Explore the vast range of titles available.

Major advances in the quantum theory of macroscopic systems, in combination with stunning experimental achievements, have brightened the field and brought it to the attention of the general community in natural sciences. Today, working knowledge of dissipative quantum mechanics is an essential tool for many physicists. This book — originally published in 1990 and republished in 1999 as an enlarged second edition — delves much deeper than ever before into the fundamental concepts, methods, and applications of quantum dissipative systems, including the most recent developments.

We investigate the work fluctuations in an overdamped non-equilibrium process that is stopped at a stochastic time. The latter is characterised by a first passage event that marks the completion of the non-equilibrium process. In particular, we consider a particle diffusing in one dimension in the presence of a time-dependent potential U ( x , t ) = k | x − v t | n / n , where k  > 0 is the stiffness and n  > 0 is the order of the potential. Moreover, the particle is confined between two absorbing walls, located at L ± ( t ) , that move with a constant velocity v and are initially located at L ± ( 0 ) = ± L . As soon as the particle reaches any of the boundaries, the process is said to be completed and here, we compute the work done W by the particle in the modulated trap upto this random time. Employing the Feynman–Kac path integral approach, we find that the typical values of the work scale with L with a crucial dependence on the order n . While for n  > 1, we show that ⟨ W ⟩ ∼ L 1 − n   exp k L n / n − v L / D for large L , we get an algebraic scaling of the form ⟨ W ⟩ ∼ L n for the n  < 1 case. The marginal case of n  = 1 is exactly solvable and our analysis unravels three distinct scaling behaviours: (i) ⟨ W ⟩ ∼ L for v  >  k , (ii) ⟨ W ⟩ ∼ L 2 for v  =  k and (iii) ⟨ W ⟩ ∼ exp − ( v − k ) L for v  <  k . For all cases, we also obtain the probability distribution associated with the typical values of W . Finally, we observe an interesting set of relations between the relative fluctuations of the work done and the first-passage time for different n —which we argue physically. Our results are well supported by the numerical simulations.

The dissociative photodetachment dynamics of the oxalate anion, C2O4H− + hν → CO2 + HOCO + e−, were theoretically studied using the on-the-fly path-integral and ring-polymer molecular dynamics methods, which can account for nuclear quantum effects at the density-functional theory level in order to compare with the recent experimental study using photoelectron–photofragment coincidence spectroscopy. To reduce computational time, the force acting on each bead of ring-polymer was approximately calculated from the first and second derivatives of the potential energy at the centroid position of the nuclei beads. We find that the calculated photoelectron spectrum qualitatively reproduces the experimental spectrum and that nuclear quantum effects are playing a role in determining spectral widths. The calculated coincidence spectrum is found to reasonably reproduce the experimental spectrum, indicating that a relatively large energy is partitioned into the relative kinetic energy between the CO2 and HOCO fragments. This is because photodetachment of the parent anion leads to Franck–Condon transition to the repulsive region of the neutral potential energy surface. We also find that the dissociation dynamics are slightly different between the two isomers of the C2O4H− anion with closed- and open-form structures.

A quantum model of a free-electron laser (FEL) is considered. Two different approaches for the exploration of the the FEL system are considered. In the first case, the Heisenberg equations of motion are mapped on the basis of the initial wave functions, which consists of the photon coherent states and many-dimensional electron coherent states. This mapping is an exact procedure, which makes it possible to obtain an exact equation of motion for the intensity of the laser field in a closed form. The obtained equation is controlled by a Koopman operator. The analytical expression for the evolution of the FEL intensity is obtained in the framework of a perturbation theory, which is constructed for a small time scale. The second way of the consideration is based on the construction of the many-dimensional path integrals for the evolution of the wave function. This method also makes it possible to estimate the time evolution and the gain of the FEL intensity.

