This paper deals with applications of non-Gaussian method to treat spatial fluctuations and their manifestations in the near-critical region. The basis for calculations is the celebrated Landau-Ginzburg-Wilson Hamiltonian. The results obtained are largely analytical, but numerical computations have also been done for further support and illustration. We have calculated expressions for the partition function, dynamic susceptibility, and total scattering intensity, as well as the correlation function and the correlation range. It is demonstrated that, in the limit of the volume V\ensuremath{\rightarrow}\ensuremath{\infty} or the reduced temperature \ensuremath{\tau}==(T-
Tc)/
Tc being large enough, the obtained results all tend to their Gaussian estimates. However, for finite volumes or small but nonzero \ensuremath{\epsilon}, asymptotic behaviors are found that are mostly governed by exponential functional relations. Two illustrative examples are provided for experimental support: (1) Brillouin scattering in ferroelastics and (2) Raman scattering in simple fluids.
We have studied the amplitude and temporal correlations of intensity fluctuations of multiply scattered light transmitted through a colloidal suspension. When the coherence length
lc of the incident light becomes comparable to the width of the probability distribution of photon-path lengths, both the amplitude and dynamics of intensity fluctuations become functions of
lc. In this article we present a simple theory that relates these coherence effects to the photon path-length distribution P(L). We have tested this theory using P(L) calculated from computer simulations and from the diffusion approximation, and find that our theory qualitatively accounts for the observed dependences on coherence length. Our results indicate when the effects of finite laser coherence must be taken into account in quasielastic multiple scattering. In the few-scattering regime of quasielastic multiple scattering, we find that the intensity autocorrelation function generally has a stretched-exponential form, and we discuss the origin of this behavior.
Recently the authors have derived kinetic equations describing the behavior of the spin autocorrelation function ${\ensuremath{\Gamma}}_{q}(t)$ in a Heisenberg system at infinite temperature. In the present paper, this derivation is extended to finite temperatures (above the critical point). It is shown that, in the Weiss limit where the number of neighbors $Z\ensuremath{\rightarrow}\ensuremath{\infty}$, the effects of the equilibrium (Ornstein-Zernicke) correlations present in the system at the initial time can be entirely incorporated within an effective temperature-dependent interaction which governs the temporal behavior of the autocorrelation function (af). A non-Markoffian kinetic equation is obtained in which the kernel is highly nonlinear in the complete af ${\ensuremath{\Gamma}}_{q}(t)$; this contrasts with the infinite-temperature case, where the kernel was a functional of the direct af only. This new feature leads to simple approximations near the critical point, as will be discussed in detail in the next paper of this series.
We derive analytical expressions for the spectral moments of the dynamical response functions of the Hubbard model using the high-temperature series expansion. We consider generic dimension
d as well as the infinite-
d limit, arbitrary electron density
n, and both finite and infinite repulsion
U. We use moment-reconstruction methods to obtain the one-electron spectral function, the self-energy, and the optical conductivity. They are all smooth functions at high temperature and, at large
U, they are featureless with characteristic widths of the order of the lattice hopping parameter
t. In the infinite-
d limit, we compare the series expansion results with accurate numerical renormalization group and interaction expansion quantum Monte Carlo results. We find excellent agreement down to surprisingly low temperatures, throughout most of the bad-metal regime, which applies for $T\ensuremath{\gtrsim}(1\ensuremath{-}n)D$, the Brinkman-Rice scale. The resistivity increases linearly in
T at high temperature without saturation. This results from the
1/T behavior of the compressibility or kinetic energy, which play the role of the effective carrier number. In contrast, the scattering time (or diffusion constant) saturates at high
T. We find that $\ensuremath{\sigma}(n,T)\ensuremath{\approx}(1\ensuremath{-}n)\ensuremath{\sigma}(n=0,T)$ to a very good approximation for all
n, with $\ensuremath{\sigma}(n=0,T)\ensuremath{\propto}t/T$ at high temperatures. The saturation at small
n occurs due to a compensation between the density dependence of the effective number of carriers and that of the scattering time. The
T dependence of the resistivity displays a kneelike feature which signals a crossover to the intermediate-temperature regime where the diffusion constant (or scattering time) starts increasing with decreasing
T. At high temperatures, the thermopower obeys the Heikes formula, while the Wiedemann-Franz law is violated with the Lorenz number vanishing as
1/T2. The relevance of our calculations to experiments probing high-temperature transport in materials with strong electronic correlations or ultracold atomic gases in an optical lattice is briefly discussed.
The dynamics of spin fluctuations in nonequilibrium and equilibrium states is studied in the one-phase region above
Tc. The equal-time structure factor is computed after quenching the system from an infinite temperature to T>
Tc. The result shows growing spatial patterns which can be scaled with temperature-dependent scaling functions. The different-time (dynamic) structure factor is computed by taking time averages while the system is fluctuating in equilibrium states. The equilibrium autocorrelation decays following a diffusive exponential term and a slower but transient term. The relaxation rate due to the diffusion supports the dynamic scaling.