Renormalization Group Method Applied to Kinetic Equations: Roles of Initial Values and Time

The so-called renormalization group (RG) method is applied to derive kinetic and transport equations from the respective microscopic equations. The derived equations include the Boltzmann equation in classical mechanics, the Fokker-Planck equation, and a rate equation in a quantum field theoretical model. Utilizing the formulation of the RG method which elucidates the important role played by the choice of the initial conditions, the general structure and the underlying assumptions in the derivation of kinetic equations in the RG method are clarified. It is shown that the present formulation naturally leads to the choice for the initial value of the microscopic distribution function at arbitrary time t₀ to be on the averaged distribution function to be determined. The averaged distribution function may be thought of as an integral constant of the solution of the microscopic evolution equation; the RG equation gives the slow dynamics of the would-be initial constant, which is actually the kinetic equation governing the averaged distribution function. It is further shown that the averaging as given above gives rise to a coarse-graining of the time-derivative which is expressed with the initial time t₀, and thereby leads to time-irreversible equations even from a time-reversible equation. It is shown that a further reduction of the Boltzmann equation to fluid dynamical equations and the adiabatic elimination of fast variables in the Fokker-Planck equation are also performed in a unified way in the present method.     

Fermi liquid theory: a renormalization group approach

We show how Fermi liquid theory results can be systematically recovered using a renormalization group (RG) approach. Considering a two-dimensional system with a circular Fermi surface, we derive RG equations at one-loop order for the two-particle vertex function in the limit of small momentum () and energy () transfer and obtain the equation which determines the collective modes of a Fermi liquid. The density-density response function is also calculated. The Landau function (or, equivalently, the Landau parameters F _l ^s and F _l ^a ) is determined by the fixed point value of the -limit of the two-particle vertex function (). We show how the results obtained at one-loop order can be extended to all orders in a loop expansion. Calculating the quasi-particle life-time and renormalization factor at two-loop order, we reproduce the results obtained from two-dimensional bosonization or Ward Identities. We discuss the zero-temperature limit of the RG equations and the difference between the Field Theory and the Kadanoff-Wilson formulations of the RG. We point out the importance of n-body () interactions in the latter. Received: 27 June 1997 / Received in final form: 17 December 1997 / Accepted: 26 January 1998     

The field-theoretic description of dynamics of interfaces in disordered media

The time evolution of an interface in a disordered media is described by using the propagator method. The method enables one to represent the perturbation expansions of different quantities characterizing the interface by means of diagrams which are familiar from the field theory. By the analysis of the divergences in the vicinity of the critical dimension d_c = 4 we found that the regularization of the theory demands the renormalization of the mobility and all moments of the disorder correlator excepting the zero one. The renormalization group (RG) calculations of the average velocity of the interface, the roughness, and the functional RG equation of the disorder correlator are presented to order ? = 4 - d. The latter coincides with the result obtained by D. S. Fisher in the equilibrium case. The RG equations have a pole at the value of the driving force, which coincides with the value of the threshold below which the interface becomes pinned as predicted by Bruinsma and Aeppli. The behavior of the mobility in the vicinity of the pole is discussed.     

Exact renormalization group and running Newtonian coupling in higher-derivative gravity

We discuss exact renormalization group (RG) in R ² gravity using the effective average action formalism. The truncated evolution equation for such a theory against the de Sitter background leads to a system of nonperturbative RG equations for cosmological and gravitational coupling constants. An approximate solution of these RG equations shows that antiscreening or screening behavior of the Newtonian coupling arises, depending on the higher-derivative coupling constants. Pis’ma Zh. éksp. Teor. Fiz. 65, No. 8, 571–575 (25 April 1997) Published in English in the original Russian journal. Edited by Steve Torstveit.     

