Schedule for: 18w5059 - Quantum Transport Equations and Applications

One of the main problems in statistical mechanics is the mathematical derivation, through space-time scaling limits, of macroscopic conservation laws, like compressible Euler equations or the heat equation, from the microscopic dynamics of molecules. Different space-time scalings, that we denote hydrodynamic limits, can lead to different macroscopic equations: the Euler system of equations governs the convergence to mechanical equilibrium (towards constant pressure) in a hyperbolic scaling, while at diffusive scaling (larger time) heat equation governs the convergence to thermal equilibrium (constant temperature), if thermal conductivity is finite. Deriving macroscopic limits from a deterministic (classical Hamiltonian or quantum) dynamics is a famous arduous problem. Some results can be obtained by adding to the dynamics some random terms such that momentum, energy and volume are still conserved but all other integrals of the motion are destroyed. The harmonic chain, despite the fact to be completely integrable, is an exception and, for the deterministic dynamics in thermal (global) equilibrium, the Euler equation are valid in the hyperbolic scaling. If the masses are random, thanks to Anderson localization, Euler equations are valid even out of thermal equilibrium. Furthermore, in the random mass case, the temperature profile remains constant in time at every later time scale, including the diffusive one, giving a further proof that thermal conductivity of this system is null. This results on the harmonic chain are extended to the corresponding quantum dynamics. (Works in collaboration with Cedric Bernardin, Francois Huveneers, Amirali Hannani)

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I will introduce some general properties of the generators mentioned in the title. This general properties will be illustrated in the case of low density type generators in the talk by Fernando Guerrero-Poblete

We search for the structure of quantum Markov generators describing the reduced dynamics of a test particle interacting with a diluted Bose gas in the low density limit. We also search for the structure of invariant states in the generic case.

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We study the notion of quantum Kac's chaos which was implicitly introduced by Spohn and explicitly formulated by Gottlieb. We prove the analogue of a result of Sznitman which gives the equivalence of Kac's chaos to 2-chaoticity and to convergence of empirical measures. Finally we give a simple, different proof of a result of Spohn which states that chaos propagates with respect to certain Hamiltonians that define the evolution of the mean field limit for interacting quantum systems. The talk is based on joint work with Rade Musulin.

The class of Meixner random variables can be described in terms of a Lie Algebra structure generated by their quantum operators. This Lie Algebra structure is useful in computing first the kernels that give the second quantization operators, and then the Wick products generated by these random variables. We restrict our attention to three of the most important types of Meixner random variables: Gaussian, Poisson, and Gamma, and present some H ̈older inequalities for the norms of the Wick products generated by them. We show that these inequalities are related to sharp inequalities from classic Harmonic Analysis concerning the norms of some convolution-type products

In this talk the notion of circulant c.p. map will be extended to the Toeplitz case together with the usual technique of invariant subspaces in order to study the operator itself and its spectrum. Some results and simple cases will be reviewed. Finally, the relation between Toeplitz maps and a family of (non degenerate) WCLT generators will be described together with some simple results regarding the spectrum. This is a work in progress.

We will discuss on the general structure of the stationary states of excitation energy transport models and weak coupling limit type generators. Starting with the case of a non-degenerate reference Hamiltonian, we study the structure of stationary states belonging to the annihilator of all Krauss operators, including detailed balance and local detailed balance states. \\ (Joint work with \'Alvaro Hern\'andez-Cervantes)

The mechanisms of quantum transport have attracted the attention of researchers from different areas (physics, chemistry, mathematics), who have produced an important amount of scientific, experimental or theoretical, work focused on describing and modeling these complex mechanisms. \\ Due to their characteristics as complex systems, semigroups of weak coupling limit type, with degenerate Hamiltonian, are suitable for modeling these phenomena. In this talk we will discuss the mathematical structure of the infinitesimal generator of some of these semigroups and the set of their stationary states. We will also present what is Free-Decoherence Algebra of a dynamic quantum semigroup of Markov, some characterizations of it and its application in the quantum transport model of $[1]$. \\ \\ \noindent{\bf References} \\ $[1]$ Aref'eva, Y., Volovich, I. and Kozyrev, S. 2015. Stochastic Limit Method and Interference in Quantum Many-particles Systems. Theoretical and Mathematical Physics 183(3) (2015) 782-799

We describe the structure of generators of norm-continuous quantum Markov semigroups with atomic decoherence-free subalgebra providing a natural decomposition of a Markovian quantum open system into its irreducible components and noiseless components. As an application, we discuss the structure of invariant states of certain quantum Markov semigroups of weak coupling limit type

In this talk we want to explain the relationships between the atomicity of the decoherence-free subalgebra, environmental decoherence, ergodic decomposition of the trace class operators, and the structure of fixed points. In particular, we show that, for a Quantum Markov Semigroup (QMS) with a faithful normal invariant state, the atomicity of the decoherence-free subalgebra and environmental decoherence are equivalent. Moreover, we characterize the set of reversible states and explicitly describe the relationship between the decoherence-free subalgebra and the fixed point subalgebra for QMSs with the above equivalent properties. Loosely speaking one can say that, for QMSs with a faithful invariant state, the same conclusions can be drawn replacing finite dimensionality of the system Hilbert space by atomicity of the decoherence-free subalgebra.

In recent years, digraph induced generators of quantum dynamical semigroups have been introduced and studied, particularly in the context of unique relaxation. In this talk we consider the converse construction to show every generator $\mathcal{L}$ naturally gives rise to a digraph $G_\mathcal{L}$. In specialized cases we use properties of $G_\mathcal{L}$ to explicitly compute all invariant states of the semigroup. In the general case we discuss relaxation and the number of invariant states of the semigroup in terms of several other generator induced graphs

Attal and Joye (J. Funct. Anal. 247, 2007) studied a repeated-interactions model driven by particles in a faithful normal state. As they observed, the limit dynamics obey a Langevin equation driven by quantum noises which satisfy the commutation relations for a quasifree state. Inspired by this example, we describe a general framework to handle such quasifree random walks. The theory of quasifree quantum stochastic integration required to describe the limit cocycles builds on early work of Hudson and Lindsay, together with more recent work of Lindsay, Margetts and Weatherall. \\ (Joint work with Micha³ Gnacik, Martin Lindsay and Ping Zhong)

We show that the quantum dynamical entropy introduced by Slomczynski and Zyczkowski is nonlinear in the time interval between successive measurements of a quantum dynamical system. This is in contrast to Kolmogorov-Sinai dynamical entropy for classical dynamical systems, which is linear in time. We also compute the exact values of quantum dynamical entropy for the Hadamard walk with varying Lüders-von Neumann instruments and partitions

D. Bures defined a metric on states of a C*-algebra as the infimum of the distance between associated vectors in common GNS representations. Now there are modifications and extensions of this notion to completely positive maps. We present some recent results in the area. This is based on joint works with K. Sumesh and Mithun Mukherjee.