Dynamical Self-consistency Research Articles

Atom-Field Interaction: From Vacuum Fluctuations to Quantum Radiation and Quantum Dissipation or Radiation Reaction

In this paper, we dwell on three issues: (1) revisit the relation between vacuum fluctuations and radiation reaction in atom-field interactions, an old issue that began in the 1970s and settled in the 1990s with its resolution recorded in monographs; (2) the fluctuation–dissipation relation (FDR) of the system, pointing out the differences between the conventional form in linear response theory (LRT) assuming ultra-weak coupling between the system and the bath, and the FDR in an equilibrated final state, relaxed from the nonequilibrium evolution of an open quantum system; (3) quantum radiation from an atom interacting with a quantum field: We begin with vacuum fluctuations in the field acting on the internal degrees of freedom (idf) of an atom, adding to its dynamics a stochastic component which engenders quantum radiation whose backreaction causes quantum dissipation in the idf of the atom. We show explicitly how different terms representing these processes appear in the equations of motion. Then, using the example of a stationary atom, we show how the absence of radiation in this simple cases is a result of complex cancellations, at a far away observation point, of the interference between emitted radiation from the atom and the local fluctuations in the free field. In so doing we point out in Issue 1 that the entity which enters into the duality relation with vacuum fluctuations is not radiation reaction, which can exist as a classical entity, but quantum dissipation. Finally, regarding issue 2, we point out for systems with many atoms, the co-existence of a set of correlation-propagation relations (CPRs) describing how the correlations between the atoms are related to the propagation of their (retarded non-Markovian) mutual influence manifesting in the quantum field. The CPR is absolutely crucial in keeping the balance of energy flows between the constituents of the system, and between the system and its environment. Without the consideration of this additional relation in tether with the FDR, dynamical self-consistency cannot be sustained. A combination of these two sets of relations forms a generalized matrix FDR relation that captures the physical essence of the interaction between an atom and a quantum field at arbitrary coupling strength.

A new dynamically self-consistent version of the Besançon Galaxy model

Context. Dynamically self-consistent galactic models are necessary for analysing and interpreting star counts, stellar density distributions, and stellar kinematics in order to understand the formation and the evolution of our Galaxy. Aims. We modify and improve the dynamical self-consistency of the Besançon Galaxy model in the case of a stationary and axisymmetric gravitational potential. Methods. Each stellar orbit is modelled by determining a Stäckel approximate integral of motion. Generalised Shu distribution functions (DFs) with three integrals of motion are used to model the stellar distribution functions. Results. This new version of the Besançon model is compared with the previous axisymmetric BGM2014 version and we find that the two versions have similar densities for each stellar component. The dynamically self-consistency is improved and can be tested by recovering the forces and the potential through the Jeans equations applied to each stellar distribution function. Forces are recovered with an accuracy better than one per cent over most of the volume of the Galaxy.

Erratum: Screened Coulomb interaction calculations: cRPA implementation and applications to dynamical screening and self-consistency in uranium dioxide and cerium [Phys. Rev. B 89, 125110 (2014)

Erratum: Screened Coulomb interaction calculations: cRPA implementation and applications to dynamical screening and self-consistency in uranium dioxide and cerium [Phys. Rev. B 89, 125110 (2014)

