Singularity of the “swallow-tail” type and the glass-glass transition in a system of collapsing hard spheres

In the mode-coupling approximation, we consider the transition to the glass state in a system of collapsing hard spheres (a system with the hard-core potential to which a repulsive step is added). We propose an approximation for the structure factor of the system, which we use to construct the phase diagram of the transition to the glass state. We show that there exists a maximum on the liquid-glass curve corresponding to the reentrant transition to the glass state in the system. In the framework of the proposed model, we consider bifurcations of solutions of the equations describing the transition to the glass state and show that there exist bifurcations of the “swallow-tail” type corresponding to the glass-glass transition.     

On the Renormalization Group Approach to Perturbation Theory for PDEs

We investigate the rigorous application of the renormalization group method to (singular) perturbation theory for nonlinear partial differential equations. As a paradigm, we consider the concrete example of the nonlinear Schrödinger equation with quadratic nonlinearity in three spatial dimensions. We obtain an approximate solution using the RG method together with an estimate of the difference between the true and approximate solutions. Our analysis applies to cases where (space–time) resonances are present.     

Monodromization and Difference Equations with Meromorphic Periodic Coefficients

We consider a system of two first-order difference equations in the complex plane. We assume that the matrix of the system is a 1-periodic meromorphic function having two simple poles per period and bounded as Im z → ±∞. We prove the existence and uniqueness of minimal meromorphic solutions, i.e., solutions having simultaneously a minimal set of poles and minimal possible growth as Im z → ±∞. We consider the monodromy matrix representing the shift-byperiod operator in the space of meromorphic solutions and corresponding to a basis built of two minimal solutions. We check that it has the same functional structure as the matrix of the initial system of equations and, in particular, is a meromorphic periodic function with two simple poles per period. This implies that the initial equation is invariant with respect to the monodromization procedure, that is, a natural renormalization procedure arising when trying to extend the Floquet–Bloch theory to difference equations defined on the real line or complex plane and having periodic coefficients. Our initial system itself arises after one renormalization of a self-adjoint difference Schrödinger equation with 1-periodic meromorphic potential bounded at ±i∞ and having two poles per period.     

Temperature Green’s functions in Fermi systems: The superconducting phase transition

We consider the model of an equilibrium Fermi system of arbitrary-spin particles with the density-densitytype interaction. Based on the microscopic Hamiltonian in the formalism of temperature Green’s functions, we find critical modes and construct an effective action describing a neighborhood of the phase transition point. A renormalization group analysis of the obtained model leads to the standard critical behavior indices for spin-1/2 fermions but shows that in the system of higher-spin fermions, a first-order phase transition occurs whose temperature exceeds the standard estimates for the temperature of a second-order phase transition.     

Directed-bond percolation subjected to synthetic compressible velocity fluctuations: Renormalization group approach

We study the directed-bond percolation process (sometimes called the Gribov process because it formally resembles Reggeon field theory) in the presence of irrotational velocity fluctuations with long-range correlations. We use the renormalization group method to investigate the phase transition between an active and an absorbing state. All calculations are in the one-loop approximation. We calculate stable fixed points of the renormalization group and their regions of stability in the form of expansions in three parameters (ε, y, η). We consider different regimes corresponding to the Kraichnan rapid-change model and a frozen velocity field.     

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

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.     

Ultraviolet divergences in <Emphasis Type="Italic">D</Emphasis>=8 <Emphasis Type="Italic">N</Emphasis>=1 supersymmetric Yang–Mills theory

We consider the leading and subleading UV divergences for the four-point on-shell scattering amplitudes in the D=8 N=1 supersymmetric Yang–Mills theory in the planar limit for ladder-type diagrams. We obtain recurrence relations that allow obtaining the leading and subleading divergences in all loops purely algebraically starting from the one-loop diagrams (for the leading poles) and the two-loop diagrams (for the subleading poles). We sum the leading and subleading divergences over all loops using differential equations that are generalizations of the renormalization group equations to nonrenormalizable theories. We discuss the properties of the obtained solutions and the dependence of the constructed counterterms on the scheme.     

