Maurizio Verri

Properties of Quantum Markovian Master Equations

In this paper we give an essentially self-contained account of some general structural properties of the dynamics of quantum open Markovian systems. We review some recent results regarding the problem of the classification of quantum Markovian master equations and the limiting conditions under which the dynamical evolution of a quantum open system obeys an exact semigroup law (weak coupling limit and singular coupling limit). We discuss a general form of quantum detailed balance and its relation to thermal relaxation and to microreversibility.

Analytical and numerical study of photocurrent transients in organic polymer solar cells

Abstract This article is an attempt to provide a self consistent picture, including existence analysis and numerical solution algorithms, of the mathematical problems arising from modeling photocurrent transients in organic polymer solar cells (OSCs). The mathematical model for OSCs consists of a system of nonlinear diffusion–reaction partial differential equations (PDEs) with electrostatic convection, coupled to a kinetic ordinary differential equation (ODE). We propose a suitable reformulation of the model that allows us to prove the existence of a solution in both stationary and transient conditions and to better highlight the role of exciton dynamics in determining the device turn-on time. For the numerical treatment of the problem, we carry out a temporal semi-discretization using an implicit adaptive method, and the resulting sequence of differential subproblems is linearized using the Newton–Raphson method with inexact evaluation of the Jacobian. Then, we use exponentially fitted finite elements for the spatial discretization, and we carry out a thorough validation of the computational model by extensively investigating the impact of the model parameters on photocurrent transient times.

MULTISCALE MODELING AND SIMULATION OF ORGANIC SOLAR CELLS

Abstract In this article, we continue our mathematical study of organic solar cells (OSCs) and propose a two-scale (micro- and macro-scale) model of heterojunction OSCs with interface geometries characterized by an arbitrarily complex morphology. The microscale model consists of a system of partial and ordinary differential equations in an heterogeneous domain, that provides a full description of excitation/transport phenomena occurring in the bulk regions and dissociation/recombination processes occurring in a thin material slab across the interface. The macroscale model is obtained by a micro-to-macro scale transition that consists of averaging the mass balance equations in the normal direction across the interface thickness, giving rise to nonlinear transmission conditions that are parametrized by the interfacial width. These conditions account in a lumped manner for the volumetric dissociation/recombination phenomena occurring in the thin slab and depend locally on the electric field magnitude and orientation. Using the macroscale model in two spatial dimensions, device structures with complex interface morphologies, for which existing data are available, are numerically investigated showing that, if the electric field orientation relative to the interface is taken into due account, the device performance is determined not only by the total interface length but also by its shape.

A multiscale approach in the computational modeling of the biophysical environment in artificial cartilage tissue regeneration

Tissue Engineering is a strongly interdisciplinary scientific area aimed at understanding the principles of tissue growth to produce biologically functional replacements for clinical use. To achieve such an ambitious goal, complex biophysical phenomena must be understood in order to provide the appropriate environment to cells (nutrient delivery, fluid-mechanical loading and structural support) in the bioengineered device. Such a problem has an inherent multiphysics/multiscale nature, as it is characterized by material heterogeneities and interplaying processes occurring within a wide range of temporal and spatial scales. In this context, computational models are useful to gain a quantitative and comprehensive understanding of phenomena often difficult to be accessed experimentally. In this paper, we propose a mathematical and computational model that represents, to our knowledge, the first example of a self-consistent multiscale description of coupled nutrient mass transport, fluid-dynamics and biomass production in bioengineered constructs. We specifically focus on articular cartilage regeneration based on dynamically perfused bioreactors, and we investigate by numerical simulations three issues critical in this application. First, we study oxygen distribution in the construct, since achieving an optimal level throughout the construct is a main control variable to improve tissue quality. Second, we provide a quantitative evaluation of how interstitial perfusion can enhance nutrient delivery and, ultimately, biomass production, compared with static culture. Third, we perform a sensitivity analysis with respect to biophysical parameters related to matrix production, assessing their role in tissue regeneration.

