Provide a complete theoretical explanation for the observed scaling of active bath diffusivity with confinement and bacterial density

Establish a comprehensive theoretical explanation for the empirically observed linear scaling of the active bath diffusivity, defined as D_b = u_b^2 τ_b, with bacterial concentration n and a confinement factor quantified by the ratio of available space to tracer size (R/R_i) in motile Escherichia coli suspensions confined within spherical droplets, clarifying the mechanisms by which confinement (outer droplet radius R_o and tracer radius R_i) modulates D_b.

Background

The paper tracks buoyant tracers inside droplets containing motile Escherichia coli and models the active bath as an Ornstein–Uhlenbeck process characterized by a memory time τ_b and a speed u_b. From fits to experimental mean squared displacements, the authors extract τ_b and u_b and define the active bath diffusivity as D_b = u_b^2 τ_b.

Across a wide range of droplet and tracer sizes and bacterial concentrations, the data for D_b collapse when plotted against n R/R_i (with R = R_o − R_i), revealing a linear scaling at small n R/R_i. While the empirical scaling is robust, the paper explicitly notes that a complete theoretical explanation for why D_b scales in this way is still lacking.