It is commonly accepted that reaction-diffusion equations cannot be obtained by a Lagrangian field theory. Guided by the well known connection between quantum and diffusion equations, we implement here a Lagrangian approach valid for totally general nonlinear reacting-diffusing systems which allows the definition of global conserved observables derived using Nöther’s theorem.

The small-time asymptotic properties of the reaction front formed by a reaction $A+B\to C$ coupled to diffusion are considered. Reactants $A$ and $B$ are initially separately dissolved in two identical solvents. The solvents are brought into contact and the reactants meet through diffusion. The small-time asymptotic position of the center of mass of the reaction rate is obtained analytically. When one of the reactants diffuses much faster than the other reactant then the position of the local maximum in the reaction rate travels on a length scale related to the diffusion coefficient of the slowest diffusing reactant while the first moment of the reaction rate and the width of the reaction front are on a length scale related to the diffusion coefficient of the fastest diffusing reactant. If the sum of the initial reactant concentrations is fixed, then the fastest reaction rate is obtained when equal concentrations are used. The first-order solutions are analytically obtained, however, each solution involves an integral which requires numerical evaluation. Various small-time asymptotic analytical reaction front properties are obtained. In particular, one finds that the position of the center of mass of the product concentration distribution is initially located at three quarters of the position of the center of mass of the reaction rate.

We analyze the dynamics of a periodically forced oscillator ensemble with global nonlinear coupling. Without forcing, the system exhibits complicated collective dynamics, even for the simplest case of identical phase oscillators: due to nonlinearity, the synchronous state becomes unstable for certain values of the coupling parameter, and the system settles at the border between synchrony and asynchrony, what can be denoted as partial synchrony. We find that an external common forcing can result in two synchronous states: (i) a weak forcing entrains only the mean field, whereas the individual oscillators remain unlocked to the force and, correspondingly, to the mean field; (ii) a strong forcing fully synchronizes the system, making the phases of all oscillators identical. Analytical results are confirmed by numerics.

Time delay in the kinetic terms of reaction-diffusion systems has been investigated. It has been shown that short delay beyond a critical threshold may induce spatiotemporal instabilities. For unequal diffusivities and appropriate parameter space delay may induce Turing instability resulting in stationary patterns and also interesting Turing-Hopf transition with the formation of spirals. The theoretical scheme has been numerically explored in two different prototypical reaction-diffusion systems.

The stability problem of hydromagnetic Taylor-Couette flows with toroidal magnetic fields is considered for various magnetic Prandtl numbers. Only the most uniform (but not current-free) field has been treated. For high enough Hartmann numbers, the toroidal field is always unstable due to the magnetic kink-type instability, which is stabilized by rigid basic rotation. The axial electric current, which drives the instability, is reduced by the electromotive force induced by the instability itself. Numerical simulations show that this electromotive force only depends on the molecular magnetic diffusivity rather than the viscosity. The resulting eddy diffusivity should be on the order of the molecular diffusivity for all the considered magnetic Prandtl numbers. If this is true also for very small magnetic Prandtl numbers (not possible to simulate) then one can use this effect to measure the eddy diffusivity ${\eta }_{\text{T}}$ in a laboratory. In a sodium experiment (without rotation), a detectable potential difference of $\sim 16\text{mV}$ between top and bottom will result for a container of 1 m length and a gap width of 10 cm.

The body-centered-cubic Coulomb crystal of ions in the presence of a uniform magnetic field is studied using the rigid electron background approximation. The phonon mode spectra are calculated for a wide range of magnetic-field strengths and for several orientations of the field in the crystal. The phonon spectra are used to calculate the phonon contribution to the crystal energy, entropy, specific heat, Debye-Waller factor of ions, and the rms ion displacements from the lattice nodes for a broad range of densities, temperatures, chemical compositions, and magnetic fields. Strong magnetic field dramatically alters the properties of quantum crystals. The phonon specific heat increases by many orders of magnitude. The ion displacements from their equilibrium positions become strongly anisotropic. The results can be relevant for dusty plasmas, ion plasmas in Penning traps, and especially for the crust of magnetars (neutron stars with superstrong magnetic fields $B\gtrsim {10}^{14}\text{G}$). The effect of the magnetic field on ion displacements in a strongly magnetized neutron star crust can suppress the nuclear reaction rates and make them extremely sensitive to the magnetic-field direction.

