7.1 Diffusion Monte Carlo (DMC)
First suggested in the Forties (see [MEYERS 56]);
rediscovered in the eighties ([CEPERLEY 80]).
Originally applied to the ground state of a bosonic system
such as $\scriptstyle 4$He
[KALOS 74,
WHITLOCK 79].
Later: extended to fermions and excited states
(
[BARNETT 86,
CEPERLEY 88]); see also
[CEPERLEY 96].
Time-dependent Schroedinger equation for a particle of mass
$m$ in a potential $U(r)$
$
i \hbar \frac{\textstyle \partial \Psi(r,t)}{\textstyle \partial t} = {\rm H} \, \Psi(r,t)
$
(7.1)
with the energy operator
$
{\rm H} \equiv
- \frac{\textstyle \hbar^{2}}{\textstyle 2m} \nabla^{2} + \left[ U(r) - E_{T} \right]
$
(7.2)
and a trial energy $E_{T}$
Define an imaginary time variable $s \equiv it/\hbar$; then
$
\frac{\textstyle \partial \Psi(r,s)}{\textstyle \partial s} = D \nabla^{2} \Psi(r,s)
-\left[ U(r) - E_{T} \right] \Psi(r,s)
$
(7.3)
$\Longrightarrow$ Diffusion with autocatalysis!
Visualize $\Psi$ as describing the density of bacteria diffusing in a fluid
with locally varying nutrient concentration.
Expand $\Psi$ in eigenfunctions $\Psi_{n}$ of ${\rm H}$:
$\Longrightarrow$
Solve 7.3 for various values of $E_{T}$:
find that $E_{T}$ for which $I(s)$ remains stationary. Then
$E_{T}=E_{0}$ and $\Psi = \Psi_{0}$.
How to solve 7.3?
Consider the diffusion and the autocatalysis parts of the equation
separately.
Diffusion part:
$
\frac{\textstyle \partial n(r,t)}{\textstyle \partial t} = D\, \nabla^{2} n(r,t)
$
(7.5)
May be solved by a random walk procedure:
Start
$N$ Brownian walkers
$
r_{i}(t_{n+1})=r_{i}(t_{n})+\$\bf\xi _{i}\,,\;\;\;i=1,\dots N
$
(7.6)
where
$\xi_{x,y,z}$ are drawn from a Gauss distribution with
$\sigma^{2}=2D\, \Delta t$.
Now consider an ensemble of
$M$ such $N$-particle systems and write the local distribution density
at time $t$ as
$
p(r,t) \equiv \langle \delta \left[ r_{i}(t)-r \right]
rangle = \frac{\textstyle 1}{\textstyle M}
\frac{\textstyle 1}{\textstyle N}
\sum \limits_{l=1}^{M} \sum \limits_{i=1}^{N}
\delta \left[ r_{i,l}(t)-r \right]
$
(7.7)
Autocatalysis part:
$
\frac{\textstyle \partial n(r,t)}{\textstyle \partial t} = f(r) \, n(r,t)
$
(7.8)
Instead of writing the formal solution as
$
n(r,t)=n(r,0)\,exp \left[f(r)t\right]
$
(7.9)
Consider an ensemble of $M$ systems of $N$ particles each, at fixed positions.
Let the number $M$ of systems in the ensemble be allowed to vary:
- Systems that contain many particles located at positions with
high values of $f(r)$ are replicated
- Systems with unfavorable configurations are deleted
Procedure:
Let the ensemble be given at step $t_{n}$.
- For each of the $M(t_{n})$ systems, determine the
multiplicity (see equ. 7.9)
$
K_{l}=exp\left[ \sum \limits_{i=1}^{N} f(r_{i,l}) \, \Delta t \right]\,,
\;\;\;l=1, \dots M(t_{n})
$
(7.10)
- Replicate the $l$-th system such that on the average
$K_{l}$ copies are present. To achieve this, produce first
$int (K_{l})-1$ copies (where $ int(..)$= next smaller integer)
and then, with probability $w \equiv K_{l}- int(K_{l})$,
one additional copy. (In practice, draw $\xi$ equidistributed
$\in [0,1]$ and check whether $\xi \leq w$.) If
$K_{l} < 1$, remove, with probability $1-K_{l}$, the
$l$-th system from the ensemble.
Again, the distribution density
7.7 is an estimate of the density at position
$r$
Combining the two stochastic techniques for solving the
diffusion and autocatalytic equations we obtain the following
procedure.
