Dynamical response to perturbation of critical Boolean networks

Boolean networks can be used as simple but general models for complex self-organizing systems. The freedom to choose different rules and structures of interactions makes this model applicable to a wide variety of complex phenomena. It is known that the damage dynamics in annealed Boolean systems should fall in the same universality class of the directed percolation model. In this work we present results about the behavior of this model at and near the critically ordered condition for both the annealed and the quenched versions of the model. Our study concentrates on the way the system responds to a small perturbation. We show that the characteristic correlation time, i.e., the time in which any memory of this perturbation is lost, diverges as one moves towards criticality. Exactly at the critical point, we observe that the time for returning to the natural state after the perturbation follows a power-law distribution. This indicates that most perturbations are quickly restored, while few events may have a global effect on the system, suggesting a mechanism that assures at the same time robustness and adaptability. The critical exponents obtained are in agreement with the values expected for the universality class of mean-field directed percolation both in the annealed and in the quenched Boolean network model. This gives further evidence that annealed Boolean networks may in certain conditions provide a good model for understanding the behavior of regulatory systems. Our results may give insight into the way real self-organizing systems respond to external stimuli, and why critically ordered systems are often observed in Nature.     

Adiabatic theory,Liapunov exponents,and rotation number for quadratic Hamiltonians

We consider the adiabatic problem for general time-dependent quadratic Hamiltonians and develop a method quite different from WKB. In particular, we apply our results to the Schrödinger equation in a strip. We show that there exists a first regular step (avoiding resonance problems) providing one adiabatic invariant, bounds on the Liapunov exponents, and estimates on the rotation number at any order of the perturbation theory. The further step is shown to be equivalent to a quantum adiabatic problem, which, by the usual adiabatic techniques, provides the other possible adiabatic invariants. In the special case of the Schrödinger equation our method is simpler and more powerful than the WKB techniques.     

An exact solution to the multi-dimensional line transfer equation

The theory of generalized analytic functions is used to obtain an exact closed form analytical solution to a transfer problem for spectral line radiation in a multi-dimensional atmosphere. The multi-dimensional full-space and half-space Green's functions so obtained are quite general and may be used, along with the corresponding orthogonality relationships, to obtain solutions to any general multi-dimensional radiative transfer problem involving model two-level atoms. An application of the method using perturbation techniques is illustrated.     

Stochastic Perturbation of Integrable Systems: A Window to Weakly Chaotic Systems

Integrable non-linear Hamiltonian systems perturbed by additive noise develop a Lyapunov instability, and are hence chaotic, for any amplitude of the perturbation. This phenomenon is related, but distinct, from Taylor’s diffusion in hydrodynamics. We develop expressions for the Lyapunov exponents for the cases of white and colored noise. The situation described here being ‘multi-resonance’—by nature well beyond the Kolmogorov–Arnold–Moser regime, it offers an analytic glimpse on the regime in which many near-integrable systems, such as some planetary systems, find themselves in practice. We show with the aid of a simple example, how one may model in some cases weakly chaotic deterministic systems by a stochastically perturbed one, with good qualitative results.     

γγ Angular correlations perturbed by randomly reorienting hyperfine fields

The stochastic model of Blume for time-dependent hyperfine interactions is extended to randomly reorienting fields on the base of the work of Lynden-Bell. Two limiting cases are considered, the so-called diffusion and the strong collision model. Formulas are given for the perturbation factors of the angular correlation function which are valid for any fluctuation rate. Numerical calculations for the nuclear spinI=5/2 are presented.     

The region of validity of perturbation theory

We present a study of the region of validity of perturbation theory applied to rough surface scattering. We solve numerically the case of a periodic surface or grating varying in one dimension. For a statistical ensemble of gratings with a sufficiently long period one may obtain a good approximation of rough surface scattering. We use this to test the validity of perturbation theory.

