Journal of Statistical Physics, Vol. 13, No. 5, 1975 Nonlinear Momentum Relaxation of an Impurity in a Harmonic Chain James T. Hynes, 1 Raymond Kapral, 2 and Michael Weinberg 8 Received March 21, 1975 A microscopic derivation of the generalized Langevin equation for arbitrary powers of the momentum of an impurity in a harmonic chain is presented. As a direct consequence of the Gaussian character of the conditional momentum distribution function, nonlinear momentum coupling effects are absent for this system and the Langevin equation takes on a particularly simple form. The kernels which characterize the decay of higher powers of the impurity momentum depend on the ratio of the masses of the impurity and bath particles, in contrast to the situation for the momentum Langevin equation for this system. The simplicity of the harmonic chain dynamics is exploited in order to investigate several features of the relaxation, such as the factorization approximation for time-dependent correlation functions and the decay of the kinetic energy autocorrelation function. KEY WORDS: Brownian motion; linear harmonic chain; Langevin equation ; energy relaxation; Gaussian non-Markovian process. 1. INTRODUCTION In the past the study of the Brownian motion of an impurity particle sus- pended in a fluid has relied heavily upon a phenomenological approach. (1) This work was supported in part by the National Science Foundation (J. T. H.) and the National Research Council of Canada (R.K). 1 Department of Chemistry, University of Colorado, Boulder, Colorado. Alfred P. Sloan Fellow. 2 Department of Chemistry, University of Toronto, Toronto, Ontario, Canada. 3 Department of Physics, Toledo University, Toledo, Ohio. 427 9 1975 Plenum Publishing Corporation, 227 West 17th Street, New York, N.Y. 10011. No part of this pub- lication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording, or otherwise, without written permission of the publisher.