The Number of Master Integrals is Finite

For a fixed Feynman graph one can consider Feynman integrals with all possible powers of propagators and try to reduce them, by linear relations, to a finite subset of integrals, the so-called master integrals. Up to now, there are numerous examples of reduction procedures resulting in a finite number of master integrals for various families of Feynman integrals. However, up to now it was just an empirical fact that the reduction procedure results in a finite number of irreducible integrals. It this paper we prove that the number of master integrals is always finite.     

Differential Equations Compatible with KZ Equations

We define a system of dynamical differential equations compatible with the KZ differential equations. The KZ differential equations are associated to a complex simple Lie algebra g. These are equations on a function of n complex variables z _i taking values in the tensor product of n finite dimensional g-modules. The KZ equations depend on the dual variable in the Cartan subalgebra of g. The dynamical differential equations are differential equations with respect to the dual variable. We prove that the standard hypergeometric solutions of the KZ equations also satisfy the dynamical equations. As an application we give a new determinant formula for the coordinates of a basis of hypergeometric solutions.     

Homogeneous Generalized Master Equations

It is shown that the method proposed in V. F. Los [J. Phys. A: Math. Gen. 34: 638–6403 (2001)], which allows for turning the inhomogeneous time-convolution generalized master equation (TC-GME) into homogeneous (while retaining initial correlations) time-convolution generalized master equation (TC-HGME) for the relevant part of a distribution function, is fully applicable to the quantum case and to the time-convolutionless GME (TCL-GME). It is demonstrated by rederiving the TC-HGME and showing that it works in both the classical and quantum physics cases. The time-convolutionless HGME (TCL-HGME) retaining initial correlations, which is formally the same for both the classical and quantum physics, has also been derived. Both the TC-HGME and TCL-HGME are exact equations applicable on any timescale and allow for consecutive treating the initial correlations and collisions on the equal footing. A new equation for a momentum distribution function retaining initial correlations has been obtained in the linear in the density of quantum particles approximation. Connection of this equation to the quantum Boltzmann equation is discussed.     

From Convolutionless Generalized Master to Pauli Master Equations

Previously we have proved that time integrals of memory functions (i.e. markovian transfer rates from Pauli Master Equations — PME) in Time-Convolution Generalized Master Equations (TC-GME) for probabilities of finding a state of an asymmetric system interacting with a bath with a continuous spectrum are exactly zeroprovided that no approximation was involved. This is irrespective of the usual finite-pertubational-order correspondence with the Golden Rule transition rates. Here, attention is turned to an alternative way to derive the rigorous PME from the Time-Convolutionless Generalized Master Equations (TCL-GME). Arguments are given that the long-time limit of coefficients in TCL-GME for the above probabilities is, under the same assumption and presuming that this limit exists, equal to zero.The author should like to acknowledge a support of grants GAUK-292 of the Grant Agency of Charles University and GAR-2428 of the Czech Grant Agency.     

Triangular Newton Equations with Maximal Number of Integrals of Motion

Abstract

We study two-dimensional triangular systems of Newton equations (acceleration = velocity-independent force) admitting three functionally independent quadratic integrals of motion. The main idea is to exploit the fact that the first component M ₁(q ₁) of a triangular force depends on one variable only. By using the existence of extra integrals of motion we reduce the problem to solving a simultaneous system of three linear ordinary differential equations with nonconstant coefficients for M ₁(q ₁). With the help of computer algebra we have found and solved these ordinary differential equations in all cases. A complete list of superintegrable triangular equations in two dimensions is been given. Most of these equations were not known before.     

On the Gaussian Approximation for Master Equations

We analyze the Gaussian approximation as a method to obtain the first and second moments of a stochastic process described by a master equation. We justify the use of this approximation with ideas coming from van Kampen’s expansion approach (the fact that the probability distribution is Gaussian at first order). We analyze the scaling of the error with a large parameter of the system and compare it with van Kampen’s method. Our theoretical analysis and the study of several examples shows that the Gaussian approximation turns out to be more accurate than van Kampen’s expansion at first order. This could be specially important for problems involving stochastic processes in systems with a small number of particles.     

Differential Constraints Compatible with Linearized Equations

Abstract

Differential constraints compatible with the linearized equations of partial differential equations are examined. Recursion operators are obtained by integrating the differential constraints.     

Differential Geometry from Differential Equations

We first show how, from the general 3rd order ODE of the form , one can construct a natural Lorentzian conformal metric on the four-dimensional space . When the function satisfies a special differential condition the conformal metric possesses a conformal Killing field, , which in turn, allows the conformal metric to be mapped into a three dimensional Lorentzian metric on the space ) or equivalently, on the space of solutions of the original differential equation. This construction is then generalized to the pair of differential equations, z _ss =S(z,z _s ,z _t ,z _st ,s,t) and z _tt =T(z,z _s ,z _t ,z _st ,s,t), with z _s and z _t the derivatives of z with respect to s and t. In this case, from S and T, one can again, in a natural manner, construct a Lorentzian conformal metric on the six dimensional space (z,z _s ,z _t ,z _st ,s,t). When the S and T satisfy differential conditions analogous to those of the 3^rd order ode, the 6-space then possesses a pair of conformal Killing fields, and which allows, via the mapping to the four-space of (z,z _s ,z _t ,z _st ) and a choice of conformal factor, the construction of a four-dimensional Lorentzian metric. In fact all four-dimensional Lorentzian metrics can be constructed in this manner. This construction, with further conditions on S and T, thus includes all (local) solutions of the Einstein equations. Received: 10 October 2000 / Accepted: 26 June 2001     

Simple Equations Method and Non-Linear Differential Equations with Non-Polynomial Non-Linearity