Only the perfect conductor case was considered. We find that as the grazing angle becomes small the perturbation result for the TE (E horizontal) polarization remains valid, while for the TM (E vertical) polarization it breaks down. The results show that the perturbation results should be used carefully when being compared with experimental data at grazing angles.     

γγ Angular correlations perturbed by randomly reorienting hyperfine fields

The stochastic model of Blume for time-dependent hyperfine interactions is extended to randomly reorienting fields on the base of the work of Lynden-Bell. Two limiting cases are considered, the so-called diffusion and the strong collision model. Formulas are given for the perturbation factors of the angular correlation function which are valid for any fluctuation rate. Numerical calculations for the nuclear spinI=5/2 are presented.     

Perturbation theory and the parton model in QCD

We prove that for any process which admits a parton-model interpretation, the naive parton model can be modified to include the effects of QCD interactions to all orders in perturbation theory. This requires that the mass singularities in quark and gluon inclusive cross sections factor into universal functions which renormalize the naive parton model distribution and decay functions. We prove that this factorization takes place for all leading and non-leading logs and thus check consistency of the parton model to all orders in perturbation theory.     

Cosmographic study of the universe’s specific heat: a landscape for cosmology?

Non-abelian gauge field inflation is studied in the context of warm inflation scenario. We introduce this scenario as a mechanism that gives an end for gauge-flation model. Slow-roll parameters and perturbation parameters are presented for this model. We find the general conditions which are required for this model to be realizable in slow-roll approximation. We also develop our model in the context of intermediate and logamediate scenarios which are exact solutions of inflationary field equation in the Einstein theory. General expressions of slow-roll parameters, tensor–scalar ratio and scalar spectral index are presented in terms of inflaton field for these two cases. Our model is compatible with recent observational data from Planck satellite.     

Brillouin-Wigner and Feenberg perturbation methods in many-body theory

Brillouin-Wigner (BW) perturbation formulae can be rearranged into a form first proposed by Feenberg. Feenberg's perturbation formulae also follow from a variational principle. They are successfully tested at two typical problems, for which ordinary perturbation techniques completely fail. The first is to find conditions for the bosonization of the Tomonaga model perturbatively. The second is to clarify whether non-Fermi liquid behavior of the momentum distribution function of the Luttinger model can be achieved perturbatively.     

A Linearization Approach for Rational Nonlinear Models in Mathematical Physics

In this paper,a novel method for linearization of rational second order nonlinear models is discussed.In particular,we discuss an application of the δ expansion method(created to deal with problems in Quantum Field Theory) which will enable both the linearization and perturbation expansion of such equations.Such a method allows for one to quickly obtain the order zero perturbation theory in terms of certain special functions which are governed by linear equations.Higher order perturbation theories can then be obtained in terms of such special functions.One benefit to such a method is that it may be applied even to models without small physical parameters,as the perturbation is given in terms of the degree of nonlinearity,rather than any physical parameter.As an application,we discuss a method of linearizing the six Painleve equations by an application of the method.In addition to highlighting the benefits of the method,we discuss certain shortcomings of the method.     

A model for the packing of irregularly shaped grains

In this paper we will attempt to address the problem of the packing properties of granular materials composed of irregularly shaped grains (using configurational statistical mechanics). In particular, we will develop a model for a system of irregular grains based upon perturbing a packing of mono- or poly-disperse spheres. In the mono-disperse case we will show that the system packs less densely than a packing of perfect spheres, except when local correlations between configurations of grains are taken into account. The opposite is found to be true for a perturbation expansion based upon poly-disperse spheres. Finally we will show that for a bi-disperse packing of spheres phase segregation occurs for any size ratio and discuss whether this is to be expected.     

