The most fundamental characteristic of a physical system can often be deduced from its behaviour under discrete symmetry transformations, such as time reversal, parity and chirality. Here, we review some of the basic symmetry properties of the relativistic quantum theories for free electrons in ($2+1$)- and ($1+1$)-dimensional spacetime. Additional flavour degrees of freedom are necessary to properly define symmetry operations in ($2+1$) dimensions, and are generally present in physical realizations of such systems, for example in single sheets of graphite. We find that there exist two possibilities for defining any flavour-coupling discrete symmetry operation of the two-flavour ($2+1$)-dimensional Dirac theory. Some physical implications of this previously unnoticed duplicity are discussed.

Safety issues for the use of products containing nanoparticles need to be considered, since these nanoparticles may break through human skin to damage cells. In this paper, applied mathematical techniques are used to model the penetration of a spherical gold nanoparticle into an assumed circular hole in a lipid bilayer. The 6–12 Lennard-Jones potential is employed, and the total molecular interaction energy is obtained using the continuous approximation. Nanoparticles of three different radii, namely, 10, 15 and 20 Å, are studied, which are initiated at rest, confined to the axis of the hole. A similar behaviour for these three cases is observed. The critical hole radii at which these nanoparticles enter the bilayer are 12.65, 17.62 and 22.60 Å, respectively. Further, once the hole radii become larger than 20.79, 23.14 and 27.02 Å, respectively, the gold nanoparticles tend to remain at the mid-plane of the bilayer, and do not pass through the bilayer.

We investigate the mechanics of a nano logic gate, comprising a metallofullerene which is located inside a square-shaped single-walled carbon nanotorus involving non-metallic, single-walled carbon nanotubes with perfect nanotoroidal corners. These are highly novel and speculative nanodevices whose construction, no doubt, involves many technical challenges. The energy for the system is obtained from the 6–12 Lennard-Jones potential with the continuous approximation. Our approach shows that there is not much difference between the energy when the metallofullerene is located in the tubes compared to when it is at the corners, and therefore the metallofullerene may be controlled by a small voltage. By applying two voltage inputs to produce external electric fields, one for the left–right motion and the other for the top–bottom motion, the metallofullerene can be moved to one of the four corners. Assuming that at the four corners there are charge detectors, the proposed device can be designed as a logic gate with different signals corresponding to particular gates.

Nanospace governs the dynamics of physical, chemical, material and biological systems, and the facility to model it with analytical formulae provides an essential tool to address some of the worlds’ key problems such as gas purification, separation and storage. This paper aims to provide some analytical models to exploit building blocks representing various geometric shapes that describe nanostructures. In order to formulate the various building blocks, we use the continuous approximation which assumes a uniform distribution of atoms on their surfaces. We then calculate the potential energy of the van der Waals interaction between an atom and the structure to evaluate the location of the atom where the potential energy is at its minimum. We provide applications of the analytical models for some real structures where more than one type of building block is required.

Recently, organic nanostructures have attracted much attention, and amongst them peptide nanotubes are of interest in many fields of application including medicine and nanobiotechnology. Peptide nanotubes are formed by self-assembly of cyclic peptides with alternating L- and D-amino acids. Due to their biodegradability, flexible design and easy synthesis, many applications have been proposed such as artificial transmembrane ion channels, templates for nanoparticles, mimicking pore structures, nanoscale testing tubes, biosensors and carriers for targeted drug delivery. The mechanisms of an ion, a water molecule and an ion–water cluster entering into a peptide nanotube of structure cyclo[(-D-Ala-L-Ala-)$_{4}$] are explored here. In particular, the Lennard-Jones potential and a continuum approach are employed to determine three entering mechanisms: (i) through the tube open end, (ii) through a region between each cyclic peptide ring and (iii) around the edge of the tube open end. The results show that while entering the nanotube by method (i) is possible, an ion or a molecule requires initial energy to overcome an energetic barrier to be able to enter the nanotube through positions (ii) and (iii). Due to its simple structure, the D-, L-Ala cyclopeptide nanotube is chosen in this model; however, it can be easily extended to include more complicated nanotubes.

More and more experimental evidence demonstrates that the slip boundary condition plays an important role in the study of nano- or micro-scale fluid. We propose a homogenization approach to study the effective slippage problem. We show that the effective slip length obtained by homogenization agrees with the results obtained by the traditional method in the literature for the simplest Stokes flow; then we use our approach to deal with two examples which seem quite hard by other analytical methods. We also include some numerical results to validate our analytical results.