Michal Hnatič

Field-theoretic technique for irreversible reaction processes

The single-species annihilation reaction A + A → O/ is studied in the presence of random advecting field. In order to determine possible infrared behaviour of the system all stable fixed points are presented to two-loop approximation in double (∈, Δ) expansion with the corresponding regions of stability. The main result of this paper is the calculation of all the renormalization constants and the decay exponent to the second-order precision as well as calculation of scaling function the mean particle number to the first order. Effects of random sources and sinks on reaction kinetics in the master-equation description have been investigated in the framework of a field-theoretic model, obtained by the “second quantization” a la Doi of the corresponding master equation. It has been demonstrated that random sources and sinks have a significant effect on the asymptotic behaviour of the model and two universality classes for their description have been identified by the scaling analysis. Results are compared with the Langevin-equation description of the same process.

Two-loop calculation of anomalous kinetics of the reaction A + A → Ø in randomly stirred fluid

The single-species annihilation reaction A + A → Ø is studied in the presence of a random velocity field generated by the stochastic Navier-Stokes equation. The renormalization group is used to analyze the combined influence of the density and velocity fluctuations on the long-time behavior of the system. The direct effect of velocity fluctuations on the reaction constant appears only from the two-loop order, therefore, all stable fixed points of the renormalization group and their regions of stability are calculated in the two-loop approximation in the two-parameter (ε, Δ) expansion. A renormalized integro-differential equation for the number density is put forward which takes into account the effect of density and velocity fluctuations at next-to-leading order. Solution of this equation in perturbation theory is calculated in a homogeneous system.

Superfluid phase transition with activated velocity fluctuations: Renormalization group approach.

A quantum field model that incorporates Bose-condensed systems near their phase transition into a superfluid phase and velocity fluctuations is proposed. The stochastic Navier-Stokes equation is used for a generation of the velocity fluctuations. As such this model generalizes model F of critical dynamics. The field-theoretic action is derived using the Martin-Siggia-Rose formalism and path integral approach. The regime of equilibrium fluctuations is analyzed within the perturbative renormalization group method. The double (ε,δ)-expansion scheme is employed, where ε is a deviation from space dimension 4 and δ describes scaling of velocity fluctuations. The renormalization procedure is performed to the leading order. The main corollary gained from the analysis of the thermal equilibrium regime suggests that one-loop calculations of the presented models are not sufficient to make a definite conclusion about the stability of fixed points. We also show that critical exponents are drastically changed as a result of the turbulent background and critical fluctuations are in fact destroyed by the developed turbulence fluctuations. The scaling exponent of effective viscosity is calculated and agrees with expected value 4/3.

Numerical Solution of a Nonlinear Integro-Differential Equation

A discretization algorithm for the numerical solution of a nonlinear integrodifferential equation modeling the temporal variation of the mean number density a (t ) in the single-species annihilation reaction A + A → 0 is discussed. The proposed solution for the two-dimensional case (where the integral entering the equation is divergent) uses regularization and then finite differences for the approximation of the differential operator together with a piecewise linear approximation of a (t ) under the integral. The presented numerical results point to basic features of the behavior of the number density function a(t) and suggest further improvement of the proposed algorithm.

On the mathematical modelling of the annihilation process

Using field-theoretic approach the reaction process A+A→∅ is studied in the vicinity of space dimension dc =2 by means of the perturbative renormalization group. Dimensional regularization with the use of minimal subtraction scheme is applied and fixed points with corresponding regions of stability are calculated to the two-loop approximation in double (e,

Magnetic-Field Induced Isotropic to Nematic Phase Transition in Ferronematics

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