P HYSICAL R EVIEW
LETTERS
VOLUME 77 9 DECEMBER 1996 NUMBER 24
Algebraic Relaxation Laws for Classical Particles in 1D Anharmonic Potentials
Surajit Sen,
1
Robert S. Sinkovits,
2
and Soumya Chakravarti
3
1
Physics Department, State University of New York,
Buffalo, New York 14260-1500
2
Code 6410, Naval Research Laboratory, Washington, D.C. 20375
3
Physics Department, California State Polytechnic University,
Pomona, California 91768
(Received 8 August 1995)
Using extensive numerical analysis and exact calculations we show that the relaxation of a classical
particle in 1D anharmonic potential landscapes with a leading quartic term follows a 1t decay law
at all temperatures, leading to a logarithmically increasing mean square displacement. For leading
anharmonic terms of form x
2n
we find that the asymptotic relaxation is consistent with 1t
f
, where
f 1n 2 1 at all temperatures. We briefly comment on the possible implications of this result in
the study of displacive structural transitions and in complex systems. [S0031-9007(96)01811-X]
PACS numbers: 05.20.Gg
A variety of problems of considerable interest in physics
is closely related to the behavior of a single classical parti-
cle relaxing in an anharmonic single or multiwell potential
landscape while in thermal contact [1]. A detailed micro-
scopic understanding of relaxation in 1D multiwell poten-
tials could give important clues to the problems associated
with thermally activated dynamics in the glass transition
[2] and of relaxation in related complex systems [3].
In this Letter we show both numerically and analytically
that the relaxation of any dynamical variable of a classical
particle in a variety of anharmonic potentials follows a
1t
f
asymptotic decay law. This relaxation represents
the loss of knowledge of the initial conditions of the
anharmonic oscillator due to thermal effects. We find that
f is sensitive to the exponent of the leading anharmonic
term in the potential. To our knowledge, these are the
simplest systems for which a wide variety of slow algebraic
decay is found. We conclude with a discussion of the
possible implications of our findings.
We start with a classical particle in an anharmonic
potential, V
anh
x and consider several distinct forms for
V
anh
x. Initially, we focus on potentials with leading
quartic anharmonicity, perhaps the most common form of
anharmonicity encountered in simple physical problems.
These potentials are V
anh
x,
I
A
2
x
2
1
B
4
x
4
1 Cx, II coshx, III cosx .
(1)
Our objective is to study the time-dependent behavior
of some chosen dynamical variable Ct (e.g., position
[xt ], velocity [yt ], etc.). We assume that the system
is in thermal contact which may or may not lead to relax-
ation to equilibrium after the system has been subjected
to an infinitesimal perturbation (i.e., no assumptions re-
garding ergodicity are made [4]). The study of the asymp-
totic behavior of such a relaxation process, after the
perturbation has been removed, which can generally be
characterized by the normalized relaxation function (RF)
Ct C0C0
2
, where ··· represents canonical en-
semble averages, is the focus of this Letter.
Cases (I) and (II) in Eq. (1) describe systems in which
the particle is spatially localized for all energies. To study
the relaxation of the particle in (I) and (II) numerically, we
integrate the equations of motion for each case at fixed to-
tal energy E from some initial time t to some final time
t 1t
E
, where t
E
refers to the period of motion for the
chosen E. xt , yt , and at (i.e., acceleration) and their
corresponding microcanonical ensemble RF’s are calcu-
lated and tabulated at 1024 equally spaced time intervals
spanning the period t
E
. The canonical ensemble RF is
0031-9007 96 77(24) 4855(5)$10.00 © 1996 The American Physical Society 4855