Self-force via Green functions and worldline integration

A compact object moving in curved spacetime interacts with its own gravitational field. This leads to both dissipative and conservative corrections to the motion, which can be interpreted as a self-force acting on the object. The original formalism describing this self-force relied heavily on the Green function of the linear differential operator that governs gravitational perturbations. However, because the global calculation of Green functions in non-trivial black hole spacetimes has been an open problem until recently, alternative methods were established to calculate self-force effects using sophisticated regularization techniques that avoid the computation of the global Green function. We present a method for calculating the self-force that employs the global Green function and is therefore closely modeled after the original self-force expressions. Our quantitative method involves two stages: (i) numerical approximation of the retarded Green function in the background spacetime; (ii) evaluation of convolution integrals along the worldline of the object. This novel approach can be used along arbitrary worldlines, including those currently inaccessible to more established computational techniques. Furthermore, it yields geometrical insight into the contributions to self-interaction from curved geometry (back-scattering) and trapping of null geodesics. We demonstrate the method on the motion of a scalar charge in Schwarzschild spacetime. This toy model retains the physical history-dependence of the self-force but avoids gauge issues and allows us to focus on basic principles. We compute the self-field and self-force for many worldlines including accelerated circular orbits, eccentric orbits at the separatrix, and radial infall. This method, closely modeled after the original formalism, provides a promising complementary approach to the self-force problem.Comment: 18 pages, 9 figure

The Uniform Continuity of Characteristic Function from Convoluted Exponential Distribution with Stabilizer Constant

It is constructed convolution of generated random variable from independent and identically exponential distribution with stabilizer constant. The characteristic function of this distribution is obtained by using Laplace-Stieltjes transform. The uniform continuity property of characteristic function from this convolution is obtained by using analytical methods as basic properties

Universality classes of interaction structures for NK fitness landscapes

Kauffman's NK-model is a paradigmatic example of a class of stochastic models of genotypic fitness landscapes that aim to capture generic features of epistatic interactions in multilocus systems. Genotypes are represented as sequences of $L$ binary loci. The fitness assigned to a genotype is a sum of contributions, each of which is a random function defined on a subset of $k \le L$ loci. These subsets or neighborhoods determine the genetic interactions of the model. Whereas earlier work on the NK model suggested that most of its properties are robust with regard to the choice of neighborhoods, recent work has revealed an important and sometimes counter-intuitive influence of the interaction structure on the properties of NK fitness landscapes. Here we review these developments and present new results concerning the number of local fitness maxima and the statistics of selectively accessible (that is, fitness-monotonic) mutational pathways. In particular, we develop a unified framework for computing the exponential growth rate of the expected number of local fitness maxima as a function of $L$, and identify two different universality classes of interaction structures that display different asymptotics of this quantity for large $k$. Moreover, we show that the probability that the fitness landscape can be traversed along an accessible path decreases exponentially in $L$ for a large class of interaction structures that we characterize as locally bounded. Finally, we discuss the impact of the NK interaction structures on the dynamics of evolution using adaptive walk models.Comment: 61 pages, 9 figure

On the evaluation of sunset-type Feynman diagrams

We introduce an efficient configuration space technique which allows one to compute a class of Feynman diagrams which generalize the scalar sunset topology to any number of massive internal lines. General tensor vertex structures and modifications of the propagators due to particle emission with vanishing momenta can be included with only a little change of the basic technique described for the scalar case. We discuss applications to the computation of $n$-body phase space in $D$-dimensional space-time. Substantial simplifications occur for odd space-time dimensions where the final results can be expressed in closed form through rational functions. We present explicit analytical formulas for three-dimensional space-time.Comment: 45 pages in LaTeX, 3 PostScript figures included in the tex