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An exactly solvable macroscopic fluctuation theory of single-file diffusion

This paper demonstrates that the macroscopic fluctuation theory for single-file diffusion, modeled as a gas of extended Brownian hard rods, is exactly solvable via a canonical transformation, enabling the explicit computation of large-deviation statistics for tracer position and integrated current in both continuum and lattice exclusion models.

Original authors: Sandeep Jangid, Soumyabrata Saha, Kapil Sharma, Jitendra Kethepalli, Benjamin Guiselin, Jacopo De Nardis, Tridib Sadhu

Published 2026-07-16

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CC BY 4.0

Original authors: Sandeep Jangid, Soumyabrata Saha, Kapil Sharma, Jitendra Kethepalli, Benjamin Guiselin, Jacopo De Nardis, Tridib Sadhu

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a crowded hallway where everyone is trying to walk in the same direction, but the hallway is so narrow that no one can step aside to let someone else pass. If you are in the middle of the pack, you are stuck. You can only move if the person in front of you moves, and they can only move if the person in front of them moves. This is the essence of "single-file diffusion," a phenomenon that happens everywhere from water molecules zipping through tiny carbon nanotubes to ions squeezing through pores in a cell membrane. In the chaotic world of physics, where particles usually bounce around randomly and forget their past, these single-file systems are stubborn. They remember where they started for a very long time, and their movement is weirdly slow and correlated, behaving nothing like the smooth, predictable flow of a normal gas.

For decades, scientists have tried to predict exactly how these crowded particles behave, especially when they get pushed or pulled in strange ways. They have built mathematical models, like "exclusion processes" where particles are treated as tiny points that simply cannot occupy the same spot. While these models have given us some clues, they are often like trying to solve a puzzle with half the pieces missing; the math gets incredibly messy, and we can only calculate the answer for a few specific, simple situations. The big question has been: Is there a way to solve the full, complex story of these crowded particles, including how they fluctuate and behave when they are far from their usual state?

This is where a new study by Sandeep Jangid, Soumyabrata Saha, and their colleagues steps in. They decided to look at the problem through a different lens, treating the particles not as tiny points, but as "Brownian hard rods"—imagine them as little rigid sticks with a specific length, rather than dimensionless dots. This might sound like a small change, but it turns out to be a game-changer. The researchers discovered that by using a clever mathematical "magic trick" called a canonical transformation, they could turn this messy, interacting system of rods into a much simpler system of non-interacting point particles. It's like realizing that a tangled ball of yarn can be perfectly straightened out if you just look at it from the right angle.

By making this switch, the team was able to solve the equations that describe the system's behavior exactly. They didn't just guess; they calculated the precise statistics for two key things: where a specific "tracer" particle ends up after a long time, and how many particles cross a certain line (the "integrated current"). They did this for two different scenarios: one where the starting positions of the particles are random and averaged out (the "annealed" ensemble), and another where the starting positions are fixed and known (the "quenched" ensemble). The results are strikingly clear. They found that while the position of a single particle depends on the size of the rods in a specific way, the flow of particles across a line has an even more complex relationship with the rod size.

To make sure their math wasn't just a pretty theory, the authors ran computer simulations that act like a high-speed replay of these crowded hallways. They used a special technique called "importance sampling" to watch rare events—like a particle moving much further than expected—happen over and over again. The computer data matched their new formulas perfectly, confirming that their "magic trick" works. They also showed that this method isn't just for smooth, continuous space; it works for grid-based models too, where particles hop from spot to spot. This work provides a rare, fully solvable map for a complex system, offering a clear window into how collective behavior emerges when particles are forced to move in a single file. It suggests that even in the most crowded, constrained environments, there is an underlying order that can be unlocked with the right mathematical key.

Problem Statement
Single-file diffusion, a phenomenon where particles in narrow channels move in one dimension without overtaking, is a ubiquitous feature in low-dimensional transport systems ranging from ion conduction in cell membranes to molecular diffusion in porous media. While the symmetric simple exclusion process (SSEP) on a lattice has served as a paradigmatic model for this behavior, yielding exact results via integrability methods, these results are often fragmented and rely on sophisticated microscopic techniques tailored to specific observables or initial ensembles. Furthermore, the discrete nature of lattice models limits their direct connection to natural continuum realizations. A major recent advance involved the formal solution of macroscopic fluctuation theory (MFT) for the exclusion process; however, explicit solutions for generic macroscopic observables remain scarce. The specific challenge addressed in this work is the lack of an exactly solvable continuum model for single-file diffusion that can provide explicit large-deviation statistics for key observables, particularly for extended particles with finite size.

