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.

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• 10 X 9 lattice with one vacancy, 89 particles

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Outline

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

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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
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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
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• 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

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Brownian Movement
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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
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• ????????????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)

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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
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Brownian Bridges
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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?

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

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• 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

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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
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Start
End
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Energy Considerations
45
Start
End
36
Total Energy
54
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• 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

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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
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• 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

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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
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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.

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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.

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• 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

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• 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)?