Dynamic bifurcation of a modified Kuramotoben Sivashinsky equation with higher-order nonlinearity

Under the periodic boundary condition, dynamic bifurcation and stability in the modified Kuramoto—Sivashinsky equation with a higher-order nonlinearity μ(u_x)^pu_xx are investigated by using the centre manifold reduction procedure. The result shows that as the control parameter crosses a critical value, the system undergoes a bifurcation from the trivial solution to produce a cycle consisting of locally asymptotically stable equilibrium points. Furthermore, for cases in which the distances to the bifurcation points are small enough, one-order approximations to the bifurcation solutions are obtained.     

Differential real-space renormalization of thed-Dimensional Gaussian model

With the aid of the differential real-space method we derive exact renormalization group (RG) equations for the Gaussian model ind dimensions. The equations involved + 1 spatially dependent nearest-neighbor interactions. We locate a critical fixed point and obtain the exact thermal critical indexy _T = 2. A special trajectory of the full nonlinear RG transformation is found and the free energy of the corresponding initial state calculated.Supported by Deutsche Forschungsgemeinschaft through Sonderforschungsbereich 130 Ferroelektrika.     

Causality and the renormalization group

We suppose that for the invariant coupling constant (ICC) the spectral representation of the Källen-Lehmann type is valid. By combining this conjecture with the general solution of the functional renormalization group (RG) equation it is possible to analyze the type of singularity in the coupling constant at g=0. For logarithmic models it is of the form exp (-1/g).     

Smoothness of Invariant Manifolds for Nonautonomous Equations

For semiflows generated by ordinary differential equations v’=A(t)v admitting a nonuniform exponential dichotomy, we show that for any sufficiently small perturbation f there exist smooth stable and unstable manifolds for the perturbed equation v’=A(t)v+f(t,v). As an application, we establish the existence of invariant manifolds for the nonuniformly hyperbolic trajectories of a semiflow. In particular, we obtain smooth invariant manifolds for a class of vector fields that need not be C^1+α for any α ∈ (0,1). To the best of our knowledge no similar statement was obtained before in the nonuniformly hyperbolic setting. We emphasize that we do not need to assume the existence of an exponential dichotomy, but only the existence of a nonuniform exponential dichotomy, with sufficiently small nonuniformity when compared to the Lyapunov exponents of the original linear equation. Furthermore, for example in the case of stable manifolds, we only need to assume that there exist negative Lyapunov exponents, while we also allow zero exponents. Our proof of the smoothness of the invariant manifolds is based on the construction of an invariant family of cones.Supported by the Center for Mathematical Analysis, Geometry, and Dynamical Systems, and through Fundação para a Ciência e a Tecnologia by Program POCTI/FEDER, Program POSI, and the grant SFRH/BPD/14404/2003.     

Construction of the Hierarchical φ4-Trajectory

We study the invariant unstable manifold of the trivial renormalization-group fixed point tangent to the ⁴-vertex in the hierarchical approximation. We parametrize it by a running ⁴-coupling with linear step -function. The manifold is studied as a fixed point of the renormalization group composed with a flow of the running coupling. We present a rigorous construction of it beyond perturbation theory by means of a contraction mapping. Starting from a perturbative approximant of order seven, we obtain a convergent representation in dimensions 2 < D < 28/9 with certain restrictions. The perturbative approximant is logarithmically divergent in D = 3 dimensions.     

Invariant manifolds associated to nonresonant spectral subspaces

We show that, if the linearization of a map at a fixed point leaves invariant a spectral subspace which satisfies certain nonresonance conditions, the map leaves invariant a smooth manifold tangent to this subspace. This manifold is as smooth as the map—when the smoothness is measured in appropriate scales—but is unique amongC ^L invariant manifolds, whereL depends only on the spectrum of the linearization or on some more general smoothness classes that we detail. We show that if the nonresonance conditions are not satisfied, a smooth invariant manifold need not exist, and we also establish smooth dependence on parameters. We also discuss some applications of these invariant manifolds and briefly survey related work.     

QCD corrections at the Z0

All QCD corrections in hadron-initiated lepton-pair production are questionable for Q²?100 GeV² since a perturbative expansion is not justified in this kinematical range. Application of the renormalization group (RG) program at Z₀ energies is meaningful. We exhibit the size of the non-log correction terms, introduce experimental tests, and point to the assumptiona inherent in Drell-Yan (DY) type analyses.     