The Besançon Galaxy model renewed

The understanding of Galaxy evolution can be facilitated by the use of population synthesis models, which allows us to test hypotheses on the star formation history, star evolution, and chemical and dynamical evolution of the Galaxy. The new version of the Besancon Galaxy model (hereafter BGM) aims to provide a more flexible and powerful tool to investigate the initial mass function (IMF) and star formation rate (SFR) of the Galactic disc. We present a new strategy for the generation of thin disc stars, which assumes the IMF, SFR and evolutionary tracks as free parameters. We have updated most of the ingredients for the star count production and, for the first time, binary stars are generated in a consistent way. The local dynamical self-consistency is maintained in this new scheme. We then compare simulations from the new model with Tycho-2 data and the local luminosity function, as a first test to verify and constrain the new ingredients. The effects of changing thirteen different ingredients of the model are systematically studied. For the first time, a full sky comparison is performed between BGM and data. This strategy allows us to constrain the IMF slope at high masses, which is found to be close to 3.0 and excludes a shallower slope such as Salpeter's one. The SFR is found decreasing whatever IMF is assumed. The model is compatible with a local dark matter density of 0.011 Mo/pc^3 implying that there is no compelling evidence for the significant amount of dark matter in the disc. While the model is fitted to Tycho-2 data, which is a magnitude limited sample with V<11, we check that it is still consistent with fainter stars. The new model constitutes a new basis for further comparisons with large scale surveys and is being prepared to become a powerful tool for the analysis of the Gaia mission data.

Screened Coulomb interaction calculations: cRPA implementation and applications to dynamical screening and self-consistency in uranium dioxide and cerium

We report an implementation of the constrained random phase approximation (cRPA) method within the projector augmented-wave framework. It allows for the calculation of the screened interaction in the same Wannier orbitals as our recent DFT+U and DFT+DMFT implementations. We present calculations of the dynamical Coulomb screened interaction in uranium dioxide and alpha and gamma cerium on Wannier functions. We show that a self-consistent calculation of the static screened interaction in DFT+U together with a consistent Wannier basis is mandatory for gamma cerium and uranium dioxide. We emphasize that a static approximation for the screened interaction in alpha cerium is too drastic.

Reaction coordinates, one-dimensional Smoluchowski equations, and a test for dynamical self-consistency

We propose a method for identifying accurate reaction coordinates among a set of trial coordinates. The method applies to special cases where motion along the reaction coordinate follows a one-dimensional Smoluchowski equation. In these cases the reaction coordinate can predict its own short-time dynamical evolution, i.e., the dynamics projected from multiple dimensions onto the reaction coordinate depend only on the reaction coordinate itself. To test whether this property holds, we project an ensemble of short trajectory swarms onto trial coordinates and compare projections of individual swarms to projections of the ensemble of swarms. The comparison, quantified by the Kullback-Leibler divergence, is numerically performed for each isosurface of each trial coordinate. The ensemble of short dynamical trajectories is generated only once by sampling along an initial order parameter. The initial order parameter should separate the reactants and products with a free energy barrier, and distributions on isosurfaces of the initial parameter should be unimodal. The method is illustrated for three model free energy landscapes with anisotropic diffusion. Where exact coordinates can be obtained from Kramers-Langer-Berezhkovskii-Szabo theory, results from the new method agree with the exact results. We also examine characteristics of systems where the proposed method fails. We show how dynamical self-consistency is related (through the Chapman-Kolmogorov equation) to the earlier isocommittor criterion, which is based on longer paths.

