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Quantum Master Equations

Heinz-Peter Breuer and Francesco Petruccione

in The Theory of Open Quantum Systems

Published in print:
2007
Published Online:
February 2010
ISBN:
9780199213900
eISBN:
9780191706349
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199213900.003.03
Subject:
Physics, Theoretical, Computational, and Statistical Physics

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 ... More


The master equation

J. C. Garrison and R. Y. Chiao

in Quantum Optics

Published in print:
2008
Published Online:
September 2008
ISBN:
9780198508861
eISBN:
9780191708640
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198508861.003.0019
Subject:
Physics, Atomic, Laser, and Optical Physics

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 ... More


Stochastic Modeling and Analysis

Richard M. Murray

in Biomolecular Feedback Systems

Published in print:
2014
Published Online:
October 2017
ISBN:
9780691161532
eISBN:
9781400850501
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691161532.003.0004
Subject:
Biology, Biochemistry / Molecular Biology

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 ... More


The Theory of Open Quantum Systems

Heinz-Peter Breuer and Francesco Petruccione

Published in print:
2007
Published Online:
February 2010
ISBN:
9780199213900
eISBN:
9780191706349
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199213900.001.0001
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This book treats the central physical concepts and mathematical techniques used to investigate the dynamics of open quantum systems. To provide a self-contained presentation, the text begins with a ... More


Properties of Langevin and Fokker–Planck equations

Helmut Hofmann

in The Physics of Warm Nuclei: with Analogies to Mesoscopic Systems

Published in print:
2008
Published Online:
September 2008
ISBN:
9780198504016
eISBN:
9780191708480
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198504016.003.0025
Subject:
Physics, Nuclear and Plasma Physics

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 ... More


Introduction

Pierre Cardaliaguet, François Delarue, Jean-Michel Lasry, and Pierre-Louis Lions

in The Master Equation and the Convergence Problem in Mean Field Games: (AMS-201)

Published in print:
2019
Published Online:
May 2020
ISBN:
9780691190716
eISBN:
9780691193717
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691190716.003.0001
Subject:
Mathematics, Applied Mathematics

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 ... More


The environment is watching

Serge Haroche and Jean-Michel Raimond

in Exploring the Quantum: Atoms, Cavities, and Photons

Published in print:
2006
Published Online:
January 2010
ISBN:
9780198509141
eISBN:
9780191708626
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198509141.003.0004
Subject:
Physics, Atomic, Laser, and Optical Physics

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 ... More


Non-Markovian Dynamics in Physical Systems

Heinz-Peter Breuer and Francesco Petruccione

in The Theory of Open Quantum Systems

Published in print:
2007
Published Online:
February 2010
ISBN:
9780199213900
eISBN:
9780191706349
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199213900.003.10
Subject:
Physics, Theoretical, Computational, and Statistical Physics

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 ... More


Dissipative Processes

STEPHEN M. BARNETT and PAUL M. RADMORE

in Methods in Theoretical Quantum Optics

Published in print:
2002
Published Online:
February 2010
ISBN:
9780198563617
eISBN:
9780191714245
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198563617.003.0005
Subject:
Physics, Theoretical, Computational, and Statistical Physics

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 ... More


Gauge Theories: Master Equation And Renormalization

JEAN ZINN-JUSTIN

in Quantum Field Theory and Critical Phenomena

Published in print:
2002
Published Online:
January 2010
ISBN:
9780198509233
eISBN:
9780191708732
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198509233.003.0021
Subject:
Physics, Theoretical, Computational, and Statistical Physics

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 ... More


Projection Operator Techniques

Heinz-Peter Breuer and Francesco Petruccione

in The Theory of Open Quantum Systems

Published in print:
2007
Published Online:
February 2010
ISBN:
9780199213900
eISBN:
9780191706349
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199213900.003.09
Subject:
Physics, Theoretical, Computational, and Statistical Physics

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 ... More


Presentation of the Main Results

Pierre Cardaliaguet, François Delarue, Jean-Michel Lasry, and Pierre-Louis Lions

in The Master Equation and the Convergence Problem in Mean Field Games: (AMS-201)

Published in print:
2019
Published Online:
May 2020
ISBN:
9780691190716
eISBN:
9780691193717
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691190716.003.0002
Subject:
Mathematics, Applied Mathematics

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 ... More


A Starter: The First-Order Master Equation

Pierre Cardaliaguet, François Delarue, Jean-Michel Lasry, and Pierre-Louis Lions

in The Master Equation and the Convergence Problem in Mean Field Games: (AMS-201)

Published in print:
2019
Published Online:
May 2020
ISBN:
9780691190716
eISBN:
9780691193717
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691190716.003.0003
Subject:
Mathematics, Applied Mathematics

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 ... More


The Second-Order Master Equation

Pierre Cardaliaguet, François Delarue, Jean-Michel Lasry, and Pierre-Louis Lions

in The Master Equation and the Convergence Problem in Mean Field Games: (AMS-201)

Published in print:
2019
Published Online:
May 2020
ISBN:
9780691190716
eISBN:
9780691193717
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691190716.003.0005
Subject:
Mathematics, Applied Mathematics

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 ... More


Master equations and simulation algorithms for the discrete‐stochastic approach

Daniel T. Gillespie and Effrosyni Seitaridou

in Simple Brownian Diffusion: An Introduction to the Standard Theoretical Models

Published in print:
2012
Published Online:
January 2013
ISBN:
9780199664504
eISBN:
9780191748516
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199664504.003.0006
Subject:
Physics, Soft Matter / Biological Physics

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 ... More


COHERENCE AND DECOHERENCE

Dante Gatteschi, Roberta Sessoli, and Jacques Villain

in Molecular Nanomagnets

Published in print:
2006
Published Online:
September 2007
ISBN:
9780198567530
eISBN:
9780191718298
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198567530.003.0011
Subject:
Physics, Condensed Matter Physics / Materials

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 ... More


Numerical methods

Hidetoshi Nishimori and Gerardo Ortiz

in Elements of Phase Transitions and Critical Phenomena

Published in print:
2010
Published Online:
January 2011
ISBN:
9780199577224
eISBN:
9780191722943
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199577224.003.0011
Subject:
Physics, Theoretical, Computational, and Statistical Physics

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 ... More


Numerical Simulation for Biochemical Kinetics

Daniel T. Gillespie and Linda R. Petzold

in System Modeling in Cellular Biology: From Concepts to Nuts and Bolts

Published in print:
2006
Published Online:
August 2013
ISBN:
9780262195485
eISBN:
9780262257060
Item type:
chapter
Publisher:
The MIT Press
DOI:
10.7551/mitpress/9780262195485.003.0016
Subject:
Mathematics, Mathematical Biology

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 ... More


Pre-equilibrium reactions

Helmut Hofmann

in The Physics of Warm Nuclei: with Analogies to Mesoscopic Systems

Published in print:
2008
Published Online:
September 2008
ISBN:
9780198504016
eISBN:
9780191708480
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198504016.003.0009
Subject:
Physics, Nuclear and Plasma Physics

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 ... More


Mean Field Game System with a Common Noise

Pierre Cardaliaguet, François Delarue, Jean-Michel Lasry, and Pierre-Louis Lions

in The Master Equation and the Convergence Problem in Mean Field Games: (AMS-201)

Published in print:
2019
Published Online:
May 2020
ISBN:
9780691190716
eISBN:
9780691193717
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691190716.003.0004
Subject:
Mathematics, Applied Mathematics

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 ... More


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