Extended Data Fig. 3: Minimal model of mass cascade with scale selection.

(a) In the model, two clusters M_n of size 2ⁿ⁻¹ can merge into a cluster M_n+1 of size 2ⁿ. This aggregation process occurs with a rate \({k}_{n}^{+}\). Conversely, a cluster M_n+1 can split into two clusters M_n (except for monomers M₁). This fragmentation process occurs with a rate \({k}_{n+1}^{-}\). (b) The rates \({k}_{n}^{\pm }\) of aggregation/fragmentation depend on the size of the cluster, so that (i) large clusters are more likely to fragment that small ones (blue curve) and (ii) small clusters are more likely to aggregate than large ones (red curve). (c-e) Equations (29) are numerically solved starting from the initial condition c_n = 0 for all n. We have set \({\kappa }_{0}^{\pm }=2\) and \({\kappa }_{1}^{\pm }=1\), as well as J_in = J_out = 1. The number distribution c_n is plotted in panel c, while the mass distribution ρ_n = 2ⁿ⁻¹m₀c_n (normalized by its maximum value) is plotted in panel d. In panel e, we show the fluxes \({J}_{n}^{+}\) (red) and \({J}_{n}^{-}\) (blue) defined in Eqs. (31)–(32). The inset of panel e shows the total flux \({J}_{n}={J}_{n}^{+}+{J}_{n}^{-}\), which is constant away from the boundaries. (f) Entropy production in the mass cascade. We plot the rate of entropy production (computed within the mean-field model) as a function of the flux J_in through the system, in logarithmic scale. (As the distribution is stationary, there is a constant flux equal to the input flux). We have set \(N=15,{\kappa }_{0}^{\pm }=0.1,{\kappa }_{1}^{\pm }=1,{J}_{{\rm{out}}}=1\) and c_n(t = 0) = 0 for all n. (g) Comparison between mean-field and Monte-Carlo simulations. The red curve (labelled ODE) shows the mean-field solution of Eq. (29), while the blue curve (labelled MC) shows the solution of the kinetic Monte-Carlo simulations (average plus or minus half a standard deviation over 1000 samples). We have set \(N=15,{\kappa }_{0}^{\pm }=0.1,{\kappa }_{1}^{\pm }=1,{J}_{{\rm{in}}}={J}_{{\rm{out}}}=1\) and c_n(t = 0) = 0 for all n. (h) Equations (29) with no influx and outflux (J_in = J_out = 0) are numerically solved starting from the initial condition with only monomers c_n(t = 0) = c₀(0)δ_n,0, with different values of the number of monomers c₀(0). We observe that (i) a peak in the steady-state distribution only arises when the initial number of monomers c₀(0) is large enough and (ii) the position of the peak moves as c₀(0) increases. We have set \({\kappa }_{0}^{\pm }=2\) and \({\kappa }_{1}^{\pm }=1\).