Piotr Garbaczewski

Dynamics of dissipation

Nonequilibrium Dynamics.- Some Recent Advances in Classical Statistical Mechanics.- Deterministic Thermostats and Flctuation Relations.- What Is the Microscopic Response of a System Driven Far From Equilibrium?.- Non-equilibrium Statistical Mechanics of Classical and Quantum Systems.- Dynamics of Relaxation and Chaotic Behaviour.- Dynamical Theory of Relaxation in Classical and Quantum Systems.- Relaxation and Noise in Chaotic Systems.- Fractal Structures in the Phase Space of Simple Chaotic Systems with Transport.- Dynamical Semigroups.- Markov Semigroups and Their Applications.- Invitation to Quantum Dynamical Semigroups.- Finite Dissipative Quantum Systems.- Complete Positivity in Dissipative Quantum Dynamics.- Quantum Stochastic Dynamical Semigroup.- Driving, Dissipation and Control in Quantum Systems.- Driven Chaotic Mesoscopic Systems, Dissipation and Decoherence.- Quantum State Control in Cavity QED.- Solving Schrodingers Equation for an Open System and Its Environment.- Dynamics of Large Systems.- Thermodynamic Behavior of Large Dynamical Systems.- Coherent and Dissipative Transport in Aperiodic Solids: An Overview.- Scaling Limits of Schrodinger Quantum Mechanics.

Ornstein–Uhlenbeck–Cauchy process

We combine earlier investigations of linear systems subject to Levy fluctuations with recent attempts to give meaning to so-called Levy flights in external force fields. We give a complete construction of the Ornstein–Uhlenbeck–Cauchy process as a fully computable paradigm example of Doob’s stable noise-supported Ornstein–Uhlenbeck process. Despite the nonexistence of all moments, we determine local characteristics (forward drift) of the process, generators of forward and backward dynamics, and relevant (pseudodifferential) evolution equations. The induced nonstationary spatial process is proved to be Markovian and quite apart from its inherent discontinuity defines an associated velocity process in a probabilistic sense.

The method of Boson expansions in quantum theory

Abstract We give a review of Boson expansion methods applied in the quantum theory as e.g. expansions of spin, bifermion and Fermion operators in cases of the finite and infinite number of degrees of freedom. The basic purpose of the paper is to formulate the most general criterions allowing to get so called finite spin approximation of any given Bose field theory, and the class of Fermion theories associated with it. Quite the converse, we need also to be able to reconstruct the primary Bose field theory while any finite spin or Fermi systems are given.

Impenetrable barriers and canonical quantization

We address an apparent conflict between the traditional canonical quantization framework of quantum theory and the spatially restricted quantum dynamics, when the translation invariance of the otherwise free quantum system is broken by boundary conditions. By invoking an exemplary case of a particle in an infinite well, we analyze spectral problems for related, confined and global, observables. In particular, we show how one can make sense of various operators pertaining to trapped particles by not ignoring the rest of the real line (e.g., that space which is never occupied by the particle in question).We address an apparent conflict between the traditional canonical quantization framework of quantum theory and spatially restricted quantum dynamics when the translation invariance of an otherwise free quantum system is broken by boundary conditions. By considering the example of a particle in an infinite well, we analyze spectral problems for related confined and global observables. In particular, we show how we can interpret various operators related to trapped particles by not ignoring the rest of the real line that is never occupied by a particle.

Representations of the CAR generated by representations of the CCR. Fock case

We present a method of constructing the Fock representation of the canonical anti-communtation relations in the Fock representation of the canonical commutation relations. An explicit formula for Fermi creation and annihilation operators in terms of Bose ones is given.

Brownian motion in a magnetic field.

We derive explicit forms of Markovian transition probability densities for the velocity-space, phase-space, and the Smoluchowski configuration-space Brownian motion of a charged particle in a constant magnetic field. By invoking a hydrodynamical formalism for those stochastic processes, we quantify a continual (net on the local average) heat transfer from the thermostat to diffusing particles.

Derivation of the quantum potential from realistic Brownian particle motions

Abstract We present in this Letter a detailed analysis of mechanisms by which the phase space Brownian motion of an ensemble of massive particles, in the diffusion regime, is governed by the Schrodinger equation. It is explicitly shown how the pressure of the diffusing ensemble is linked to the quantum potential, known to appear in the Hamiltonian-Jacobi-Madelung formulation of the corresponding quantum dynamics. The quantum state vector (wave function) corresponds in this picture to the physically real diffusing medium governing the collective evolution of the particle ensemble.

Quantization of spinor fields

Influenced by Klauder’s investigations on the same subject, we study the question of correspondence principle for Dirac fields, looking for its formulation without use of Grassman algebras. We prove that with each Fermi operator (the series with respect to asymptotic free fields): Ω (ψ,ψ): one can associate the functional ΩC(ψC, ψC) with respect to classical spinor fields. Here the projector 1F and the Hilbert (Fock) space FF=1FFB are given such that the identity 1F: ΩC(ψB, ψB): 1FFF = :Ω (ψ, ψ):FF defines the mediating boson level, where coherent state expectation values of operator expressions are in order: 〈:ΩC(ψB, ψB):〉=ΩC(ψC, ψC). For proofs we employ functional differentiation (resp. integration) methods, especially in connection with the use of functional representations of the CCR and CAR algebras.

NATURAL BOUNDARIES FOR THE SMOLUCHOWSKI EQUATION AND AFFILIATED DIFFUSION-PROCESSES

The Schrodinger problem of deducing the microscopic dynamics from the input-output statistics data is known to admit a solution in terms of Markov diffusion processes. The uniqueness of the solution is found to be linked to the natural boundaries respected by the underlying random motion. By choosing a reference Smoluchowski diffusion process, we automatically fix the Feynman-Kac potential and the field of local accelerations it induces. We generate the family of affiliated diffusion processes with the same local dynamics but different inaccessible boundaries on finite, semi-infinite, and infinite domains. For each diffusion process a unique Feynman-Kac kernel is obtained by the constrained (Dirichlet boundary data) Wiener path integration. As a by-product of the discussion, we give an overview of the problem of inaccessible boundaries for the diffusion and bring together (sometimes viewed from unexpected angles) results which are little known and dispersed in publications from scarcely communicating areas of mathematics and physics.

Differential entropy and time

We give a detailed analysis of the Gibbs-type entropy notion and its dynamical behavior in case of time-dependent continuous probability distributions of varied origins: related to classical and quantum systems. The purpose-dependent usage of conditional Kullback-Leibler and Gibbs (Shannon) entropies is explained in case of non-equilibrium Smoluchowski processes. A very different temporal behavior of Gibbs and Kullback entropies is confronted. A specific conceptual niche is addressed, where quantum von Neumann, classical Kullback-Leibler and Gibbs entropies can be consistently introduced as information measures for the same physical system. If the dynamics of probability densities is driven by the Schrodinger picture wave-packet evolution, Gibbs-type and related Fisher information functionals appear to quantify nontrivial power transfer processes in the mean. This observation is found to extend to classical dissipative processes and supports the view that the Shannon entropy dynamics provides an insight into physically relevant non-equilibrium phenomena, which are inaccessible in terms of the Kullback-Leibler entropy and typically ignored in the literature.