Phase behaviour and structure of a superionic liquid in nonpolarized nanoconfinement

The ion-ion interactions become exponentially screened for ions confined in ultranarrow metallic pores. To study the phase behaviour of an assembly of such ions, called a superionic liquid, we develop a statistical theory formulated on bipartite lattices, which allows an analytical solution within the Bethe-lattice approach. Our solution predicts the existence of ordered and disordered phases in which ions form a crystal-like structure and a homogeneous mixture, respectively. The transition between these two phases can potentially be first or second order, depending on the ion diameter, degree of confinement and pore ionophobicity. We supplement our analytical results by three-dimensional off-lattice Monte Carlo simulations of an ionic liquid in slit nanopores. The simulations predict formation of ionic clusters and ordered snake-like patterns, leading to characteristic close-standing peaks in the cation-cation and anion-anion radial distribution functions

Monte-Carlo study of anisotropic scaling generated by disorder

We analyze the critical properties of the three-dimensional Ising model with linear parallel extended defects. Such a form of disorder produces two distinct correlation lengths, a parallel correlation length $\xi_\parallel$ in the direction along defects, and a perpendicular correlation length $\xi_\perp$ in the direction perpendicular to the lines. Both $\xi_\parallel$ and $\xi_\perp$ diverge algebraically in the vicinity of the critical point, but the corresponding critical exponents $\nu_\parallel$ and $\nu_\perp$ take different values. This property is specific for anisotropic scaling and the ratio $\nu_\parallel/\nu_\perp$ defines the anisotropy exponent $\theta$. Estimates of quantitative characteristics of the critical behaviour for such systems were only obtained up to now within the renormalization group approach. We report a study of the anisotropic scaling in this system via Monte Carlo simulation of the three-dimensional system with Ising spins and non-magnetic impurities arranged into randomly distributed parallel lines. Several independent estimates for the anisotropy exponent $\theta$ of the system are obtained, as well as an estimate of the susceptibility exponent $\gamma$. Our results corroborate the renormalization group predictions obtained earlier.Comment: 22 pages, 9 figure

Cross-over between diffusion-limited and reaction-limited regimes in the coagulation-diffusion process

The change from the diffusion-limited to the reaction-limited cooperative behaviour in reaction-diffusion systems is analysed by comparing the universal long-time behaviour of the coagulation-diffusion process on a chain and on the Bethe lattice. On a chain, this model is exactly solvable through the empty-interval method. This method can be extended to the Bethe lattice, in the ben-Avraham-Glasser approximation. On the Bethe lattice, the analysis of the Laplace-transformed time-dependent particle-density is analogous to the study of the stationary state, if a stochastic reset to a configuration of uncorrelated particles is added. In this stationary state logarithmic corrections to scaling are found, as expected for systems at the upper critical dimension. Analogous results hold true for the time-integrated particle-density. The crossover scaling functions and the associated effective exponents between the chain and the Bethe lattice are derived.Comment: 21 pages, 5 figures; v3: Scaling arguments at beginning of Section 4 were correcte

Critical behavior of the 2D Ising model with long-range correlated disorder

We study critical behavior of the diluted 2D Ising model in the presence of disorder correlations which decay algebraically with distance as $\sim r^{-a}$. Mapping the problem onto 2D Dirac fermions with correlated disorder we calculate the critical properties using renormalization group up to two-loop order. We show that beside the Gaussian fixed point the flow equations have a non trivial fixed point which is stable for $0.995<a<2$ and is characterized by the correlation length exponent $\nu= 2/a + O((2-a)^3)$. Using bosonization, we also calculate the averaged square of the spin-spin correlation function and find the corresponding critical exponent $\eta_2=1/2-(2-a)/4+O((2-a)^2)$.Comment: 14 pages, 3 figures, revtex

Survival in two-species reaction-superdiffusion system: Renormalization group treatment and numerical simulations

We analyze the two-species reaction-diffusion system including trapping reaction $A + B \to A$ as well as coagulation/annihilation reactions $A + A \to (A,0)$ where particles of both species are performing L\'evy flights with control parameter $0 < \sigma < 2$, known to lead to superdiffusive behaviour. The density, as well as the correlation function for target particles $B$ in such systems, are known to scale with nontrivial universal exponents at space dimension $d \leq d_{c}$. Applying the renormalization group formalism we calculate these exponents in a case of superdiffusion below the critical dimension $d_c=\sigma$. The numerical simulations in one-dimensional case are performed as well. The quantitative estimates for the decay exponent of the density of survived particles $B$ are in good agreement with our analytical results. In particular, it is found that the surviving probability of the target particles in a superdiffusive regime is higher than that in a system with ordinary diffusion.Comment: 24 pages, 11 figure

Effective and asymptotic criticality of structurally disordered magnets

Changes in magnetic critical behaviour of quenched structurally-disordered magnets are usually exemplified in experiments and in MC simulations by diluted systems consisting of magnetic and non-magnetic components. By our study we aim to show, that similar effects can be observed not only for diluted magnets with non-magnetic impurities, but may be implemented, e.g., by presence of two (and more) chemically different magnetic components as well. To this end, we consider a model of the structurally-disordered quenched magnet where all lattice sites are occupied by Ising-like spins of different length $L$. In such random spin length Ising model the length $L$ of each spin is a random variable governed by the distribution function $p(L)$. We show that this model belongs to the universality class of the site-diluted Ising model. This proves that both models are described by the same values of asymptotic critical exponents. However, their effective critical behaviour differs. As a case study we consider a quenched mixture of two different magnets, with values of elementary magnetic moments $L_1=1$ and $L_2=s$, and of concentration $c$ and $1-c$, correspondingly. We apply field-theoretical renormalization group approach to analyze the renormalization group flow for different initial conditions, triggered by $s$ and $c$, and to calculate effective critical exponents further away from the fixed points of the renormalization group transformation. We show how the effective exponents are governed by difference in properties of the magnetic components.Comment: 17 pages, 5 figures, 1 tabl

Self-averaging in the random 2D Ising ferromagnet

We study sample-to-sample fluctuations in a critical two-dimensional Ising model with quenched random ferromagnetic couplings. Using replica calculations in the renormalization group framework we derive explicit expressions for the probability distribution function of the critical internal energy and for the specific heat fluctuations. It is shown that the disorder distribution of internal energies is Gaussian, and the typical sample-to-sample fluctuations as well as the average value scale with the system size $L$ like $\sim L \ln\ln(L)$. In contrast, the specific heat is shown to be self-averaging with a distribution function that tends to a $\delta$-peak in the thermodynamic limit $L \to \infty$. While previously a lack of self-averaging was found for the free energy, we here obtain results for quantities that are directly measurable in simulations, and implications for measurements in the actual lattice system are discussed.Comment: 12 pages, accepted versio