Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

QP-PQ, quantum probability and white noise analysis

Quantum Interacting Particle Systems

Luigi Accardi, Franco Fagnola

Published Jul 1, 2002• 27 citations
Publisher

Abstract

The problem of extending ideas and results on the dynamics of infinite classical lattice systems to the quantum domain naturally arises in different branches of physics (nonequilibrium statistical mechanics, quantum optics, solid state, …) and new momentum from the development of quantum computer and quantum neural networks (which are in fact interacting arrays of binary systems) has been found. The stochastic limit of quantum theory allowed to deduce, as limits of the usual Hamiltonian systems, a new class of quantum stochastic flows which, when restricted to an appropriate Abelian subalgebra

DOI
10.1142/5055

Related papers

Quantum interacting particle systems : lecture Notes of the Volterra-CIRM International School, Trento, Italy, 23-29 September 2000

Luigi Accardi, Franco Fagnola

The problem of extending ideas and results on the dynamics of infinite classical lattice systems to the quantum domain naturally arises in different branches of physics (nonequilibrium statistical mechanics, quantum optics, solid state, ...) and new momentum from the development of quantum computer and quantum neural networks (which are in fact interacting arrays of binary systems) has been found. The stochastic limit of quantum theory allowed to deduce, as limits of the usual Hamiltonian systems, a new class of quantum stochastic flows which, when restricted to an appropriate Abelian subalgebra, produces precisely those interacting particle systems studied in classical statistical mechanics. Moreover, in many interesting cases, the underlying classical process "drives" the quantum one, at least as far as ergodicity or convergence to equilibrium are concerned. Thus many deep results concerning classical systems can be directly applied to carry information on the corresponding quantum system. The thermodynamic limit itself is obtained thanks to a technique (the four-semigroup method, new even in the classical case) which reduces the infinitesimal structure of a stochastic flow to that of four semigroups canonically associated to it (Chap. 1). Simple and effective methods to analyze qualitatively the ergodic behavior of quantum Markov semigroups are discussed in Chap. 2. Powerful estimates used to control the infinite volume limit, ergodic behavior and the spectral gap (Gaussian, exponential and hypercontractive bounds, classical and quantum logarithmic Sobolev inequalities, ...) are discussed in Chap. 3

WORLD SCIENTIFIC eBooks•Jan 1, 2002•0 citations

Quantum Mechanics of Finite Degrees of Freedom

Fabio Benatti

Quantum dynamical systems are described by means of non-commutative algebras of observables, by means of their time-evolution and by means of the expectation functionals that assign mean values to them.

Theoretical and mathematical physics•Jan 1, 2023•0 citations

Nonequilibrium dynamical mean-field theory and its applications

Hideo Aoki, Naoto Tsuji, Martin Eckstein +3 more

The study of nonequilibrium phenomena in correlated lattice systems has developed into one of the most active and exciting branches of condensed matter physics. This research field provides rich new insights that could not be obtained from the study of equilibrium situations, and the theoretical understanding of the physics often requires the development of new concepts and methods. On the experimental side, ultrafast pump-probe spectroscopies enable studies of excitation and relaxation phenomena in correlated electron systems, while ultracold atoms in optical lattices provide a new way to control and measure the time evolution of interacting lattice systems with a vastly different characteristic time scale compared to electron systems. A theoretical description of these phenomena is challenging because, first, the quantum-mechanical time evolution of many-body systems out of equilibrium must be computed and second, strong-correlation effects which can be of a nonperturbative nature must be addressed. This review discusses the nonequilibrium extension of the dynamical mean field theory (DMFT), which treats quantum fluctuations in the time domain and works directly in the thermodynamic limit. The method reduces the complexity of the calculation via a mapping to a self-consistent impurity problem, which becomes exact in infinite dimensions. Particular emphasis is placed on a detailed derivation of the formalism, and on a discussion of numerical techniques, which enable solutions of the effective nonequilibrium DMFT impurity problem. Insights gained into the properties of the infinite-dimensional Hubbard model under strong nonequilibrium conditions are summarized. These examples illustrate the current ability of the theoretical framework to reproduce and understand fundamental nonequilibrium phenomena, such as the dielectric breakdown of Mott insulators, photodoping, and collapse-and-revival oscillations in quenched systems. Furthermore, remarkable novel phenomena have been predicted by the nonequilibrium DMFT simulations of correlated lattice systems, including dynamical phase transitions and field-induced repulsion-to-attraction conversions.

