Large deviations of multiple absorption currents of interacting diffusing particles: additivity, vorticity, and extensions beyond the linearized regime

Agranov and Meerson (Phys. Rev. E 95, 062124 (2017)) computed, within macroscopic fluctuation theory (MFT) combined with the additivity assumption and linearization around the steady state, the Gaussian statistics of particle absorption by multiple patches on a domain immersed in a gas of interacting diffusing particles: the covariance matrix of the absorption currents is governed by an electrostatic capacitance matrix, the sign of the cross-correlations is fixed by the transport coefficients through the sign of $[\sigma'/D]'$, and the optimal flux field is generically vortical, with vorticity vanishing only when one conditions on the total absorption current. The problem posed in their Discussion - and left open by subsequent work, as confirmed by the 2025-2026 surveys of Hurtado and of Bodineau-Derrida - is to extend this multiple-current fluctuation theory beyond typical, small fluctuations: determine the joint large-deviation function of the multiple absorption currents; establish whether the additivity principle (time-independent optimal path) holds for multiple coupled currents, and whether its failure, where it occurs, is a dynamical phase transition accompanied by a change of the optimal-flux topology; decide whether the optimal flux conditioned on the total current remains vortex-free or becomes vortical at a critical total current; and extend the theory to two-dimensional infinite systems, where no steady state exists and logarithmic corrections to the action are expected.

Background

Diffusive lattice gases - for example the symmetric simple exclusion process (SSEP), in which particles hop to neighboring lattice sites subject to at most one particle per site - admit a universal large-scale description by fluctuating hydrodynamics: the coarse-grained density $\rho(\mathbf{x},t)$ obeys $\partial_t\rho+\nabla\cdot\mathbf{J}=0$ with a fluctuating current $\mathbf{J}=-D(\rho)\nabla\rho+\sqrt{\sigma(\rho)}\,\boldsymbol{\eta}$, where $D(\rho)$ and $\sigma(\rho)$ are the model-dependent diffusivity and mobility and $\boldsymbol{\eta}$ is Gaussian white noise. Macroscopic fluctuation theory (MFT; Bertini, De Sole, Gabrielli, Jona-Lasinio and Landim, Rev. Mod. Phys. 87, 593 (2015)) makes this a variational calculus for rare events: the probability of a history decays as $\exp(-\mathcal{S})$, with $\mathcal{S}$ an action functional minimized by the optimal (most probable) history. For the archetypal observable - the total current through a one-dimensional system coupled to two reservoirs - the large-deviation function (LDF), the exponential decay rate of the probability of a given time-averaged current, is generally accessible only under the additivity principle of Bodineau and Derrida (2004), which postulates that the optimal density and current profiles are time independent at long times; it is proven to hold when the transport coefficients satisfy $D(\rho)\sigma''(\rho)\le D'(\rho)\sigma'(\rho)$ (true for the SSEP, violated by the Kipnis-Marchioro-Presutti (KMP) model of stochastic heat conduction, for which $D$ is constant and $\sigma(\rho)\propto\rho^2$). In one dimension the principle is known to break down in some models through dynamical phase transitions (DPTs): non-analytic points of the current LDF associated with symmetry breaking or with time-dependent optimal paths; Shpielberg (2017) gave such DPTs a Hamiltonian-geometric interpretation, and Perez-Espigares, Garrido and Hurtado (2016) extended additivity to $d>1$ for a single current (the weak additivity principle), finding that a spatially structured - generically vortical - optimal current vector field is essential. Vortical optimal currents in two dimensions were in fact observed earlier by Bodineau, Derrida and Lebowitz (2008) for fluctuations of a partial current through a slit.