For a specific structure of trypsin inhibitor, starting from a $5\text{Å}$ phase set, about 80% of the $2\text{Å}$ phases are correctly determined within an error of $18°$ by applying a multiresolution refinement procedure. The refinement proceeds both in real and reciprocal spaces. In extending the structure from 5 to $2\text{Å}$, the amplitudes of the reflections are the only requirements for this procedure. In contrast to the conventional phase extension scheme, the amplitudes are used not only in the Fourier synthesis but also in the real-space density modifications.

The dependence of film surface roughness and porosity on lattice size in a porous thin film deposition process is studied via kinetic Monte Carlo simulations on a triangular lattice. For sufficiently large lattice size the steady-state value of the expected film porosity has a weak dependence on the lattice size and the steady-state value of the expected surface roughness square varies linearly with lattice size. An analysis of the film morphology based on a stochastic partial differential equation description of the film surface morphology supports and explains the findings of the numerical simulations.

We present simulation results for the contact process on regular cubic networks that are composed of a one-dimensional lattice and a set of long edges with unbounded length. Networks with different sets of long edges are considered that are characterized by different shortest-path dimensions and random-walk dimensions. We provide numerical evidence that an absorbing phase transition occurs at some finite value of the infection rate and the corresponding dynamical critical exponents depend on the underlying network. Furthermore, the time-dependent quantities exhibit log-periodic oscillations in agreement with the discrete scale invariance of the networks. In case of spreading from an initial active seed, the critical exponents are found to depend on the location of the initial seed and break the hyperscaling law of the directed percolation universality class due to the inhomogeneity of the networks. However, if the cluster-spreading quantities are averaged over initial sites, the hyperscaling law is restored.

We follow here the freezing of the orientational degrees of freedom of strongly interacting magnetic and charged nanoparticles, as the colloidal glass transition is approached. Using a magnetoinduced birefringence technique, we show that the rotational dynamics drastically slows down following a Vogel-Fulcher law. More precisely, this slowing down occurs above a volume fraction threshold ${\varphi }^{\ast }$, the value of which depends on the range of electrostatic repulsion between nanoparticles. An interpretation in terms of effective spheres, slightly anisotropic, is proposed. The aging of the rotational dynamics of the more concentrated samples is reported on long time scales, with an exponential growth of the rotational characteristic time with the age $t_w$ of the sample. An attempt of age rescaling at different volume fractions leads us to introduce a $\varphi$-dependent “birth age” $t_w^0\left(\varphi \right)$, which diverges analytically at the Vogel-Fulcher volume fraction.

We revisit the classic problem of the structure of the isotropic-nematic interface within Ginzburg-Landau-de Gennes theory, refining previous analytic treatments of biaxiality at the interface. We compare our analysis with numerical results obtained through a highly accurate spectral collocation scheme for the solution of the Landau-Ginzburg-de Gennes equations. In comparison to earlier work, we obtain improved agreement with numerics for both the uniaxial and biaxial profiles, accurate asymptotic results for the decay of biaxial order on both nematic and isotropic sides of the interface, and accurate fits to data from density-functional approaches to this problem.

The conformational free energy of ring polymers in a blend of ring and linear polymers is investigated using the bond-fluctuation model. Previously established scaling relationships for the free energy of a ring polymer are shown to be valid only in the mean-field sense, and alternative functional forms are investigated. It is shown that it may be difficult to accurately express the total free energy of a ring polymer by a simple scaling argument, or in closed form.

