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Physical Review A

Long-Time Behavior of Space- and Time-Dependent Correlation Functions

James W. Dufty

Published May 1, 1972• 41 citations
Publisher

Abstract

The long-time behavior of space- and time-dependent correlation functions is obtained including the ring contributions or correlated two-body effects. The latter give rise to anomalous hydrodynamic effects and persistent correlations in the nonhydrodynamic part.
DOI
10.1103/physreva.5.2247

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The low-frequency, long-wavelength properties of a paramagnet in d dimensions are studied within the framework of a non-linear Langevin equation. The frequency and wave-vector dependent transport coefficient Gamma (k, omega ) has a singular part which varies as omega d/2 at k=0 so its time Fourier transform decays as t -(1+d 2)/, in contrast to the assumptions of linearised hydrodynamics according to which Gamma (k,t) decays rapidly in a microscopic time. Both reversible and irreversible processes give contributions to this effect but with opposite signs. The latter contribution is non vanishing as k to 0 only because we include a drift term in the Langevin equation which depends explicitly on the energy density. The consequences of this generalisation of the usual Langevin equation are investigated in detail in a perturbation expansion.
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We construct a quantum Schwinger-Keldysh (SK) effective field theory for the diffusive hydrodynamics of a conserved scalar field. Quantum corrections within the SK framework are guided by fluctuation-dissipation relations, enforced via a dynamical Kubo-Martin-Schwinger (KMS) symmetry. We find that the KMS symmetry necessarily generates fluctuation contributions in the SK effective action at all orders in the noise field, thereby giving rise to intrinsically non-Gaussian noise. We use our results to compute one-loop quantum corrections to the two-point density-density retarded correlation function, leading to a quantum generalization of hydrodynamic long-time tails. Our results apply at arbitrarily high orders in ℏ\hbarℏ. The one-loop results for retarded correlation functions have been expressed in terms of a family of polynomials. We also provide a closed-form expression for the one-loop results at leading order in the wavevector expansion.