Persistence effects in first-passage time characteristics of liquids
Abstract Book of the XXIII IUPAP International Conference on Statistical Physics, Genova, Italy, (9-13 July 2007)Abstract
We analyse diffusive molecular motion in several liquids, obtained
from molecular dynamics simulations, by a first passage time analysis.
Within the simulation box, a slab of thickness $L$, with two (virtual)
parallel absorbing boundaries, normal to the $z$ axis, is defined. A
particle, initially present at a position $z=z_0$ in the slab, is
monitored until it crosses one of the boundaries, and the elapsed time
is recorded. From the diffusion equation, with Smoluchowski boundary
conditions $c(0)=c(L)=0$, one may derive 1 that the mean exit time
for an ensemble of such particles is given by the parabolic equation
$T(z_0)=12Dz_0(L-z_0)$, where $D$ is the diffusion
coefficient. Our simulation data, however, can only be reconciled with
this equation if we assume that each boundary is shifted outwards by
an amount $łambda_M$, i.e., we replace
$z_0\rightarrowz_0=z_0+łambda_M$ and
$L\rightarrowL=L+2łambda_m$.
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The quantity $łambda_M$ is identified as the Milne extrapolation
length 23. It is related to the presence of a kinetic boundary layer
near an absorbing interface, where large deviations from the
Maxwellian velocity distribution occur. This phenomenon can be
derived from a Fokker-Planck equation, which accounts for positions
and velocities, in contrast to the diffusion equation, where only
positions are present. Far away from the interface, however, the
system can still be described by a diffusion equation, however with an
apparently shifted absorbing boundary. The Milne length was found
analytically 3 as $łambda_M = |\zeta(12)|\,l_v$. Here
$\zeta(12)=- 1.4603\dots$ is a Riemann zeta function and
$l_v=D\,m/k_BT$ is a velocity correlation length, with $m$ the
molecular mass, $T$ the temperature and $k_B$ the Boltzmann constant.
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Our simulations show deviations from this theoretical prediction. For
simple liquids (Ar, O$_2$, CO$_2$) we obtain values approximately 1.5
times larger. For water (several models) approximately a factor 3 is
found. This is interpreted in terms of memory effects in the molecular
motion. It appears to be stronger in a hydrogen bonding liquid like
water than in simple liquids, in accordance with current understanding
of such systems.\\
1) S. Redner, A Guide to First-Passage Processes (Cambridge University Press, 2001).\\
2) M. Burschka and U. Titulaer, J. Stat. Phys. 25, 569 (1981).\\
3) T. W. Marshall and E. J. Watson, J. Phys. A: Math. Gen.
20, 1345 (1987).