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This chapter introduces the fundamentals of the description of the quantum dynamics of open systems in terms of quantum master equations, together with its most important applications. Special emphasis is laid on the theory of completely positive quantum dynamical semigroups, which leads to the concept of a quantum Markov process. It discusses the relaxation to equilibrium and the multi-time structure of quantum Markov processes, as well as their irreversible nature which is characterized by an appropriate entropy functional. Microscopic derivations for various quantum master equations are presented, such as the quantum optical master equation and the master equation for quantum Brownian motion. As a further application, the master equation describing continuous measurements is derived and used to study the quantum Zeno effect. The chapter also contains a treatment of non-linear, mean field quantum master equations together with applications to laser theory and super-radiance.

This chapter examines the evolution of an open system — the sample — with the quantum Liouville equation for the world density operator. The fundamental approximation is that the action of the sample on the environment is negligible compared to the action of the environment on the sample. This leads to the master equation for the (reduced) sample density operator. Photons in a cavity and a two-level atom are presented as examples. The P-function representation of the sample density operator yields the Fokker-Planck equation. This is used to show the robustness of coherent states, and to describe a driven mode in a lossy cavity. The discussion next turns to quantum jumps and their experimental observation. Quantum jumps are related to the master equations by means of the Monte Carlo wavefunction algorithm, quantum trajectories, and quantum state diffusion.

This chapter explores stochastic behavior in biomolecular systems. It does so by first building on the preliminary discussion of stochastic modeling laid out in Chapter 2. The chapter reviews methods for modeling stochastic processes, including the chemical master equation (CME), the chemical Langevin equation (CLE), and the Fokker–Planck equation (FPE). Given a stochastic description, the chapter then analyzes the behavior of the system using a collection of stochastic simulation and analysis tools. This chapter makes use of a variety of topics in stochastic processes; readers should have a good working knowledge of basic probability and some exposure to simple stochastic processes.

This chapter introduces Langevin- and Fokker-Planck equations by way of a heuristic approach to the physics of Brownian particles. Variances of physical quantities are defined and related to cumulants. General properties of stochastic processes are described, with special emphasis on Markov processes and the Chapman-Kolmogorov equation. From the latter, Fokker-Planck equations are derived by the Kramers-Moyal expansion and connections with Langevin equations are established. The meaning of additive and multiplicative noise is clarified. Mathematical problems implied by the latter are studied for a one-dimensional system. Ito's and Stratonivich's suggestions of how to handle stochastic differential equations are presented. The concept of the mean first passage time is elucidated and basic formulas for its evaluation are derived. A short excursion is devoted to the Master equation and to detailed balance. Microscopic approaches to transport problems are reviewed. The perturbative Nakajima-Zwanzig equation is derived and compared with equations used in quantum optics.

This chapter provides a background on recent advances in the theory of mean field games (MFGs). MFGs has met an amazing success since pioneering works of more than ten years ago. It gives a self-contained study of the so-called master equation and an answer to the convergence problem. MFGs should be understood as games with a continuum of players, each of them interacting with the whole statistical distribution of the population. In this regard, they are expected to provide an asymptotic formulation for games with finitely many players with mean field interaction. This chapter focuses on the converse problem, which may be formulated by confirming whether the equilibria of the finite games converge to a solution of the corresponding MFG as the number of players becomes very large.

This chapter presents the general relaxation theory which describes the dynamical evolution of an open system A coupled to a large environment E. Section 4.1 reviews the main properties of the density matrix, including ‘density matrix purification’. Section 4.2 describes a general representation of a ‘quantum process’, mapping a density matrix onto another one. With a few simple and reasonable hypotheses on E, Section 4.3 casts the master equation under the generic Lindblad form. Section 4.4 introduces the Monte Carlo description of quantum relaxation. Section 4.5 considers a cavity mode resonantly coupled to a single two-level atom and describes how the Rabi oscillation phenomenon is altered by atomic and cavity relaxation. In Section 4.6, the two-level atoms are coupled sequentially to the spring, providing a simple model of a ‘micromaser’. Finally, in Section 4.7, the spins are simultaneously coupled to the spring, realizing a simple model for co-operative emission phenomena.

