In this work, we consider the spread of a ‘civilisation’ in an idealised homogeneous isotropic universe where all the planets of interest are habitable. Following a framework that goes beyond the usual idea of percolation in common undergraduate computational physics textbooks, we investigate the behaviour of the number of colonised planets with time, and the total colonisation time for three types of universes. These include static, dark energy-dominated, and matter-dominated universes. For all these types of universes, we find a remarkable fit with the Logistic Growth Function for the number of colonised planets with time. This is in spite of the fact that for the matter- and dark-energy dominated universes, the space itself is expanding. For the total colonisation time, T, the case for a dark energy-dominated universe is marked with divergence beyond the linear regime characterised by small values of the Hubble parameter, H. Not all planets in a spherical section of this universe can be ‘colonised’ due to the presence of a shrinking Hubble sphere. In other words, the recession speeds of other planets go beyond the speed of light making them impossible to reach. On the other hand, for a matter-dominated universe, while there is an apparent horizon, the Hubble sphere is growing instead of shrinking. This leads to a finite total colonisation time that depends on the Hubble parameter characterising the Universe; in particular, we find T ∼ H for small H and T ∼ H 2 for large H.

(Avoided) crossings are ubiquitous in physics and are connected to many physical phenomena such as hidden symmetries, the Berry phase, entanglement, Landau–Zener processes, the onset of chaos, etc. A pedagogical approach to cataloging (avoided) crossings has been proposed in the past, using matrices whose eigenvalues avoid or cross as a function of some parameter. The approach relies on the mathematical tool of the discriminant, which can be calculated from the characteristic polynomial of the matrix, and whose roots as a function of the parameter being varied yield the locations as well as degeneracies of the (avoided) crossings. In this article we consider matrices whose symmetries force two or more eigenvalues to be degenerate across the entire range of variation of the parameter of interest, thus leading to an identically vanishing discriminant. To show how this case can be handled systematically, we introduce a perturbation to the matrix and calculate the roots of the discriminant in the limit as the perturbation vanishes. We show that this approach correctly generates a nonzero ‘reduced’ discriminant that yields the locations and degeneracies of the (avoided) crossings. We illustrate our technique using the matrix Hamiltonian for benzene in Hückel theory, which has recently been discussed in the context of (avoided) crossings in its spectrum.

When considering the motion of an incompressible fluid, it is common practice to take the curl on both sides of the Navier–Stokes (or Euler) equations and cancel the pressure force. The governing equations are sufficient to derive the velocity field of the fluid without any knowledge of the pressure. In fact, the pressure is only calculated after obtaining the velocity field. This raises a number of conceptual problems. For instance, why is the pressure unnecessary for obtaining the velocity field? Traditionally, forces have been considered as the ‘causes’ of motion, and the resulting acceleration as the ‘effect’. However, the acceleration (the effect) and the resulting velocity field can be obtained without any recourse to the pressure (the cause), seemingly violating the principle of ‘cause’ and ‘effect’. We address these questions by deriving the pressure force of an incompressible fluid, starting from d’Alembert’s principle of virtual work, as a ‘reaction force’ that maintains the incompressibility condition. Next, we show that taking the curl on both sides of the Navier–Stokes (or Euler) equations is equivalent to using d’Alembert’s principle of virtual work, which cancels out the virtual work of the pressure gradient. This shows that abstract procedures, such as taking the curl on both sides of an equation, can actually be tacit applications of rich physical principles, without one realizing it. This can be quite instructive in a classroom of undergraduate students.

Abstract In Radiation Physics classes, point sources are typically used, for which it is relatively easy to describe the signal obtained by a radiation detector, such as the NaI(Tl) scintillation detector. The use of large extended radiation sources is generally avoided due to the mathematical complexity that their description may involve. However, the use of Monte Carlo simulation methods allows this limitation to be overcome. Potassium chloride, containing the 40K isotope, is an ideal candidate for carrying out this type of experiment. The source activity is obtained through the detection of the 1460.8 keV gamma- photon emitted in the 40K decay. In the first experiment, a cylindrical container is used, placing the NaI(Tl) detector in its center and filling the remaining space with potassium chloride. In a second, more complex case, a large radioactive source consisting of a container filled with a mixture of sand and potassium chloride, with the NaI(Tl) detector placed in the center of the mixture, is used. In this case, the mass of potassium chloride is approximately 1/5 of the sand mass. In both experiments, the detection efficiency is obtained by Monte Carlo simulation. A careful analysis of the experimental data allows to obtain a good agreement between the measured and calculated value of the activity.

