RIT “Stochastic Dynamics:
Models, Analysis and Numerics”
Lecture 2: Problems and challenges in stochastic dynamics
Maria Cameron
Why do we use the Brownian Motion?
dxj =pjmj
dt
dpj = (−∇V (x) − γmj−1pj )dt + 2β−1γ dwj ,
j = 1...M N
dxidt
=pimi
dpidt
= −∇V (x)
i = 1...N
Two main effects:- slowing down
-random change of the direction of motion
Alanine-Dipeptide in the water(by Maddalena Venturoli (U. Rome, Italy))
Are there other mathematically-tractable
options?Gaussian Process with Correlation in time
Levy Flights
Difference of two Poisson processes
A process with heavy tails (e.g. with Student’s pdf)
Backward and Forward Kolmogorov Equations
dx = b(x)dt +σ (x)dw
(1) Suppose we are interested in some function f(x,t)
df (x(t),t) = ftdt +∇f idx +12dxT∇∇fdx + ...
= ftdt +∇f ib(x)dt +∇f iσ (x)dw +12
σ (x)dw( )T ∇∇fσ (x)dw + ...
Now we take the expected value
E f (x,t)[ ] = ft dt '0
t
∫ + ∇f ib(x) + 12a(x) :∇∇f⎛
⎝⎜⎞⎠⎟dt '
0
t
∫
Let f=f(x(t)) only. We define the generator of the random process as
Lf = limt→0
E[ f (x(t))]− f (x(0))t
= ∇f ib(x) + 12a(x) :∇∇f
a = σ Tσ
dx = b(x)dt +σ (x)dw
Example 1. Committor functiondx = −∇V (x)dt + 2β−1dw The committor function
q(x) = the probabilityto reach first B rather than A starting from
the point x.
Δq − β∇V i∇q = 0, x ∈n \ (A∪ B)q(∂A) = 0, q(∂B) = 1
Example 2. First passage time
ΔT − β∇V i∇T = −1, x ∈AT (∂A) = 0
A
Backward Kolmogorov Equation:
ddtE f (x(t))[ ] = ∇f ib(x) + 1
2a(x) :∇∇f
(2) How does the probability density change?
E f (x,t)[ ] = f (x0 ,t0 )p(x,t | x0 ,t0 )dx∫ddtE f (x,t)[ ] = f (x0 ,t0 )pt (x,t | x0 ,t0 )dx∫
ddtE f (x,t)[ ] = E Lf[ ] = (Lf , p) = ( f ,L * p)
pt = L * p = −∇ b(x)p( ) + 12
∂2
∂xi∂x ji, j∑ aij p( )
a = σ Tσdx = b(x)dt +σ (x)dw
Forward Kolmogorov Equation: pt = −∇ b(x)p( ) + 12
∂2
∂xi∂x ji, j∑ aij p( )
Lecture by Leonid Koralov on Feb. 23
Transition Path TheoryThe language for describing reactive events
W. E and E. Vanden-Einden, 2006
• Committor function q(x) = probability to hit first B rather than A starting at x
Δq − β∇V i∇q = 0, x ∈n \ (A∪ B)q(∂A) = 0, q(∂B) = 1
• Reactive current JR(x)
JR (x) = Z−1β−1e−βV (x )∇q(x)
vAB = Z−1β−1 e−βV (x ) ∇q 2 dx
Rn \ A∪B( )∫
• Transition rate νAB from A to B
Lecture by Maria Cameron on March 9
Kramers-Smolukhowsky approximation
dx = −∇V (x)dt + 2β−1dw,
If the γm-1 is large (the mass is small and/or the friction coefficient is large)
one can assume that the dp is small and set it to zero.
Lecture by Sandra Cerrai on March 30
dxj =pjmj
dt
dpj = (−∇V (x) − γmj−1pj )dt + 2β−1γ dwj
Illustration on theDouble-Well potential V(x)=x4-2x2
β=∞β=10γ=1
β=10γ=3
dxj = vjdt
dvj = (−∇V (x) − γmj−1vj )dt + 2β−1γmj
−1dwj
Difficulty
We can solve the Backward and Forward Kolmogorov Equations only if
the dimension of the phase space is low
the entries of the matrix preceding dw are not too small
Monte-Carlo(1) Propose a move.
E.g. pick a spin and flip it.
(2) Calculate the quantity
e-βH(new state)/e -βH(old state) = e -β∆H
(3) if e -β∆H > 1 do the proposed moveotherwise do it with probability e -β∆H
and remain in the old state with probability 1 - e -β∆H.
Monte-Carlo allows to calculate average values using the ergodicity. Important feature: One does not need to know the normalization constant for the pdf.
Problems: (1) There can be high rate of rejectance of the proposed states(2) Equilibration might not happen for very long time due to correlation
Lecture by Jonathan Weare (U. Chicago) on April 6
2D Ising model
Stochastic intergatorsOverdamped dynamics: dx = -∇V(x)dt + (2β-1)1/2 dw
Euler-Mariyama: xn+1 = xn - ∇V(xn) h + (2β-1)1/2 wn
Shortcomings: requires small time step, unstable if V(x) grows faster than x2
MALA (Metropolis-Adjusted Langevin Algorithm, Roberts and Tweedie, 1996)
The full Langevin Dynamics
Verlet
Analog of MALA for Verlet (Bou-Rabee and Vanden-Eijnden, 2010)
Lecture by Maria Cameron on April 13
Large Deviation Theory
• SDE
• F.-W. action functional
• Transition probability along the path
• If two minima are separated by a single mountain pass then the path that satisfies is the MEP, and the minimum is achieved in the limit of infinite time
Freidlin and Wentzell, 1979, 1998
dx = −∇V (x)dt + 2β−1dw
ST (ϕ) = ϕ t +∇V (ϕ(t)) 2 dt0
T
∫∝ e−βST
minT ,ϕ (t )
ST (ϕ )
Lecture by Sandra Cerrai on March 2
Methods for finding transitions paths
• Methods for computing transition paths at zero temperature limit:
• String method (E, Ren and V.-E., 2002)
• Nudged elastic band method (Jonsson et al, 1998)
• MaxFlux functional (Cameron and V.-E., 2010)
• Methods for finding saddle points:
• Dimer method (Henkelman and Jonsson, 1999)
• Activation-relaxation technique (Barkema, Mousseau, 2001 )
• Methods for computing rare events at finite temperature:
• Finite temperature string method (E, Ren and V.-E., 2005)
• MaxFlux Functional (Berkowitz, 1983; Huo and Straub, 1997)
Lecture by Maria Cameron
on March 16
Challengies
High dimensionality of the phase space
Complex and expensive-to-evaluate potential
Multiple transition paths
Dimensional reduction.Collective variables.
dx = −∇V (x)dt + 2β−1dw
dz = −M (z)∇U + β−1divM (z)( )dt + 2β−1M (z)dw
M = diffusion matrixsymmetric pozitive definite matrix
Maragliano, Fischer, Vanden-Eijnden, Ciccotti, 2006
Mij (z) = mk−1 ∂θi
∂xk
⎛⎝⎜
⎞⎠⎟k=1
n
∑θ= z
∂θ j
∂xk
⎛⎝⎜
⎞⎠⎟θ= z
θ1(x1,..., xn ),...,θm (x1,..., xn )Collective variables:
U(z) = −β −1 ln Z −1 e−βV (x ) δ (zi −θi (x)
i=1
m
∏⎡⎣⎢
⎤⎦⎥dx
Rn∫
⎛
⎝⎜
⎞
⎠⎟U = free energy