arXiv:1310.5329•Reviews of Modern Physics•Jun 24, 2014•775 citations

Quantum Integrable Systems

Asesh Roy Chowdhury, Aninlya Ghose Choudhury

The study of integrable systems has opened new horizons in classical physics over the past few decades, particularly in the subatomic world. Yet despite the field now having reached a level of maturity, very few books provide an introduction to the field accessible to specialists and nonspecialists alike, and none offer a systematic survey of the m

Chapman & Hall/CRC eBooks•Jan 28, 2004•11 citations

Statistical Field Theory of Interacting Nambu Dynamics

Tamiaki Yoneya

Abstract We develop a statistical field theory for classical Nambu dynamics by partially employing the quantum field-theory method. One of the unsolved problems in Nambu dynamics has been to extend it to interacting systems without violating a generalized canonical structure characterized by the presence of multiple Hamiltonians, which together govern the dynamics of time evolution on an equal footing through a Nambu bracket as an analog to the Poisson bracket of ordinary Hamiltonian dynamics with a single Hamiltonian. In the present paper, we propose to include interactions from the standpoint of classical statistical dynamics by formulating it as a field theory on Nambu’s generalized phase space in an operator formalism. We first construct a general framework for such a field theory and its probabilistic interpretation. Then, on the basis of this framework, we give an exactly solvable model of self-interactions in a many-body Nambu system treated as a closed dynamical system satisfying the H-theorem. It is shown that a generalized microcanonical ensemble and a generalized canonical ensemble characterized by many temperatures are reached dynamically as equilibrium states starting with certain classes of initial nonequilibrium states via a continuous Markov process, which lies beyond the usual framework based on the Liouville theorem. Compared with the usual classical statistical mechanics on the basis of standard Hamiltonian dynamics, some important new features associated with Nambu dynamics will emerge, with respect to the symmetries underlying the dynamics of the nonequilibrium as well as the equilibrium states and also to some conceptual properties, such as a formulation of a generalized Kubo–Martin–Schwinger-like condition characterizing the generalized canonical equilibrium states and the “relative” nature of the temperatures.

arXiv:2512.09965•Progress of Theoretical and Experimental Physics•Apr 2, 2026•0 citations

Statistical mechanics of quantum systems of particles

David Pierre Ruelle

ABS>It is shown that methods applied earlier to classical systems may be extended to quantum systems of particles interacting by a two-body potential. In particular, using the canonical ensemble, for a large class of potentials, the existence of a limit for the free energy per particle when the system becomes infinite was proven. (auth)