Continuous-time quantum Monte Carlo impurity solvers

Continuous-time quantum Monte Carlo impurity solvers are algorithms that sample the partition function of an impurity model using diagrammatic Monte Carlo techniques. The present paper describes codes that implement the interaction expansion algorithm originally developed by Rubtsov, Savkin, and Lichtenstein, as well as the hybridization expansion method developed by Werner, Millis, Troyer, et al. These impurity solvers are part of the ALPS-DMFT application package and are accompanied by an implementation of dynamical mean-field self-consistency equations for (single orbital single site) dynamical mean-field problems with arbitrary densities of states. Program summary Program title: dmft Catalogue identifier: AEIL_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEIL_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: ALPS LIBRARY LICENSE version 1.1 No. of lines in distributed program, including test data, etc.: 899 806 No. of bytes in distributed program, including test data, etc.: 32 153 916 Distribution format: tar.gz Programming language: C++ Operating system: The ALPS libraries have been tested on the following platforms and compilers: • Linux with GNU Compiler Collection (g++ version 3.1 and higher), and Intel C++ Compiler (icc version 7.0 and higher) • MacOS X with GNU Compiler (g++ Apple-version 3.1, 3.3 and 4.0) • IBM AIX with Visual Age C++ (xlC version 6.0) and GNU (g++ version 3.1 and higher) compilers • Compaq Tru64 UNIX with Compq C++ Compiler (cxx) • SGI IRIX with MIPSpro C++ Compiler (CC) • HP-UX with HP C++ Compiler (aCC) • Windows with Cygwin or coLinux platforms and GNU Compiler Collection (g++ version 3.1 and higher) RAM: 10 MB–1 GB Classification: 7.3 External routines: ALPS [1], BLAS/LAPACK, HDF5 Nature of problem: (See [2].) Quantum impurity models describe an atom or molecule embedded in a host material with which it can exchange electrons. They are basic to nanoscience as representations of quantum dots and molecular conductors and play an increasingly important role in the theory of “correlated electron” materials as auxiliary problems whose solution gives the “dynamical mean field” approximation to the self-energy and local correlation functions. Solution method: Quantum impurity models require a method of solution which provides access to both high and low energy scales and is effective for wide classes of physically realistic models. The continuous-time quantum Monte Carlo algorithms for which we present implementations here meet this challenge. Continuous-time quantum impurity methods are based on partition function expansions of quantum impurity models that are stochastically sampled to all orders using diagrammatic quantum Monte Carlo techniques. For a review of quantum impurity models and their applications and of continuous-time quantum Monte Carlo methods for impurity models we refer the reader to [2]. Additional comments: Use of dmft requires citation of this paper. Use of any ALPS program requires citation of the ALPS [1] paper. Running time: 60 s–8 h per iteration.

Self-consistent asset pricing models

We discuss the foundations of factor or regression models in the light of the self-consistency condition that the market portfolio (and more generally the risk factors) is (are) constituted of the assets whose returns it is (they are) supposed to explain. As already reported in several articles, self-consistency implies correlations between the return disturbances. As a consequence, the alphas and betas of the factor model are unobservable. Self-consistency leads to renormalized betas with zero effective alphas, which are observable with standard OLS regressions. When the conditions derived from internal consistency are not met, the model is necessarily incomplete, which means that some sources of risk cannot be replicated (or hedged) by a portfolio of stocks traded on the market, even for infinite economies. Analytical derivations and numerical simulations show that, for arbitrary choices of the proxy which are different from the true market portfolio, a modified linear regression holds with a non-zero value α i at the origin between an asset i's return and the proxy's return. Self-consistency also introduces “orthogonality” and “normality” conditions linking the betas, alphas (as well as the residuals) and the weights of the proxy portfolio. Two diagnostics based on these orthogonality and normality conditions are implemented on a basket of 323 assets which have been components of the S&P500 in the period from January 1990 to February 2005. These two diagnostics show interesting departures from dynamical self-consistency starting about 2 years before the end of the Internet bubble. Assuming that the CAPM holds with the self-consistency condition, the OLS method automatically obeys the resulting orthogonality and normality conditions and therefore provides a simple way to self-consistently assess the parameters of the model by using proxy portfolios made only of the assets which are used in the CAPM regressions. Finally, the factor decomposition with the self-consistency condition derives a risk-factor decomposition in the multi-factor case which is identical to the principal component analysis (PCA), thus providing a direct link between model-driven and data-driven constructions of risk factors. This correspondence shows that PCA will therefore suffer from the same limitations as the CAPM and its multi-factor generalization, namely lack of out-of-sample explanatory power and predictability. In the multi-period context, the self-consistency conditions force the betas to be time-dependent with specific constraints.

