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Equilibrium statistics of a slave estimator in Langevin processes

David S. Dean1,2, Ian T. Drummond1, Ron R. Horgan1, and Satya N. Majumdar2,3

  • 1DAMTP, CMS, University of Cambridge, Cambridge, CB3 0WA, United Kingdom
  • 2Laboratoire de Physique Théorique, UMR CNRS 5152, IRSAMC, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 04, France
  • 3Laboratoire de Physique Théorique et Modèles Statistiques, UMR 8626, Université Paris Sud, Bâtiment 100, 91045 Orsay Cedex, France

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Phys. Rev. E 71, 031103 – Published 11 March, 2005

DOI: https://doi.org/10.1103/PhysRevE.71.031103

Abstract

We analyze the statistics of an estimator, denoted by ξt and referred to as the slave, for the equilibrium susceptibility of a one dimensional Langevin process xt in a potential ϕ(x). The susceptibility can be measured by evolving the slave equation in conjunction with the original Langevin process. This procedure yields a direct estimate of the susceptibility and avoids the need, when performing numerical simulations, to include applied external fields explicitly. The success of the method, however, depends on the statistical properties of the slave estimator. The joint probability density function for xt and ξt is analyzed. In the case where the potential of the system has a concave component the probability density function of the slave acquires a power law tail characterized by a temperature dependent exponent. Thus we show that while the average value of the slave, in the equilibrium state, is always finite and given by the fluctuation-dissipation relation, higher moments and indeed the variance may show divergences. The behavior of the power law exponent is analyzed in a general context and it is calculated explicitly in some specific examples. Our results are confirmed by numerical simulations and we discuss possible measurement discrepancies in the fluctuation dissipation relation which could arise due to this behavior.

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Phys. Rev. E 71, 031103– Published 11 March, 2005
  • Received 22 September 2004

© 2005 American Physical Society

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