We consider a driven quantum harmonic oscillator strongly coupled to a heat bath. Starting from the exact quantum Langevin equation, we use a Green’s function approach to determine the corresponding semiclassical equation for the Wigner phase space distribution. In the limit of high friction and high temperature, we apply Brinkman’s method to derive the quantum Smoluchowski equation for the probability distribution in position space. We further determine the range of validity of the equation and discuss the special case of a Brownian parametric oscillator.

A bent-shaped molecule exhibiting the mesophase sequence of B6-B1 was studied to understand the structural characteristics of the B6 phase compared with those of the B1 phase. A well-oriented sample was prepared in a magnetic field and examined by wide-angle x-ray diffraction measurements through the B6-B1 phase transition. The B6 phase has been considered to show only the (002) reflection, but this B6 phase showed broad scattering just inside the (002) reflection. The broad scattering has an intensity maximum at a very similar position to that of the (101) reflection in B1 and evolves into the well-defined (101) reflection on cooling into B1. Thus, B6 has a similar frustrated structure to B1, but the size of the antidomain in B6 may be dynamically distributed while B1 possesses an antidomain of definite size.

We study critical properties of the continuous Abelian sandpile model with anisotropies in toppling rules that produce ordered patterns on it. Also, we consider the continuous directed sandpile model perturbed by a weak quenched randomness, study critical behavior of the model using perturbative conformal field theory, and show that the model has a random fixed point.

Many real-life networks exhibit a spatial dependence; i.e., the probability to form an edge between two nodes in the network depends on the distance between them. We investigate the influence of spatial dependence on the spectral density of the network. When increasing spatial dependence in Erdös-Rényi, scale-free, and small-world networks, it is found that the spectrum changes. Due to the spatial dependence the degree of clustering and the number of triangles increase. This results in a higher asymmetry (skewness). Our results show that the spectrum can be used to detect and quantify clustering and spatial dependence in a network.

We apply a recently developed renormalization-group (RG) method to study synchronization in a one-dimensional chain of phase-coupled oscillators in the regime of weak randomness. The RG predicts how oscillators with randomly distributed frequencies and couplings form frequency-synchronized clusters. Although the RG was originally intended for strong randomness, i.e., for distributions with long tails, we find good agreement with numerical simulations even in the regime of weak randomness. We use the RG flow to derive how the correlation length scales with the width of the coupling distribution in the limit of large coupling. This leads to the identification of a universality class of distributions with the same critical exponent $\nu$. We also find universal scaling for small coupling. Finally, we show that the RG flow is characterized by a universal approach to the unsynchronized fixed point, which provides physical insight into low-frequency clusters.

We present a numerical solution of the parquet approximation, a conserving diagrammatic approach which is self-consistent at both the single-particle and the two-particle levels. The fully irreducible vertex is approximated by the bare interaction thus producing the simplest approximation that one can perform with the set of equations involved in the formalism. The method is applied to the Hubbard model on a half-filled $4\times 4$ cluster. Results are compared to those obtained from determinant quantum Monte Carlo (DQMC), FLuctuation EXchange (FLEX), and self-consistent second-order approximation methods. This comparison shows a satisfactory agreement with DQMC and a significant improvement over the FLEX or the self-consistent second-order approximation.

The influence of the heating mechanism on the kinetic energy densities of the components of a vibrated granular mixture is investigated. Collisions of the particles with the vibrating wall are inelastic and characterized by two coefficients of normal restitution, one for each of the two species. By means of molecular-dynamics simulations, it is shown that the nonequipartition of kinetic energy is not affected by the differential mechanism of energy injection aside the usual boundary layer around the wall. The macroscopic state of the mixture in the bulk is defined by intensive variables that do not include the partial granular temperatures of the components.

We present a study of the continuous-time equations governing the dynamics of a susceptible-infected-susceptible model on heterogeneous metapopulations. These equations have been recently proposed as an alternative formulation for the spread of infectious diseases in metapopulations in a continuous-time framework. Individual-based Monte Carlo simulations of epidemic spread in uncorrelated networks are also performed revealing a good agreement with analytical predictions under the assumption of simultaneous transmission or recovery and migration processes.