This chapter applies the time-convolutionless projection operator technique to some typical systems featuring strong non-Markovian behaviour. The first example models the decay of a two-level system into a reservoir with arbitrary spectral density. For this problem the expansion of the master equation is carried out to all orders in the system-reservoir coupling. As specific cases the damped Jaynes–Cummings model and the spontaneous decay into a photonic band gap are studied. The model is used to exemplify the breakdown of the expansion in the strong coupling regime, and to illustrate the emergence of completely positive master equations which are not in Lindblad form and involve negative decay rates. The chapter also treats the non-Markovian dynamics in quantum Brownian motion and in the spin-boson model, and illustrates the stochastic unravelling of non-Markovian quantum processes.

This chapter treats dissipation arising from coupling to the environment. Problems treated include photoionization, spontaneous emission, and cavity damping. A short-time perturbative analysis leads to decay of a single state coupled to a continuum at a rate determined by Fermi's golden rule. For longer times a non-perturbative solution of the amplitude equations leads to Weisskopf–Wigner decay. The final value theorem provides a simple method for obtaining the final state spectrum without first solving for the dynamics. A Heisenberg picture analysis leads to operator Langevin equations in which environment operators introduce quantum noise. A parallel analysis is possible working in the Schroedinger picture and leads to a master equation for the density operator of the damped system. This is solved to provide a complete description of the influence of dissipation on the system of interest. It is sometimes simpler to solve directly for the evolution of the field statistics.

To discuss the renormalization of gauge theories in the non-abelian case in its full generality, it is necessary to use a rather abstract formalism, which allows one to understand the algebraic structure of the renormalization procedure without being overwhelmed by the notational complexity. There is, however, a price to pay: the translation of the general identities which then appear into usual and more concrete notation becomes a non-trivial exercise. This chapter is organized as follows. It first quantizes the theory in the temporal gauge. Using a simple identity, it shows the equivalence with a quantization in a general class of gauges. This identity automatically implies a BRS symmetry, and, therefore, a set of WT identities for correlation functions. It shows that WT identities are also direct consequences of a quadratic master equation satisfied the quantized action, equation in which gauge and BRS symmetries are no longer explicit. It shows that in the case of renormalizable gauges the master equation is stable under renormalization. This is solved to determine the structure of the renormalized action. The chapter verifies that the master equation encodes in a subtle way the gauge properties of the quantized action.

This chapter is devoted to the foundations and to the basic mathematical structure of the non-Markovian quantum dynamics of open systems. It gives a survey of the Nakajima–Zwanzig projection operator methods with the help of which one derives so-called generalized master equations for the reduced system dynamics. In the non-Markovian regime, these master equations involve a retarded memory kernel, i.e. a time-convolution integral taken over the history of the reduced system. The chapter also describes an alternative method of particular relevance in many applications, which is based on a time-local quantum master equation, and which is known as the time-convolutionless projection operator method. This method serves as a starting point for a systematic expansion in terms of ordered cumulants and forms the basis for a non-Markovian stochastic unravelling of the reduced system dynamics.

This chapter collects the main results of the master equation for the convergence of the Nash system. It explains the notation used, specifies the notion of derivatives in the space of measures, and describes the assumptions on the data. One of the striking features of the master equation is that it involves derivatives of the unknown with respect to the measure. This chapter also discusses the link between the two notions of derivatives, which have been used in the mean field game (MFG) theory. The main result states that the master equation has a unique classical solution under the regularity and monotonicity assumptions on H, F, and G. Once the master equation has a solution, this solution can be used to build approximate solutions for the Nash system with N-players.

This chapter contains a preliminary analysis of the master equation in the simpler case when there is no common noise. Some of the proofs given in this chapter consist of a sketch only. One of the reasons is that some of the arguments used to investigate the mean field games (MFGs) system have been already developed in the literature. Another reason is that the chapter constitutes a starter only, specifically devoted to the simpler case without common noise. It provides details of the global Lipschitz continuity of H. The solutions of the MFG system are uniformly Lipschitz continuous, which are independently of initial conditions.

This chapter investigates the second-order master equation with common noise, which requires the well-posedness of the mean field game (MFG) system. It also defines and analyzes the solution of the master equation. The chapter explains the forward component of the MFG system that is recognized as the characteristics of the master equation. The regularity of the solution of the master equation is explored through the tangent process that solves the linearized MFG system. It also analyzes first-order differentiability and second-order differentiability in the direction of the measure on the same model as for the first-order derivatives. This chapter concludes with further description of the derivation of the master equation and well-posedness of the stochastic MFG system.

