ABSTRACT
Individual behaviors are at the origin of any global trend or collective property. In this chapter, the authors present the analytical and mathematical tools that allow the reader to characterize these two worlds and understand their interplay. In a first approach, the individual constituents of matter, i.e., atoms and molecules, can be naturally thought of as minute replicas of simple shapes surrounding the reader. Quantum mechanics gives rise to a distribution of discrete energy levels for quantum systems. The system’s evolution can then be analyzed in terms of changes in its defining quantities or changes in the macroscopic state of the system. Maxwell was one of the first scientists to factor in the concepts of distribution and probability to determine the properties of macroscopic systems, such as gases. Indeed, Maxwell postulated that not all particles in a gas have the same velocity and that a distribution of these velocities should be used to model such a system.
