Partial differential equations in complexity science
Partial differential equations (PDEs), that is to say equations relating partial derivatives of functions of more than one variable, are part of the bedrock of most quantitative disciplines and complexity science is no exception. They invariably arise when continuous fields are introduced into models. The classic example is the distribution of heat in a thermal conductor which typically varies continuously with time and with position. The continuous field in this case is T(x,t) the temperature at position x and time t which, in this case, satisfies a linear PDE called the diffusion equation.
In complexity science, PDEs often result from the process of coarse-graining whereby microscopically discrete processes are averaged over small scales to produce an effective continuous description of larger scales. Since much of complexity science focuses on the emergent properties of such coarse-grained descriptions, the analysis of PDEs is a key part of our complexity science toolkit. Some examples of coarse-graining as applied to interacting particle systems were discussed in Chapter 3. Another well-known example is the effective description of traffic flow using a coarse-grained fluid description described by a PDE known as Burgers' equation and variants of it, cf. Chapter 4. We will revisit this application in more detail later. Real fluids also provide a wealth of examples of complex behaviour, all described by the well-known Navier-Stokes equations or variants of them which are famous for being among the most mathematically intractable PDEs of classical physics.