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Brownian dynamics with absorbing boundary conditions
Abstract Book of the XXIII IUPAP International Conference on Statistical Physics, Genova, Italy, (9-13 July 2007)
Аннотация
The one-dimensional motion of a Brownian particle, either free or in a linear force field,
in an infinite medium is well understood both analytically and numerically. The
associated Fokker-Planck equation may be solved analytically, for $\delta$-function
initial conditions, and a number of numerical schemes has been used to simulate the
corresponding stochastic differential equation. However, the presence of
a boundary, herein considered to be perfectly absorbing, introduces subtleties both
in the analytical solution and the numerical simulations.
For a free, inertialess Brownian particle the Green's function of the associated
Fokker-Planck equation may
obtained via the method of images: the physical source at $x_0$ is reflected with
respect to the absorbing boundary (considered to be at $x_wall =0$) to
create an anti-particle (hole) at $-x_0$. The absorbing-wall Green's function is, then,
obtained by propagating both particles and anti-particles via the infinite-medium
Green's function: particles and anti-particles annihilate at the boundary as required.
However, the naive numerical simulation of the corresponding Wiener process,
via, e.g., the Euler scheme, introduces an error at the boundary: the numerically
obtained particle density does not vanish at the absorbing boundary. The non-zero
value of the density arises from the stochastic nature of the particle trajectories. As
particle motion is not ballistic, a particle may recross the boundary.
Therefore, there exists a finite probability~1 that a particle starting at $x_0$ and
ending at $x_1$ touched the wall during the simulation time step
$P_abs = \Big - x_0 x_1 /(D t ) \Big $ with
$D$ the diffusion coefficient and $t$ the simulation time step.
If the simulation procedure is corrected accordingly numerical results agree
with the analytical solution.
Moreover, for a continuous injection of particles at, for example, the origin the naive numerical scheme introduces an
unphysical boundary layer close to the injection point--the concentration close to the
injection point is depleted. The fictitious boundary
layer stems from the usual implementation of the numerical simulation that considers
particle injection occurring exactly at the injection point. However,
a finite probability exists that a particle enters the simulation domain by crossing the boundary,
namely, the injected particle does not necessarily originate at the injection point. The spurious boundary
layer disappears~2 when particles are injected with an entrance probability
distribution $P_inject (x) =
\Big \pi/ (2 \sigma) \Big ^1/2 erfc \Big x/(2 \sigma) \Big $ where
$\sigma^2 = 2 D t$. Both corrections, particle recrossing and proper entrance distribution,
were used in numerical simulations of continuous injection of
particles with an absorbing wall. Results are shown in Fig. (1).
The average concentration obtained from the naive simulations exhibits the spurious
boundary layer close to the injection point, and it does not vanish at the
absorbing boundary, whereas the concentration determined
from the modified simulations is in excellent agreement with the steady-state
analytical solution, a linear concentration profile.
For inertial particles in a linear force field, Kramer's equation, the boundary condition
at the absorbing wall is $f(0,v,t) = 0$ for $v>0$ with $f(x,v,t)$ the position
and velocity probability distribution of the Brownian particles.
Accordingly, the reflection of a physical source at $(x_0,v_0)$
creates a source of anti-particles in physical space, suggesting that a modification
of the usual method of images is required. Possible extensions of the method of
images for Kramer's problem will be presented.\\
1) T.\ M.\ A.\ O.\ M.\ Barenbrug, E.\ A.\ J.\ F.\ Peters, and J.\ D.\ Schieber, J. Chem. Phys. 117, 9202 (2002).\\
2) A.\ Singer and Z.\ Schuss, Phys. Rev. E 71, 026115 (2005).
