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J. Honkonen

Publications and source records attributed to J. Honkonen.

17 recordsLinked to original sources
arXiv2021

Fixed $d$ Renormalization Group Analysis of Conserved Surface Roughening

Conserved surface roughening represents a special case of interface dynamics where the total height of the interface is conserved. Recently, it was suggested [F. Caballero et al., Phys. Rev. Lett. 121, 020601 (2018)] that the original continuum model known as `Conserved Kardar-Parisi-Zhang'(CKPZ) equation is incomplete, as additional non-linearity is not forbidden by any symmetry in $d > 1$. In this work, we perform detailed field-theoretic renormalization group (RG) analysis of a general stochastic model describing conserved surface roughening. Systematic power counting reveals additional marginal interaction at the upper critical dimension, which appears also in the context of molecular beam epitaxy. Depending on the origin of the surface particle's mobility, the resulting model shows two different scaling regimes; If the particles move mainly due to the gravity, the leading dispersion law is $\omega \sim k^2$, and the mean-field approximation describing a flat interface is exact in any spatial dimension. On the other hand, if the particles move mainly due to the surface curvature, the interface becomes rough with the mean-field dispersion law $\omega \sim k^4$ , and the corrections to scaling exponents must be taken into account. We show, that the latter model consist of two sub-class of models that are decoupled in all orders of perturbation theory. Moreover, our RG analysis of the general model reveals that the universal scaling is described by a rougher interface than CKPZ universality class. The universal exponents are derived within the one-loop approximation in both fixed $d$ and $\varepsilon$-expansion schemes, and their relation is discussed. We point out all important details behind these two schemes which are often overlooked in the literature, and their misinterpretation might lead to inconsistent results.

cond-mat.stat-mech↗
arXiv2018

Modelling turbulence via numerical functional integration using Burgers' equation

We investigate the feasibility of modelling turbulence via numeric functional integration. By transforming the Burgers' equation into a functional integral we are able to calculate equal-time spatial correlation of system variables using standard methods of multidimensional integration. In contrast to direct numerical simulation, our method allows for simple parallelization of the problem as the value of the integral within any region can be calculated separately from others. Thus the calculations required for obtaining one correlation data set can be distributed to several supercomputers and/or the cloud simultaneously. We present the mathematical background of our method and its numerical implementation. We are interested in a steady state system with isotropic and homogeneous turbulence, for which we use a lattice version of the functional integral used in the perturbative analysis of stochastic transport equations. The numeric implementation is composed of a fast serial program for evaluating the integral over a given volume and a parallel Python wrapper that divides the problem into subvolumes and distributes the work among available processes. The code is available at https://github.com/iljah/hdintegrator for anyone to download, use, study, modify and redistribute. We present velocity cross correlation for a 10x2 lattice in space and time respectively, and analyse the computational resources required for the integration. We also discuss potential improvements to the presented method.

physics.comp-ph↗
arXiv2017

Influence of turbulent mixing on critical behavior of directed percolation process : effect of compressibility

Universal behavior is a typical emergent feature of critical systems. A paramount model of the non-equilibrium critical behavior is the directed bond percolation process that exhibits an active- to-absorbing state phase transition in the vicinity of a percolation threshold. Fluctuations of the ambient environment might affect or destroy the universality properties completely. In this work we assume that the random environment can be described by means of compressible velocity fluctu- ations. Using field-theoretic models and renormalization group methods we investigate large-scale and long-time behavior. Altogether eleven universality classes are found, out of which four are stable in the infrared limit and thus macroscopically accessible. In contrast to the model without veloc- ity fluctuations a possible candidate for a realistic three-dimensional case, a regime with relevant short-range noise, is identified. Depending on the dimensionality of space and the structure of the turbulent flow we calculate critical exponents of the directed percolation process. In the limit of the purely transversal velocity field random force critical exponents comply with the incompressible results obtained by previous authors. We have found intriguing non-universal behavior related to the mutual effect of compressibility and advection.

cond-mat.stat-mech↗
arXiv2016

Numerical Solution of a Nonlinear Integro-Differential Equation

An algorithm for the numerical solution of a nonlinear integro-differential equation arising in the single-species annihilation reaction $A + A \rightarrow\varnothing$ modeling is discussed. Finite difference method together with the linear approximation of the unknown function is considered. For divergent integrals presented in the equation for dimension $d=2$ a regularization is used. Some numerical results are presented.

math.NA↗
arXiv2011

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 $ε$ and $Δ$, where $ε$ is the deviation from the Kolmogorov scaling parameter and $Δ$ 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.

