In this chapter, we explore the concept of information in living organisms in its broadest sense. Biological organisms perceive the external environment, alter their own state, and take action (selection among possibilities). To capture these properties intrinsic to the organisms, we begin by discussing the “information quantity” that quantifies such situations. Starting with the definition of information quantity, we introduce Shannon entropy and provide an overview of Shannon’s information theory framework. We also discuss Kullback–Leibler divergence and mutual information. Next, moving on to information in DNA sequences, we cover various aspects such as differences in the frequency of AT and GC occurrence, the structure of genetic codes, long-range correlations in DNA sequences, and recent findings in intergenic sequences. Additionally, we explain kinetic proofreading as one candidate for achieving high accuracy in molecular recognition from a combination of unreliable elements. Furthermore, we explore the relationship between entropy in statistical mechanics and information, elucidating the connection between Maxwell’s demon and information using the Szilard engine as a mediator. Finally, we introduce intriguing points from the perspective of dynamics and information, highlighting the dynamic interplay between the two.

Chien-Shiung Wu (1912–1997) is often referred to as “the Chinese Marie Curie” even though she conducted most of her research in the US. She is best known for her discovery of the non-conservation of parity for weakly interacting particles – a finding for which she is widely regarded as having been passed over for the 1957 Nobel Prize in Physics. Seven years earlier, though, in a one-page letter to Physical Review, Wu and her graduate student also quietly reported what has come to be understood as the first conclusive evidence of entangled photons. Twenty years later, as quantum foundations research emerged from shadow, Wu revisited her 1949 experiment with a more refined approach. Wu shared the new results with Stuart Freedman, a collaborator of John Clauser. Clauser et al. would rigorously critique Wu’s experiments through at least 1978. In 2022, the Nobel Committee honored Clauser, Alain Aspect, and Anton Zeilinger, each of whom had produced increasingly convincing proof of entanglement beginning in the 1970s. Wu’s foundational work from almost seventy years earlier, however, was not mentioned. This chapter aims to help bring Wu’s entangled photons back into the light.

This chapter explores what we could do with a computer whose operating system is quantum mechanics, rather than classical mechanics. One of the answers is: factorise primes really quickly. We will explain why this is interesting.

This chapter quantitatively examines molecule numbers and reaction rates within a cell, along with thermal fluctuations and Brownian motion, from a mesoscopic perspective. Thermal fluctuations of molecules are pivotal in chemical reactions, protein folding, molecular motor systems, and so on. We introduce estimations of cell size and molecule numbers within cells, highlighting the possible significance of the minority of molecules. Describing their behaviors necessitates dealing with stochastic fluctuations, and the Gillespie algorithm, widely employed in Monte Carlo simulations for stochastic chemical reactions, is described. We elaborate on extrinsic and intrinsic noise in cells, and on why understanding how cells process fluctuations for sensing is crucial. To facilitate this comprehension, we revisit the fundamentals of statistics, including the law of large numbers and the central limit theorem. We derive the diffusion equation from random walk and confirm the dimensionality dependence of random walks, and elucidate Brownian motion as the continuous limit of random walk and explain the Einstein relation. As examples of the physiological significance of fluctuations in cell biology, we estimate the diffusion constant of proteins inside cells, diffusion-limited reactions, and introduce bacterial random walks and chemotaxis, and amoeboid movements of eukaryotic cells.

Molecular motors that transform chemical energy into mechanical motion can be modeled in different ways. Thermodynamically consistent ratchet-type models lead to transport against an external force in a periodic potential that switches between different shapes. In a second class of models, the motor is described by a set of internal states that leads to discrete steps along a filamentous track. In the class of hybrid models, the motor cycles through internal states while pulling a cargo particle that follows a Langevin dynamics. For these motors, the first and second laws and the thermodynamic efficiency are discussed and illustrated with experimental data for a rotary motor, the F1-ATPase.

Cells regulate their proliferation, differentiation, and motility in response to external stimuli. Often, these responses involve a complex interplay of association, dissociation, and catalytic reactions, characterized by highly specific intermolecular interactions. This chapter examines cellular responses arising from such chemical reactions from a mathematical standpoint. As examples of input–output relationships, we introduce the Hill equation, Adair equation, and the MWC model concerning allosteric regulation, which describe cooperative behaviors. We discuss the Michaelis–Menten equation in enzyme reactions, covering activation, inactivation, push–pull reactions, zero-order ultrasensitivity, and positive feedback switches. Furthermore, we present the formation of a bell-shaped input–output curve by feed-forward loops, and the mechanisms of adaptation and fold-change detection utilizing feed-forward loops, or negative feedback. We explore bacterial chemotaxis mechanisms through models such as the Asakura–Honda model and the Barkai–Leibler model.