The effect of strongly anisotropic turbulent mixing on critical behavior: Renormalization group analysis of two nonstandard systems

We consider the effect of strongly anisotropic turbulent mixing on the critical behavior of two systems: a φ ³ critical dynamics model describing universal properties of metastable states in the vicinity of a firstorder phase transition and a reaction-diffusion system near the point of a second-order transition between fluctuation and absorption states (a simple epidemic process or the Gribov process). In both cases, we demonstrate the existence of a new strongly nonequilibrium, anisotropic scaling regime (universality class) for which both the mixing and the nonlinearity in the order parameter are relevant. We evaluate the corresponding critical dimensions in the one-loop approximation of the renormalization group.     

Renormalization scenario for the quantum Yang–Mills theory in four-dimensional space–time

We consider the renormalization of the Yang–Mills theory in four-dimensional space–time using the background-field formalism.     

Space–time chaos in a system of reaction–diffusion equations

We find conditions for the bifurcation of periodic spatially homogeneous and spatially inhomogeneous solutions of a three-dimensional system of nonlinear partial differential equations describing a soil aggregate model. We show that the transition to diffusion chaos in this model occurs via a subharmonic cascade of bifurcations of stable limit cycles in accordance with the universal Feigenbaum–Sharkovskii–Magnitskii bifurcation theory.     

Lotka–Volterra system and KCC theory: Differential geometric structure of competitions and predations

We consider the differential geometric structure of competitions and predations in the sense of the Lotka–Volterra system based on KCC theory. For this, we visualise the relationship between the Jacobi stability and the linear stability as a single diagram. We find the following. (I) Ecological interactions such as competition and predation can be described by the deviation curvature. In this case, the sign of the deviation curvature depends on the type of interaction, which reflects the equilibrium point type. (II) The geometric quantities in KCC theory can be expressed in terms of the mean and Gaussian curvatures of the potential surface. In this particular case, the deviation curvature can be interpreted as the Willmore energy density of the potential surface. (III) When the equations of the system have nonsymmetric structure for the species (e.g. a predation system), each species also has nonsymmetric geometric structure in the nonequilibrium region, but symmetric structure around the equilibrium point. These findings suggest that KCC theory is useful to establish the geometrisation of ecological interactions.     

Bootstrap Equations in Effective Theories

In this paper, we consider general properties of effective field theories. We note that the freedom of fixing renormalization conditions in the effective field theory is not as large as it seems. Consideration of the minimal set of correctness requirements of the perturbative scheme based on the Dyson's formula for the S-matrix leads to severe restrictions on essential parameters of the theory and hence on the possible set of renormalization conditions. In the first part of this paper, we give a short review of the structure of localizable effective field theories. We discuss necessary conditions which ensure the correctness of the first step of the iterative scheme of calculation of the S-matrix, i.e., the construction of tree-level amplitudes. In the second part, we discuss examples which demonstrate the main stages of acquisition and analysis of the system of bootstrap equations. Bibliography: 17 titles.     

Incommensurable state of a spin density wave and superconductivity in quasi-two-dimensional systems with an anisotropic energy spectrum

We investigate phase transitions in quasi-two-dimensional systems with an anisotropic energy spectrum and a deviation from the half-filling of the energy band (μ ≠ 0). We demonstrate the possibility of the transition of an insulator into a half-metallic state when the nesting condition is violated because the parameter μ ≠ 0 and of taking the umklapp processes into account. We obtain the basic equations for the parameters of the superconducting (Δ) and magnetic (M) orders and determine the conditions for the emergence of superconductivity on the background of a spin-density-wave state and also for the coexistence of superconductivity and magnetism. We show that the transition of a magnetic system into a superconducting state as the parameter μ increases can be a first-order phase transition at low temperatures. We also obtain an expression for the heat capacity jump C _S -C _N at T = T _c , which depends on M and μ and differs essentially from the case of the Bardeen-Cooper-Schrieffer theory. We also consider the transformations related to the density of electron states of the relevant anisotropic system, which can undergo essential changes under pressure or doping.     

