In this article we investigate the spectral properties of the infinitesimal generator of an infinite system of master equations arising in the analysis of the approach to equilibrium in statistical mechanics. The system under investigation thus consists of infinitely many first-order differential equations governing the time evolution of probabilities susceptible of describing jumps between the eigenstates of a differential operator with a discrete point spectrum. The transition rates between eigenstates are chosen in such a way that the so-called detailed balance conditions are satisfied, so that for a large class of initial conditions the given system possesses a global solution which converges exponentially rapidly toward a time independent probability of Gibbs type. A particular feature and a challenge of the problem under investigation is that in the natural functional space where the initial-value problem is well-posed, the infinitesimal generator is realized as a non normal and non dissipative compact operator, whose spectrum therefore does not exhibit a spectral gap around the zero eigenvalue in contrast to the finite-dimensional case.
We consider the Burgers-type system studied by Foursov, w_t &=& w_{xx} + 8 w w_x + (2-4α)z z_x, z_t &=& (1-2α)z_{xx} - 4αz w_x + (4-8α)w z_x - (4+8α)w^2 z + (-2+4α)z^3, (*) for which no recursion operator or master symmetry was known so far, and prove that the system (*) admits infinitely many local generalized symmetries that are constructed using a nonlocal {\em two-term} recursion relation rather than from a recursion operator.
We provided a class of legitimate memory kernels leading to completely positive trace preserving dynamical maps. Our construction is based on a simple normalization procedure. Interestingly, when applied to the celebrated Wigner-Weisskopf theory it gives the standard Markovian evolution governed by the local master equation.
We start by considering infinite dimensional Markovian dynamics in R^m generated by operators of hypocoercive type and for such models we obtain short and long time pointwise estimates for all the derivatives, of any order and in any direction, along the semigroup. We then look at infinite dimensional models (in (Rm)^{Z ^d}) produced by the interaction of infinitely many finite dimensional dissipative dynamics of the type indicated above. For these infinite dimensional models we study finite speed of propagation of information, well-posedness of the semigroup, time behaviour of the derivatives…
We formulate the dynamics of an infinitely extended open dissipative quantum system, ${Σ]$,in the Schroedinger picture.The generic model on which this is based comprises a C*-algebra,$[\cal A}$,of observables, a folium, ${\cal F}$, of states on this algebra and a one parameter semigroup,$τ$, of linear transformations of ${\cal F}$ that represents its dynamics and is given by a natural infinite volume limit of the corresponding semigroup for a finite system. On this basis, we establish that the dynamic of $Σ$ is given by a one parameter semigroup of completely positive transformations of the W*…
We consider the class of quantum mechanical master equations defined on a generic Banach space, arising by projecting weakly perturbed one-parameter groups of isometries. We show that the possible semigroup approximations are far from unique. However, uniqueness can be reestablished through the introduction of a dynamical time averaging map. The generator of the resulting Contraction Semigroup is always well defined, irrespective of the dimensions of the projected subspace, and of the spectral properties of its free dynamics. We show how our approach includes and generalizes the preexisting li…
We analyze two different confining mechanisms for Lévy flights in the presence of external potentials. One of them is due to a conservative force in the corresponding Langevin equation. Another is implemented by Levy-Schroedinger semigroups which induce so-called topological Levy processes (Levy flights with locally modified jump rates in the master equation). Given a stationary probability function (pdf) associated with the Langevin-based fractional Fokker-Planck equation, we demonstrate that generically there exists a topological Lévy process with the very same invariant pdf and in the rever…
We introduce quantum versions of the Kac Master Equation and the Kac Boltzmann Equation. We study the steady states of each of these equations, and prove a propagation of chaos theorem that relates them. The Quantum Kac Master Equation (QKME) describes a quantum Markov semigroup, while the Kac Boltzmann Equation describes a non-linear evolution of density matrices on the single particle state space. All of the steady states of the $N$ particle quantum system described by the QKME are separable, and thus the evolution described by the QKME is entanglement breaking. The results set the stage for…
We investigate confined Lévy flights under premises of the principle of detailed balance. The master equation admits a transformation to Lévy - Schrödinger semigroup dynamics (akin to a mapping of the Fokker-Planck equation into the generalized diffusion equation). We solve a stochastic targeting problem for arbitrary stability index $0<μ<2$ of Lévy drivers: given an invariant probability density function (pdf), specify the jump - type dynamics for which this pdf is a long-time asymptotic target. Our ("$μ$-targeting") method is exemplified by Cauchy family and Gaussian target pdfs. We solve th…
