Single-file diffusion — the motion of particles confined so they cannot pass one another — is common in narrow channels. The authors present an exact solution of the macroscopic fluctuation theory (MFT) for a natural continuum model of this physics: a one-dimensional gas of Brownian hard rods (rods of finite length that diffuse and cannot overlap). Using a canonical transformation they reduce the interacting problem to a solvable form and obtain explicit formulas for the rare (large) fluctuations of two standard observables: the tracer particle’s position and the integrated current across a point.

Concretely, the paper studies rods of length a that perform Brownian motion between collisions (the bare diffusion coefficient in their units is D0 = 1). The initial condition is a domain-wall, or step, profile with different average densities to the left and right of the origin. The authors treat two common ways of averaging over initial conditions: the annealed ensemble (where initial positions are allowed to fluctuate) and the quenched ensemble (where initial positions are fixed). Working in the long-time, large-scale regime where fluctuations grow like the square root of time, they solve the MFT variational problem and produce explicit scaled cumulant generating functions and large-deviation functions for the tracer position and the net number of particles that cross the origin.

At a high level, MFT describes the coarse-grained density field with a noisy diffusion equation characterized by a density-dependent diffusivity and mobility. For the hard-rod gas these transport coefficients are explicit (the diffusivity behaves as D(ρ)=1/(1−aρ)^2 and the mobility as σ(ρ)=2ρ in the paper’s units). The key technical step is a canonical transformation that maps the extended, interacting rod problem to a dual, simpler problem of non-interacting Brownian objects. That mapping converts the MFT variational problem into one that can be solved explicitly. From that solution the authors extract the full shape of the large-deviation functions, the scaling of their tails (for example, certain tail behaviors scale like λ3/2 in generating-function variables), and the optimal density trajectories—the most probable ways the system arranges itself to produce a rare fluctuation. In the annealed case they also identify time-reversal symmetries of the optimal trajectories.