75 research outputs found
Stochastic magnetohydrodynamic turbulence in space dimensions
Interplay of kinematic and magnetic forcing in a model of a conducting fluid
with randomly driven magnetohydrodynamic equations has been studied in space
dimensions by means of the renormalization group. A perturbative
expansion scheme, parameters of which are the deviation of the spatial
dimension from two and the deviation of the exponent of the powerlike
correlation function of random forcing from its critical value, has been used
in one-loop approximation. Additional divergences have been taken into account
which arise at two dimensions and have been inconsistently treated in earlier
investigations of the model. It is shown that in spite of the additional
divergences the kinetic fixed point associated with the Kolmogorov scaling
regime remains stable for all space dimensions for rapidly enough
falling off correlations of the magnetic forcing. A scaling regime driven by
thermal fluctuations of the velocity field has been identified and analyzed.
The absence of a scaling regime near two dimensions driven by the fluctuations
of the magnetic field has been confirmed. A new renormalization scheme has been
put forward and numerically investigated to interpolate between the
expansion and the double expansion.Comment: 12 pages, 4 figure
Functional Methods in Stochastic Systems
Field-theoretic construction of functional representations of solutions of
stochastic differential equations and master equations is reviewed. A generic
expression for the generating function of Green functions of stochastic systems
is put forward. Relation of ambiguities in stochastic differential equations
and in the functional representations is discussed. Ordinary differential
equations for expectation values and correlation functions are inferred with
the aid of a variational approach.Comment: Plenary talk presented at Mathematical Modeling and Computational
Science. International Conference, MMCP 2011, Star\'a Lesn\'a, Slovakia, July
4-8, 201
Study of Anomalous Kinetics of The Annihilation Reaction A+A->0
Using the perturbative renormalization group, we study the influence of a
random velocity field on the kinetics of the single-species annihilation
reaction A+A->0 at and below its critical dimension d_c=2. We use the
second-quantization formalism of Doi to bring the stochastic problem to a
field-theoretic form. We investigate the reaction in the vicinity of the space
dimension d=2 using a two-parameter expansion in and , where
is the deviation from the Kolmogorov scaling parameter and
is the deviation from the space dimension d=2. We evaluate all the necessary
quantities, including fixed points with their regions of stability, up to the
second order of the perturbation theory.Comment: Presented in the Third International Conference "Models in QFT: In
Memory of A.N. Vasiliev" St. Petersburg - Petrodvorez, October 201
Operator Approach to the Master Equation for the One-Step Process
Presentation of the probability as an intrinsic property of the nature leads
researchers to switch from deterministic to stochastic description of the
phenomena. The procedure of stochastization of one-step process was formulated.
It allows to write down the master equation based on the type of of the kinetic
equations and assumptions about the nature of the process. The kinetics of the
interaction has recently attracted attention because it often occurs in the
physical, chemical, technical, biological, environmental, economic, and
sociological systems. However, there are no general methods for the direct
study of this equation. Leaving in the expansion terms up to the second order
we can get the Fokker-Planck equation, and thus the Langevin equation. It
should be clearly understood that these equations are approximate recording of
the master equation. However, this does not eliminate the need for the study of
the master equation. Moreover, the power series produced during the master
equation decomposition may be divergent (for example, in spatial models). This
makes it impossible to apply the classical perturbation theory. It is proposed
to use quantum field perturbation theory for the statistical systems (the
so-called Doi method). This work is a methodological material that describes
the principles of master equation solution based on quantum field perturbation
theory methods. The characteristic property of the work is that it is
intelligible for non-specialists in quantum field theory. As an example the
Verhulst model is used because of its simplicity and clarity (the first order
equation is independent of the spatial variables, however, contains
non-linearity). We show the full equivalence of the operator and combinatorial
methods of obtaining and study of the one-step process master equation.Comment: in Russian; in Englis
Advection of a passive scalar field by turbulent compressible fluid: renormalization group analysis near
The field theoretic renormalization group (RG) and the operator product
expansion (OPE) are applied to the model of a density field advected by a
random turbulent velocity field. The latter is governed by the stochastic
Navier-Stokes equation for a compressible fluid. The model is considered near
the special space dimension . It is shown that various correlation
functions of the scalar field exhibit anomalous scaling behaviour in the
inertial-convective range. The scaling properties in the RG+OPE approach are
related to fixed points of the renormalization group equations. In comparison
with physically interesting case , at additional Green function
has divergences which affect the existence and stability of fixed points. From
calculations it follows that a new regime arises there and then by continuity
moves into . The corresponding anomalous exponents are identified with
scaling dimensions of certain composite fields and can be systematically
calculated as series in (the exponent, connected with random force) and
. All calculations are performed in the leading one-loop
approximation.Comment: 11pages, 6 figures, LATEX2e. arXiv admin note: substantial text
overlap with arXiv:1611.00327; text overlap with arXiv:1611.0130
Anomalous scaling of a passive scalar advected by the turbulent velocity field with finite correlation time and uniaxial small-scale anisotropy
The influence of uniaxial small-scale anisotropy on the stability of the
scaling regimes and on the anomalous scaling of the structure functions of a
passive scalar advected by a Gaussian solenoidal velocity field with finite
correlation time is investigated by the field theoretic renormalization group
and operator product expansion within one-loop approximation. Possible scaling
regimes are found and classified in the plane of exponents ,
where characterizes the energy spectrum of the velocity field in the
inertial range , and is related to the
correlation time of the velocity field at the wave number which is scaled
as . It is shown that the presence of anisotropy does not disturb
the stability of the infrared fixed points of the renormalization group
equations which are directly related to the corresponding scaling regimes. The
influence of anisotropy on the anomalous scaling of the structure functions of
the passive scalar field is studied as a function of the fixed point value of
the parameter which represents the ratio of turnover time of scalar field
and velocity correlation time. It is shown that the corresponding one-loop
anomalous dimensions, which are the same (universal) for all particular models
with concrete value of in the isotropic case, are different (nonuniversal)
in the case with the presence of small-scale anisotropy and they are continuous
functions of the anisotropy parameters, as well as the parameter . The
dependence of the anomalous dimensions on the anisotropy parameters of two
special limits of the general model, namely, the rapid-change model and the
frozen velocity field model, are found when and ,
respectively.Comment: revtex, 25 pages, 37 figure
- ā¦