N G van Kampen 1981 Amsterdam: North-Holland xiv + 419 pp price Dfl 180 This is a book which, at a lower price, could be expected to become an essential part of the library of every physical scientist concerned with problems involving fluctuations and stochastic processes, as well as those who just enjoy a beautifully written book. It provides an extensive graduate-level introduction which is clear, cautious, interesting and readable.

https://iopscience.iop.org/article/10.1088/0031-9112/34/4/040/pdf

Stochastic Processes in Physics and Chemistry

We study the effects of turbulent mixing on the critical behaviour of a nonequilibrium system near its second-order phase transition between the absorbing and fluctuating states. The model describes the spreading of an agent (e.g., infectious disease) in a reaction-diffusion system and belongs to the universality class of the directed bond percolation process, also known as the simple epidemic process, and is equivalent to the Reggeon field theory. The turbulent advecting velocity field is modelled by the Obukhov?Kraichnan's rapid-change ensemble: Gaussian statistics with the correlation function vv ? ?(t ? t')k?d??, where k is the wave number, and 0 < ? < 2 is a free parameter. Using the field theoretic renormalization group we show that, depending on the relation between the exponent ? and the spatial dimension d, the system reveals different types of large-scale, long-time asymptotic behaviour, associated with four possible fixed points of the renormalization group equations. In addition to known regimes (ordinary diffusion, ordinary directed percolation process and passively advected scalar field), the existence of a new nonequilibrium universality class is established, and the corresponding critical dimensions are calculated to the first order of the double expansion in ? and ? = 4 ? d (one-loop approximation). It turns out, however, that the most realistic values ? = 4/3 (Kolmogorov's fully developed turbulence) and d = 2 or 3 correspond to the case of a passive scalar field, when the nonlinearity of the Reggeon model is irrelevant, and the spreading of the agent is completely determined by the turbulent transfer.

Effects of turbulent mixing on the nonequilibrium critical behaviour

Critical behaviour of a fluid (binary mixture or liquid crystal), subjected to strongly anisotropic turbulent mixing, is studied by means of the field theoretic renormalization group. As a simplified model, relaxational stochastic dynamics of a non-conserved scalar order parameter, coupled to a random velocity field with prescribed statistics, is considered. The velocity is taken Gaussian, white in time, with a correlation function of the form ∝δ(t − t')/|k⊥|d+ξ, where k⊥ is the component of the wave vector, perpendicular to the distinguished direction ('direction of the flow')—the d-dimensional generalization of the ensemble introduced by Avellaneda and Majda (1990 Commun. Math. Phys. 131 381) within the context of passive scalar advection. It is shown that, depending on the relation between the exponent ξ and the space dimensionality d, the system exhibits various types of large-scale self-similar behaviour, associated with different infrared attractive fixed points of the renormalization group equations. In addition to well-known asymptotic regimes (model A of equilibrium critical dynamics and a passively advected scalar with no self-interaction), the existence of a new, non-equilibrium and strongly anisotropic type of critical behaviour (universality class) is established, and the corresponding critical dimensions are calculated to the second order of the double expansion in ξ and e = 4 − d (two-loop approximation). The most realistic values of the model parameters (for example, d = 3 and the Kolmogorov exponent ξ = 4/3) belong to this class. The scaling behaviour appears anisotropic in the sense that the critical dimensions related to the directions parallel and perpendicular to the flow are essentially different. The results are in qualitative agreement with the results, obtained in experiments and simulations of fluid systems subjected to various kinds of regular and chaotic anisotropic flows.

Critical behaviour of a fluid in a random shear flow: Renormalization group analysis of a simplified model

