Quantum–Classical Correspondence Principle for Heat Distribution in Quantum Brownian Motion

Quantum Brownian motion, described by the Caldeira–Leggett model, brings insights to the understanding of phenomena and essence of quantum thermodynamics, especially the quantum work and heat associated with their classical counterparts. By employing the phase-space formulation approach, we study the heat distribution of a relaxation process in the quantum Brownian motion model. The analytical result of the characteristic function of heat is obtained at any relaxation time with an arbitrary friction coefficient. By taking the classical limit, such a result approaches the heat distribution of the classical Brownian motion described by the Langevin equation, indicating the quantum–classical correspondence principle for heat distribution. We also demonstrate that the fluctuating heat at any relaxation time satisfies the exchange fluctuation theorem of heat and its long-time limit reflects the complete thermalization of the system. Our research study justifies the definition of the quantum fluctuating heat via two-point measurements.

Interaction among neighbouring rectangular finite strike slip faults in a linear viscoelastic half space representing lithosphere-asthenosphere system

There are seismically active regions consisting of fault system with a number of neighbouring earthquake faults. A movement across any one of them may affect the nature of stress accumulation near the others. Mathematical models may be developed to study these interactions and the pattern of interseismic stresses during the aseismic period in between two consecutive seismic events. In this paper, the lithosphere-asthenosphere system is being represented by a linear viscoelastic half space. The material of the half space is expected to possess the properties of both Maxwell and Kelvin type materials. It is assumed that the system is under a steady shear stress generated by some tectonic phenomena. For obtaining the solution for displacement, strain and stresses from the resulting boundary value problem, Integral transform, Green’s function techenique and correspondence principle have been used. Appropriate estimates of the model parameters were used in carrying out the numerical computations for investigating the nature of interactions among the faults.

Fundamentals of Diatomic Molecular Spectroscopy

The interpretation of optical spectra requires thorough comprehension of quantum mechanics, especially understanding the concept of angular momentum operators. Suppose now that a transformation from laboratory-fixed to molecule-attached coordinates, by invoking the correspondence principle, induces reversed angular momentum operator identities. However, the foundations of quantum mechanics and the mathematical implementation of specific symmetries assert that reversal of motion or time reversal includes complex conjugation as part of anti-unitary operation. Quantum theory contraindicates sign changes of the fundamental angular momentum algebra. Reversed angular momentum sign changes are of heuristic nature and are actually not needed in analysis of diatomic spectra. This work addresses sustenance of usual angular momentum theory, including presentation of straightforward proofs leading to falsification of the occurrence of reversed angular momentum identities. This review also summarises aspects of a consistent implementation of quantum mechanics for spectroscopy with selected diatomic molecules of interest in astrophysics and in engineering applications.

Hilbert’s First Problem and the New Progress of Infinity Theory

In the 19th century, Cantor created the infinite cardinal number theory based on the "1-1 correspondence" principle. The continuum hypothesis is proposed under this theoretical framework. In 1900, Hilbert made it the first problem in his famous speech on mathematical problems, which shows the importance of this question. We know that the infinitesimal problem triggered the second mathematical crisis in the 17-18th centuries. The Infinity problem is no less important than the infinitesimal problem. In the 21st century, Sergeyev introduced the Grossone method from the principle of "whole is greater than part", and created another ruler for measuring infinite sets. At the same time, 16 the development of the infinity theory provides new ideas for solving Hilbert's first problem, and provides a new mathematical foundation for Cosmic Continuum Theory

3. The elements of a crime: mens rea

This chapter examines the mens rea or mental fault of the accused. Because an actus reus is treated in law as a bad thing, an intention to cause it is, in law, a bad intention, a guilty mind. Similarly, consciously taking an unjustified risk of causing an actus reus—that is, recklessness—is also a bad state of mind. Unintentionally causing an actus reus by negligence may also be regarded as legally blameworthy. Each of these implies different degrees of ‘fault’. The chapter also discusses subjective and objective fault, intention in crimes other than murder, the distinction between motive and intention, subjective recklessness and malice, wilful blindness, suspicion and reasonable grounds to suspect, the correspondence principle and constructive crime, coincidence in time of actus reus and mens rea, ignorance of the law, absence of a ‘claim of right’ as an element in mens rea and proof of intention and foresight.