This is a part of the Advanced Mechanics course MA2341. These notes are partially based on the textbook “Mechanics” by L.D. Landau and E.M. Lifshitz. ∗Email: [email protected] † Correspondent fellow at Steklov Mathematical Institute, Moscow.

In this paper, we present the mathematical analysis of a covolume method for the approximations of the Ginzburg–Landau (GL) model for superconductivity. A nice feature of this approach is that the gauge invariance properties are retained in discrete approximations based on triangular grids. We also use properties of discrete vector fields to study issues such as the gauge choices and their enfo...

A generalized Ginzburg–Landau–Devonshire (GLD) theory is used to study the properties of ferroelectric thin films sandwiched between two metal electrodes with a first-order phase transition. By taking into account the effect of the imperfect surface layer and a natural boundary condition, the temperature and film-thickness dependence of the spontaneous polarization and the dielectric susceptibi...

The various uncommon properties of magnetic nanoparticles (Stoner nanoparticles) and molecular magnets have attracted much attention in industry as well as in the academic community. In this review we begin with going through important experiments in this field and possible applications of these materials. Then we introduce the theoretic models currently used for Stoner nanoparticles and molecu...

Let Ω be a 2D domain with a hole ω. In the domain A = Ω \ ω consider a class J of complex valued maps having degrees 1 and 1 on ∂Ω, ∂ω respectively. In a joint work with P. Mironescu we show that if cap(A) ≥ π (subcritical domain), minimizers of the Ginzburg-Landau energy E κ exist for each κ. They are vortexless and converge in H(A) to a minimizing S-valued harmonic map as the coherency length...

We discuss how to use Wang-Landau simulations in an efficient manner to investigate the statistical mechanics of individual lattice polymers and peptides adsorbed at a planar surface. For nearest neighbor interactions, we show that a single Wang-Landau simulation, recording the density of states as a function of numbers of internal contacts and of surface beads, is sufficient to give a full des...

Using mirror pairs (M3,W3) in type II superstring compactifications on Calabi-Yau threefols, we study, geometrically, F-theory duals of M-theory on seven manifolds with G2 holonomy. We first develop a new way for getting Landau Ginzburg (LG) CalabiYau threefols W3, embedded in four complex dimensional toric varieties, mirror to sigma model on toric Calabi-Yau threefolds M3. This method gives di...

In order to evaluate the energy distribution (due to matter and fields including gravitation) associated with a space-time model of Szekeres class I and II metrics, we consider the Einstein, Bergmann-Thomson and LandauLifshitz energy and/or momentum definitions in the tele-parallel gravity (the tetrad theory of gravitation). We find the same energy distribution using Einstein and Bergmann-Thoms...

Adhesion of microorganisms to various interfaces has been explained by the classical Derjaguin–Landau–Verwey– Overbeek (DLVO) theory of colloid stability. The theory has been used as a qualitative model, but also in a quantitative way to calculate adhesion free energy changes involved in microbial adhesion. In this paper some important investigations will be review that show how the DLVO theory...