American physicist Freda Friedman Salzman (1927–1981) became an active feminist after her faculty position at the University of Massachusetts Boston was not renewed, under the university’s misogynistic anti-nepotism policy. Whereas her long-lasting struggle and eventual reappointment has already been expounded to some extent, her contributions to physics have not been given proper historical consideration. It is easier to learn about Friedman Salzman’s “weight of being a woman” – as she put it – than about her academic work. This chapter remedies that omission by shedding light on one of her key accomplishments. In 1956, Geoffrey Chew and Francis Low established the well-known Chew–Low model to put the understanding of nuclear interactions on a sounder theoretical basis. The model, however, leads to a daunting nonlinear integral equation. Friedman Salzman and her husband managed to solve the integral equation numerically. Stanley Mandelstam soon recognized the achievement of “Salzman and Salzman” (as he wrote) by naming their approach the “Chew–Low–Salzman method.”

Trapping of the RRR algorithm on nonsolutions can be avoided by modifying the constraint sets and also the metric. This chapter also covers general good practice on the use of RRR.

Drop some ink in a glass of water. It will slowly spread through the whole glass, moving in a manner known as diffusion. This process is so common that it gets its own chapter. We will describe the basics of diffusion, as captured by the heat equation, before understanding how diffusion comes about from an underlying randomness. We will see this through the eyes of the Langevin and Fokker-Planck equations.

An isolated system is described by a classical Hamiltonian dynamics. In the long-time limit, the trajectory of such a system yields a histogram, i.e., a distribution for any observable. With one plausible assumption, introduced here as a fundamental principle, this histogram is shown to lead to the microcanonical distribution. Pressure, temperature, and chemical potential can then be identified microscopically. This dynamical approach thus recovers the results that are often obtained for equilibrium by minimizing a postulated entropy function.

For time-dependent driving, the key concepts of time-reversed and backward protocols are introduced. The reversibility of Hamiltonian dynamics is shown to imply that work is antisymmetric with respect to time-reversal. Integral fluctuation relations are introduced as a general property of certain distributions. For the work distributions, this yields the Jarzynski relation, which expresses free-energy differences as a particular nonlinear average over nonequilibrium work. Various limiting cases such as slow driving and the apparent counterexample of free expansion of a gas are discussed. The Bochkov–Kuzovlev relation is shown to be another variant of such an integral fluctuation relation. The Crooks fluctuation relation yields a symmetry of the work distributions for a forward and a backward process. As an important application, free energy differences and a free energy landscape based on exploiting the Hummer–Szabo relation are recovered as illustrated with experimental data for the unfolding of biopolymers.

Take anything in the universe, put it in a box, and heat it up. Regardless of what you start with, the motion of the substance will be described by the equations of fluid mechanics. This remarkable universality is the reason why fluid mechanics is important.

The key equation of fluid mechanics is the Navier-Stokes equation. This textbook starts with the basics of fluid flows, building to the Navier-Stokes equation while explaining the physics behind the various terms and exploring the astonishingly rich landscape of solutions. The book then progresses to more advanced topics, including waves, fluid instabilities, and turbulence, before concluding by turning inwards and describing the atomic constituents of fluids. It introduces ideas of kinetic theory, including the Boltzmann equation, to explain why the collective motion of 1023 atoms is, under the right circumstances, always governed by the laws of fluid mechanics.

The asymmetric random walk is introduced as a simple model for a molecular motor. Thermodynamic consistency imposes a condition on the ratio between the forward and the backward rate. Fluctuations in finite time can be derived analytically and are used to illustrate the thermodynamic uncertainty relation. For the long-time limit, concepts from large deviation theory like a rate function and a contraction can be determined explicitly.

Take water and push it through a pipe. If the flow is slow, then everything proceeds in a nice, orderly fashion. But as you force the water to move faster and faster, it starts to wobble. And then those wobbles get bigger until, at some point the flow loses all coherence as it tumbles and turn, tripping over itself in an attempt to push forwards. This is turbulent flow.

Understanding turbulence remains one of the great outstanding questions of classical physics. Why does it occur? How does it occur? How should we characterise such turbulent flows? The purpose of this chapter is to take the first tiny steps towards addressing these questions.

Projecting to sets A and B are the elementary operations used by the RRR algorithm to find solutions in their intersection. This chapter covers all the projections that arise in this book.

The periodic table is one of the most iconic images in science. All elements are classified in groups, ranging from metals on the left that go bang when you drop them in water through to gases on the right that don’t do very much at all. The purpose of this chapter is to start to look at the periodic table from first principles, to understand the structure and patterns that lie there.