# Monte Carlo Simulations in Aviation Contrail Study: A Review  
[PERSON]  
1 Italian Aerospace Research Centre (CIRA), Via Maiorise, snc, 81043 Capua, Italy; e.maarenna@cira. and crystals. Traditionally, Eulerian schemes have been employed in the field of climate studies and weather prediction modeling. In the last fifteen years, however, due to the incredible increase in available computational power, Lagrangian parameterization has gained attention within the scientific community. These particle-tracking approaches have the advantage of conceptually reproducing the system, made up of particles with different sizes and shapes that move and interact within the fluid. Beyond the need for dedicated algorithms, reducing the overwhelming amount of computational resources that would be otherwise needed for simulating each particle in a cloud, the critical points that still remain to be addressed concern the physics of the interactions between particles and their initial distribution within a condensation trail, which depends on atmospheric, as well as exhaust chemical and physical, parameters. Coupling direct observation using airborne and remote-sensing probes in space with laboratory experiments would, in principle, help shed some light on the physics of forming condensation trails and their transition into persistent clouds. However, going beyond the actual state-of-the-art will require advances in laboratory and ground-based equipment as well as a new generation of airborne and space observational devices. In fact, the major challenge in observing both laboratory and natural clouds lays in the difficulty of separating the contributions of the different processes that add to the development of the phenomenon. Here, the role of simulations and statistics becomes crucial, conceptually providing the possibility of performing a deconvolution of the observed results with respect to the different acting processes. Numerical sampling and statistics are intimately linked to the development of computing machines. Applications in this field were pioneered at the beginning of the contemporary era of electronic computers, in particular, during theoretical fission reaction studies carried out during the Second World War. The name of the statistical sampling method, \"Monte Carlo\", recovered by [PERSON] and [PERSON] immediately after the war, comes from a joke by [PERSON], in reference to [PERSON]'s passion for card games and some anecdotes on his family [3]. A Monte Carlo simulation is based on the capability of reproducing randomness using an algorithm. This can be accomplished using one of the so-called pseudo-random number generators, a class of algorithms invented by [PERSON], that have now reached a high degree of mathematical sophistication [4]. The term \"pseudo\" accounts for the impossibility of reproducing, with a deterministic algorithm, a real random distribution. However, pseudo-random methods are devised to provide a set of numbers whose distribution is statistically indistinguishable from the realization of a random variable uniformly distributed in [0, 1]. The availability of uniformly distributed numbers allows reproduction of a generic probability density function (pdf) through an inverse transformation. In fact, the uniform distribution U(0,1) can be associated with the cumulative probability function of a generic stochastic variable, X, whose values are bounded between 0 and 1 by definition. A random process characterized by a vector of random variables, \(\mathrm{V}=\{X_{1},X_{2},\ldots,X_{n}\}\), can be then simulated using the inverse transformation approach. If the random variables are independent, this is trivially possible using n-times the pdf generation algorithm. In the case of dependent random components of the vector X, one can start generating the first random variable, X\({}_{1}\). Knowledge of the conditional pdf, \(f_{2}(x_{2}\,|\,x_{1})\), allows the generation of X\({}_{2}\). The availability of the remaining n-2 conditional pdf can be exploited for simulation of the whole random process.