Directed Polymers in Restricted Geometries
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DIRECTED POLYMERS IN RESTRICTED GEOMETRIES G. ZUMOFENS, J. KLAFTER#, AND A. BLUMEN& S Physical Chemistry Laboratory, ETH-Zentrum, CH-8092 ZMrich, Switzerland # School of Chemistry, Tel-Aviv University, Tel-Aviv, 69978 Israel & Theoretical Polymer Physics, University of Freiburg, W-7800 Freiburg, Germany ABSTRACT We study numerically directed polymers in random potential fields for one-dimensional and fractal substrates. For fractal substrates the time evolution of the mean transverse fluctuations depends besides on the randomness of the potential also on the fractal nature of the substrate. The two effects enter in a subordinated way, i.e. the corresponding characteristic exponents due to the potential and the substrate combine multiplicatively. For a one-dimensional substrate the propagator P(x, t), the probability distribution of the
transverse displacement x(t), follows the scaling form P(x, t) ,
(x2(t))-'/
2
where ý is
f(ý),
the scaling variable ý = x/(x2(t))1/ 2 . The numerical results support the scaling function f() - exp(-c 6 ) with 6 > 2 which indicates an "enhanced" Gaussian behavior. These results are compared with those of a related "toy model". INTRODUCTION The problem of directed polymers (DP) in random media [1] has been a subject of extensive research due to its intimate relationship to several fundamental problems such as the dynamics of surface growth [2], interfaces in random spin systems [3] and the driven Burgers equation [4] and has been shown to have properties previously encountered in the study of spin glasses [5,6]. Of special interest have been the enhanced transverse fluctuations of DPs [1] and the low temperature (strong coupling) to high temperature (weak coupling) phase transition [7,8]. The DP problem is defined by walks directed in d space-time dimensions with coordinates (x, t), where d - 1 is the substrate dimension. The problem obeys a partial differential equation for which an equivalent representation is given by the path integral
[1,7,9,101 WO(x, t) =
J (xt)
Dx'(t')exp f- J0 dt'[(1/4D)('dX'/dt')
2
+ r7(x',
,
(1)
where W(x,t) is the weight of all DPs with one end at (x,t) = (0,0) and the other at (x, t). D is the diffusion coefficient and 77(x, t) is a random potential field which is usually assumed to be either Gaussian or white noise with delta function correlation:
(t7(x, t)) = 0
and
(j(x, t)r7(x', t')) = A2 d-I(x - x')6(t - t').
(2)
The randomness term given by Eq.(2) causes the DPs to be stretched in the transverse direction so that low energy sites, even far from the origin, are visited. This randomnessinduced stretching contributes to transverse fluctuations which have been shown to grow with time, at low temperatures, faster than expected for a simple Brownian motion, (x 2 (t)) -,, t', v > 1. Some analytical results which have been derived for DP in dimensions d = (1 + 1) demonstrate that v = 4/3 , a result which has been confirmed by the
Mat. Res. Soc. Symp. Proc. Vol. 290. @1993 Materials Research Society
20
simulations. The exponent is assumed
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