Applied Mathematics Research Centre

The AMRC hosts a series of seminars on subjects in statistical physics and fluid dynamics.

**Useful maths you will never use***Wednesday December 10, 2014, 19:00 h, ECG24*

There is useful maths all around us that make our modern lives possible. From rescuing your lost words in text messages to protecting your Facebook profile, we rely on numbers to transmit and protect information every day. In this this highly engaging session, Matt Parker (aka the Standup Mathematician) will open your eyes to the ubiquitous sea of numbers we all live in but don't need to use ourselves.

This is the BCS Christmas lecture 2014.

**Universal amplitudes in the canonical ensemble***Wednesday December 10, 2014, 14:00 h, DH Seminar Room*

Universal amplitudes play an important role in numerical study of critical phenomena. We study the finite-size scaling of universal amplitudes in the q-state random-cluster model under the constraint that the total number of occupied bonds is fixed, so that the canonical ensemble applies. We observe that at criticality, new finite-size corrections with exponent y_{can}=-|2y_t-d| are induced, where y_t=1/\nu is the thermal renormalization exponent and d is the spatial dimension. Moreover, we find that universal values of dimensionless parameters like Binder ratios and wrapping probabilities are modified for systems with $2y_t-d>0$. For the bond percolation model where thermal fluctuations are absent, correction exponent y_{can} still occurs, and universal amplitudes like the excess cluster number are not only modified but become non-universal. A full explanation should take into account fluctuation-suppression effects, in addition to the well-known Fisher renormalization mechanism.

**Universal free energy distribution in the critical point of a random Ising ferromagnet**

It is well known that the presence of weak quenched disorder in a ferromagnetic system can essentially modify its critical properties in the vicinity of the phase transition point such that new universal critical exponents may set in. On the other hand in recent years it is argued that due to the presence of disorder the statistical properties of some thermodynamical quantities at the critical point can become non-self-averaging. The aim of the present study is to demonstrate that due to the presence of weak disorder the

statistics of the free energy fluctuations in the critical point of the Ising ferromagnet is described by a nontrivial universal distribution function. We discuss the non-self-averaging phenomena in the critical point of weakly disordered Ising ferromagnet. In terms of the renormalized replica Ginzburg-Landau Hamiltonian in dimensions D<4 we derive an explicit expression for the probability distribution function (PDF) of the critical free energy fluctuations. In particular, using known fixed-point values for the renormalized coupling parameters we obtain the universal curve for such PDF in the dimension D=3. It is demonstrated that this function is strongly asymmetric: its left tail is much more slow than the right one.

The work was done in collaboration with Victor Dotsenko.

**Spin Dynamics: ****Plateaus, ****Bifurcations****, Chaos, ****Period-3 Window Trajectories, Superstable Points***Wednesday December 3, 2014, 14:00 h, DH Seminar Room*

Dynamical technique is applied for the third layer of 3He absorbed on the surface of graphite (kagome lattice) with 1D and multidimensional mapping. The existence of different magnetization plateaus are obtained for solid 3He. Maximal Lyapunov exponent exhibit plateaus also and it coincide magnetization one. Considering the superstable cycles for non-integer values of Q-state Potts model describing RNA-like polymers, universality class as gelation processes in branched polymers, percolation, earthquakes and antiferromagnetic multisite interaction Ising model efficient in the analysis of magnetic properties of solid 3He are considered. A particular attention is devoted to the period three window. The phase transition is observed in the supercritical point for multi-dimensional rational mapping of the spin-1/2 Ising-Heisenberg model on a diamond chain characterizing natural azurite, at T -> 0 and the absence of a magnetic field in the antiferromagnetic case.

**Modelling of photo-sensitive polymers**

Photo-induced deformations in azobenzene-containing polymers (azo-polymers) are central to a number of applications, such as optical storage and fabrication of diffractive elements. The microscopic nature of the underlying opto-mechanical coupling is yet not clear. In this study, we address the experimental finding that the scenario of the effects depends on molecular architecture of the used azo-polymer. Typically, opposite deformations in respect to the direction of light polarization are observed for liquid crystalline and amorphous azo-polymers. In this study, we undertake molecular dynamics simulations of two different models that mimic these two types of azo-polymers. We employ hybrid force field modeling and consider only trans-isomers of azobenzene, represented as Gay-Berne sites. The effect of illumination on the orientation of the chromophores is considered on the level of orientational hole burning and emphasis is given to the resulting deformation of the polymer matrix. We reproduce deformations of opposite sign for the two models being considered here and discuss the relevant microscopic mechanisms in both cases.

**A kinematic explanation for gamma-ray bursts**

Gamma-ray bursts are flashes of light observed from all directions in space, lasting from milliseconds to a few minutes, which start as gamma rays then soften progressively to X-rays and ultimately to radio waves. They have been attributed to cataclysmic events. Colin Rourke and I propose, however, that many of them may be optical illusions, simply the result of our entry into the region illuminated by a continuously emitting object. At such an entry, the emitter appears infinitely blue-shifted and infinitely bright. We demonstrate the phenomenon in de Sitter space, where much can be calculated explicitly, and then extend the idea to more general space-times.

**Parallel tempering and 3D spin-glass models**

We first present a brief review of Parallel Tempering Monte Carlo, and then we summarize the results of the study of two different versions of anisotropic three-dimensional spin glass models. In the first transverse case, the random exchange is applied only in the xy planes with the z interactions ferromagnetic. In the second longitudinal case, only z interactions are random with x and y ferromagnetic. Using parallel tempering and finite-size analysis, we determine their phase diagrams and compare them with the isotropic model. In particular, at their symmetry points, we report a striking coincidence of the transitions temperatures of the isotropic and the transverse anisotropic case, while the longitudinal anisotropic case yields a significantly higher critical temperature. For the ferromagnetic-spin glass transition line, we find forward behavior for the longitudinal case, in contrast to the reentrant behavior of the isotropic and the transverse models. We estimate the critical behavior along the transition lines, and conclude that all three models share the same universality classes, indicating the irrelevance of the introduced spatial anisotropy for the critical exponents.

**Nano-Polymeric Materials. Synthesis, Properties and Applications**

TBA

**Dynamical transition in driven DNA under oscillatory force: Hysteresis and Scaling**

We propose an unrestricted length model to show the existence of a dynamical transition in a system of driven DNA under the influence of an oscillatory force of amplitude F and frequency \omega. In the true thermodynamic limit, for the infinite length we find that the area of the hysteresis loops does not show any transition. The high-frequency scaling regime extends to lower frequencies for larger chain length L, and the system has only one scaling, where the area of the hysteresis loop scales with $\omega^{-1}F^2$. We also find that the system feels infinite length and the scaling becomes length independent, if the length $L$ is greater than the critical length L_c. For a chain of finite length, we observe that the area of hysteresis loops scales with the same exponents as proposed in a recent study [Phys. Rev. Lett. 110, 258102 (2013)]. The scaling for large but finite $L$ at temperature T = 0 and > 0 remains invariant.