•Oct 1, 1963•13 citations
The problem of extending ideas and results on the dynamics of infinite classical lattice systems to the quantum domain naturally arises in different branches of physics (nonequilibrium statistical mechanics, quantum optics, solid state, …) and new momentum from the development of quantum computer and quantum neural networks (which are in fact interacting arrays of binary systems) has been found. The stochastic limit of quantum theory allowed to deduce, as limits of the usual Hamiltonian systems, a new class of quantum stochastic flows which, when restricted to an appropriate Abelian subalgebra
The problem of extending ideas and results on the dynamics of infinite classical lattice systems to the quantum domain naturally arises in different branches of physics (nonequilibrium statistical mechanics, quantum optics, solid state, ...) and new momentum from the development of quantum computer and quantum neural networks (which are in fact interacting arrays of binary systems) has been found. The stochastic limit of quantum theory allowed to deduce, as limits of the usual Hamiltonian systems, a new class of quantum stochastic flows which, when restricted to an appropriate Abelian subalgebra, produces precisely those interacting particle systems studied in classical statistical mechanics. Moreover, in many interesting cases, the underlying classical process "drives" the quantum one, at least as far as ergodicity or convergence to equilibrium are concerned. Thus many deep results concerning classical systems can be directly applied to carry information on the corresponding quantum system. The thermodynamic limit itself is obtained thanks to a technique (the four-semigroup method, new even in the classical case) which reduces the infinitesimal structure of a stochastic flow to that of four semigroups canonically associated to it (Chap. 1). Simple and effective methods to analyze qualitatively the ergodic behavior of quantum Markov semigroups are discussed in Chap. 2. Powerful estimates used to control the infinite volume limit, ergodic behavior and the spectral gap (Gaussian, exponential and hypercontractive bounds, classical and quantum logarithmic Sobolev inequalities, ...) are discussed in Chap. 3
The study of nonequilibrium phenomena in correlated lattice systems has developed into one of the most active and exciting branches of condensed matter physics. This research field provides rich new insights that could not be obtained from the study of equilibrium situations, and the theoretical understanding of the physics often requires the development of new concepts and methods. On the experimental side, ultrafast pump-probe spectroscopies enable studies of excitation and relaxation phenomena in correlated electron systems, while ultracold atoms in optical lattices provide a new way to control and measure the time evolution of interacting lattice systems with a vastly different characteristic time scale compared to electron systems. A theoretical description of these phenomena is challenging because, first, the quantum-mechanical time evolution of many-body systems out of equilibrium must be computed and second, strong-correlation effects which can be of a nonperturbative nature must be addressed. This review discusses the nonequilibrium extension of the dynamical mean field theory (DMFT), which treats quantum fluctuations in the time domain and works directly in the thermodynamic limit. The method reduces the complexity of the calculation via a mapping to a self-consistent impurity problem, which becomes exact in infinite dimensions. Particular emphasis is placed on a detailed derivation of the formalism, and on a discussion of numerical techniques, which enable solutions of the effective nonequilibrium DMFT impurity problem. Insights gained into the properties of the infinite-dimensional Hubbard model under strong nonequilibrium conditions are summarized. These examples illustrate the current ability of the theoretical framework to reproduce and understand fundamental nonequilibrium phenomena, such as the dielectric breakdown of Mott insulators, photodoping, and collapse-and-revival oscillations in quenched systems. Furthermore, remarkable novel phenomena have been predicted by the nonequilibrium DMFT simulations of correlated lattice systems, including dynamical phase transitions and field-induced repulsion-to-attraction conversions.
Abstract We develop a statistical field theory for classical Nambu dynamics by partially employing the quantum field-theory method. One of the unsolved problems in Nambu dynamics has been to extend it to interacting systems without violating a generalized canonical structure characterized by the presence of multiple Hamiltonians, which together govern the dynamics of time evolution on an equal footing through a Nambu bracket as an analog to the Poisson bracket of ordinary Hamiltonian dynamics with a single Hamiltonian. In the present paper, we propose to include interactions from the standpoint of classical statistical dynamics by formulating it as a field theory on Nambu’s generalized phase space in an operator formalism. We first construct a general framework for such a field theory and its probabilistic interpretation. Then, on the basis of this framework, we give an exactly solvable model of self-interactions in a many-body Nambu system treated as a closed dynamical system satisfying the H-theorem. It is shown that a generalized microcanonical ensemble and a generalized canonical ensemble characterized by many temperatures are reached dynamically as equilibrium states starting with certain classes of initial nonequilibrium states via a continuous Markov process, which lies beyond the usual framework based on the Liouville theorem. Compared with the usual classical statistical mechanics on the basis of standard Hamiltonian dynamics, some important new features associated with Nambu dynamics will emerge, with respect to the symmetries underlying the dynamics of the nonequilibrium as well as the equilibrium states and also to some conceptual properties, such as a formulation of a generalized Kubo–Martin–Schwinger-like condition characterizing the generalized canonical equilibrium states and the “relative” nature of the temperatures.