Transport of soft and charged matter at small scales is at the heart of many modern scientific challenges. For instance, the design of new nanomaterials or nanofluidic devices depends on the fundamental understanding of transport of fluids and suspensions [1]. From a biological perspective, intracellular self-organization is governed by the transport of proteins, ions, organelles, or vesicles [2]. Many precise tasks are performed with great efficiency, in spite of the structural complexity of these media, which originates from the strong thermal fluctuations which are typically felt at such scales, and from the interactions between the different agents (crowding, electrostatics, hydrodynamics...). From a thermodynamic point of view, the kinetics of such processes are essentially controlled by the nonequilibrium nature of the system: the continuous conversion of energy at the microscale typically governs spatial structures [3, 4].

In theoretical approaches, such systems are usually represented by suspensions of interacting particles, embedded in a solvent that causes their stochastic motion. The suspended particles generally evolve very far from equilibrium, either because they are driven by an external constraint (electric field, pressure gradient...), or because they are ‘active’, in the sense that they locally convert the chemical energy available in their environment into mechanical work [5]. In addition, in real systems, the suspensions of interest typically experience spatial confinement, whose interplay with nonequilibrium dynamics makes them particularly difficult to model.

The average behavior of these systems, that can be read for instance into particle currents, transport coefficients such as effective mobilities, or density profiles, is often well established, and constitutes the most direct way to study their properties. However, the fluctuations in these systems, i.e. their behavior beyond average, also govern a wealth of physical properties. For instance, fluctuation-induced effects, such as Casimir interactions, originate from the interplay between thermal noise and the boundary conditions imposed on the system [6]. As an additional example, the individual properties of tagged particles (or ‘tracers’), are closely related to the behavior of the suspension beyond average, and are of great importance [7]. Indeed, their diffusivity or the statistics of their first-passage to a target are key observables to understand the kinetics of biological processes, or molecular transport at the nanoscale.

From a computational and theoretical perspective, such dynamical and beyond-average effects are probably the most important challenges of the field. These research topics are naturally at the crossroads between the scientific interests of different communities. First, from a conceptual perspective, studying systems of interacting particles beyond average properties require specific theoretical tools, such as descriptions in terms of fluctuating fields [8] or large deviations principles [9], which are central topics in contemporary statistical mechanics. Second, the nonequilibrium aspects are at the heart of active matter, whose static and dynamical properties have been the object of a significant research effort during the past years [10, 11]. Finally, the methodological approaches from the field of chemical physics, that are currently shifting with the introduction of machine learning techniques [12, 13], are key to describe fluctuations at the microscopic scale and in complex geometries.