On precise center stable manifold theorems for certain reaction–diffusion and Klein–Gordon equations

We consider positive, radial and exponentially decaying steady state solutions of the general reaction–diffusion and Klein–Gordon type equations and present an explicit construction of infinite-dimensional invariant manifolds in the vicinity of these solutions. The result is a precise stable manifold theorem for the reaction–diffusion equation and a precise center-stable manifold theorem for the Klein–Gordon equation, which include the co-dimension of the manifolds and the decay rates for even perturbations.     

Renormalization group approach to interacting fermion systems in the two-particle-irreducible formalism

We describe a new formulation of the functional renormalization group (RG) for interacting fermions within a Wilsonian momentum-shell approach. We show that the Luttinger-Ward functional is invariant under the RG transformation, and derive the infinite hierarchy of flow equations satisfied by the two-particle-irreducible (2PI) vertices. In the one-loop approximation, this hierarchy reduces to two equations that determine the self-energy and the 2PI two-particle vertex Φ⁽²⁾. Susceptibilities are calculated from the Bethe-Salpeter equation that relates them to Φ⁽²⁾. While the one-loop approximation breaks down at low energy in one-dimensional systems (for reasons that we discuss), it reproduces the exact results both in the normal and ordered phases in single-channel (i.e. mean-field) theories, as shown on the example of BCS theory. The possibility to continue the RG flow into broken-symmetry phases is an essential feature of the 2PI RG scheme and is due to the fact that the 2PI two-particle vertex, contrary to its 1PI counterpart, is not singular at a phase transition. Moreover, the normal phase RG equations can be directly used to derive the Ginzburg-Landau expansion of the thermodynamic potential near a phase transition. We discuss the implementation of the 2PI RG scheme to interacting fermion systems beyond the examples (one-dimensional systems and BCS superconductors) considered in this paper.     

Regge behaviour of distribution functions and t and x-evolutions of gluon distribution function at low-x

In this paper, t and x-evolutions of gluon distribution function from Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution equation in leading order (LO) at low-x are presented assuming the Regge behaviour of quarks and gluons at this limit. We compare our results of gluon distribution function with MRST 2001, MRST 2004 and GRV 1998 parametrizations and show the compatibility of Regge behaviour of quark and gluon distribution functions with perturbative quantum chromodynamics (PQCD) at low-x. We also discuss the limitations of Taylor series expansion method used earlier to solve DGLAP evolution equations in the Regge behaviour of distribution functions.      

A new family of evolution water-wave equations possessing two-soliton solutions

We find a new family of fifth-order water-wave equations having common invariant manifold of the fourth order. These evolution equations are nonintegrable except for two cases corresponding to the Sawada–Kotera and Kaup–Kupershmidt equations. The invariant manifold of the family is an autonomous equation F-VI from the Cosgrove's classification of fourth-order ODEs having the Painlevé property. Two-parameter solutions of the equation F-VI allow to find two-soliton solutions for this family of evolution equations.     

Specific heat behavior of high temperature superconductors in the pseudogap regime

We consider the competition between the one dimensionalization effect due to a magnetic field and the hopping parameters in quasi-one-dimensional conductors. Our study is based on a perturbative renormalization group method with three cut-off parameters, the bandwidth E₀, the 1D-2D crossover temperature T^*₁, which is related to the hopping process t₁, and the magnetic energy . We have found that the renormalized crossover temperatures T^*₁ and T^*₂, at which the respectively hopping processes t₁ and t₂ become coherent, are reduced compared to the bare values as the field is increased. We discuss the consequences of these renormalization effects on the temperature-field phase diagram of the organic conductors.Received: 5 March 2003, Published online: 23 July 2003PACS: 64.60.-i General studies of phase transitions - 75.30.Fv Spin-density waves - 72.15.Gd Galvanomagnetic and other magnetotransport effects - 74.70.Kn Organic superconductors     

Why one needs a functional renormalization group to survive in a disordered world

In this paper, we discuss why functional renormalization is an essential tool to treat strongly disordered systems. More specifically, we treat elastic manifolds in a disordered environment. These are governed by a disorder distribution, which after a finite renormalization becomes non-analytic, thus overcoming the predictions of the seemingly exact dimensional reduction. We discuss how a renormalizable field theory can be constructed even beyond 2-loop order. We then consider an elastic manifold embedded inN dimensions, and give the exact solution forN →ɛ This is compared to predictions of the Gaussian replica variational ansatz, using replica symmetry breaking. Finally, the effective action at order 1/N is reported.     

