Our goal in this chapter is to look more closely at the underlying mathematical formalism of quantum mechanics. We will look at the quantum state, how it evolves in time, and what it means to interrogate the state by performing a measurement. It is here that we meet the famed Heisenberg uncertainty principle.

Optimal protocols transform a given initial distribution into a given final one in finite time with a minimal amount of work or entropy production. We first analyze this optimization paradigmatically for a driven harmonic oscillator for which analytical results can be obtained. For a general Langevin dynamics, it is shown that the optimal protocol can be realized through a time-dependent potential with no need to use a nonconservative force. In contrast for discrete systems, nonconservative driving decreases the thermodynamic costs. For a broader perspective, we introduce concepts from information geometry which deals with the statistical manifold of distributions. The Fisher information provides a metric on this manifold from which the distance between two distributions as the minimal length connecting them can be derived. Speed limits yield relations between these quantities referring to an initial and a final distribution and the entropy production associated with the transformation of the former into the latter. For slow processes, cost along the optimal protocol or path is bounded by the distance between these distributions and the inverse of the allocated time.

The bipartisan problem formulation is introduced, where relatively simple sets A and B are designed that hold solutions in their intersection.

This chapter deals with processes both from a macroscopic, thermodynamic point of view and from a dynamical perspective. For the latter, a class of processes is introduced that can be described through a Hamiltonian description with a time-dependent external control parameter. It is shown how the expressions of work and heat from classical thermodynamics can be obtained as an appropriate average over an initial distribution. The second law inequality relating work and free energy can then be proven as a consequence of a master inequality. With well-specified additional assumptions, second law inequalities for heat exchange and entropy production are derived.

Until now, we’ve only considered the motion of a single particle. If our goal is to understand everything in the universe, that’s a little limiting. In this section, we take a small step forwards: we will describe the dynamics of multiple interacting particles. Among other things, this will highlight the importance of the conservation of momentum and angular momentum.

Classical mechanics starts with Newtons three laws, among them the famous F=ma. But these laws are not quite as transparent as they may seem. In this chapter, we introduce the laws and provide some commentary. We will also learn about Galileos ideas of relativity, a precursor to the much more shocking ideas of Einstein that come later.

Laura M. Chalk (later, Laura Rowles, 1904−1996) was the first woman to complete a PhD in physics at McGill University in Montreal, Canada. Her doctoral research on the quantum phenomenon called the Stark effect, under the supervision of J. Stuart Foster, produced the earliest experimental test of Erwin Schrödinger’s wave mechanics. After a brief stint as a postdoctoral fellow at King’s College London, she chose to return home and dedicate herself to teaching and marriage. This paper aims to fully recover Chalk’s work and explore why the Foster−Chalk experiment was overlooked in physics historiography. It considers the Stark effect’s significance in quantum physics and the impact of gender on her personal trajectory. Shaped by personal choice, systemic discrimination, and acceptance of societal norms, Chalk Rowles’ story highlights the paradoxes faced by women in a culturally disembodied yet male-dominated field, and reflects broader themes of gender and identity in the history of women in physics.

Quantum particles, like happy families, are all the same. In fact, not only are they the same. They are literally indistinguishable. This has deep and important consequences that are fleshed out in this chapter.

The real fun of the Maxwell equations comes when we understand the link between electricity and magnetism. A changing magnetic flux can induce currents to flow. This is Faraday’s law of induction. We start this chapter by understanding this link and end this chapter with one of the great unifying discoveries of physics: that the interplay between electric and magnetic fields is what gives rise to light.

In this chapter, we explore how electric and magnetic fields behave inside materials. The physics can be remarkably complicated and messy but the end result are described by a few, very minor, changes to the Maxwell equations. This allows us to understand various properties of materials, such as conductors.

What is the essence of quantum mechanics? What makes the quantum world truly different from the classical one? Is it the discrete spectrum of energy levels? Or the inherent lack of determinism? The purpose of this chapter is to go back to basics in an attempt to answer this question. We will look at the framework of quantum mechanics in an attempt to get a better understanding of what we mean by a “state”, and what we mean by a “measurement”. A large part of our focus will be on the power of quantum entanglement.

In this chapter, we discuss dynamical system approaches for cellular differentiation. We explain how intracellular reaction dynamics can give rise to various attractors using a simple discrete-time and discrete-state reaction model known as a Boolean network. Subsequently, we outline the behavior of a simple stochastic differentiation model of stem cells, where the scaling law discovered therein aligns well with that observed in the distribution of clonal cell populations generated by epidermal stem cells. To integate both approaches, we introduce a theory wherein cell–cell interactions induce transitions between attractors, and stability at the cell-population level emerges through the regulation of these dynamic transitions. Such a circular relationship satisfies the consistency between the cell and the cell population. We expound on three types of differentiation processes, that by Turing instability, transition from an oscillatory state (limit-cycle) to a fixed point, and retaining oscillatory expression dynamics. Additionally, we analyze stability at the cell population level through the regulation of differentiation ratios and the differentiation dynamics of stem cells. Finally, we engage in a discussion of unresolved issues in the field.

Large temporal fluctuations or oscillations in cellular states are widely observed in biological systems, for instance, in neural firing, circadian rhythms, and collective motion of amoebae. These phenomena arise from the interplay between positive and negative feedback mechanisms, as discussed in previous chapters. In this chapter, we focus on such dynamic changes in cellular states. Using trajectories of oscillatory dynamics in phase planes such as the Brusselator, we provide detailed explanations of conditions for oscillation through the use of nullcline and Jacobian matrix analyses. We confirm the existence of two mechanisms: the activator-inhibitor system and the substrate-depletion system. Furthermore, we extensively introduce the Hodgkin–Huxley equations concerning membrane potential and excitability, which represent a significant milestone in the fields of biophysics, theoretical biology, and electrophysiology. Through quantitative comparison with experimental data, we elucidate the mechanisms underlying its dynamics, which are explained by the reduction of variables leading to the FitzHugh–Nagumo equations.