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Physical Review A

Behavior of Autocorrelation Functions

Noel R. Corngold

Published Oct 1, 1972• 32 citations
Publisher

Abstract

This paper is concerned with the behavior of the autocorrelation and memory functions of statistical mechanics as $t\ensuremath{\rightarrow}\ensuremath{\infty}$. A simple calculation shows that for a large class of models leading to power-law behavior, the two functions have opposite sign and common functional behavior in the limit $t\ensuremath{\rightarrow}\ensuremath{\infty}$.

DOI
10.1103/physreva.6.1570

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This paper is concerned with the behavior of the autocorrelation and memory functions of statistical mechanics as t\ensuremath→\ensuremath∞t\ensuremath{\rightarrow}\ensuremath{\infty}t\ensuremath→\ensuremath∞. A simple calculation shows that for a large class of models leading to power-law behavior, the two functions have opposite sign and common functional behavior in the limit t\ensuremath→\ensuremath∞t\ensuremath{\rightarrow}\ensuremath{\infty}t\ensuremath→\ensuremath∞.
We present a study of electric transport at high temperature in a model of strongly interacting spinless fermions without disorder. We use exact diagonalization to study the statistics of the energy eigenvalues, eigenstates, and the matrix elements of the current. These suggest that our nonrandom Hamiltonian behaves like a member of a certain ensemble of Gaussian random matrices. We calculate the conductivity $\ensuremath{\sigma}(\ensuremath{\omega})$ and examine its behavior, both in finite-size samples and as extrapolated to the thermodynamic limit. We find that $\ensuremath{\sigma}(\ensuremath{\omega})$ has a prominent nondivergent singularity at $\ensuremath{\omega}=0$ reflecting a power-law long-time tail in the current autocorrelation function that arises from nonlinear couplings between the long-wavelength diffusive modes of the energy and particle number.
We formulate a single-cluster Monte Carlo algorithm for the simulation of the random-cluster model. This algorithm is a generalization of the Wolff single-cluster method for the q-state Potts model to noninteger values q>1. Its results for static quantities are in a satisfactory agreement with those of the existing Swendsen-Wang-Chayes-Machta (SWCM) algorithm, which involves a full-cluster decomposition of random-cluster configurations. We explore the critical dynamics of this algorithm for several two-dimensional Potts and random-cluster models. For integer q, the single-cluster algorithm can be reduced to the Wolff algorithm, for which case we find that the autocorrelation functions decay almost purely exponentially, with dynamic exponents z(exp)=0.07 (1), 0.521 (7), and 1.007 (9) for q=2, 3, and 4, respectively. For noninteger q, the dynamical behavior of the single-cluster algorithm appears to be very dissimilar to that of the SWCM algorithm. For large critical systems, the autocorrelation function displays a range of power-law behavior as a function of time. The dynamic exponents are relatively large. We provide an explanation for this peculiar dynamic behavior.
The universality class, even the order of the transition, of the two-dimensional Ising model depends on the range and the symmetry of the interactions (Onsager model, Baxter-Wu model, Turban model, etc.), but the critical temperature is generally the same due to self-duality. Here we consider a sudden change in the form of the interaction and study the nonequilibrium critical dynamical properties of the nearest-neighbor model. The relaxation of the magnetization and the decay of the autocorrelation function are found to display a power law behavior with characteristic exponents that depend on the universality class of the initial state.