In this chapter we provide an overview of data modeling and describe the formulation of probabilistic models. We introduce random variables, their probability distributions, associated probability densities, examples of common densities, and the fundamental theorem of simulation to draw samples from discrete or continuous probability distributions. We then present the mathematical machinery required in describing and handling probabilistic models, including models with complex variable dependencies. In doing so, we introduce the concepts of joint, conditional, and marginal probability distributions, marginalization, and ancestral sampling.

At the macroscale, thermodynamics rules the balances of energy and entropy. In nonisolated systems, the entropy changes due to the contributions from the internal entropy production, which is always nonnegative according to the second law, and the exchange of entropy with the environment. The entropy production is equal to zero at equilibrium and positive out of equilibrium. Thermodynamics can be formulated either locally for continuous media or globally for systems in contact with several reservoirs. Accordingly, the entropy production is expressed in terms of either the local or the global affinities and currents, the affinities being the thermodynamic forces driving the system away from equilibrium. Depending on the boundary and initial conditions, the system can undergo relaxation towards equilibrium or nonequilibrium stationary or time-dependent macrostates. As examples, thermodynamics is applied to diffusion, electric circuits, reaction networks, and engines.

When thinking of city maps, we instinctively envision a network of links along which an ever-changing flow of traffic is carried. Such an idealised description, however, is not limited to the maps we are all familiar with. From the interactions between atoms and subatomic particles to the gravitational forces which act between the billions of galaxies stretching across the known universe, from the transmission of electrical signals in our brains to the complexity of social interactions between people, most if not all phenomena we encounter, consciously or not, find a natural representation in the form of networks. Indeed, it can be argued that the abstract notion of interacting objects resides at the very heart of our conceptual understanding of nature as it touches upon the very fabric of physical reality with its finite and discrete makeup. How can we leverage the mathematical study of interconnected objects, the theory of networks and graphs, in our quest of understanding nature, and what are its limitations?

We define the single queue, introduce notation and some relations and properties, and present simple examples of queues. We also discuss simulation of queues.