Renormalizable asymptotically free quantum theory of gravity

We prove that higher derivative quantum gravity is asymptotically free in all essential coupling constants by the calculation of one-loop counterterms (correcting the previous result of Julve and Tonin) and the solution of the corresponding renormalization group (RG) equations. Strong arguments are presented in favour of the possibility that renormalizable asymptotically free gravity establishes asymptotic freedom for the effective mass parameters and non-gauge couplings in grand unified gauge theories. We also analyse the RG equations in the Einstein theory with Λ term and in the higher derivative conformal invariant theories. Among other topics discussed are the algorithm for the divergences of the determinant of the fourth-order differential operator, the consistent renormalization of the boundary terms in the action, the one-loop β-function in the fourth derivative vector gauge theory and the RG equations in the $\text{“gφ}{}^4\text{+ ηRφ}{}^2\text{”}$ theory.     

The modified Korteweg-de Vries equation on the quarter plane with t-periodic data

We study the modified Korteweg-de Vries equation posed on the quarter plane with asymptotically t-periodic Dirichlet boundary datum u(0,t) in the sense that u(0,t) tends to a periodic function g?₀ (t) with period τ as t → ∞. We consider the perturbative expansion of the solution in a small ε > 0. Here we show that if the unknown boundary data u_x(0,t) and u_xx(0,t) are asymptotically t-periodic with period τ which tend to the functions g?₁ (t) and g?₂ (t) as t → ∞, respectively, then the periodic functions g?₁ (t) and g?₂ (t) can be uniquely determined in terms of the function g?₀ (t). Furthermore, we characterize the Fourier coefficients of g?₁ (t) and g?₂ (t) to all orders in the perturbative expansion by solving an infinite system of algebraic equations. As an illustrative example, we consider the case of a sine-wave as Dirichlet datum and we explicitly determine the coefficients for large t up to the third order in the perturbative expansion.     

A perturbative nonequilibrium renormalization group method for dissipative quantum mechanics

We study a generic problem of dissipative quantum mechanics, a small local quantum system with discrete states coupled in an arbitrary way (i.e. not necessarily linear) to several infinitely large particle or heat reservoirs. For both bosonic or fermionic reservoirs we develop a quantum field-theoretical diagrammatic formulation in Liouville space by expanding systematically in the reservoir-system coupling and integrating out the reservoir degrees of freedom. As a result we obtain a kinetic equation for the reduced density matrix of the quantum system. Based on this formalism, we present a formally exact perturbative renormalization group (RG) method from which the kernel of this kinetic equation can be calculated. It is demonstrated how the nonequilibrium stationary state (induced by several reservoirs kept at different chemical potentials or temperatures), arbitrary observables such as the transport current, and the time evolution into the stationary state can be calculated. Most importantly, we show how RG equations for the relaxation and dephasing rates can be derived and how they cut off generically the RG flow of the vertices. The method is based on a previously derived real-time RG technique [1-4] but formulated here in Laplace space and generalized to arbitrary reservoir-system couplings. Furthermore, for fermionic reservoirs with flat density of states, we make use of a recently introduced cutoff scheme on the imaginary frequency axis [5] which has several technical advantages. Besides the formal set-up of the RG equations for generic problems of dissipative quantum mechanics, we demonstrate the method by applying it to the nonequilibrium isotropic Kondo model. We present a systematic way to solve the RG equations analytically in the weak-coupling limit and provide an outlook of the applicability to the strong-coupling case.