A new form of the redfield equation for dissipative systems related to the matrix of correlation functions

In the Redfield theory framework, we consider the problem of the vibrational dynamics in dissipative systems. We decompose the Hamiltonian of interaction of the observed system with a thermal bath into terms that are products of system transition operators and bath transition operators. Using the decomposition, we construct a correlation function matrix containing all the information about the interaction of the system with the bath and obtain a new operator form of the Redfield equation. We consider the procedure for factoring the interaction operator and constructing the correlation function matrix. Using the diagonalization of the obtained matrix, we give correlated dissipation operators, whose introduction simplifies the form of the Redfield equations. We show that for several problems in which fundamental transition frequencies can be chosen, this procedure significantly reduces the dimensionality of the dissipative dynamics problem.     

Electron-phonon system with strong electronic correlations

The properties of the Hubbard-Holstein model for an electron-phonon system with strong electron correlations are investigated on the basis of a new diagram technique. The equations of the main dynamical quantities and optical dispersionless phonons are established. The problem of excluding partially or completely the phonon's coordinates and the corresponding renormalization of the physical parameters of the theory is discussed.Published in Teoreticheskaya i Matematicheskaya Fizika, Vol. 103, No. 1, pp. 138–160, April, 1995.     

Minimizing transient times in a coupled solid‐state laser model

We consider a system of ordinary differential equations modelling the dynamics of two coupled solid‐state lasers. Under the dynamics, this system may execute transitions between in‐phase and out‐of‐phase states. For satellite communications and high‐speed data transfer the transition times should be reduced to their shortest possible duration. In this paper, we apply optimal control theory to find the values of various laser parameters (e.g. the amplitude of the injected field, detunings, and coupling constants) which minimize the transient times between out‐of‐phase and in‐phase states. The effect of each parameter is shown to be independent of the other two, and the transient time is shown to be a strictly increasing function of detuning and a strictly decreasing function of the coupling constant and amplitude of the injected field. The effect of initial conditions on transient times is also analysed. Published in 2005 by John Wiley & Sons, Ltd.     

A Theory of Quantized Fields Based on Orthogonal and Symplectic Clifford Algebras

The transition from a classical to quantum theory is investigated within the context of orthogonal and symplectic Clifford algebras, first for particles, and then for fields. It is shown that the generators of Clifford algebras have the role of quantum mechanical operators that satisfy the Heisenberg equations of motion. For quadratic Hamiltonians, the latter equations are obtained from the classical equations of motion, rewritten in terms of the phase space coordinates and the corresponding basis vectors. Then, assuming that such equations hold for arbitrary path, i.e., that coordinates and momenta are undetermined, we arrive at the equations that contains basis vectors and their time derivatives only. According to this approach, quantization of a classical theory, formulated in phase space, is replacement of the phase space variables with the corresponding basis vectors (operators). The basis vectors, transformed into the Witt basis, satisfy the bosonic or fermionic (anti)commutation relations, and can create spinor states of all minimal left ideals of the corresponding Clifford algebra. We consider some specific actions for point particles and fields, formulated in terms of commuting and/or anticommuting phase space variables, together with the corresponding symplectic or orthogonal basis vectors. Finally we discuss why such approach could be useful for grand unification and quantum gravity.     

On the asymptotic transition to complexity in quantum chromodynamics

Quantum chromodynamics (QCD) is a renormalizable gauge theory that successfully describes the fundamental interaction of quarks and gluons. The rich dynamical content of QCD is manifest, for example, in the spectroscopy of complex hadrons or the emergence of quark–gluon plasma. There is a fair amount of uncertainty regarding the behavior of perturbative QCD in the infrared and far ultraviolet regions. Our work explores these two domains of QCD using non-linear dynamics and complexity theory. We find that local bifurcations of the renormalization flow destabilize asymptotic freedom and induce a steady transition to chaos in the far ultraviolet limit. We also conjecture that, in the infrared region, dissipative non-linearity of the renormalization flow supplies a natural mechanism for confinement.     

Evolution of a quantum system of many particles interacting via the generalized Yukawa potential

We study the evolution of a system of N particles that have identical masses and charges and interact via the generalized Yukawa potential. The system is placed in a bounded region. The evolution of such a system is described by the Bogoliubov–Born–Green–Kirkwood–Yvon (BBGKY) chain of quantum kinetic equations. Using semigroup theory, we prove the existence of a unique solution of the BBGKY chain of quantum kinetic equations with the generalized Yukawa potential.