Uniform and Completely Nonequilibrium Invariant States for Weak Coupling Limit Type Quantum Markov Semigroups Associated with Eulerian Cycles

2021 ◽  
Vol 28 (02) ◽  
Author(s):  
M. A. Cruz de la Rosa ◽  
J. C. García-Corte ◽  
F. Guerrero-Poblet
Keyword(s):  
Weak Coupling ◽  
Sufficient Conditions ◽  
Weak Coupling Limit ◽  
Invariant State ◽  
Markov Semigroups ◽  
Invariant States

We define the uniform and completely nonequilibrium invariant states, which are associated with Eulerian cycles; once we did this, we use the Hierholzer’s algorithm to obtain a canonical Euler-Hierholzer cycle, and for it, characterize the invariant state. For the simplest case of nonequilibrium, we give sufficient conditions for these states to be invariant and write its eigenvalues explicitly.

Generic Quantum Markov Semigroups with Degenerate Ground States: The Fock Case

2018 ◽  
Vol 25 (02) ◽  
pp. 1850010 ◽  
Author(s):  
Skander Hachicha ◽  
Ikbel Nasraoui
Keyword(s):  
Weak Coupling ◽  
Weak Coupling Limit ◽  
Markov Semigroup ◽  
Zero Temperature ◽  
Invariant State ◽  
Markov Semigroups ◽  
Initial State ◽  
Arbitrary Initial State ◽  
Invariant States ◽  
Quantum Markov Semigroup

We consider quantum Markov semigroups arising from the weak coupling limit of a system with generic Hamiltonian coupled to a boson Fock zero temperature reservoir. We find all the invariant states of a generic quantum Markov semigroup and compute explicitly the limit invariant state explicitly starting from an arbitrary initial state. We also show that convergence is exponentially fast under some natural assumptions.

scholarly journals Entropy Production and Detailed Balance for a Class of Quantum Markov Semigroups

2015 ◽  
Vol 22 (03) ◽  
pp. 1550013 ◽  
Author(s):  
F. Fagnola ◽  
R. Rebolledo
Keyword(s):  
Entropy Production ◽  
Weak Coupling ◽  
Detailed Balance ◽  
Weak Coupling Limit ◽  
Invariant State ◽  
Markov Semigroups ◽  
Detailed Balance Condition ◽  
Balance Condition ◽  
Do So

We give an explicit entropy production formula for a class of quantum Markov semigroups, arising in the weak coupling limit of a system coupled with reservoirs, whose generators [Formula: see text] are sums of other generators [Formula: see text] associated with positive Bohr frequencies [Formula: see text] of the system. As a consequence, we show that any such semigroup satisfies the quantum detailed balance condition with respect to an invariant state if and only if all semigroups generated by each [Formula: see text] do so with respect to the same invariant state.

Quantum Markov Semigroups of Low Density Limit: the Generic Case

2019 ◽  
Vol 26 (04) ◽  
pp. 1950021
Author(s):  
Luigi Accardi ◽  
Fernando Guerrero-Poblete
Keyword(s):  
Test Particle ◽  
Weak Coupling ◽  
Weak Coupling Limit ◽  
General Definition ◽  
Low Density ◽  
Markov Semigroups ◽  
Density Limit ◽  
Stochastic Limit ◽  
Invariant States ◽  
Low Density Limit

We investigate the structure of quantum Markov generators that describe the reduced dynamics of a test particle interacting with a dilute Bose gas in the low density limit. These generators, called low density limit type generators (LDL), differ from the weak coupling limit type generators (WCL) studied in [4, 5] because of the presence of the T-operator that describes the change in momentum of the gas particle due to collisions with the test particle. We propose a general definition of Markov generator of stochastic limit type that includes (almost) all generators arising in the stochastic limit approach and we prove that the associated semigroups as well as their invariant states (or weights) have a very special structure which extends the explicit representation for the generic quantum Markov semigroups of weak coupling limit type, due to Accardi, Hachicha and Ouerdiane [7]. We also investigate the structure of invariant states and give conditions on the T - operator for the existence of a unique invariant state and of equilibrium states.

