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A geometric derivation of the linear Boltzmann equation for a particle interacting with a Gaussian random field, using a Fock space approach

Sébastien Breteaux (2014)

Annales de l’institut Fourier

In this article the linear Boltzmann equation is derived for a particle interacting with a Gaussian random field, in the weak coupling limit, with renewal in time of the random field. The initial data can be chosen arbitrarily. The proof is geometric and involves coherent states and semi-classical calculus.

Complete positivity and the neutral kaon system

Fabio Benatti, roberto Floreanini (1998)

Banach Center Publications

New experiments on neutral K-mesons might turn out to be promising tests of the hypothesis of Complete Positivity in the physics of open quantum systems. In particular, a consistent dynamical description of correlated neutral kaons seems to ask for Complete Positivity.

Many-body aspects of approach to equilibrium

Eric Carlen, M. C. Carvalho, Michael Loss (2000)

Journées équations aux dérivées partielles

Kinetic theory and approach to equilibrium is usually studied in the realm of the Boltzmann equation. With a few notable exceptions not much is known about the solutions of this equation and about its derivation from fundamental principles. In 1956 Mark Kac introduced a probabilistic model of N interacting particles. The velocity distribution is governed by a Markov semi group and the evolution of its single particle marginals is governed (in the infinite particle limit) by a caricature of the spatially...

On the transient and recurrent parts of a quantum Markov semigroup

Veronica Umanità (2006)

Banach Center Publications

We define the transient and recurrent parts of a quantum Markov semigroup 𝓣 on a von Neumann algebra 𝓐 and we show that, when 𝓐 is σ-finite, we can write 𝓣 as the sum of such semigroups. Moreover, if 𝓣 is the countable direct sum of irreducible semigroups each with a unique faithful normal invariant state ρₙ, we find conditions under which any normal invariant state is a convex combination of ρₙ's.

On two quantum versions of the detailed balance condition

Franco Fagnola, Veronica Umanità (2010)

Banach Center Publications

Quantum detailed balance conditions are often formulated as relationships between the generator of a quantum Markov semigroup and the generator of a dual semigroup with respect to a certain scalar product defined by an invariant state. In this paper we survey some results describing the structure of norm continuous quantum Markov semigroups on ℬ(h) satisfying a quantum detailed balance condition when the duality is defined by means of pre-scalar products on ℬ(h) of the form ⟨ x , y ⟩ s : = t r ( ρ 1 - s x * ρ s y ) (s ∈ [0,1]) in order...

Quantum detailed balance conditions with time reversal: the finite-dimensional case

Franco Fagnola, Veronica Umanità (2011)

Banach Center Publications

We classify generators of quantum Markov semigroups on (h), with h finite-dimensional and with a faithful normal invariant state ρ satisfying the standard quantum detailed balance condition with an anti-unitary time reversal θ commuting with ρ, namely t r ( ρ 1 / 2 x ρ t 1 / 2 ( y ) ) = t r ( ρ 1 / 2 θ y * θ ρ t 1 / 2 ( θ x * θ ) ) for all x,y ∈ and t ≥ 0. Our results also show that it is possible to find a standard form for the operators in the Lindblad representation of the generators extending the standard form of generators of quantum Markov semigroups satisfying the usual...

Quantum Euler-Poisson systems: Existence of stationary states

Ansgar Jüngel, Hailiang Li (2004)

Archivum Mathematicum

A one-dimensional quantum Euler-Poisson system for semiconductors for the electron density and the electrostatic potential in bounded intervals is considered. The existence and uniqueness of strong solutions with positive electron density is shown for quite general (possibly non-convex or non-monotone) pressure-density functions under a “subsonic” condition, i.e. assuming sufficiently small current densities. The proof is based on a reformulation of the dispersive third-order equation for the electron...

Stochastic Dynamics of Quantum Spin Systems

Adam Majewski, Robert Olkiewicz, Bogusław Zegarliński (1998)

Banach Center Publications

We show that recently introduced noncommutative L p -spaces can be used to constructions of Markov semigroups for quantum systems on a lattice.

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