-scale two-point density correlations.

Full Counting Statistics

The cumulant generating function for the integrated charge crossing a point over time $T$,

$$\tilde Q_T = \int_0^\infty\!dx\int d\theta\,h(\theta)\left[\rho_T(x,\theta)-\rho_0(x,\theta)\right],$$

is given in closed form by a BMFT saddle-point that modifies the initial profile by a $\lambda$-dependent deformation. All cumulants scale as $T$, and the distribution is generically non-Gaussian beyond the second cumulant (Kethepalli et al., 23 May 2025).

Two-Point and Multi-Point Correlations

BMFT predicts that while the initial state possesses only short-range ($\delta$-correlated) fluctuations, ballistic propagation by the deterministic Euler flow generically generates long-range $O(1/\ell)$ correlations at macroscopic separations. The connected two-point function at equal times and positions $x_1$, $x_2$,

$$S_{ij}(x_1, t; x_2, t) = C_{ij}(x_1, t)\,\delta(x_1 - x_2) + E_{ij}(x_1, x_2; t),$$

exhibits a nonlocal "regular" part $E_{ij}$ when $x_1\neq x_2$ due to convected initial fluctuations. BMFT provides explicit evolution equations for $E_{ij}$, typically hyperbolic PDEs dictated by the local flux Jacobian (Doyon et al., 2022, Hübner et al., 2024).

These non-clustered, long-range correlations represent a violation of the usual local equilibrium assumption and have been corroborated by both microscopic calculations and direct molecular-dynamics simulations (notably in the hard-rod gas) (Kundu, 12 Apr 2025, Mukherjee et al., 8 Jan 2026).

4. BMFT in Integrable and Hard-Rod Gas Models

In the context of integrable models, particularly the hard-rod gas, BMFT has been formulated explicitly at both the microscopic and hydrodynamic levels. The microscopic mapping (“Jepsen mapping”) translates hard rods to free points, allowing for rigorous derivations of the Euler hydrodynamics and large-deviation actions.

For the hard-rod gas, the effective velocity and phase-space density satisfy

$$\partial_t f(x, v, t) + \partial_x [v_\text{eff}(v; [f])\, f(x, v, t)] = 0$$

with

$$v_\text{eff}(v; [f]) = \frac{v - a \int dv' v' f(x, v', t)}{1 - a \rho(x, t)}$$

and mass density $\rho(x, t) = \int dv\, f(x, v, t)$. BMFT provides an action principle for the large deviations of $f$, yielding numerically tractable expressions for macroscopic correlation functions, and explicit analytic formulae for the mass-density two-point function at arbitrary space-time separations (Kundu, 12 Apr 2025, Mukherjee et al., 8 Jan 2026).

The exact agreement between the macroscopic BMFT correlator and microscopic correlation functions, upon systematic coarse-graining, validates the hydrodynamic and large-deviation assumptions underlying BMFT (Mukherjee et al., 8 Jan 2026).

5. Nonlocality, Diffusion from Convection, and Breakdown of Gradient Expansion

BMFT reveals that in one-dimensional, linearly degenerate (integrable) fluids, corrections to the Euler equations at diffusive order are entirely determined by the nonlocal, convected initial correlations. The standard Navier–Stokes gradient expansion, as obtained through the Kubo formula and the assumption of local equilibrium, fails beyond linear response.

BMFT shows that the correction to the continuity equation is of nonlocal integral kernel type, not a local second derivative. Schematically,

$$\partial_t q_i + A_i{}^k \partial_x q_k + \partial_x \left[\frac{1}{2} \frac{\delta^2 j_i}{\delta q_r \delta q_k} \ell^{-1} E^{\text{sym}}_{rk}(x, t)\right] = 0$$

with $E^{\text{sym}}$ the symmetrized nonlocal part of the two-point function. This framework describes "diffusion from convection": all hydrodynamic fluctuations arise from deterministic ballistic redistribution of initial $\delta$-correlated noise, yielding a fully reversible hydrodynamic theory with no entropy production at the described order (Hübner et al., 2024).

6. Extensions, Applications, and Connections

BMFT’s mathematical structure, based on the deterministic convection of initial fluctuation profiles, extends to broad classes of systems admitting Euler hydrodynamics. In integrable quantum systems, BMFT generalizes to calculations of entanglement Rényi entropies, where the large-scale behaviour of entanglement growth matches that predicted by the quasi-particle picture, itself a consequence of hydrodynamic fluctuation theory. The BMFT path-integral with analytic continuation to imaginary bias computes Rényi entropy dynamics, unifying the understanding of entanglement growth and charge fluctuation statistics (Vecchio et al., 2023).

For non-integrable (chaotic) systems, BMFT describes the propagation of fluctuations up to ballistic timescales; however, discrete-mode structures and subleading diffusive terms must be incorporated to capture full behaviour, especially near sound-ray singularities or under breaking of integrability (Vecchio et al., 2023, Hübner et al., 2024).

A summary of key features is organized below:

Context	Main BMFT Prediction	Reference
Integrable models	All macrosopic correlations stem from initial noise convected by Euler flow	(Kethepalli et al., 23 May 2025)
Hard-rod gas	Exact matching of BMFT and microscopic coarse-grained correlators; emergence of long-range $O(1/\ell)$ correlations	(Mukherjee et al., 8 Jan 2026)
Diffusive regime	Correction to Euler hydrodynamics is nonlocal, set by BMFT convected kernel, not gradient expansion	(Hübner et al., 2024)

BMFT underpins a shift in hydrodynamic fluctuation theory, providing a rigorous, non-dissipative framework for the emergence of universal nonlocal correlations and large deviations in ballistic many-body systems (Kethepalli et al., 23 May 2025, Doyon et al., 2022, Kundu, 12 Apr 2025, Mukherjee et al., 8 Jan 2026, Hübner et al., 2024).