Dynamical solutions of a quantum Heisenberg spin glass model

We consider quantum-dynamical phenomena in the $\mathrm{SU}(2)$, $S=1/2$ infinite-range quantum Heisenberg spin glass. For a fermionic generalization of the model we formulate generic dynamical self-consistency equations. Using the Popov-Fedotov trick to eliminate contributions of the non-magnetic fermionic states we study in particular the isotropic model variant on the spin space. Two complementary approximation schemes are applied: one restricts the quantum spin dynamics to a manageable number of Matsubara frequencies while the other employs an expansion in terms of the dynamical local spin susceptibility. We accurately determine the critical temperature $T_c$ of the spin glass to paramagnet transition. We find that the dynamical correlations cause an increase of $T_c$ by 2% compared to the result obtained in the spin-static approximation. The specific heat $C(T)$ exhibits a pronounced cusp at $T_c$. Contradictory to other reports we do not observe a maximum in the $C(T)$-curve above $T_c$.

A plausible Galactic spiral pattern and its rotation speed

We report calculations of the stellar and gaseous response to a Milky Way mass distribution model including a spiral pattern with a locus as traced by K-band observations, over imposed on the axisymmetric components in the plane of the disk. The stellar study extends calculations from previous work concerning the self-consistency of the pattern. The stellar response to the imposed spiral mass is studied via computations of the central family of periodic and nearby orbits as a function of the pattern rotation speed, $\Omega_p$, among other parameters. A fine grid of values of $\Omega_p$ was explored ranging from 12 to 25 $km s^{-1} kpc^{-1}$. Dynamical self-consistency is highly sensitive to $\Omega_p$, with the best fit appearing at 20 $km s^{-1} kpc^{-1}$. We give an account of recent independent pieces of theoretical and observational work that are dependent on the value of $\Omega_p$, all of which are consistent with the value found here; the recent star formation history of the Milky Way, local inferences of cosmic ray flux variations and Galactic abundance patterns. The gaseous response, which is also a function of $\Omega_p$, was calculated via 2D hydrodynamic simulations with the ZEUS code. For $\Omega_p = 20 km s^{-1} kpc^{-1}$, the response to a two-armed pattern is a structured pattern of 4 arms, with bifurcations along the arms and interarm features. The pattern resembles qualitatively the optical arms observed in our Galaxy and other galaxies. The complex gaseous pattern appears to be linked to resonances in stellar orbits. Among these, the 4:1 resonance plays an important role, as it also determines the extent over which the imposed spiral pattern is dynamically self-consistent.

Dynamically consistent magnetic fields produced by differential rotation

We investigate the dynamical consequences of an axisymmetric velocity field with a poloidal magnetic field driven by a prescribed e.m.f. E. The problem is motivated by previous investigations of dynamically driven dynamos in the magnetostrophic range. A geostrophic zonal flow field is added to a previously described velocity, and determined by the requirement that Taylor's constraint (Taylor 1963) (guaranteeing dynamical self-consistency of the fields) be satisfied. Several solutions are exhibited, and it is suggested that self-consistent solutions can always be found to this ‘forced’ problem, whereas the usual α-effect dynamo formalism in which E is a linear function of the magnetic field leads to a difficult transcendentally nonlinear characteristic value problem that may not always possess solutions.

Transport of an excess electron in molecular solids: Effect of the dynamical self-consistency of the polarization

In a simple model, the influence of formation of the electronic polaron on the form of the electronic dispersion law in molecular systems is calculated in a way which treats the electronic polarization self-consistently with respect to the electronic motion. Corrections to the polarization energy in the Lyons model calculated using the microelectrostatic approach are found which are of the order of the experimental half-band-width (≲0·1 eV) in current molecular crystals.

