For the Brownian motion of a particle in a fluid, the Langevin equation for its momentum is introduced phenomenologically. The strength of the noise is shown to be related to friction, and, in a second step, to the diffusion coefficient. Excellent agreement with experiments on a levitated particle in gas is demonstrated. This phenomenological Langevin equation is then shown to follow from a general projection approach to the underlying Hamiltonian dynamics of the full system in the limit of an infinite mass ratio between Brownian particles and fluid molecules. For Brownian motion in liquids, additional time-scales enter that are discussed phenomenologically and illustrated with experiments.

There are two great post-Newtonian steps in classical mechanics. The first is the Lagrangian formulation and the accompanying principle of least action. The second is the Hamiltonian formulation, which is yet another way of writing Newtons equation of motion that uncovers what is really going on. This is where we start to see the deep and beautiful mathematical structure that underlies classical mechanics. It is also where we can make connections to what comes next, with quantum mechanics following very naturally from the Hamiltonian formulation.

If youre going to understand one thing in physics then it should be the harmonic oscillator. It is simple system that underlies nearly everything else that we do. This chapter studies the quantum harmonic oscillator, solving it several times in different ways to highlight different features.

The canonical description of an aqueous solution with an embedded enzyme is introduced. The mesostates of this enzyme comprise different conformations which are affected by binding and release of solute molecules. Thermodynamic potentials of these mesostates are identified. Heat and entropy production associated with transitions between these mesostates are determined both for a simple toy model and in the general case.

Our discussion in early chapters captures the spirit of quantum mechanics but is restricted to particles moving along a line. Thats not very unrealistic. In this chapter we breathe some life into quantum particles and allow them to roam in three-dimensional space. This entails an understanding of angular momentum. We will pay particular attention to the hydrogen atom, whose quantum solution was one of the first great triumphs of quantum mechanics and still underlies all of atomic physics.

The two body problem is the question of how two objects – say the Sun and the Earth – move under their mutual gravitational attraction. The problem is, happily, fully solvable and the purpose of this chapter is to fully solve it. We will understand how Keplers laws of planetary motion arise from the more fundamental Newtonian law of gravity. Because the electrostatic force has exactly the same form as the force of gravity, we can also use our solutions to understand how electrons scatter off atoms, a famous experiment performed by Rutherford that led to an understanding of the structure of matter.

For a system in contact with a heat bath, it is shown how the distribution of any observable follows from a microcanonical description for the isolated system consisting of the system of interest and heat bath. The weak coupling approximation then leads to the standard expression for the canonical distribution. Free energy, canonical entropy, and pressure are introduced. For large systems, the equivalence of this canonical description with the microcanonical one is shown. For systems in contact with a particle reservoir, the grand-canonical distribution is derived. If the weak coupling approximation does not hold, the corrections due to strong coupling are determined. In particular, internal energy, free energy ,and entropy are identified such that the usual relations for these thermodynamic potentials hold true even in strong coupling.

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.