Methodology
The authors model single-file diffusion as a one-dimensional gas of extended Brownian hard rods (BHR) of length $a$ on an infinite line, subject to mutual hard-core repulsion. The system is prepared in a domain-wall state with step-like average density profiles. The study focuses on two primary observables: the tracer-position ( $z_t$ ) and the integrated-current ( $Q_t$ ).

The core methodological innovation is the application of MFT to the BHR system. The authors demonstrate that the MFT equations for this interacting continuum model, which are typically non-linear and coupled, become exactly solvable through a specific canonical transformation. This transformation involves:

Coordinate Transformation: A change of variables from the physical coordinate $x$ to a "free-volume" coordinate $X$ , defined by $X = x - a \int_{z(s)}^x \rho(x', s) dx'$ , where $z(s)$ is the rescaled tracer position.
Field Transformation: A mapping of the density field $\rho$ and the response field $\hat{\rho}$ to renormalized fields $r$ and $\hat{r}$ . Specifically, $r(X, s) = \rho(x, s) / (1 - a\rho(x, s))$ .

This transformation maps the BHR system onto a system of non-interacting point particles (zero-length limit, $a \to 0$ ) with renormalized transport coefficients. The authors utilize this mapping to solve the MFT variational problem explicitly. They compute the scaled cumulant-generating functions (SCGFs) for both annealed (averaging over initial positions) and quenched (fixed initial positions) ensembles. The validity of the analytical results is confirmed through rare-event importance-sampling simulations for both the continuum BHR model and a corresponding lattice gas with finite-volume exclusion.

Key Results

Exact Solvability: The paper establishes that the MFT for the extended BHR gas is exactly solvable. The canonical transformation reduces the complex interacting problem to a solvable point-particle problem, allowing for the explicit computation of SCGFs and optimal density trajectories.
SCGFs for Tracer and Current: The authors derive closed-form expressions for the SCGFs of the tracer-position ( $\mu^{tp}$ $μ^{tp}$ ) and integrated-current ( $\mu^{ic}$ $μ^{i c}$ ) in both annealed and quenched ensembles.
- The tracer-position SCGF depends on the rod length $a$ only through renormalized densities.
- The integrated-current SCGF retains an explicit dependence on $a$ .
- The results satisfy a fluctuation symmetry relation in the annealed ensemble, extending known universality between Hamiltonian hard-rods and Brownian hard-rods to annealed current statistics.
Optimal Trajectories: The paper provides explicit parametric expressions for the optimal density profiles $\rho(x, s)$ that realize large deviations of the tracer and current. Unlike previous results for SSEP where optimal profiles were only explicit at initial and final times, this solution provides the full time-evolution of the optimal trajectory.
Large-Deviation Properties:
- Typical fluctuations are Gaussian, obeying the annealed-quenched relation $\langle O_t^2 \rangle_A = \sqrt{2} \langle O_t^2 \rangle_Q$ .
- Far-tail behaviors differ: the integrated-current and quenched tracer SCGFs scale as $\lambda^{3/2}$ , while the annealed tracer SCGF has finite support, leading to a linear tail in its large-deviation function.
Extension to Lattice Gases: The authors show that the same canonical transformation logic applies to lattice gases with finite-range exclusion (multi-site exclusion processes), mapping them to the SSEP and allowing for the calculation of their SCGFs.

Significance and Claims
The paper claims to provide a "full statistical description" of central observables in single-file diffusion within a continuum framework. Its primary significance lies in offering a rare example of a non-trivial, interacting continuum model that is explicitly solvable within MFT.

Theoretical Advancement: By solving the MFT variational problem for the BHR, the work overcomes the "perceived intractability" of continuum single-file models. It provides a tractable counterpart to the phenomenology of exclusion processes, offering explicit results for observables that have remained challenging to compute for the SSEP.
Physical Relevance: The BHR model retains continuum space and finite particle extent, making it a more direct description of physically realizable single-file systems (e.g., colloidal assemblies, water in nanotubes) compared to lattice models.
Methodological Impact: The demonstration that a canonical transformation can render the MFT of an interacting system exactly solvable opens new avenues for studying macroscopic fluctuations. The authors suggest this approach could be extended to broader initial conditions, geometries, and multi-time statistics, potentially revealing connections to KPZ scaling in Brownian settings.

The work does not propose new experimental setups but validates its theoretical predictions against numerical simulations, confirming the accuracy of the derived large-deviation functions and optimal trajectories.

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