Title: Sampling in Path Space with Application to Vacancy Diffusion
1Sampling in Path Space with Application to
Vacancy Diffusion
- (a work in progress)
- Frank J. Pinski
- University of Warwick and
- University of Cincinnati
- Andrew Stuart and Jochen Voss
- University of Warwick
- 24/11/2005
2Abstract
- A mathematical framework for the sampling of path
space, in which the Onsager-Machlup functional
plays a central role. - We would like to understand phenomena that occur
on a variety of time scales. The time scales of
the motion are a reflection of the free-energy
landscape. - We would like to incorporate the effects of the
fast degrees of freedom, as we describe the
evolution of the slower-moving variables. - The approach taken here is to employ a Langevin
equation in path space, and look at paths that
are conditioned to cross a relevant free-energy
barrier.
3- 10 X 9 lattice with one vacancy, 89 particles
4Outline
- Movie Diffusion in 2-D
- Motivation and Background
- Brownian Movement
- Path Space
- Formalism
- 1-D Results and Interpretation
- 2-D Results
- Future Directions
5Robert Brown Brownian movement
- In a pamphlet, published in
- 1828, Robert Brown reported
- moving particles suspended
- in the fluid within living
- pollen grains of Clarkia
- pulchella.
- http//www.whonamedit.com/doctor.cfm/2539.html
http//upload.wikimedia.org/wikipedia/en/4/43/Brow
n.robert.jpg
6Albert Einstein 1905
- By balancing the diffusion current with the drift
current, he extracted the relationship between
transport coefficients - Brownian motion is not ballistic
- From the measurement of the diffusion constant,
one can extract the Avogadro-Locschmidt number,
and determine the size of molecules - P. Hanggi and F. Marchesoni, Chao 15, 026101
(2005).
www.plexoft.com/SBF/images/AEin.gif
7- The Langevin Approach
- ? kBT dri/dt pi/m dpi/dt
Fi -?pi fi - where fi is a random force and ? is the damping.
- Over-damped, terminal velocity ?mi dri/dt Fi
fi - Rescale time use the Fluctuation-dissipation
theorem - dri ?/du Fi?(r) ?????1/2 dW/du
- Eulers method rn i? rn-1 i? Fi?(r) ?u
???u???1/2? ?in - Ito Calculus
- Set Fi?(r)0 to obtain the equation for Brownian
motion
8Brownian Movement
9The SPDE the evolution of paths
The original equation (SDE) dX/du f(X)
? dW/du Eulers method Xn Xn-1
f(Xn-1) ?u ????u1/2 ?n The probability
distribution p(x) C Exp -?u ?(x)
the Onsager-Machlup functional where
?(x) (4 ? ?u)-2 ?n (xn-xn-1 - ?u
f(xn-1) )2 Thus the product ?uV(x) can viewed
as a potential energy divided by kBT, and we can
write a new Langevin equation ?xn/?t
-?n ?(x) (2/?u)1/2 ?Wn/?t Expand and take the
continuum limit, we arrive at a SPDE ???xn/?t
?-2( ?2xn/?u2 - g(xn,?)) 21/2 ?wn/?t Where
g(x,?)?G/?x, and G(x,?) 1/2 ( f(x)2 ?2
f(x) )
Stuart, Voss, and Wilberg
10- ????????????xn/?t ?-2( ?2xn/?u2 - g(xn,?))
21/2 ?wn/?t - Where g(x,?)?G/?x, and G(x,?) 1/2 ( f(x)2
?2 f(x) ) - The path variable is u the length of the path is
U. - The bridge path evolves as a function of
(algorithmic) time, t. - Boundary conditions x(u0,t) x- and
x(uU,t) x - Initial condition x(u,t0) x0(u)
- Discrete version u n?u t
i?t - ???xn/?t ? (xni1 - xni )/?t ?wn/?t ? 21/2
(?u ?t)-1/2 ?ni1 - ?2x/?u2 ? 1/2 ?u-2 (xn1i1-2xni1xn-1i1)
(xn1i-2xnixn-1i)
11V(x) 2x4 - 2x2 - x/4 kBT1/2
??1 V(xo) -2 near /- 1 Path Length
U10, 1000 divisions, ?u 0.01 Evolution of
path ?t 0.0001 Movie T 100 ?t
12Brownian Bridges
13Parameter Estimation
- The original equation (SDE) dx/du f(x)
? dW/du - Expand f(x) in a basis f(x) ?k ak qk(x)
- Define I 1/2 ?0,U ?x/?u - ?k ak qk(x) 2
- Minimizing this sum, one obtains a set of
equations - A a b with a
A-1 b - where A is a matrix, and a and b are column
vectors - Akl ?n qk(xn) ql(xn) bk ?n qk(xn)
(xn1 - xn) /?u - Note the use of the Ito Calculus
- Turn to 1-d test case can we extract a known
potential?
