Reaction Pathways of Metastable Markov Chains - LJ ...gentz/pages/WS12/RDS12/slides/RDS12... · Reaction Pathways of Metastable Markov Chains - LJ clusters Reorganization • Spectral

Reaction Pathways of Metastable Markov Chains - LJ clusters Reorganization

• Spectral approach to metastability - sets, currents, pathways;

• Transition path theory or how to focus on a specific `reactive’ event; • Application to Lennard-Jones clusters reorganization (with Masha Cameron).

Eric Vanden-EijndenCourant Institute

Monday, October 8, 12

Reaction Pathways of Metastable Markov Chains - LJ clusters Reorganization

• Spectral approach to metastability - sets, currents, pathways;

• Transition path theory or how to focus on a specific `reactive’ event; • Application to Lennard-Jones clusters reorganization (with Masha Cameron).

Eric Vanden-EijndenCourant Institute


• Metastability associated with presence of one or more groups of small eigenvalues of the generator, corresponding to slow relaxation processes in the system.

• Eigenvectors/eigenfunctions associated with these small eigenvalues indicate what the metastable sets are and what the mechanism of transition (reaction) between them is.

Spectral approach to metastability

0 2 4 6 8 10

x 104

−1.5

−1

−0.5

0

0.5

1

1.5

time

positio

n

x

y

−1 −0.5 0 0.5 1−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2



• Overdamped Langevin equation:

• Generator:

• Eigenvalue/eigenfunction:

• Spectral representation of transition probability distribution:

L = rV ·r� "�

L� = ��

0 = �0 < �1 �2 · · ·

p

x

t

(dy) =1X

k=0

e

��kt�

k

(x)�k

(y)C�1e

�V (y)/"dy,

C =

Z

⌦e�V (y)/"dy,

dX = �rV (X)dt+p2" dW (X 2 ⌦ ✓ Rn)



• Suppose there exists a group of K small eigenvalues:

• Indicative of metastability. More precisely, on timescales such that

the processes described by the eigenvectors of index K or higher will have decayed and the remaining part of the dynamics will be described by the slow first K eigenvectors

Note that the transition probability relaxes slowly if there is metastability, but the process itself is fast - rare events are infrequent but when they occur they typically do so fast!

0 = �0 < �1 �2 · · · �K�1 ⌧ �K · · ·

�K�1t ⇠ 1, �Kt � 1

0 2 4 6 8 10

x 104

−1.5

−1

−0.5

0

0.5

1

1.5

timepositio

nx

y

−1 −0.5 0 0.5 1−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

p

x

t

(dy) =1X

k=0

e

��kt�

k

(x)�k

(y)C�1e

�V (y)/"dy,


Example on 2D potential with 4 wells

Simulations by Masha Cameron



• Forward Kolmogorov equation as a conservation law:

• Spectral decomposition of currents:

Flowlines of current indicative of mechanisms by which slow relaxation occurs

pxt

(dy) = ⇢xt

(y)dy

j

x

t

(y) =1X

k=0

e

��kt�

k

(x)jk

(y), j

k

(y) = C

�1e

�V (y)/"r�

k

(y)

@t

⇢xt

= �div jxt

, jxt

= �rV ⇢xt

� "r⇢xt

dx

d⌧

= jk(x)

Analysis of 1-form in Witten complex


Slowest modes in 2D potential with 4 wells

Simulations by Masha Cameron


• How to make spectral approach practical as a computational tool?

In applications, one is typically in high dimensional systems whose spectrum is enormously complicated and cannot be calculated explicitly (even numerically).

• Global viewpoint of metastability also problematic - the longest timescales may not be the relevant ones (i.e. they could be associated with presence of deadends or dynamical traps), there may be many of them (subgroups into groups), etc..

• Can we focus on a single `reaction’ rather than having to analyze them all thru calculation of the spectrum? (Indeed in a given system there may be specific reactions of interest and we don’t know a priori to which part of the spectrum they are associated)

• Can this all be done even if there is no metastability (i.e. no small parameter)?

Transition Path Theory


Transition Path Theory

• Main idea: focus on `reactive’ trajectories associated with a given transition (reaction)

• Probability distribution of reactive trajectories

• Probability current of reactive trajectories

• Committor function (capacitor):

A B

µR(dy) = C�1e�V (y)/"q(y)(1� q(y))dy

jR(y) = C�1e�V (y)/"rq(y)

q(y) = Ey(⌧B < ⌧A)

Related to Bovier’s potential theoretic approach, but exact (no small parameter)!Can be generalized to non-reversible processes.


Some low D examples

−1 0 1

−0.5

0

0.5

1

1.5

2

−1 0 1

−0.5

0

0.5

1

1.5

2

−4

−2

0

2

4

TRANSITION PATH THEORY : ILLUSTRATION VIA SIMPLE EXAMPLES 7

!1.5 !1 !0.5 0 0.5 1 1.5

!1.5

!1

!0.5

0

0.5

1

1.5

2

2.5

Figure 10. Probability current. For sake of visualization I com-puted the current in a low resolution

!1.5 !1 !0.5 0 0.5 1 1.5

!1.5

!1

!0.5

0

0.5

1

1.5

2

2.5

Figure 11. Streamlines colored according to the restricted inten-sities on the committor at ”low temperature” � = 0.2.