Gaussian Quantum Markov Semigroups on a One-Mode Fock Space: Irreducibility and Normal Invariant States

2021 ◽  
Vol 28 (01) ◽  
pp. 2150001
Author(s):  
J. Agredo ◽  
F. Fagnola ◽  
D. Poletti
Keyword(s):  
Existence And Uniqueness ◽  
Fock Space ◽  
Sufficient Conditions ◽  
Quantum Oscillator ◽  
Markov Semigroup ◽  
Markov Semigroups ◽  
Type Condition ◽  
Invariant States ◽  
Quantum Markov Semigroup ◽  
Necessary And Sufficient

We consider the most general Gaussian quantum Markov semigroup on a one-mode Fock space, discuss its construction from the generalized GKSL representation of the generator. We prove the known explicit formula on Weyl operators, characterize irreducibility and its equivalence to a Hörmander type condition on commutators and establish necessary and sufficient conditions for existence and uniqueness of normal invariant states. We illustrate these results by applications to the open quantum oscillator and the quantum Fokker-Planck model.

Decomposition and Classification of Generic Quantum Markov Semigroups: The Gaussian Gauge Invariant Case

2012 ◽  
Vol 19 (02) ◽  
pp. 1250010 ◽  
Author(s):  
Franco Fagnola ◽  
Skander Hachicha
Keyword(s):  
Irreducible Component ◽  
Discrete System ◽  
Jump Process ◽  
Invariant State ◽  
Markov Semigroups ◽  
Markov Jump ◽  
Gauge Invariant ◽  
Irreducible Components ◽  
Invariant States

We study the structure of generic quantum Markov semigroups, arising from the stochastic limit of a discrete system with generic Hamiltonian interacting with a Gaussian gauge invariant reservoir. We show that they can be essentially written as the sum of their irreducible components determined by closed classes of states of the associated classical Markov jump process. Each irreducible component turns out to be recurrent, transient or have an invariant state if and only if its classical (diagonal) restriction is recurrent, transient or has an invariant state, respectively. We classify invariant states and study convergence towards invariant states as time goes to infinity.

scholarly journals On three new principles in non-equilibrium statistical mechanics and Markov semigroups of weak coupling limit type

2016 ◽  
Vol 19 (02) ◽  
pp. 1650009 ◽  
Author(s):  
Luigi Accardi ◽  
Franco Fagnola ◽  
Roberto Quezada
Keyword(s):  
Statistical Mechanics ◽  
Master Equation ◽  
Weak Coupling ◽  
Detailed Balance ◽  
Weak Coupling Limit ◽  
Markov Semigroups ◽  
Equilibrium Statistical Mechanics ◽  
Equation Formulation ◽  
The Third ◽  
Kms Condition

We introduce three new principles: the nonlinear Boltzmann–Gibbs prescription, the local KMS condition and the generalized detailed balance (GDB) condition. We prove the equivalence of the first two under general conditions and we discuss a master equation formulation of the third one.

scholarly journals A characterization of quantum Markov semigroups of weak coupling limit type

2019 ◽  
Vol 22 (02) ◽  
pp. 1950008
Author(s):  
Franco Fagnola ◽  
Roberto Quezada
Keyword(s):  
Weak Coupling ◽  
Weak Coupling Limit ◽  
Markov Semigroups ◽  
Natural Way

We characterize generators of quantum Markov semigroups leaving invariant a maximal abelian purely atomic algebra and certain operator subspaces associated with it in a natural way. From this result, we also establish a characterization of generators of quantum Markov semigroups of weak coupling limit type associated with a nondegenerate Hamiltonian.

scholarly journals An obstacle to sub-AdS holography for SYK-like models

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
Pengfei Zhang ◽  
Yingfei Gu ◽  
Alexei Kitaev
Keyword(s):  
Lyapunov Exponent ◽  
Weak Coupling ◽  
Kinetic Coefficient ◽  
Weak Coupling Limit ◽  
Branching Time ◽  
Quasiparticle Lifetime

Abstract We argue that “stringy” effects in a putative gravity-dual picture for SYK-like models are related to the branching time, a kinetic coefficient defined in terms of the retarded kernel. A bound on the branching time is established assuming that the leading diagrams are ladders with thin rungs. Thus, such models are unlikely candidates for sub-AdS holography. In the weak coupling limit, we derive a relation between the branching time, the Lyapunov exponent, and the quasiparticle lifetime using two different approximations.

Sign in / Sign up

Export Citation Format

Share Document