Critical behaviour of two systems, subjected to the turbulent mixing, is studied by means of the field theoretic renormalization group. The first system, described by the equilibrium model A, corresponds to the relaxational dynamics of a non-conserved order parameter. The second one is the strongly non-equilibrium reaction–diffusion system known as Gribov process and equivalent to the Reggeon field theory. The turbulent mixing is modelled by the Kazantsev–Kraichnan 'rapid-change' ensemble: time-decorrelated Gaussian velocity field with the power-like spectrum ∝k−d − ξ. Effects of compressibility of the fluid are studied. It is shown that, depending on the relation between the exponent ξ and the spatial dimension d, both the systems exhibit four different types of critical behaviour, associated with four possible fixed points of the renormalization group equations. Three fixed points correspond to known regimes: Gaussian fixed point, original model without mixing and passively advected scalar field. The most interesting fourth point corresponds to a new type of critical behaviour, in which both nonlinearity and turbulent mixing are relevant, and the critical exponents depend on d, ξ and the degree of compressibility. The critical exponents and regions of stability for all the regimes are calculated in the leading order of the double expansion in two parameters ξ and e = 4 − d. For both models, compressibility enhances the role of the nonlinear terms in the dynamical equations: the region in the e–ξ plane, where the new nontrivial regime is stable, is getting much wider as the degree of compressibility increases. For the incompressible fluid, the most realistic values d = 3 and ξ = 4/3 (Kolmogorov turbulence) lie in the region of stability of the passive scalar regime. If the compressibility becomes strong enough, the crossover in the critical behaviour occurs, and these values of d and ξ fall into the region of stability of the new regime, where both advection and the nonlinearity are important. In its turn, turbulent transfer becomes more efficient due to combined effects of the mixing and the nonlinear terms.

https://iopscience.iop.org/article/10.1088/1751-8113/43/40/405001/pdf

Effects of turbulent mixing on critical behaviour in the presence of compressibility: renormalization group analysis of two models

Using the field theory renormalization group, we study the critical behavior of two systems subjected to turbulent mixing. The first system, described by the equilibrium model A, corresponds to the relaxational dynamics of a nonconserved order parameter. The second system is the strongly nonequilibrium reaction-diffusion system, known as the Gribov process or directed percolation process. The turbulent mixing is modeled by the stochastic Navier-Stokes equation with a random stirring force with the correlator ∞ δ(t − t′)p4−d−y, where p is the wave number, d is the space dimension, and y is an arbitrary exponent. We show that the systems exhibit various types of critical behavior depending on the relation between y and d. In addition to known regimes (original systems without mixing and a passively advected scalar field), we establish the existence of new strongly nonequilibrium universality classes and calculate the corresponding critical dimensions to the first order of the double expansion in y and ɛ = 4 − d (one-loop approximation).

Effects of turbulent transfer on 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 $\propto\delta(t-t') k^{4-d-y}$ 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 $\propto\delta(t-t') k^{4-d-y'}$. 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 $\epsilon=4-d$).

https://arxiv.org/pdf/cond-mat/0604434.pdf

Effects of mixing and stirring on the critical behavior

We use analytic methods to analyze the discrete spectrum for the problem (Z1eZ2)2 in the united-atom limit (R ≪ 1) and obtain asymptotic expansions for the quantum defect and energy terms of the system (Z1eZ2)2 at small intercenter distances R up to terms of the order O(R6). We investigate the effect of the dimensionality factor on the energy spectrum of the hydrogen molecular ion H2+.

Two-dimensional problem of two Coulomb centers at small intercenter distances

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.

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

Using the perturbation renormalization group, we investigate the influence of a random velocity field on the critical behavior of the directed-bond percolation process near its second-order phase transition between the absorbing and active phases. We use the Antonov-Kraichnan model with a finite correlation time to describe the advecting velocity field. To obtain information about the large-scale asymptotic behavior of the model, we use the field theory renormalization group approach. We analyze the model near its critical dimension via a three-parameter expansion in ∈, δ, and η, where ∈ is the deviation from the Kolmogorov scaling, δ is the deviation from the critical space dimension, and η is the deviation from the parabolic dispersion law for the velocity correlator. We find the fixed points with the corresponding stability regions in the leading order in the perturbation scheme.

Critical behavior of percolation process influenced by a random velocity field: One-loop approximation

In the framework of a field theory model obtained by “second quantization” of a 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.

M. Hnatich

Papers

Effects of mixing and stirring on the critical behavior

Two-dimensional problem of two Coulomb centers at small intercenter distances

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

Critical behavior of percolation process influenced by a random velocity field: One-loop approximation

Field theory approach in kinetic reaction: Role of random sources and sinks