14Parameter Estimation 1-D
Forward
Backward
V(x)2x4 - 2x2 - x/4 Red - Estimate Black -
Actual
Both
15Parameter Estimation from time series
SPDE ????xn/?t ?-2( ?2xn/?u2 - g(xn,?)) 21/2
?wn/?t where g(x,?)?G/?x, and G(x,?) 1/2 (
f(x)2 ?2 f(x) )
Evolution of path U10 ?u 0.01 ?t
0.0001 Movie ?T 100 ?t V(x)2x4 - 2x2 -
x/4 Fitted and exact g
lie on top of one another
16- Application to a collection of particles
- Two-body potential V?(r) ?(R/r)p
p12 - Total Energy W ?ij V?(rij)
- G(?, x) 1/2 ?j??????(?W/?xj?)2 - ?2
(?2W/?xj?2) - gj? ?G/?xj????????? ?2 2 kBT
- The set of SPDEs??
- ?xj?/?t ?-2(?2xj?/?u2 - gj?(x,?)) 21/2 ?wj?/?t
17- gi? ?G/?xi?
- ?j p2 (?-2(p2)-2(p1)Vij) Vij xij ? /rij4
- ??jk p2 (rij2xij?-(p2)xjk???? xij?xjk?)
- Vij Vjk xij ?
/rij4 rjk2 - 1/2 ?jk p2 (rij2rjk2(xij? xjk?)-(p2)
- (rij2xik?rik2xij? )??? xij?xjk?) Vij Vjk xij ?
/rij4 rjk4 - where Vij V?(rij) rij-12 and ?2 2 kBT
18Reaction Coordinate
- Use the Fourier Transform of the one-body
distribution to determine the crystal planes - Shift the origin of each configuration to align
the planes - Place the origin at the center of the crystal
X
19Start
End
20Energy Considerations
45
Start
End
36
Total Energy
54
21- The set of SDEs??
- ?xj?/?u - ?W/?xj? ? ?wj?/?u
- Reduced to a single pair of SDEs
- ?????X/?u FX(X,Y,?) ? ?wX/?u
- ?????Y/?u FY(X,Y,?) ? ?wY/?u
- FX(X,Y,?) - ?Veff/?X
- FY(X,Y,?) - ?Veff/?Y
- Extract Veff(X,Y,?) from the data along each path
22 Relaxation a 0.9755 a/20.4877 x
0.4403 ?x 0.0474
Fitted Potential ? 2 kBT 1/2
Barrier Height ?V 2.02
X-axis Slice
Perpendicular Slice
23- The set of SPDEs??
- ?xj?/?t ?-2(?2xj?/?u2 - gj?(x,?)) 21/2 ?wj?/?t
- Reduced to a single pair of SPDEs
- ?X/?t ?-2(?2X/?u2 - gX(X,Y,?)) 21/2 ?wX/?t
- ?Y/?t ?-2(?2Y/?u2 - gY(X,Y,?)) 21/2 ?wY/?t
- where X and Y are the Reaction Coordinates,
- and G2(X,Y,?)1/2 ( ??Veff??Veff - ?2 ?2Veff
) - gX(X,Y,?) ?G2/?X gY(X,Y,?) ?G2/?Y
- Extract G from the times series data -- and then
find an approximate Veff using a least-squares
method
24 Relaxation a 0.9755 a/20.4877 x
0.50
Fitted Potential (not converged) ? 2
kBT 1/2 Nu1500 du 2 x 10-4 Nt2500
dt 8 x 10-8 Barrier Height ?V 9.8
X-axis Slice
Perpendicular Slice
25Summary
- Described the motivation for looking at paths --
paths that move from one basin of the free energy
to another - Described the ideas behind SDEs
- Explained the origins of the SPDE that describes
the evolution of paths - From the information contained in the particle
movements (as seen in the SPDE), extracted
estimates of the potential parameters (1 2-D) - Need to develop a better understanding of how to
form the SDE for the slow DOF. - Need to see what happens in 3-D.
26The End
- In many applications, the physical picture is
similar. - Our goal is to describe the relatively slow
degrees of freedom of a particle as it is
buffeted by the (rapidly varying) thermal motions
of the other particles in the system. - Our goal is to understand how a system moves from
one basin in the free-energy landscape to
another. - Starting from the many-body problem, we would
like to extract a single-particle view by
averaging over the thermal fluctuations. - To accurately determine the behavior of the slow
degrees of freedom (DOF), we are devising a
practical scheme to average over the fast ones.
27- The set of SPDEs??
- ?xj?/?t ?-2(?2xj?/?u2 - gj?(x,?)) 21/2 ?wj?/?t
- Reduced to a single pair of SPEs
- ?X/?t ?-2(?2X/?u2 - gX(X,Y,?)) 21/2 ?wX/?t
- ?Y/?t ?-2(?2Y/?u2 - gY(X,Y,?)) 21/2 ?wY/?t
- Next step to look at
- ?X, eff2 ?t /T ?n X( (n1)?t)- X( n?t)2
- ??Y, eff2 ?t /T ?n Y( (n1)?t)- Y( n?t)2
28 - Determine the characteristic time, tc, of the
auto-correlation function at each point - Auto-correlation function
- q(x, t) lt ?(x - X(0)) (X(t)-X(0)) gt
- The variance (with tc N ?t)
- Q2(x, tc) ?t ?n q(x, n?t)?