Ana María Cetto Kramis (born 1946) studied physics at Universidad Nacional Autónoma de México and biophysics at Harvard University. As a faculty member back in Mexico, she spent over half a century delving into the fundamentals of quantum physics, with a singular focus on its stochastic interpretation. In addition to her theoretical work, she founded Latindex and has become a key figure in the open access movement. She has also had a long and influential contribution to international scientific cooperation. Her professional and personal journeys culminate with the dynamization of the International Year of Quantum Science and Technologies 2025, aiming to shed light on her understanding of quantum science and of science as a whole. This chapter is mostly based on an oral history, which is here also revisited as a historiographical methodology from its early use at the origins of the history of quantum physics.

In recent years, Grete Hermann (1901–1984) has been rediscovered as a principal figure in the history and philosophy of quantum physics. In particular, her criticism of Johann von Neumann’s so-called “no hidden variables” proof is a focal point of interest. Did she really find a mistake in this proof? We argue that the whole debate is misleading. It fits too well with the image of a forgotten woman who disproved a result of a mathematical genius, but it is neither historically nor systematically justified. Despite Hermann’s challenging thoughts on quantum physics, her impressive and important achievements were in ethics and politics. We offer a new and broader reading of Hermann’s interpretation of quantum physics and try to build a bridge between her works on quantum physics and ethics. In doing so, we focus on her interpretation of Heisenberg’s cut as a methaphorical device to argue against Leonard Nelson’s theory of free will and for freedom and responsibility as cornerstones of any democratic society.

We all know what a wave is. But you may not know just how many different kinds of waves there are and what strange and interesting properties they have. We start this chapter with something very familiar from everyday life: waves on the surface of an ocean. While they may be familiar, their mathematical description is surprisingly subtle. This can be traced, like so many other things in fluid mechanics, to the boundary conditions.

The fundamentals of electromagnetism are simple. Moving electric charges set up electric and magnetic fields. In turn, these fields make the charges move. This dance between charges and fields is described by the Maxwell equations. This brief chapter describes how this comes about. It is, in a sense, everything you need to know about electromagnetism, enshrined in these simple equations. The rest of the book is mere commentary.

The laws of classical mechanics are valid in so-called inertial frames. Roughly speaking, these are frames that are at rest. But what if you, one day, find yourself in a frame that is not in- ertial? For example, suppose that every 24 hours you happen to spin around an axis which is 2500 miles away. What would you feel? Or what if every year you spin around an axis 36 million miles away? Would that have any effect on your everyday life? In this chapter, we describe what happens if you sit in a rotating reference frame and the effects of the resulting centrifugal and Coriolis forces.

Forthcoming

To understand what the Maxwell equations are telling us, it’s useful to dissect them piece by piece. The simplest piece comes from looking at stationary electric charges and how they give rise to electric fields. A consequence of this is the Coulomb force law between charges. This, and much more, will be described in this chapter.

The chapter then goes on to explore many other different kinds of waves that arise in different situations, from the atmosphere, to supersonic aircraft to traffic jams.

Theres a lot of interesting physics to be found if you subject an atom to an electric or magnetic field. This chapter explores this physics. It covers the Stark effect and the Zeeman effect and Rabi oscillations. it also looks at what happens when coherent states of photons in a cavity interact with atoms.

In this chapter, we ease in gradually by thinking about a quantum particle moving along a line. This provides an opportunity for us to learn about the properties of the wavefuntion and how it encodes properties such as the position and momentum of the particle. We will also see how the physics of a system is described by the Schrodinger equation.

Sonja Ashauer (1923–1948) trained as a physicist at the University of São Paulo in Brazil and obtained a PhD in theoretical and mathematical physics from the University of Cambridge, under the guidance of Paul Dirac. Acknowledged as the first Brazilian woman with a physics PhD, her life was brief: She passed away six months after defending her thesis. In her few contributions, she explored the non-physical consequences of classical equations for point electrons, reformulated by Dirac in the late 1930s to address divergence issues in quantum electrodynamics. This chapter traces Ashauer’s journey from São Paulo, where she collaborated with a small and enthusiastic group of young researchers around the Italian–Russian physicist Gleb Wataghin and focused on cosmic ray physics research, to Cambridge, where she found a more secluded research environment.