Agranov and Meerson (2017) used MFT to treat a genuinely multi-current problem, motivated by cell sensing (a cell measures its environment through receptors distributed over its surface; Berg and Purcell): a three-dimensional domain carrying $s$ absorbing patches, immersed in an infinite gas of interacting diffusing particles at density $\rho_0$, the patches acting as reservoirs at zero density. Under the additivity assumption and linearization - valid for typical, small fluctuations - they obtained the Gaussian covariance matrix of the numbers of particles absorbed by the different patches, expressed through the electrostatic capacitance matrix of the patch configuration. Two results stand out. First, the sign of the cross-correlations between currents into different patches is fixed, independently of geometry, by the sign of $[\sigma'(\rho)/D(\rho)]'$: currents anti-correlate for the SSEP (the analog of the electronic Hanbury Brown and Twiss effect in multi-terminal diffusive conductors, Sukhorukov and Loss (1999)) and correlate strictly positively for the KMP model - a prediction for mesoscopic wave energy transport, where the wave-intensity mobility obeys $\sigma(I)\propto I^2$ and experiments on intensity correlations in disordered waveguides exist (Sarma et al. (2014)). Remarkably, this is the same condition that guarantees additivity for arbitrary currents in single-current systems, which the authors noted hints at a possible relation between correlations and the dynamical phase transitions through which additivity breaks down. Second, the optimal flux field generically develops vorticity, but vorticity vanishes when one conditions on the total absorption current into all patches combined; under the resulting vortex-free assumption the unique time-independent solution of the full nonlinear MFT equations yields every cumulant of the total current equal to the corresponding one-dimensional cumulant times the capacitance. All of this, however, lives at the level of typical fluctuations. The full large-deviation regime of the multiple currents - the fate of additivity and of the vortex-free property there, and the extension to two-dimensional infinite systems - remains open, as the 2025 lecture notes of Hurtado and of Derrida and the 2026 work of Bodineau and Derrida on weak reservoir contacts make explicit.

Problem Statement

For a gas of interacting diffusing particles described by MFT transport coefficients $D(\rho)$ and $\sigma(\rho)$, in contact with $s$ absorbing patches (density zero on the patches, reflecting boundary elsewhere, density $\rho_0$ at infinity), determine the joint large-deviation statistics of the absorbed particle numbers $(N_1,\ldots,N_s)$ beyond the Gaussian, linearized regime solved by Agranov and Meerson. The three items below are independent sub-questions: each can be resolved on its own, for a specific model or geometry where appropriate, without presupposing resolution of the others. Specifically:

1. Solve the full time-dependent nonlinear MFT variational problem for the joint large-deviation function (equivalently the scaled cumulant generating function) $G(\lambda_1,\ldots,\lambda_s)$ of the multiple absorption currents, or determine the analytic structure of $G$ - its domain of analyticity, its singularities and their exponents. In particular, establish whether the additivity principle - a time-independent optimal path - holds for the multiple-current LDF; where it fails, determine the mechanism, in particular whether the failure occurs via a dynamical phase transition (a non-analyticity of $G$) accompanied by a change of topology of the optimal flux field (a change in the number/linking of its vortical structures).

2. For conditioning on the total absorption current into all patches combined: determine whether the optimal flux field remains vortex-free beyond the small-fluctuation regime, or whether a critical total current exists at which the optimal flux becomes vortical - a possible phase transition - and hence whether the result that every cumulant of the total current equals the corresponding one-dimensional cumulant times the capacitance extends beyond the linearized regime.

3. Extend the multiple-current absorption fluctuation theory to two-dimensional infinite systems, which do not reach a true steady state, so that the action is expected to acquire logarithmic corrections to its linear-in-time scaling - determine the action and these corrections. The source also notes that the formalism can be readily extended to other finite geometries sustaining multiple currents and, with some modifications, to reservoirs held at different nonzero densities (rather than zero-density absorbers); these are recorded as source-suggested routine extensions, not as required deliverables of this problem.

An answer may proceed within MFT (with controlled approximations) or by rigorous microscopic arguments; results for specific models (such as SSEP or KMP) or specific geometries are admissible partial answers provided they settle the corresponding item for that model or geometry. The source also proposes an experimental test of the predicted strictly positive cross-correlations for KMP-like mobility in disordered wave systems ($\sigma(I)\propto I^2$); this is recorded as an outstanding prediction, not as a required deliverable of this problem.

References

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