Quantum Probability and Infinite Dimensional Analysis

QUANTUM BOLTZMANN STATISTICS IN INTERACTING SYSTEMS

Luigi Accardi, Sergei Vladimirovich Kozyrev

Published Jan 1, 2003• 0 citations

Abstract

Collective operators for generic quantum systems with discrete spectrum are investigated. These operators, considered as operators in the entagled Fock space (space generated by action of collective creations of the vacuum) satisfy a particular kind of quantum Boltzmann (or free) commutational relations.

DOI: 10.1142/9789812704290_0002

Related papers

Quantum Boltzmann Statistics in Interacting Systems

Luigi Accardi, Sergei Vladimirovich Kozyrev

Collective operators for generic quantum system with discrete spectrum are investigated. These operators, considered as operators in the entangled Fock space (space generated by action of collective creations on the vacuum) satisfy a particular kind of Quantum Boltzmann (or free) commutational relations.

Birkhäuser Basel eBooksJan 1, 20030 citations

Quantum Boltzmann statistics in interacting systems

Luigi Accardi, Sergei Vladimirovich Kozyrev

Collective operators that describe interaction of generic quantum system with discrete spectrum with a quantum field are investigated. These operators, considered as operators in the entangled Fock space (space generated by action of collective creations on the vacuum) in the stochastic limit satisfy a particular kind of Quantum Boltzmann (or free) commutational relations. This clarifies a general phenomenon of arising of Quantum Boltzmann relations in interacting systems.

arXiv:quant-ph/0105009May 3, 20010 citations

q-Deformation and Free Statistics for Interaction of a Field and a Particle

Sergei Vladimirovich Kozyrev

arXiv:2001.01602Physics of Particles and NucleiJul 1, 20200 citations

Fock representations of multicomponent (particularly non-Abelian anyon) commutation relations

Alexei Daletskii, A. A. Kalyuzhnyi, Eugene Lytvynov +1 more

Nov 7, 201910 citations

On quantum statistics for ensembles with a finite number of particles

Evgenii Dmitrievich Trifonov

The well-known Bose–Einstein and Fermi–Dirac quantum distributions can be considered as stationary solutions of kinetic equations for the mean occupation numbers in an ideal gas of an arbitrary finite number of identical particles.

Physics-UspekhiJul 31, 20115 citations

Binary quantum collision operators conserving mass momentum and energy

Pierre Degond, Christian A. Ringhofer

In this Note, we generalize the Boltzmann collision operator modeling binary particle–particle collisions to a quantum framework using nonlocal quantum entropy principles.

Comptes Rendus MathématiqueApr 30, 200311 citations

Quantum Probability and Infinite Dimensional Analysis

QUANTUM BOLTZMANN STATISTICS IN INTERACTING SYSTEMS

Luigi Accardi, Sergei Vladimirovich Kozyrev

Published Jan 1, 2003• 0 citations

Publisher

Abstract

DOI: 10.1142/9789812704290_0002

Related papers

Quantum Boltzmann Statistics in Interacting Systems

Luigi Accardi, Sergei Vladimirovich Kozyrev

Birkhäuser Basel eBooksJan 1, 20030 citations

Quantum Boltzmann statistics in interacting systems

Luigi Accardi, Sergei Vladimirovich Kozyrev

arXiv:quant-ph/0105009May 3, 20010 citations

q-Deformation and Free Statistics for Interaction of a Field and a Particle

Sergei Vladimirovich Kozyrev

Abstract Emerging of free (or quantum Boltzmann) statistics for a model of quantum particle interacting with quantum field is described in the stochastic limit without dipole approximation. The quantum field is considered in a Gaussian (for example temperature) state. Entangled operators which describe interaction of the field and the particle satisfy the $q$ -deformed relations which in the stochastic limit generate free statistics.Abstract Emerging of free (or quantum Boltzmann) statistics for a model of quantum particle interacting with quantum field is described in the stochastic limit without dipole approximation. The quantum field is considered in a Gaussian (for example temperature) state. Entangled operators which describe interaction of the field and the particle satisfy the $$q$$ -deformed relations which in the stochastic limit generate free statistics.

arXiv:2001.01602Physics of Particles and NucleiJul 1, 20200 citations

Fock representations of multicomponent (particularly non-Abelian anyon) commutation relations

