arXiv subjects

Ricardo García-Pelayo

Publications and source records attributed to Ricardo García-Pelayo.

4 recordsLinked to original sources

arXiv2024

Fourier series for the three-dimensional random flight

The probability density function of the random flight with isotropic initial conditions is obtained by an expansion in the number of collisions and the in the spatial harmonics of the solution, as in a Fourier series. The method holds for any dimension and is worked out in detail for the three dimensional case. In this case the probability density functions conditional to 1 and 2 collisions are also found using a different method, which yields them in terms of elementary functions and the polylogarithm function Li$_2$. The latter method is exact in the sense that one does not have to truncate a series, as in the first method. This provides a reference to decide where to truncate the series. A link is provided to a web page where the reader may download the series truncated at 132 collisions; for times larger than 100 times the average inter-collision time, the Gaussian approximations is used. The case in which the initial condition is a particle moving along a fixed direction is briefly considered.

math-ph↗

arXiv2023

New techniques to solve the 1-dimensional random flight

We present here two new techniques to solve the one-dimensional random flight. The first one is an expansion in the number of collisions. The second one is the obtention of a Fourier series. This second technique can be applied to an arbitrary number of dimensions. Some mathematical identities are obtained.

math-ph↗

arXiv2016

Pairs of subsets of spheres and Cartesian products thereof with the same distribution of distance

We prove the following three statements: 1) Let $(A, \bar A)$ be a partition of the spherical surface $S^n$ into two measurable sets. Let $st_A$ and $st_{\bar A}$ be their measure density functions of distance. Then $|st_A - st_{\bar A}|$ depends only on the difference of their $n$-areas. 2) If the spherical surface $S^n$ is divided in two measurable subsets $A$ and $\bar A$ of equal $n$-surface, then these two subsets have the same distribution of distance. 3) Let there be a pair $(S, S')$ of subsets of a sphere $S^{n}$ such that $st_S = st_{S'}$. Then their complementary subsets satisfy $st_{\bar S} = st_{\bar S'}$ and $st_{S, \bar S} = st_{S', \bar S'}$, where $st_{A, B}$ is the measure density function of distance between a point in $A$ and a point in $B$. Furthermore, it is shown that the statements remain true when $S^n$ is substituted by the Cartesian product $S^{n_1} \times ... \times S^{n_r}$ endowed with the metric which is naturally inherited from its factors.

math.PR↗

arXiv2015

Computation of the multi-chord distribution of convex and concave polygons

Analytical expressions for the distribution of the length of chords corresponding to the affine invariant measure on the set of chords are given for convex polygons. These analytical expressions are a computational improvement over other expressions published in 2011. The correlation function of convex polygons can be computed from the results obtained in this work, because it is determined by the distribution of chords. An analytical expression for the multi-chord distribution of the length of chords corresponding to the affine invariant measure on the set of chords is found for non convex polygons. In addition we give an algorithm to find this multi-chord distribution which, for many concave polygons, is computationally more efficient than the said analytical expression. The results also apply to non simply connected polygons.

math.MG↗