When a quantum system has some external time dependence, some rather special things happen. This chapter explores this subject. Among the topics that we cover are the adiabatic theorem, Berry phase, the sudden approximation, and time-dependent perturbation theory.

There are two great equations of classical physics: one is Einstein’s equation of general relativity, the other the Navier-Stokes equation that describes how fluids flow. In this chapter, we meet Navier-Stokes.

This equation differs from the Euler equation by the addition of a viscosity term. This is not a small change and makes solutions to the Navier-Stokes equation much richer and more subtle than those of the Euler equation. In this chapter, we begin our exploration of these solutions.

Active particles self-propel through some intrinsic mechanism. First, a simple one-dimensional model is introduced for which the density profile between confining walls and the pressure exerted on these walls can be calculated analytically. In three dimensions, run-and-tumble particles, active Brownian particles, and active Ornstein–Uhlenbeck particles constitute three classes of models that can be described by Langevin equations. The identification of entropy production in the steady state is shown to be ambiguous. The continuum limit of a thermodynamically consistent discrete model shows that Langevin descriptions contain some implicit coarse-graining which prevents the recovery of the full physical entropy production.

This chapter deals with correlation and response functions in equilibrium and in nonequilibrium steady states for a Langevin dynamics. First, the harmonic oscillator in equilibrium is discussed as a paradigmatic case. In the general nonlinear case, it is shown how time-derivatives in correlation functions can be replaced by state variables. The response function is derived within the path integral formalism. It can be expressed by various forms of a correlation function. One particularly transparent version restores the form of the equilibrium fluctuation-dissipation theorem for a nonequilibrium steady state. A second strategy to derive a response function starts with the perturbed Fokker–Plank operator. Causality imposes the Kramers–Kronig relations between the real and imaginary parts of the response function. Through the Harada–Sasa relation, the deviation from the equilibrium form of the fluctuation-dissipation relation can be related to the mean entropy production.

In this chapter, we explore theoretical aspects of the origin of life problem. Firstly, we address the Chicken and Egg problem referring to the “RNA world.” We explain a mathematical model of the RNA replication system introduced by Eigen and discuss the conditions necessary for self-replication, referring “error catastrophe.” As a potential solution, we discuss the “hypercycle,” alongside its vulnerabilities and the acquisition of evolvability through compartmentalization. On another front, we examine Dyson’s catalytic reaction system as an alternative hypothesis, showing that catalytic reaction networks capable of maintaining themselves and undergoing imperfect reproduction may have appeared first. We also refer to a simple model of polymer reactions, arguing that such autocatalytic reaction networks can stochastically emerge, as proposed by Kauffman. Furthermore, we describe a cell model featuring an intracellular chemical reaction network that divides based on its state, highlighting the universal nature of reaction dynamics in replicating cells and the power-law distribution of chemical abundance (Zipf’s law), which has been verified across many organisms. Additionally, we introduce the concept of “minority control” in catalytic reaction networks, which can carry primitive genetic information. Finally, we discuss perspectives on research regarding the origin of life.

This chapter serves as an intuitive introduction to dynamical systems within the realm of biological systems, through visual representations of state space dynamics. Biological examples and experimental realizations are described to demonstrate how dynamical systems concepts are applicable in solving fundamental problems in cell biology. Differential equations are taken as typical of dynamical systems, and we explain topics such as nullcline and fixed points, linear stability analysis, and attractors, elucidating their significance using systems such as gene toggle switches. The introduction of limit cycles and the Poincaré–Bendixson theorem in two-dimensional systems is followed by examples such as the Brusselator and the repressilator system. Furthermore, we explore the basin structure in multi-attractor systems and provide detailed explanations using toggle switch systems to illustrate time-scale separation between variables and adiabatic elimination of variables. Several instances of co-dimension 1 bifurcations commonly observed in biological systems are presented, with a discussion of their biological significance in processes like cell differentiation. Finally, chaos theory is introduced.

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.