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Sampling in Path Space with Application to Vacancy Diffusion

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Sampling in Path Space with Application to Vacancy Diffusion (a work in progress) ... 10 X 9 lattice with one vacancy, 89 particles. Outline. Movie: Diffusion in 2-D ... – PowerPoint PPT presentation

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Title: Sampling in Path Space with Application to Vacancy Diffusion


1
Sampling 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

2
Abstract
  • 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

4
Outline
  • Movie Diffusion in 2-D
  • Motivation and Background
  • Brownian Movement
  • Path Space
  • Formalism
  • 1-D Results and Interpretation
  • 2-D Results
  • Future Directions

5
Robert 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
6
Albert 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

8
Brownian Movement
9
The 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)

11
V(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
12
Brownian Bridges
13
Parameter 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?

14
Parameter Estimation 1-D
Forward
Backward
V(x)2x4 - 2x2 - x/4 Red - Estimate Black -
Actual
Both
15
Parameter 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

18
Reaction 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
19
Start
End
20
Energy 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
25
Summary
  • 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.

26
The 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)?
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