Alexei Daletskii, A. A. Kalyuzhnyi, Eugene Lytvynov +1 more

Let $H$ be a separable Hilbert space and $T$ be a self-adjoint bounded linear operator on $H^{\otimes 2}$ with norm $\le1$ , satisfying the Yang--Baxter equation. Bo\.zejko and Speicher (1994) proved that the operator $T$ determines a $T$ -deformed Fock space $\mathcal F(H)=\bigoplus_{n=0}^\infty\mathcal F_n(H)$ . We start with reviewing and extending the known results about the structure of the $n$ -particle spaces $\mathcal F_n(H)$ and the commutation relations satisfied by the corresponding creation and annihilation operators acting on $\mathcal F(H)$ . We then choose $H=L^2(X\to V)$ , the $L^2$ -space of $V$ -valued functions on $X$ . Here $X:=\mathbb R^d$ and $V:=\mathbb C^m$ with $m\ge2$ . Furthermore, we assume that the operator $T$ acting on $H^{\otimes 2}=L^2(X^2\to V^{\otimes 2})$ is given by $(Tf^{(2)})(x,y)=C_{x,y}f^{(2)}(y,x)$ . Here, for a.a.\ $(x,y)\in X^2$ , $C_{x,y}$ is a linear operator on $V^{\otimes 2}$ with norm $\le1$ that satisfies $C_{x,y}^*=C_{y,x}$ and the spectral quantum Yang--Baxter equation. The corresponding creation and annihilation operators describe a multicomponent quantum system. A special choice of the operator-valued function $C_{xy}$ in the case $d=2$ determines non-Abelian anyons (also called plektons). For a multicomponent system, we describe its $T$ -deformed Fock space and the available commutation relations satisfied by the corresponding creation and annihilation operators. Finally, we consider several examples of multicomponent quantum systems.Let $H$ be a separable Hilbert space and $T$ be a self-adjoint bounded linear operator on $H^{\otimes 2}$ with norm $\le1$, satisfying the Yang--Baxter equation. Bo\.zejko and Speicher (1994) proved that the operator $T$ determines a $T$-deformed Fock space $\mathcal F(H)=\bigoplus_{n=0}^\infty\mathcal F_n(H)$. We start with reviewing and extending the known results about the structure of the $n$-particle spaces $\mathcal F_n(H)$ and the commutation relations satisfied by the corresponding creation and annihilation operators acting on $\mathcal F(H)$. We then choose $H=L^2(X\to V)$, the $L^2$-space of $V$-valued functions on $X$. Here $X:=\mathbb R^d$ and $V:=\mathbb C^m$ with $m\ge2$. Furthermore, we assume that the operator $T$ acting on $H^{\otimes 2}=L^2(X^2\to V^{\otimes 2})$ is given by $(Tf^{(2)})(x,y)=C_{x,y}f^{(2)}(y,x)$. Here, for a.a.\ $(x,y)\in X^2$, $C_{x,y}$ is a linear operator on $V^{\otimes 2}$ with norm $\le1$ that satisfies $C_{x,y}^*=C_{y,x}$ and the spectral quantum Yang--Baxter equation. The corresponding creation and annihilation operators describe a multicomponent quantum system. A special choice of the operator-valued function $C_{xy}$ in the case $d=2$ determines non-Abelian anyons (also called plektons). For a multicomponent system, we describe its $T$-deformed Fock space and the available commutation relations satisfied by the corresponding creation and annihilation operators. Finally, we consider several examples of multicomponent quantum systems.

Nov 7, 201910 citations

On quantum statistics for ensembles with a finite number of particles

Evgenii Dmitrievich Trifonov

Physics-UspekhiJul 31, 20115 citations

Binary quantum collision operators conserving mass momentum and energy

Pierre Degond, Christian A. Ringhofer

In this Note, we generalize the Boltzmann collision operator modeling binary particle–particle collisions to a quantum framework using nonlocal quantum entropy principles.

Comptes Rendus MathématiqueApr 30, 200311 citations

q

-deformed relations which in the stochastic limit generate free statistics.

Let

H

be a separable Hilbert space and

T

be a self-adjoint bounded linear operator on

H^{\otimes 2}

with norm

\le1

, satisfying the Yang--Baxter equation. Bo\.zejko and Speicher (1994) proved that the operator

T

determines a

T

-deformed Fock space

\mathcal F(H)=\bigoplus_{n=0}^\infty\mathcal F_n(H)

. We start with reviewing and extending the known results about the structure of the

n

-particle spaces

\mathcal F_n(H)

and the commutation relations satisfied by the corresponding creation and annihilation operators acting on

\mathcal F(H)

. We then choose

H=L^2(X\to V)

, the

L^2

-space of

V

-valued functions on

X

. Here

X:=\mathbb R^d

and

V:=\mathbb C^m

with

m\ge2

. Furthermore, we assume that the operator

T

acting on

H^{\otimes 2}=L^2(X^2\to V^{\otimes 2})

is given by

(Tf^{(2)})(x,y)=C_{x,y}f^{(2)}(y,x)

. Here, for a.a.\

(x,y)\in X^2

C_{x,y}

is a linear operator on

V^{\otimes 2}

with norm

\le1

that satisfies

C_{x,y}^*=C_{y,x}

and the spectral quantum Yang--Baxter equation. The corresponding creation and annihilation operators describe a multicomponent quantum system. A special choice of the operator-valued function

C_{xy}

in the case

d=2

determines non-Abelian anyons (also called plektons). For a multicomponent system, we describe its

T

-deformed Fock space and the available commutation relations satisfied by the corresponding creation and annihilation operators. Finally, we consider several examples of multicomponent quantum systems.