arXiv:1609.06078 · Acta Physica Polonica A

Renormalization Group Calculation of Dynamic Exponent in the Models E and F with Hydrodynamic Fluctuations

Michal Dančo, Michal Hnatič, Marina Vladimirovna Komarova, T. Lučivjanský, Mikhail Yu. Nalimov

Published Apr 1, 2017• 2 citations

Abstract

The renormalization group method is applied in order to analyze models E and F of critical dynamics in the presence of velocity fluctuations generated by the stochastic Navier-Stokes equation.Results are given to the one-loop approximation for the anomalous dimension γ λ and fixed-points' structure.The dynamic exponent z is calculated in the turbulent regime and stability of the fixed points for the standard model E is discussed.

DOI: 10.12693/aphyspola.131.651

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arXiv:1609.06078 · Acta Physica Polonica A

Renormalization Group Calculation of Dynamic Exponent in the Models E and F with Hydrodynamic Fluctuations

Michal Dančo, Michal Hnatič, Marina Vladimirovna Komarova, T. Lučivjanský, Mikhail Yu. Nalimov

Published Apr 1, 2017• 2 citations

arXiv HTML PDF

Abstract

DOI: 10.12693/aphyspola.131.651

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\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

^{4}\mathrm{He}

at various pressures. Predictions are made for the bulk thermal conductivity very close to ${\mathit{T}}_{\ensuremath{\lambda}}$ where the departures from dynamic scaling should be observable. Effective static and dynamic parameters are computed that can be applied to other critical phenomena above and below the \ensuremath{\lambda} line of

^{4}\mathrm{He}

q

is the wave number and $\ensuremath{\xi}$ is the correlation length) and shows a narrow lydrodynamical width as $x\ensuremath{\rightarrow}0$.

d\lesssim 4

, characterized by a Wilson-Fisher fixed point with dynamical critical exponent

z=2+ O(ε^2)

, the critical dynamics is dominated by a novel multiplicative noise fixed point. The zeroes of the beta function depend on the stochastic prescription used to define the Wiener integrals. However, the critical exponents and the anomalous dimension do not depend on the prescription used. Thus, even though each stochastic prescription produces different dynamical evolutions, all of them are in the same universality class.