# Calculations¶

The ZENO code is composed of two types of calculations: exterior and interior.

## Exterior calculation¶

The exterior calculation focuses on the computation of electrical properties including the capacitance, the electric polarizability tensor, and the intrinsic conductivity. Once the electrical properties are known, the hydrodynamic properties, including the hydrodynamic radius and the intrinsic viscosity, can be precisely estimated by invoking an electrostatic-hydrodynamic analogy as detailed in Refs. [3][4][5]. Other related properties are also determined.

To compute the aforementioned properties for an object requires the solution of Laplace’s equation outside the object with appropriate boundary conditions. This is efficiently accomplished by using a Monte Carlo method, which involves (1) creating a launch sphere that encloses the object, (2) launching random walks from the surface of the launch sphere, and (3) determining the fate of such walks—if they hit the object or go to infinity. These walks are exterior to the object, hence the name for the calculation. Each random walk is generated using a method called Walk on Spheres. This algorithm requires generating a sphere for each step in the random walk. The center of this sphere is located at the end of the current random walk; the radius of the sphere is determined by finding the shortest distance between the center of the sphere and the object. Finally, the step in the walk is taken by randomly choosing a point on the surface of the sphere. The process is then repeated. Since the size of spheres will progressively get smaller as the object is approached, a cutoff distance, known as the skin thickness, is required. Without a cutoff distance, the algorithm would continue, at least theoretically, indefinitely. As this is reminiscent of Zeno’s paradox of Tortoise and Achilles, the code is named in Zeno’s honor. For more details on this method refer to Refs. [3][8][9].

## Interior calculation¶

The interior calculation determines the volume and the gyration tensor for an object using a Monte Carlo method. Specifically, this calculation involves generating random points within the same launch sphere as in the exterior calculation. The location of these points can then be used to approximate all of the relevant properties. For example, the volume of the object is estimated by the fraction of points inside the object multiplied by the the volume of the launch sphere. The interior calculation is given its name since the points in the interior of the object are essential for computing the properties.