Exponential series expansion for correlation functions of many-body systems

Fabrizio Barocchi, Eleonora Guarini, and Ubaldo Bafile
Phys. Rev. E 90, 032106 – Published 8 September 2014

Abstract

We demonstrate that in Hamiltonian many-body systems at equilibrium, any kind of time dependent correlation function c(t) can always be expanded in a series of (complex) exponential functions of time when its Laplace transform C̃(z) has single poles. The characteristic frequencies can be identified as the eigenfrequencies of the correlation. This is done without introducing the concepts of fluctuating forces and memory functions, due to Mori and Zwanzig and extensively used in the literature in the last decades. Our method is based on a different projection technique in the Hilbert space S of the system and shows that appropriate approximations of the exponential series are related to the contraction of S to a finite, usually small, number of dimensions. The time dependence of correlation functions is always described in detail by a multiple-exponential functionality also at long times. This result is therefore also valid for correlation functions of many-body Hamiltonian systems for which a power-law dependence, observed in restricted time ranges and predicted to be the asymptotic one, can be considered at most as a useful approximate modeling of long-time behavior.

  • Received 9 January 2014
  • Revised 2 June 2014

DOI:https://doi.org/10.1103/PhysRevE.90.032106

©2014 American Physical Society

Authors & Affiliations

Fabrizio Barocchi1, Eleonora Guarini1, and Ubaldo Bafile2

  • 1Dipartimento di Fisica e Astronomia, Università di Firenze, via G. Sansone 1, I-50019 Sesto Fiorentino, Italy
  • 2Consiglio Nazionale delle Ricerche, Istituto dei Sistemi Complessi, via Madonna del Piano 10, I-50019 Sesto Fiorentino, Italy

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Vol. 90, Iss. 3 — September 2014

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