Figure 7.1:
Quantum mechanical diffusion Monte Carlo
$N$ (non-interacting) particles of mass $m$, distributed at random in
a given spatial region, are subject to the influence of a potential
$U(r)$. Determine the "diffusion constant"
$D=\hbar^{2}/2m$; choose a trial energy $E_{T}$, a time step
$\Delta s$ and an initial ensemble size $M(s_{0})$.
-
For each system $l$ ($=1, \dots \, M(s_{0})$) in the ensemble
and for each particle $i$ ($=1, \dots \, N$) perform a random
displacement step
$
r_{i,l}(s_{n+1})=r_{i,l}(s_{n})+\xi_{i,l}\,,\;\;\;
i=1,\dots N
$
where the components of the vector $\xi_{i,l}$ are picked from
a Gaussian distribution with $\sigma^{2}=2D\,\Delta s$.
-
For each system $l$ determine the multiplicity $K_{l}$
according to
$
K_{l}=exp\left\{ \left[ \sum_{i=1}^{N} U(r_{i,l})-E_{T} \right]
\Delta s \right\}
$
-
Produce $\mbox{int}(K_{l})-1$ copies of each system
($\mbox{int}(...)$ denoting the nearest smaller integer;)
with probability $w =K_{l}-\mbox{int}(K_{l})$ produce one additional
copy, such that on the average there are $K_{l}$ copies in all. If
$ K_{l}<1 $, purge the system with probability
$1-K_{l}$ from the ensemble.
-
If the number $M$ of systems contained in the ensemble increases
systematically (i.e. for several successive steps), choose a smaller
$E_{T}$; if $M$ increases, take a larger $E_{T}$.
-
Repeat until $M$ remains constant; then the ground state
energy is $E_{0}=E_{T}$ and
$
\Psi_{0}(r) = \langle \delta (r_{i,l}-r) \rangle
$
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So far: $\Psi$ must be real and $\ge 0$
$\Longrightarrow$ Bosons
($\scriptstyle 4 $He or similar).
Generalization for fermions: fixed node and
released node approximation [CEPERLEY 88].
Note: The analogy between the wave function
$\Psi(r,t)$ and the local density $n(r,t)$ is purely formal.
It must be distinguished from the physical interpretation of
$|\Psi (r)|^{2}=prob\{ quantum\;object\;to\; be\; found\; at\; r\}$
Importance sampling DFT:
If the potential $U(r)$ is highly negative in some region of space,
the autocatalytic term may get out of control and must be handled differently:
- Introduce an estimate
$\Psi_{T}(r)$ of the correct solution $\Psi_{0}(r)$;
- define an auxiliary function
$
f(r,s) \equiv \Psi_{T}(r)\,\Psi(r,s)
$
(7.11)
- insert this in 7.3 to find
$
\frac{\textstyle \partial f}{\textstyle \partial s} = D \nabla^{2} f
- \left[ \frac{\textstyle {\rm H} \Psi_{T}}{\textstyle \Psi_{T}} - E_{T} \right] \, f
-D \,\nabla \cdot \left[ f \, \nabla \ln | \Psi_{T} |^{2} \right]
$
(7.12)
$
\frac{\textstyle {\rm H}\,\Psi_{T}}{\textstyle \Psi_{T}} \approx E_{0} \approx E_{T}
$
(7.13)
the autocatalytic term is now well-behaved, and the multiplicity
$K_{l}$ will remain bounded.
Visualisation of equ. 7.12: The new term looks like an
advective contribution. In the image of a diffusing
and multiplying bacterial strain there is now an additional
driving force
$
F(r) \equiv \nabla \, \ln |\Psi_{T}(r)|^{2}
$
(7.14)
which creates a flow, or drift. This means that the
individual diffusors follow a preferred direction along $F(r)$:
$
r_{i,l}(s_{n+1})=r_{i,l}(s_{n})+ \xi_{i,l}+D \Delta s \,
F(r_{i,l}(s_{n}))
$
(7.15)
The multiplicity $K_{l}$ is now
$
K_{l}=exp \left\{ \left[
\frac{\textstyle {\rm H} \Psi_{T}}{\textstyle \Psi_{T}}
- E_{T} \right] \, \Delta s \right\}
$
(7.16)
Green's function Monte Carlo (GFMC):
Another formulation of the DMC procedure [SKINNER 85].
Recent literature:
See [CEPERLEY 96] and web sites [CEPERLEY WWW] or [CAVENDISH WWW].
vesely 2006
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