8 PHILIPP METZNER,ERIC VANDEN-EIJNDEN,CHRISTOF SCHTTE

!1.5 !1 !0.5 0 0.5 1 1.5

!1.5

!1

!0.5

0

0.5

1

1.5

2

2.5

Figure 12. Streamlines colored according to the restricted inten-sities on the committor at ”low temperature” � = 0.6.

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

epsilon

rela

tive r

ate

s

Figure 13. Here one can see the switching of the preferable tran-sition tubes.

4 PHILIPP METZNER,ERIC VANDEN-EIJNDEN,CHRISTOF SCHTTE

rate (committor) 1.912·10°2

rate (volume integral) 1.924·10°2

rate (DNS) 2.079 · 10°2

var(DNS) 1.33·10°6

Table 2. Transition Rate(� = 0.6, 400⇥ 400Grid,N = 1000, ⇥ = 10°5)

3.2. Schulten-Potential. Now we consider the Smoluchowsky dynamics in a sim-ple potential which exhibits metastability. This model system was first introducedand studied in (?).

−1.5 −1 −0.5 0 0.5 1 1.5

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

−2

0

2

4

6

8

10

12

Figure 5. Contour plot of the potentialMonday, October 8, 12

Transition Path Theory for MJP

Can again be generalized to non-reversible processes.

• Generator = transition rate matrix

• Microscopic reversibility (detailed balance)

• Probability distribution of reactive trajectories

• Probability current of reactive trajectories

• Committor function:

µiLi,j = µjLj,i

µRi = µiqi(1� qi)

qi = Ei(⌧B < ⌧A)

fRi,j = µiLi,j(qj � qi)+ LR

i,j = Li,j(qj � qi)+

Li,j i, j 2 S = {1, 2, . . . , N}

Generator of loop erased reactive paths


Transition Path Theory for MJP

A

B

Effective current Committor

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Example of a maze (not metastable!)


icosahedron

octahedron

• Double funnel landscape - ground state is not accessible directly by self-assembly and requires dynamical reorganization;

• Can be described by a MJP using the network calculated by Wales;

• ~1e4 local minima of potential; • ~1e4 saddle points between these minima; • Network of orbits (minimum energy paths); • Transition matrix ;

• Spectral analysis difficult (large network, many small eigenvalues).

• Can be analyzed by TPT. • LDT only applicable at extremely low temperatures. At small temperature, systems remains strongly metastable, but pathway and rate of transition are different than those predicted by LDT (entropic effects).

Li,j = ⌫e��Vi,j/"

Reorganization of LJ cluster after self-assemblywith Masha Cameron



The highest potential barrier

Perc

enta

ge o

f pat

hway

s

342 - 354

958 - 607

3223 - 354958 - 1

396 - 5162

Perc

enta

ge o

f pat

hway

s

The highest potential barrier

3223 - 354

3886 - 354351 - 354

3184 - 2831396 - 15021208 - 3299248 - 3299355 - 3544950 - 2933

342 -354958 - 1

958 - 607

T= 0.05

T= 0.15



T= 0.05

T= 0.15

Number of transition pathways increase with T

They become shorter and go thru higher barriers



T= 0.05

T= 0.15

Number of transition pathways increase with T

They become shorter and go thru higher barriers



T= 0.05

T= 0.15


• TPT can be used to focus and analyze a specific `reaction’.

• Gives rate (mean frequency of transition) and mechanism via analysis of current.

• No small parameter needed - reduces to LDT or Bovier’s approach in right limits, but can be used outside the range of applicability of these asymptotic theories.

• Can be and has been used in many other examples.

Conclusions


Some other applications

▷ Thermally induced magnetization reversal in submicron ferromagnetic elements

a)

b)

ï1 0 1ï1

0

1

m1

m2

S1

S2 S3

S4

S5

S6 S7

S8

C1,2

C3,4

C5,6

C7,8

V11,2

V13,4 V1

5,6

V17,8

V21

V22 V2

3

V24

V19,10

V111,12 V1

13,14

V115,16

V25

V26 V2

7

V28

with Weinan E and Weiqing Ren

Practical side of LDT - Dynamics can be reduced to a Markov jump process on energy map, whose nodes are the energy minima and whose edges are the minimum energy paths.

▷ Hydrophobic collapse of a polymeric chain by dewetting transition

Rate limiting step is entropic - creation of a water bubble

with Tommy Miller and David Chandler


Documents

Reaction Pathways of Metastable Markov Chains - LJ ...gentz/pages/WS12/RDS12/slides/RDS12... · Reaction Pathways of Metastable Markov Chains - LJ clusters Reorganization • Spectral