Diffusion plays crucial roles in cells and tissues, and the purpose of this chapter is to theoretically examine it. First, we describe the diffusion equation and confirm that its solution becomes a Gaussian distribution. Then, we discuss concentration gradients under fixed boundary conditions and the three-color flag problem to address positional information in multicellular organism morphogenesis. We introduce the possibility of pattern formation by feed-forward loops, which can transform one gradient into another or convert a chemical gradient into a stripe pattern. Next, we introduce Turing patterns as self-organizing pattern formation, outlining the conditions for Turing instability through linear stability analysis and demonstrating the existence of characteristic length scales for Turing patterns. We provide specific examples in one-dimensional and two-dimensional systems. Additionally, we present instances of traveling waves, such as the cable equation, Fisher equation, FitzHugh–Nagumo equation, and examples of their generation from limit cycles. Finally, we introduce the transformation of temporal oscillations into spatial patterns, exemplified by models like the clock-and-wavefront model.

Physicists have a dirty secret: we’re not very good at solving equations. More precisely, humans aren’t very good at solving equations. We know this because we have computers and they’re much better at solving things than we are. This means that we must develop a toolbox of methods so that, when confronted by a problem, we have some options on how to go about understanding whats going on. The purpose of this chapter is to develop this toolbox in the guise of various approximation schemes.

The Maxwell demon and the Szilard engine demonstrate that work can be extracted from a heat bath through measurement and feedback in apparent violation of the second law. A systematic analysis shows that, by including the measurement process and the subsequent erasure of a memory according to Landauer’s principle, the second law is indeed restored. For such feedback-driven processes, the Sagawa–Ueda relation provides a generalization of the Jarzynski relation. For the general class of bipartite systems, the concepts from stochastic thermodynamics are developed. This framework applies to systems where one component “learns” about the changing state of the other one, as in simple models for bacterial sensing. The chapter closes with a simple information machine that shows how the ordered sequence of bits in a tape can be used to transform heat into mechanical work. Likewise, mechanical work can be used to erase information, i.e., randomize such a tape. These processes are shown to obey a second law of information processing.

The overdamped Langevin equation for a particle in a potential and, possibly, subject to a nonconservative force is introduced. The corresponding Fokker–Planck equation, the Smoluchowski equation, is derived. In a time-independent potential, any initial distribution finally approaches the equilibrium one. For a constant external force and periodic boundary condition like the motion along a ring, a nonequilibrium steady state is established. As an application, the Kramers escape from a meta-stable well can be discussed. The mean local velocity and the path integral representation are introduced. Thermodynamic quantities like work, heat, and entropy production are identified along individual trajectories and their ensemble averages are determined. Their distributions are shown to obey detailed fluctuation relations. A master integral fluctuation relation can be specialized to yield inter alia the Jarzynski relation, the integral fluctuation relation for entropy production, and the Hatano–Sasa relation.

Hertha Sponer’s (1894-1968) early years in physics were spent at the center of the quantum revolution. Training as an experimentalist under Debye, then heading the spectroscopy labs in Göttingen uniquely situated her to contribute to the development of quantum theory and the emergence of quantum chemistry, by novel interpretations of hitherto unexplained spectrographic data using quantum mechanics, and suggesting new applications of the theory to atoms and diatomic molecules. Sponer’s name has nevertheless been largely written out of scientific accounts of these years. When mentioned in the context of quantum theory, it is usually as Franck’s “assistant” (incorrect) and second wife – descriptions that obscure her status as a world-renowned scientist who’d contributed importantly to physics and chemistry over a long and illustrious career. Extant accounts of Sponer’s life and work almost exclusively concern her postwar years as a professor at Duke. But by then quantum theory was well established, and her research had pivoted in other directions. This chapter aims to introduce Sponer into the history of early quantum theory, with appropriate attention to her achievements.

The full beauty of Maxwell equations only becomes apparent when we realise that they are consistent with Einstein’s theory of special relativity. The purpose of this chapter is to make this relationship manifest. We rewrite the Maxwell equations in relativistic notation, where the four vector calculus equations are condensed into one, simple tensor equation. Viewed through the lens of relativity and gauge theory, the Maxwell equations are forced upon us: the world can’t be any other way.

John Wheeler (1911−2008), besides being a key figure in twentieth-century physics in his own right, was also an exceptional mentor and a key witness to historical events. Little known is that his first PhD student was a woman, Katharine Way (1902−1995), who notably played an important role in the postwar organization and dissemination of nuclear data. In the 1990s, Wheeler further made the surprising claim that Way’s work while she was his student came very close to anticipating the discovery of nuclear fission. In addition to gathering the few pieces of information about Way’s early work, this chapter provides a contextualization and evaluation of Wheeler’s words by analyzing his peculiar communicative style, which often subtly mixed history, personal experience, and theoretical insights or guiding ideas. To illustrate this, Wheeler’s pages about personalities such as Marie Curie, Lise Meitner, Maria Goeppert Mayer, and Way herself are considered. It emerges how Wheeler’s original viewpoint has to be properly discussed when evaluating his claim about his former student’s work.

At the heart of classical mechanics sits the venerable equation F=ma. To solve this equation, we first need to specify the force at play. In this chapter, we start along this journey. We will look at various forces, including gravity, electromagnetism and friction, and start to understand some of their features. For each, we will solve F=ma in some simple settings.

The harmonic oscillator is, by some margin, the most important system in physics. This is partly because its easy and we can solve it. And partly because, under the right circumstances, pretty much anything else can be made to look like a bunch of coupled harmonic oscillators. In this chapter, we look at what happens when a bunch of harmonic oscillators – or springs – are connected to each other.

The essence of dimensional analysis is very simple: if you are asked how hot it is outside, the answer is never “2 o’clock”. You’ve got to make sure that the units, or “dimensions”, agree. In this chapter, we understand what it means for quantities to have dimensions and how getting to grips with this can help solve problems without doing any serious work.

Symmetries are a key idea in physics. In the classical world, they are associated to conservation laws, courtesy of Emmy Noether. The same, and more, is true in the quantum world. In this chapter we explore how symmetries manifest themselves in quantum mechanics. Special attention will be given to time evolution and the role of SU(2) and angular momentum

This chapter deals with advanced topics for a multivariate Langevin and Fokker–Planck dynamics. For systems with multiplicative noise it is shown that neither the drift term in the Langevin equation nor the discretization parameter can be determined uniquely. If one of the two is fixed, the other one is determined. In contrast, the Fokker–Planck equation, which contains the physically observable distribution is unique. Experimental data for a particle near a wall illustrate the relevance of space-dependent friction. Martingales are introduced for a Langevin dynamics with a nonlinear expression of entropy production as a prominent example that with Doob’s optimal stopping theorem leads to universal results of its statistics. Finally, underdamped Langevin dynamics is described by the Klein–Kramers equation, for which entropy production is determined by the irreversible currents. A multi-time-scale analysis recovers the Smoluchowski equation in the overdamped limit even in the presence of an inhomogeneous temperature for which an anomalous contribution to entropy production is found.

Whereas RRR has been successfully applied to a broad range of problems, the example in this chapter shows that our understanding of the algorithm is far from complete.