Non-markovian behavior in low-temperature damping

We give a description of the reduced dynamics of an open quantum system whose initial state is (in general) correlated with the surroundings. The averaging method (whose mathematical theory was developed in another paper) is used to construct a series expansion of the reduced dynamics in powers of the ratio τ1/τ0 of the characteristic relaxation times τ1 and τ0 of the surroundings and of the system respectively. To first order in τ1/τ0 we obtain the well-known Markovian approximation which corresponds to the weak coupling limit. Starting from second order in τ1/τ0 there appear various kinds of non-Markovian effects including initial slips which have attracted much attention in the recent literature. The relevance of these higher-order corrections stems from the fact that the characteristic time τ1 cannot be much smaller than hkT for quantum surroundings at temperature T.

Master equation treatment of the singular reservoir limit

Abstract We exhibit an explicit model where the generalized master equation becomes Markovian in the limit when the decay time of the correlation functions of the reservoir tends to zero.

Quantum dynamical semigroups and multipole relaxation of a spin in isotropic surroundings

We derive and discuss three different parametrizations of the generator of a dynamical semigroup which describes the Markovian relaxation of a spin j under the influence of isotropic surroundings. The relevant parametrizations that we consider are the strengths of the polarities of the interaction, the relaxation rates of the different multipoles and the transition probabilities per unit time among the Zeeman sublevels. The results are model independent and allow us to derive a set of relations and inequalities for the transition probabilities and for the relaxation rates whose validity is not bound to any specific assumption concerning the mechanisms which govern the relaxation.

A multiscale thermo-fluid computational model for a two-phase cooling system

Abstract In this paper, we describe a mathematical model and a numerical simulation method for the condenser component of a novel two-phase thermosiphon cooling system for power electronics applications. The condenser consists of a set of roll-bonded vertically mounted fins among which air flows by either natural or forced convection. In order to deepen the understanding of the mechanisms that determine the performance of the condenser and to facilitate the further optimization of its industrial design, a multiscale approach is developed to reduce as much as possible the complexity of the simulation code while maintaining reasonable predictive accuracy. To this end, heat diffusion in the fins and its convective transport in air are modeled as 2D processes while the flow of the two-phase coolant within the fins is modeled as a 1D network of pipes. For the numerical solution of the resulting equations, a Dual Mixed-Finite Volume scheme with Exponential Fitting stabilization is used for 2D heat diffusion and convection while a Primal Mixed Finite Element discretization method with upwind stabilization is used for the 1D coolant flow. The mathematical model and the numerical method are validated through extensive simulations of realistic device structures which prove to be in excellent agreement with available experimental data.

Multiscale simulation of organic heterojunction light harvesting devices

Purpose – The purpose of this paper is to develop a computational model for the simulation of heterojunction organic photovoltaic devices with a specific application to a light harvesting capacitor (LHC) consisting of a double layer of organic materials connected in series with two insulating layers and an external resistive load. Design/methodology/approach – The model is based on a coupled system of nonlinear partial and ordinary differential equations describing current flow throughout the external resistive load as the result of exciton generation in the bulk, exciton dissociation into bonded pairs at the acceptor-donor material interface, and electron/hole charge generation and drift-diffusion transport in the two device materials. Findings – Numerical simulation results are shown to be in good agreement with measured on-off transient currents and allow for novel insight on the microscopical phenomena which affect the external LHC performance, in particular, the widely different time scales at which ...

The zeroth law of thermodynamics

We discuss a convenient formulation of the most intuitive property of equilibrium, the zeroth law of thermodynamics, in terms of conditions of dynamical stability for a system which may interact (weakly) with its surroundings. This allows us to provide a completely rigorous, though elementary, justification of the Gibbs canonical ensemble as the description of thermodynamical equilibrium for a spatially confined quantum system with a fixed, finite number of particles. For quantum systems with variable particle numbers, the same kind of assumptions leads to the grand-canonical ensemble.