A model for noise processes in semiconductor laser amplifiers, part 1: the travelling-wave semiconductor laser amplifier

A new model for the noise processes in a travelling-wave semiconductor laser amplifier is presented. This model is based on applying perturbation techniques to the basic travelling-wave rate equations to deduce the noise rate equations. These equations are then used to calculate the power spectral densities of the intensity, electron density and phase noises introduced into a single signal mode propagating through the amplifier. The model can be applied to any gain profile within the amplifier, ranging from unsaturated to completely saturated. Furthermore, in contrast to most other semiconductor laser amplifier models, this model does not require the homogenization of the photon field over the length of the device.     

The region of validity of perturbation theory

Abstract

We present a study of the region of validity of perturbation theory applied to rough surface scattering. We solve numerically the case of a periodic surface or grating varying in one dimension. For a statistical ensemble of gratings with a sufficiently long period one may obtain a good approximation of rough surface scattering. We use this to test the validity of perturbation theory.

Only the perfect conductor case was considered. We find that as the grazing angle becomes small the perturbation result for the TE (E horizontal) polarization remains valid, while for the TM (E vertical) polarization it breaks down. The results show that the perturbation results should be used carefully when being compared with experimental data at grazing angles.     

$\mathcal{PT}$ Invariant Complex E8 Root Spaces

We provide a construction procedure for complex root spaces invariant under antilinear transformations, which may be applied to any Coxeter group. The procedure is based on the factorisation of a chosen element of the Coxeter group into two factors. Each of the factors constitutes an involution and may therefore be deformed in an antilinear fashion. Having the importance of the E ₈-Coxeter group in mind, such as underlying a particular perturbation of the Ising model and the fact that for it no solution could be found previously, we exemplify the procedure for this particular case. As a concrete application of this construction we propose new generalisations of Calogero-Moser-Sutherland models and affine Toda field theories based on the invariant complex root spaces and deformed complex simple roots, respectively.     

Planar theory with spin and tensor forces

We develop techniques for summing parquet diagrams for systems with two-body interactions with spin, tensor, and isospin components. For boson systems, we sum the same set of diagrams which have been used to derive the optimized hypernetted-chain variational theory when the force is independent of spin and isospin. The present derivation leads to unique local approximations in each of the six independent channels. Singularities in any of these channels at either low or high density correspond to instabilities of the physical system. In contrast to previous variational approaches, the need for commutator terms does not arise in this limit. We derive an energy functional from which the equations of motion may be obtained by functional differentiation. The equations of motion may be solved by a paired-phonon-analysis method which requires time proportional to the number of channels present. Although the primary emphasis of this theory is to suggest ways of including realistic nuclear forces in fermion theories, we present some model calculations which demonstrate the capabilities of the present approach.     

Alternative analysis to perturbation theory in quantum mechanics

We develop an alternative approach to the time independent perturbation theory in non-relativistic quantum mechanics. The method developed has the advantage to provide in one operation the correction to the energy and to the wave function; additionally we can analyze the time evolution of the system for any initial condition, which may be bothersome in the standard method. To verify our results, we apply our method to the harmonic oscillator perturbed by a quadratic potential. An alternative form of the Dyson series, in matrix form instead of integral form, is also obtained.     

Slowly varying fully nonlinear wavetrains in the Ginzburg-Landau equation

Following the ideas of Howard and Kopell [9] a perturbation theory is developed for slowly varying fully nonlinear wavetrains (i.e. solutions which appear locally as travelling waves, but with frequencies and wavelengths which may vary widely on long length and time scales). This perturbation theory is applied to the Ginzburg-Landau equation. The motion and stability of slowly varying wavetrains is shown to be governed by a simple wave equation which can develop shocks corresponding to rapid changes in wavenumber. Numerical results supporting this theory are presented. A shock structure is proposed and numerically verified. These results together with a winding invariant valid in the limit of slow variation suggest that over a large range of parameters many initial conditions relax to uniform wavetrains. The evolution of a marginally diffusively stable wavetrain is also examined; it is argued that the evolution is governed by a perturbed Korteweg-de Vries equation.