nlin.CD↗
arXiv2011

Field Theory Approach In Kinetic Reaction: Role Of Random Sources And Sinks

In the framework of a field theoretic model obtained by second quantization of Doi-type master equation, we investigate the effects of random sources and sinks on the reaction kinetics in the master-equation description. We show that random sources and sinks significantly affect the asymptotic behavior of the model and identify two universality classes when describing them using scaling analysis. We compare the results with the Langevin-equation description of the same process.

cond-mat.stat-mech↗
arXiv2006

Effects of mixing and stirring on the critical behavior

Stochastic dynamics of a nonconserved scalar order parameter near its critical point, subject to random stirring and mixing, is studied using the field theoretic renormalization group. The stirring and mixing are modelled by a random external Gaussian noise with the correlation function $\proptoδ(t-t') k^{4-d-y}$ and the divergence-free (due to incompressibility) velocity field, governed by the stochastic Navier--Stokes equation with a random Gaussian force with the correlation function $\proptoδ(t-t') k^{4-d-y'}$. Depending on the relations between the exponents $y$ and $y'$ and the space dimensionality $d$, the model reveals several types of scaling regimes. Some of them are well known (model A of equilibrium critical dynamics and linear passive scalar field advected by a random turbulent flow), but there are three new nonequilibrium regimes (universality classes) associated with new nontrivial fixed points of the renormalization group equations. The corresponding critical dimensions are calculated in the two-loop approximation (second order of the triple expansion in $y$, $y'$ and $ε=4-d$).

cond-mat.stat-mech↗
arXiv2004

Two-loop calculation of the turbulent Prandtl number

The turbulent Prandtl number has been calculated in the two-loop approximation of the $\eps$ expansion of the stochastic theory of turbulence. The strikingly small value obtained for the two-loop correction explains the good agreement of the earlier one-loop result with the experiment. This situation is drastically different from other available nontrivial two-loop results, which exhibit corrections of the magnitude of the one-loop term. The reason is traced to the mutual cancellation of additional divergences appearing in two dimensions which have had a major effect on the results of previous calculations of other quantities.

physics.flu-dyn↗
arXiv2004

Anomalous scaling of passively advected magnetic field in the presence of strong anisotropy

Inertial-range scaling behavior of high-order (up to order N=51) structure functions of a passively advected vector field has been analyzed in the framework of the rapid-change model with strong small-scale anisotropy with the aid of the renormalization group and the operator-product expansion. It has been shown that in inertial range the leading terms of the structure functions are coordinate independent, but powerlike corrections appear with the same anomalous scaling exponents as for the passively advected scalar field. These exponents depend on anisotropy parameters in such a way that a specific hierarchy related to the degree of anisotropy is observed. Deviations from power-law behavior like oscillations or logarithmic behavior in the corrections to structure functions have not been found.

nlin.CD↗
arXiv2004

Anomalous scaling of a passive scalar advected by the Navier--Stokes velocity field: Two-loop approximation

The field theoretic renormalization group and operator product expansion are applied to the model of a passive scalar quantity advected by a non-Gaussian velocity field with finite correlation time. The velocity is governed by the Navier--Stokes equation, subject to an external random stirring force with the correlation function $\propto δ(t-t') k^{4-d-2ε}$. It is shown that the scalar field is intermittent already for small $ε$, its structure functions display anomalous scaling behavior, and the corresponding exponents can be systematically calculated as series in $ε$. The practical calculation is accomplished to order $ε^{2}$ (two-loop approximation), including anisotropic sectors. Like for the well-known Kraichnan's rapid-change model, the anomalous scaling results from the existence in the model of composite fields (operators) with negative scaling dimensions, identified with the anomalous exponents. Thus the mechanism of the origin of anomalous scaling appears similar for the Gaussian model with zero correlation time and non-Gaussian model with finite correlation time. It should be emphasized that, in contrast to Gaussian velocity ensembles with finite correlation time, the model and the perturbation theory discussed here are manifestly Galilean covariant. The relevance of these results for the real passive advection, comparison with the Gaussian models and experiments are briefly discussed.

nlin.CD↗
arXiv2004

An improved $\eps$ expansion for three-dimensional turbulence: two-loop renormalization near two dimensions