In this paper, we develop new asymptotic path-independent integrals for the evaluation of the crack tip parameters in a 2D neo-Hookean material. The new integrals are of both J -integral and interaction energy integral type and rely on the separation of the asymptotic boundary value problem into independent problems for each of the deformed coordinates. Both the plane stress and plane strain cases are considered. The integrals developed are used to compute the amplitude parameters of the asymptotic crack tip fields, which allows for direct extraction of these parameters from numerical results. A long strip with an edge crack under mixed loading modes is considered for both homogeneous and biomaterial cases. It is found that the asymptotic J -integrals produce good results for the first-order parameters while the interactions integrals produce good results for both the first and second-order parameters.

\"This book presents a self-contained study of the Riemann approach to the theory of random variation and assumes only some familiarity with probability or statistical analysis, basic Riemann integration, and mathematical proofs. The author focuses on non-absolute convergence in conjunction with random variation\"--

This book provides an introductory albeit solid presentation of path integration techniques as applied to the field of stochastic processes. The subject began with the work of Wiener during the 1920's, corresponding to a sum over random trajectories, anticipating by two decades Feynman's famous work on the path integral representation of quantum mechanics. However, the true trigger for the application of these techniques within nonequilibrium statistical mechanics and stochastic processes was the work of Onsager and Machlup in the early 1950's. The last quarter of the 20th century has witnessed a growing interest in this technique and its application in several branches of research, even outside physics (for instance, in economy).

Trapped bosons exhibit fundamental physical phenomena and are at the core of emerging quantum technologies. We present a method for simulating bosons using path integral molecular dynamics. The main difficulty in performing such simulations is enumerating all ring-polymer configurations, which arise due to permutations of identical particles. We show that the potential and forces at each time step can be evaluated by using a recurrence relation which avoids enumerating all permutations, while providing the correct thermal expectation values. The resulting algorithm scales cubically with system size. The method is tested and applied to bosons in a 2-dimensional (2D) trap and agrees with analytical results and numerical diagonalization of the many-body Hamiltonian. An analysis of the role of exchange effects at different temperatures, through the relative probability of different ring-polymer configurations, is also presented.

In this second edition, a comprehensive review is given for path integration in two- and three-dimensional (homogeneous) spaces of constant and non-constant curvature, including an enumeration of all the corresponding coordinate systems which allow separation of variables in the Hamiltonian and in the path integral. The corresponding path integral solutions are presented as a tabulation. Proposals concerning interbasis expansions for spheroidal coordinate systems are also given. In particular, the cases of non-constant curvature Darboux spaces are new in this edition. The volume also contains results on the numerical study of the properties of several integrable billiard systems in compact domains (i.e. rectangles, parallelepipeds, circles and spheres) in two- and three-dimensional flat and hyperbolic spaces. In particular, the discussions of integrable billiards in circles and spheres (flat and hyperbolic spaces) and in three dimensions are new in comparison to the first edition. In addition, an overview is presented on some recent achievements in the theory of the Selberg trace formula on Riemann surfaces, its super generalization, their use in mathematical physics and string theory, and some further results derived from the Selberg (super-) trace formula.

In the excitement and rapid pace of developments, writing pedagogical texts has low priority for most researchers. However, in transforming my lecture l notes into this book, I found a personal benefit: the organization of what I understand in a (hopefully simple) logical sequence. Very little in this text is my original contribution. Most of the knowledge was collected from the research literature. Some was acquired by conversations with colleagues; a kind of physics oral tradition passed between disciples of a similar faith. For many years, diagramatic perturbation theory has been the major theoretical tool for treating interactions in metals, semiconductors, itiner- ant magnets, and superconductors. It is in essence a weak coupling expan- sion about free quasiparticles. Many experimental discoveries during the last decade, including heavy fermions, fractional quantum Hall effect, high- temperature superconductivity, and quantum spin chains, are not readily accessible from the weak coupling point of view. Therefore, recent years have seen vigorous development of alternative, nonperturbative tools for handling strong electron-electron interactions. I concentrate on two basic paradigms of strongly interacting (or con- strained) quantum systems: the Hubbard model and the Heisenberg model. These models are vehicles for fundamental concepts, such as effective Ha- miltonians, variational ground states, spontaneous symmetry breaking, and quantum disorder. In addition, they are used as test grounds for various nonperturbative approximation schemes that have found applications in diverse areas of theoretical physics.