**Appendix B. Traffic between the Larger and the Smaller Semicircular Reservoir**

Consider the setup in Figures 3 and 5 and let a homogeneous particle density *ρ* be same in both reservoirs. In this Appendix we show that, if the particles are subject to Lévy noise, accumulation in the smaller reservoir develops. Imagine a semicircular strip of width *dr* at a distance *r* from the opening. There are *ρπrdr* particles in this strip. We let the power-law approximation, cf. Equation (1), be valid for *r* > *r*0. Here *r*<sup>0</sup> is much larger than the width of the opening *d* and much smaller than the radii *R*<sup>1</sup> and *R*2. The angle *θ* is small and with *θ* expressed in radians we have *d* = *θr*. For a particle in the semicircular strip at distance *r* > *r*0, there is a probability that a Lévy jump will bring it to the other reservoir. To achieve such transition, the jump needs to be larger than *r*. For such a jump, the probability is proportional to *r*−*α*. In order to go through the opening, the jump must also be in the right direction. This leads to a factor (*d*/*r*) cos *φ* (cf. Figure 3). After the integration from *φ* = −*π*/2 to *φ* = *π*/2, we derive a "direction factor" of 2*d*/*r*, i.e., ∝ 1/*r*. Putting together all of the effects specified in this paragraph, we have the following formula for the number of transitions during a small timestep from a distance between *r* and *r* + *dr*:

$$
\rho \, dn^{tr}(r, r + dr) \propto \frac{1}{r} r^{-\alpha} r \, dr = r^{-\alpha} \, dr. \tag{A4}
$$

Next we integrate from *r*<sup>0</sup> to the boundary *Ri* (*i* = 1, 2) and find for the number of Lévyjump-associated transitions from reservoir *i*:

$$N\_i^{tr} \propto \int\_{r\_0}^{R\_i} r^{-n} \, dr \propto \text{sgn}(1 - a) \left( R\_i^{1 - a} - r\_0^{1 - a} \right). \tag{A5}$$

Care must be taken in case of *α* = 1. In that case *Ntr*,*α*=<sup>1</sup> *<sup>i</sup>* ∝ log *Ri* − log *r*0. We thus conclude that for 0 < *<sup>α</sup>* ≤ 1, the number *<sup>N</sup>tr <sup>i</sup>* diverges as *Ri* is taken to infinity. For 1 < *α* < 2, a constant value for *Ntr <sup>i</sup>* ensues if *Ri* → ∞.

The proportionality constant (associated with the ∝) and *r*<sup>0</sup> (the radius from which the power law is taken to describe the Lévy distribution) are the same for both reservoirs. Thus, if both reservoirs in Figures 3 and 5 have the same uniform *ρ*, then we find for the net number of particles Δ*Ntr* = *Ntr* <sup>1</sup> − *<sup>N</sup>tr* <sup>2</sup> that transits from the larger to the smaller reservoir:

$$
\Delta N^{tr} \propto \text{sgn}(1-\alpha) \left( R\_1^{1-a} - R\_2^{1-a} \right). \tag{A6}
$$

If 0 < *α* ≤ 1 and if values for *R*<sup>1</sup> and *R*<sup>2</sup> are large, then there is accumulation in the smaller reservoir.

For 1 < *α* < 2, there will again be accumulation in the smaller reservoir, but the effect becomes smaller as *R*<sup>2</sup> and *R*<sup>1</sup> grow and will become negligible as *R*1,2 → ∞. Effectively, the geometry of the reservoirs is irrelevant for large *R*<sup>1</sup> and *R*2. In that case it is particles near the opening that dominate the traffic through the opening.

In the main text, the above derivation is carried out for the case of a density *ρi*(*ri*) (*i* = 1, 2) that depends on the distance *ri* from the opening.

Finally, it is worth pointing out that the above derivation readily generalizes to higher dimensional reservoirs. In the 3D case, we face hemispheres. The number of particles in a hemispheric shell is *ρ*2*πr*2*dr*. For the *n*D case, the shell contains a number of particles that is proportional to *rn*−1*dr*. We thus have for *dntr nD*:

$$dn\_{nD}^{tr}(r, r+dr) \propto \frac{1}{r} r^{-n} r^{n-1} \, dr = r^{n-2-n} \, dr. \tag{A7}$$

This leads to:

$$N\_{i, \rm nD}^{tr} \propto \int\_{r\_0}^{R\_i} r^{n-2-a} \, dr \propto \left( R\_i^{n-1-a} - r\_0^{n-1-a} \right). \tag{A8}$$

and

$$
\Delta N\_{nD}^{tr} \propto \left( R\_1^{n-1-a} - R\_2^{n-1-a} \right). \tag{A9}
$$

This is an interesting result. For 3 and more dimensions, we do not need to discriminate between different ranges of *α*. Lévy noise with any *α* (0 < *α* < 2) will in that case lead to a significantly higher density in the smaller reservoir and the effect will be stronger for higher values of *R*1,2.
