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arXiv:cond-mat/0301527

Fast and simple Complex and Slow

Moshe Schwartz

Published Jan 27, 2003• 0 citations

Abstract

The decay of a general time dependent structure factors is considered. The dynamics is that of stochastic field equations of the Langevin type, where the systematic generalized force is a functional derivative of some classical field Hamiltonian with respect to the field. Equations of this type are generic and describe many physical systems. It is usually believed that simple non linear systems exhibit exponential decay in time, while non linear complex systems ,such as ferromagnets at their critical point,decay slowly as a power law or stretched exponential. A necessary condition for slow decay is that the eigenvalues of the appropriate Fokker- Planck operator accumulate at zero for each momentum q .This is a property of the system. I argue here that when the necessary condition is obeyed slow or fast (exponential) decay are both possible for simple linear systems as well as for complex non liner ones. The actual form of decay is determined by what structure factor we prefer to observe. An explicit family of linear models in which the "natural" structure factors decay exponentially is constructed. It is shown how a more complex structure factor decays in those models with a slow decay form. It is further shown how in general non linear systems it is always possible to find structure factors that decay exponentially. The question of actual observation of such structure factors in experiment and numerical simulations is discussed.

DOI: 10.48550/arxiv.cond-mat/0301527

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The authors discuss the dynamics of an observable field related to a locally conserved field. They show that for a model of phase ordering dynamics and for a variation of model C critical dynamics, that the two-time momentum space correlation function decays asymptotically as a stretched exponential. They show that under some conditions the asymptotic behaviour is experimentally observable and argue that the non-exponential behaviour is not a feature specific to these models but should rather apply for any observable coupled to a locally conserved field.

The exact equation of state, two-time correlation function, and linear response function, are calculated in the limit of infinite

N

for a classical

N

-spin system with a ferromagnetic phase transition, in a particular nonuniform magnetic field. The correlation function can be analytically continued in temperature (or magneticfield strength) from the nonferromagnetic to the ferromagnetic region of the $T\ensuremath{-}H$ plane; the result of such continuation is not, however, the correlation function for the ferromagnetic region, but a function which grows exponentially in time. The frequency-dependent linear response function has a pole at zero frequency throughout the ferromagnetic region due to a broken symmetry; the corresponding function in the nonferromagnetic region developes a pole at zero frequency as the ferromagnetic region is approached, but when the function is continued in temperature (or field strength) into the ferromagnetic region, the pole detaches itself from the origin and moves up into the complex frequency plane, signifying an exponential growth in time of the linear response. The purpose of the model is to demonstrate that this kind of behavior does not contradict any general structural properties of equilibrium thermodynamic correlation or response functions. The possible general significance of such behavior for a theory of metastable states is discussed.

Gamow's explanation of the exponential decay law uses complex "eigenvalues" and exponentially growing "eigenfunctions". This raises the question, how Gamow's description fits into the quantum mechanical description of nature, which is based on real eigenvalues and square integrable wave functions. Observing that the time evolution of any wave function is given by its expansion in generalized eigenfunctions, we shall answer this question in the most straightforward manner, which at the same time is accessible to graduate students and specialists. Moreover the presentation can well be used in physics lectures to students.

Abstract The autocorrelation function in many complex systems shows a crossover in the form of its decay: from a stretched exponential relaxation (SER) at short times to a power law at long times. Studies of the mechanisms leading to such multiple relaxation patterns are rare. Additionally, the inherent complexity of these systems makes it hard to understand the underlying mechanism leading to the crossover. Here we develop a simple one-dimensional spin model, which we call a domain wall (DW) to doublon model, that shows such a crossover as the nature of the excitations governing the relaxation dynamics changes with temperature and time. The relevant excitations are DWs and bound pairs of DWs, which we term ‘doublons’. The diffusive motion of the DWs governs the relaxation at short times, whereas the diffusive motion of the doublons yields the long-time decay. This change of excitations and their relaxation leads to a crossover from SER to a power law in the decay pattern of the autocorrelation function. We augment our numerical results with simple physical arguments and analytic derivations.

A simple quantum-mechanical model consisting of a two-level system coupled to a continuum, where the interaction can be varied from perturbative to strong, is examined for what can be learned about quantum mechanical decay using elementary methods. The decay spectrum and time dependence of the amplitudes can be calculated in this model, without resorting to perturbation theory. The spectrum is not precisely Lorentzian nor is the decay precisely exponential, features that appear to be shared by all quantum-mechanical models of decaying systems. The conditions for exponential decay and some of the reasons for deviation from this at both short and long times are discussed.

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