The master equation for a probability density function (pdf) driven by Lévy noise, if conditioned to conform with the principle of detailed balance, admits a transformation to a contractive strongly continuous semigroup dynamics. Given a priori a functional form of the semigroup potential, we address the ground-state reconstruction problem for generic Lévy-stable semigroups, for {\em all} values of the stability index $μ\in (0,2)$. That is known to resolve an invariant pdf for confined Lévy flights (e.g. the former jump-type process). Jeopardies of the procedure are discussed, with a focus on:…
In this article we investigate the long time behavior of solutions to a class of infinitely many master equations defined from transition rates that are suitable for the description of a quantum system approaching thermodynamical equilibrium with a heat bath at fixed temperature and a reservoir consisting of one species of particles characterized by a fixed chemical potential. We do so by proving a result which pertains to the spectral resolution of the semigroup generated by the equations, whose infinitesimal generator is realized as a trace-class self-adjoint operator defined in a suitable w…
In view of controlling finite dimensional open quantum systems, we provide a unified Lie-semigroup framework describing the structure of completely positive trace-preserving maps. It allows (i) to identify the Kossakowski-Lindblad generators as the Lie wedge of a subsemigroup, (ii) to link properties of Lie semigroups such as divisibility with Markov properties of quantum channels, and (iii) to characterise reachable sets and controllability in open systems. We elucidate when time-optimal controls derived for the analogous closed system already give good fidelities in open systems and when a m…
Quantum systems coupled to environments exhibit intricate dynamics. The master equation gives a Markov approximation of the dynamics, allowing for analytic and numerical treatments. It is ubiquitous in theoretical and applied quantum sciences. The accuracy of the master equation approximation was so far proven for small values of the system-environment interaction coupling strength $λ$, under the additional constraint that time $t$ must not exceed an upper bound, $λ^2 t\le$ constant. Here, we show that the Markov approximation is valid for fixed small coupling strength and for all times. We al…
Structures of quantum Fokker-Planck equations are characterized with respect to the properties of complete positivity, covariance under symmetry transformations and satisfaction of equipartition, referring to recent mathematical work on structures of unbounded generators of covariant quantum dynamical semigroups. In particular the quantum optical master-equation and the quantum Brownian motion master-equation are shown to be associated to $\mathrm{U(1)}$ and $\mathrm{R}$ symmetry respectively. Considering the motion of a Brownian particle, where the expression of the quantum Fokker-Planck equa…
The construction of the master T-operator recently suggested in Alexandrov et al. (arXiv:1112.3310) is applied to integrable vertex models and associated quantum spin chains with trigonometric R-matrices. The master T-operator is a generating function for commuting transfer matrices of integrable vertex models depending on infinitely many parameters. At the same time it turns out to be the tau-function of an integrable hierarchy of classical soliton equations in the sense that it satisfies the the same bilinear Hirota equations. The class of solutions of the Hirota equations that correspond to…
The reaction-diffusion master equation (RDME) is a lattice-based stochastic model for spatially resolved cellular processes. It is often interpreted as an approximation to spatially continuous reaction-diffusion models, which, in the limit of an infinitely large population, may be described by means of reaction-diffusion partial differential equations (RDPDEs). Analyzing and understanding the relation between different mathematical models for reaction-diffusion dynamics is a research topic of steady interest. In this work, we explore a route to the hydrodynamic limit of the RDME which uses gra…
We present a general method of solving the Cauchy problem for multidimensional parabolic (diffusion type) equation with variable coefficients which depend on spatial variable but do not change over time. We assume the existence of the $C_0$-semigroup (this is a standard assumption in the evolution equations theory, which guarantees the existence of the solution) and then find the representation (based on the family of translation operators) of the solution in terms of coefficients of the equation and initial condition. It is proved that if the coefficients of the equation are bounded, infinite…
Two widely used but distinct approaches to the dynamics of open quantum systems are the Nakajima-Zwanzig and time-convolutionless quantum master equation, respectively. Although both describe identical quantum evolutions with strong memory effects, the first uses a time-nonlocal memory kernel $\mathcal{K}$, whereas the second achieves the same using a time-local generator $\mathcal{G}$. Here we show that the two are connected by a simple yet general fixed-point relation: $\mathcal{G} = \hat{\mathcal{K}}[\mathcal{G}]$. This allows one to extract nontrivial relations between the two completely d…