An improved $\eps$ expansion in the $d$-dimensional ($d > 2$) stochastic theory of turbulence is constructed at two-loop order which incorporates the effect of pole singularities at $d \to 2$ in coefficients of the $\eps$ expansion of universal quantities. For a proper account of the effect of these singularities two different approaches to the renormalization of the powerlike correlation function of the random force are analyzed near two dimensions. By direct calculation it is shown that the approach based on the mere renormalization of the nonlocal correlation function leads to contradictions at two-loop order. On the other hand, a two-loop calculation in the renormalization scheme with the addition to the force correlation function of a local term to be renormalized instead of the nonlocal one yields consistent results in accordance with the UV renormalization theory. The latter renormalization prescription is used for the two-loop renormalization-group analysis amended with partial resummation of the pole singularities near two dimensions leading to a significant improvement of the agreement with experimental results for the Kolmogorov constant.

nlin.CD↗
arXiv2002

Large order asymptotics and convergent perturbation theory for critical indices of the $ϕ^4$ model in ${4-ε}$ expansion

Large order asymptotic behaviour of renormalization constants in the minimal subtraction scheme for the $ϕ^4$ $(4-ε)$ theory is discussed. Well-known results of the asymptotic $4-ε$ expansion of critical indices are shown to be far from the large order asymptotic value. A {\em convergent} series for the model $ϕ^4$ $(4-ε)$ is then considered. Radius of convergence of the series for Green functions and for renormalisation group functions is studied. The results of the convergent expansion of critical indices in the $4-ε$ scheme are revalued using the knowledge of large order asymptotics. Specific features of this procedure are discussed.

hep-th↗
arXiv2002

Anomalous scaling of a passive scalar advected by the turbulent velocity field with finite correlation time: Two-loop approximation

The renormalization group and operator product expansion are applied to the model of a passive scalar quantity advected by the Gaussian self-similar velocity field with finite, and not small, correlation time. The inertial-range energy spectrum of the velocity is chosen in the form $E(k)\propto k^{1-2\eps}$, and the correlation time at the wavenumber $k$ scales as $k^{-2+η}$. Inertial-range anomalous scaling for the structure functions and other correlation functions emerges as a consequence of the existence in the model of composite operators with negative scaling dimensions, identified with anomalous exponents. For $η>\eps$, these exponents are the same as in the rapid-change limit of the model; for $η<\eps$, they are the same as in the limit of a time-independent (quenched) velocity field. For $\eps=η$ (local turnover exponent), the anomalous exponents are nonuniversal through the dependence on a dimensionless parameter, the ratio of the velocity correlation time and the scalar turnover time. The universality reveals itself, however, only in the second order of the $\eps$ expansion, and the exponents are derived to order $O(\eps^{2})$, including anisotropic contributions. It is shown that, for moderate $n$, the order of the structure function, and $d$, the space dimensionality, finite correlation time enhances the intermittency in comparison with the both limits: the rapid-change and quenched ones. The situation changes when $n$ and/or $d$ become large enough: the correction to the rapid-change limit due to the finite correlation time is positive (that is, the anomalous scaling is suppressed), it is maximal for the quenched limit and monotonically decreases as the correlation time tends to zero.

nlin.CD↗
arXiv2002

Two-loop calculation of the scaling behavior of two-dimensional forced Navier-Stokes equation

Asymptotic properties of the solution of two-dimensional randomly forced Navier-Stokes equation with long-range correlations of the driving force are analyzed in the two-loop order of perturbation theory with the use of renormalization group. Kolmogorov constant of the energy spectrum is calculated for both the inverse energy cascade and the direct enstrophy cascade in the second order of the $ε$ expansion.

nlin.CD↗
arXiv2001

Stochastic magnetohydrodynamic turbulence in space dimensions $d\ge 2$

Interplay of kinematic and magnetic forcing in a model of a conducting fluid with randomly driven magnetohydrodynamic equations has been studied in space dimensions $d\ge 2$ 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 $d\ge 2$ 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 $ε$ expansion and the double expansion.

nlin.CD↗
arXiv2000

Manifestation of anisotropy persistence in the hierarchies of MHD scaling exponents

The first example of a turbulent system where the failure of the hypothesis of small-scale isotropy restoration is detectable both in the `flattening' of the inertial-range scaling exponent hierarchy, and in the behavior of odd-order dimensionless ratios, e.g., skewness and hyperskewness, is presented. Specifically, within the kinematic approximation in magnetohydrodynamical turbulence, we show that for compressible flows, the isotropic contribution to the scaling of magnetic correlation functions and the first anisotropic ones may become practically indistinguishable. Moreover, skewness factor now diverges as the Péclet number goes to infinity, a further indication of small-scale anisotropy.

nlin.CD↗