We discuss the application of the Simple Equations Method (SEsM) for obtaining exact solutions of non-linear differential equations to several cases of equations containing non-polynomial non-linearity. The main idea of the study is to use an appropriate transformation at Step (1.) of SEsM. This transformation has to convert the non-polynomial non- linearity to polynomial non-linearity. Then, an appropriate solution is constructed. This solution is a composite function of solutions of more simple equations. The application of the solution reduces the differential equation to a system of non-linear algebraic equations. We list 10 possible appropriate transformations. Two examples for the application of the methodology are presented. In the first example, we obtain kink and anti- kink solutions of the solved equation. The second example illustrates another point of the study. The point is as follows. In some cases, the simple equations used in SEsM do not have solutions expressed by elementary functions or by the frequently used special functions. In such cases, we can use a special function, which is the solution of an appropriate ordinary differential equation, containing polynomial non-linearity. Specific cases of the use of this function are presented in the second example.     

The Generalized Master equation approach to the phonon-assisted hopping

The Generalized Master Equation (GME) approach to the phonon-assisted hopping via localized single-particle states interacting with harmonic phonon continuum is presented. Starting from the Nakajima-Zwanzig form of the GME, the memory functions governing the electron kinetics are evaluated. It is proved that any direct expansion (in powers of the electron-phonon coupling constantg and overlap parameter of localized orbitalsW) of the exponentialexp {–i(1–D) L} (whereL andD are the Liouville and projection superoperators) entering the memory functions is of little use due to the presence of undamped (constant in time) terms (or terms g ²/( + i) singular at + i 0 after taking the Fourier transform). In the memory functions, however, these singular terms are verified to cancel each other at least to the fourth power ofgW. This result enables to justify the usual rate equation for the site occupation probabilities f_m in absence of an external field in the lowest order ingW. Problems with phonon-less contributions to the memory functions (W ² g ⁴) are discussed. The theory applies to the electron (exciton) transport in e.g. disordered semiconductors as well as the hopping via non-degenerated molecular orbitals in finite molecules as long as the heat bath may be approximated by a continuum harmonic phonon field.     

Virtues and Limitations of Markovian Master Equations with a Time-Dependent Generator

A Markovian master equation with time-dependent generator is constructed that respects basic constraints of quantum mechanics, in particular the von Neumann conditions. For the case of a two-level system, Bloch equations with time-dependent parameters are obtained. Necessary conditions on the latter are formulated. By employing a time-local counterpart of the Nakajima–Zwanzig equation, we establish a relation with unitary dynamics. We also discuss the relation with the weak-coupling limit. On the basis of a uniqueness theorem, a standard form for the generator of time-local master equations is proposed. The Jaynes–Cummings model with atomic damping is solved. The solution explicitly demonstrates that reduced dynamics can be described by time-local master equations only on a finite time interval. This limitation is caused by divergencies in the generator. A limit of maximum entropy is presented that corroborates the foregoing statements. A second limiting case demonstrates that divergencies may even occur for small perturbations of the weak-coupling regime.     

Generalized Master Equations and Boltzmann-like transport in solids

Solution of the linearized Generalized Master Equations and that of the linearized Boltzmann equation are compared in the vicinity of the dc limit. In addition to straightforward higher-order corrections, the former solution is shown to differ from the latter by an additional term which is made explicit for the dc conductivity to the lowest order in coupling to static random short-range impurities. Numerical estimate shows that this term (proportional to the first power of impurity concentration) may become non-negligible in realistic situations.     

A Fuchsian Matrix Differential Equation for Selberg Correlation Integrals

We characterize averages of ?_l=1^N|x - t_l|^{a- 1}{\prod_{l=1}^N|x - t_l|^{\alpha - 1}} with respect to the Selberg density, further constrained so that t_l ? [0,x] (l=1,...,q){t_l \in [0,x] (l=1,\dots,q)} and t_l ? [x,1] (l=q+1,...,N){t_l \in [x,1] (l=q+1,\dots,N)} , in terms of a basis of solutions of a particular Fuchsian matrix differential equation. By making use of the Dotsenko-Fateev integrals, the explicit form of the connection matrix from the Frobenius type power series basis to this basis is calculated, thus allowing us to explicitly compute coefficients in the power series expansion of the averages. From these we are able to compute power series for the marginal distributions of the t_j (j=1,...,N){t_j (j=1,\dots,N)} . In the case q = 0 and α < 1 we compute the explicit leading order term in the x ? 0{x \to 0} asymptotic expansion, which is of interest to the study of an effect known as singularity dominated strong fluctuations. In the case q = 0 and a ? \mathbbZ⁺{\alpha \in \mathbb{Z}^+} , and with the absolute values removed, the average is a polynomial, and we demonstrate that its zeros are highly structured.     

Path Integrals for a Class of P-Adic Schrödinger Equations

The theme of doing quantum mechanics on all Abelian groups goes back to Schwinger and Weyl. If the group is a vector space of finite dimension over a non-Archimedean locally compact division ring, it is of interest to examine the structure of dynamical systems defined by Hamiltonians analogous to those encountered over the field of real numbers. In this Letter, a path integral formula for the imaginary time propagators of these Hamiltonians is derived.     

First Integrals and Parametric Solutions for Equations Integrable Through Lie Symmetries

Abstract

We present here the explicit parametric solutions of second order differential equations invariant under time translation and rescaling and third order differential equations invariant under time translation and the two homogeneity symmetries. The computation of first integrals gives in the most general case, the parametric form of the general solution. For some polynomial functions we obtain a time parametrisation quadrature which can be solved in terms of “known” functions.