Abstract We consider the Earth moving through empty space at 30 km/s (in the sun’s frame of reference). Associated with this motion is a convective flow of kinetic and internal energy. Since there is high pressure inside the earth, and since the earth is moving, there is yet another “hydraulic” energy flow. This latter is what this article is about. Although this energy flow is huge, it is not addressed in the textbooks. The reason is that for the explanation one needs a concept which is not introduced in traditional presentations of classical gravitation: the gravitomagnetic field. The corresponding theory, gravitoelectromagnetism, was formulated in 1893 by Heaviside in analogy to Maxwell's theory of electromagnetism.

We discuss the question of what are the sources and sinks of this hydraulic, non-convective energy flow. To answer the question, we need to study the energy flow density distribution within the gravitational field. In doing so, we will make some interesting observations. The energy flow within the field is twice as large as it should be to transfer the field energy from one side of the Earth to the other. The excess flow goes back through the matter of the Earth.

Since our readers may not be familiar with Heaviside’s theory, we first treat the electromagnetic analogue of our problem and then translate the results to the gravitational situation.

Abstract Two thermodynamic processes, an adiabatic gas compression and an isothermal gas compression, taking place in a moving lab are analysed using a four-vector fundamental equation, ${\rm d} E^\mu = \delta W^\mu + \delta Q^\mu$, a relativistic generalization of the first law of thermodynamics ${\rm d}E=\delta W+ \delta Q$. These processes are first described in frame S, with the lab at rest, and then in frame ${\bar {\rm S}}$, moving with constant velocity relative to S. This formalism shows that Lorentz transformation preserves the principle of relativity in thermodynamics. The physical meaning of the norm of a four-vector is analysed, and Clausius definition of entropy variation is generalised to relativity. The classical description of the process is obtained in a moving lab by taking the low-speed limit in the four-vector fundamental equation. The formalism naturally incorporates the role of the laws of mechanics when analysing processes that are typically considered as purely thermodynamic.

For a quantum spin driven cyclically by a slowly-rotated magnetic field, geometric phases are well understood. If the cycle takes a long time the leading-order (dynamical) phase is proportional to and the geometric phase is the contribution independent of The dynamical and geometric phases are the first two terms of a series in slowness Here it is shown with an exactly solvable example that the corrections are of two types: smooth, proportional to powers of slowness, and oscillatory: essential singularities in in the form of trigonometric functions of divided by powers of The calculations are elementary and therefore suitable for presentation in graduate quantum theory courses.

The analysis of cylindrical resonators is part of standard physics curricula but, unlike for their rectangular counterpart, their mode structure is hardly ever visualized. The aim of this work is to show a way of doing it, providing a set of interactive web applications and citing potential use cases in the form of both academic courses and published research. These cover several branches of physics and engineering, showing that these materials can be useful for a broad audience.

Abstract In light of a recent direct experimental confirmation of a Lorentz contraction of Coulomb field (an electric field of a point charge in a uniform motion), we revisit some common confusions related to it, to be mindful of in teaching the subject. These include the questions about a radial nature of the field, a role of the retardation effect due to a finite speed of information transfer and some issues related to a depiction of Coulomb field by means of the Lorentz contracted field lines.

Abstract In this work, we demonstrate explicitly the unified nature of electric and
magnetic fields, from the principles of special relativity and Lorentz transformations
of the electromagnetic field tensor. Using an operational approach we construct the
tensor and its corresponding transformation law, based on the principle of relativity.
Our work helps to elucidate concepts of advanced courses on electromagnetism for
primary-level learners and shows an alternative path to derive the Lenz’s law based 
on the connection between relativity arguments and a standard electromagnetism
experiment.