Dynamical Symmetries and Symmetry Algebras

We consider the interaction of three multiplets of particles under the assumption that the members of each one of these multiplets have the same mass and the same spin. The vertex self-consistency conditions lead to an algebraic structure involving the coupling matrices. This structure, referred to as a symmetry algebra and denoted by the symbol (N,n,ν), is characterized essentially by the numbers of particles N, n, and ν belonging to each one of the three multiplets and is independent of the particular underlying dynamics. A question of particular interest is whether dynamical self-consistency implies the existence of a symmetry group that leaves the interaction invariant. We analyze this problem in detail for three particularly simple but instructive symmetry algebras. It is shown that the algebra (N,n,ν=Nn) corresponds to the case of maximal symmetry, the interaction being invariant under the group U(N)×U(n). The algebra (N,n,ν=Nn-1) is shown to have a solution only if n=N, in which case it corresponds to symmetry under the group U(N). Finally we consider the particularly instructive algebra (N=3,n=3,ν=3) which is shown to admit of three physically distinct solutions, which correspond to invariance of the interaction under a three-parameter Abelian group, the orthogonal group in three dimensions, and the 24-element permutation group S 4 , respectively.

A Dynamical Approach to the Leptonic Decays of Baryons

The leptonic decays of hyperons are analysed using the dynamical self-consistency requirement and the assumption that the leptonic decay processes originate in a driving weak interaction of the current-current type. The hadron currents are assumed to transform like an octet component of SU (3). It is shown that the axial vector amplitude can have a resonance with almost the same values of the parameters as were fixed in a previous paper4l to explain the nonleptonic decays of hyperons in the same spirit as mentioned above. The enhancement mechanism can predict the D/F ratio of the axial vector amplitude; this is found to be 2.5, which is not inconsistent with experiment.

A Dynamical Approach to Nonleptonic Decays of Hyperons

The pionic decays of hyperons are analysed under the dynamical self-consistency requirement and the assumption that the nonleptonic decay processes are originated in a driving weak interactions of the current-current type. The hadron currents are assumed to be transformed like an octet component of SU (3). It is shown that the octet enhancement of decay amplitudes can be attributed to the strong interactions. In particular, the qualitatively satisfactory numerical results can be obtained in a model where the s-wave amplitude is enhanced by two levels while the p-wave is by a single level.

A New Dynamical Model for Meson States

Reasons are advanced for considering the effect of inelastic coupling to the vector-vector state on the meson states occurring in pseudoscalar-pseudoscalar scattering. A model based on the angular-momentum properties of the box graph containing 2 vector mesons in the intermediate state is developed. In the dynamics of the model the "elementary" vector-meson octet disappears, and a vector-octet Regge trajectory is generated. A Pomeranchuk trajectory is generated in the singlet amplitude. The slopes of the trajectories are in quite reasonable agreement with experimental indications. The model predicts a $\ensuremath{\rho}\ensuremath{\pi}\ensuremath{\pi}$ coupling constant a factor of 2 or 3 above the experimental value and about the same as the value predicted by bootstrap models. The model shares with bootstrap models the feature that $S{U}_{3}$ symmetry is "predicted" as a consequence of dynamical self-consistency. The basic effect is expected to be important in more conventional dynamical schemes, and suggestions are made for further work.

Interrelations Among Dissymmetries inSU3Supermultiplets

The mass splittings within the (unitary) supermultiplets of the strongly-interacting particles are studied in the octuplet model of ${\mathrm{SU}}_{3}$ symmetry. Using arguments based on the requirement of dynamical self-consistency, qualitative properties of the first-order perturbations in masses and coupling constants are studied in the ladder approximation. Further evidence for the origin of the particular type of dissymmetry leading to the Gell-Mann-Okubo mass formula and to isotopic spin symmetry is gotten through the relations between the dynamical effects in different supermultiplets. The observed mass splittings in the pseudoscalar meson, vector meson, and the \textonehalf{}(+) baryon octuplets, as well as in the $\frac{3}{2}(+)$ baryon decuplet, are compatible with the general features expected from first-order perturbations. Some higher order perturbations in the $\frac{3}{2}(+)$ decuplet are also discussed.