The discrete-stochastic model of diffusion introduced in the preceding chapter implies that the system state evolves in time as what is known as a jump Markov process. The time evolution of such a process is generally governed by what is called a master equation. In this case, the assumption that the solute molecules move about independently of each other gives rise to two master equations, one for a single solute molecule and one for the entire collection of solute molecules. In this chapter the chapter derives both of those master equations. The chapter obtains the exact time-dependent solution of the single-molecule master equation, and the exact time-independent (equilibrium) solution of the many-molecule master equation. The chapter also derives companion stochastic simulation algorithms for the two master equations, and the chapter applies those algorithms to the microfluidic diffusion experiment of Chapter 5. The chapter uses simulation to test the validity of the discrete-stochastic version of Fick's Law.

This chapter focuses on coherence and decoherence, starting from the basis of quantum mechanics and including the classical paradox of Schrödinger’s cat and entanglement. Decoherence and relaxation are simultaneously accounted for by an evolution equation for the density matrix, which is analysed in the case of spin tunnelling and simplifies when decoherence is complete. The final section discusses the possible exploitation of coherence in quantum computing.

This chapter introduces a few typical numerical methods used in modern studies of phase transitions and critical phenomena in spin systems. The first section describes the stochastic dynamics of a generic system with discrete degrees of freedom following the master equation. This section serves as a theoretical basis for the Monte Carlo method that includes the heat bath and Metropolis algorithms of configuration updates. Another useful numerical technique is the transfer matrix method, described in the last section, and which is applied for numerically exact evaluation of the free energy and related physical quantities.

This chapter discusses concepts and techniques for mathematically describing and numerically simulating chemical systems that into account discreteness and stochasticity. The chapter is organized as follows. Section 16.2 outlines the foundations of “stochastic chemical kinetics” and derives the chemical master equation (CME)—the time-evolution equation for the probability function of the system’s state. The CME, however, cannot be solved, for any but the simplest of systems. But numerical realizations (sample trajectories in state space) of the stochastic process defined by the CME can be generated using a Monte Carlo strategy called the stochastic simulation algorithm (SSA), which is derived and discussed in Section 16.3. Section 16.4 describes an approximate accelerated algorithm known as tau-leaping. Section 16.5 shows how, under certain conditions, tau-leaping further approximates to a stochastic differential equation called the chemical Langevin equation (CLE), and then how the CLE can in turn sometimes be approximated by an ordinary differential equation called the reaction rate equation (RRE). Section 16.6 describes the problem of stiffness in a deterministic (RRE) context, along with its standard numerical resolution: implicit method. Section 16.7 presents an implicit tau-leaping algorithm for stochastically simulating stiff chemical systems. Section 16.8 concludes by describing and illustrating yet another promising algorithm for dealing with stiff stochastic chemical systems, which is called the slow-scale SSA.

In pre-equilibrium reactions, the nuclear complex breaks up before it reaches statistical equilibrium. Such reactions can be described within a time-dependent picture, in which the population and de-population of different classes of configurations is determined by master equations. This chapter describes an illustrative model for reactions with light particles, which elucidates the physics of such processes and allows the derivation of formulas for the fluctuating cross sections. The one for the last of all possible decays may be related to that of the statistical model. As an essential difference, the more general formula contains the formation cross section as a multiplicative factor, and through this depends on the initial channel. In the last section, comparisons are made with more complex theories. An approximate formula for p-h-densities is presented as an intermediate step.

This chapter talks about the unique solvability of the mean field games (MFGs) system with common noise. In terms of a game with a finite number of players, the common noise describes some noise that affects all the players in the same way, so that the dynamics of one given particle reads a certain master equation. It explains the use of the standard convention from the theory of stochastic processes that consists in indicating the time parameter as an index in random functions. Using a continuation like argument instead of the classical strategy based on the Schauder fixed-point theorem, this chapter investigates the existence and uniqueness of a solution. It discusses the effect of the common noise in randomizing the MFG equilibria so that it becomes a random flow of measures.