Mathematical Foundation of Kinetic Monte Carlo

of Kinetic Monte Carlo

Shim, Hyung Jin

Nuclear Engineering Department,

Seoul National University

1. Application of KMC for Solving the Depletion Equation 2. Mathematical Foundation of KMC

Contents

 A simple decay chain of I-135 & Xe-135 can be shown as below.

 And the corresponding deletion equations can be written by

 And from initial conditions of the clean state at t=0, the two nuclide densities can be obtained by

Problem: Depletion Eq. for Xe-135

135

I

_b ¹³⁵

Xe

g_I g_X

,

I

I f I I

dN N

dt   

g  

(A.1)

( )

X

X f I I X X X

dN N N

dt 

g

 

 







 

(A.2)

 

( ) ^{I f} 1 ^It

I

I

N t

g 

e ^



 

 



^{( )}

 

^{( )}



( ) ^{X f I f} 1 ^{X X t I f It X X t}

N tX

g  g 

e ^{  }

g 

e ^ e ^{  }

      

    

   

   

  

(A.3) (A.4)

 For simplicity, Eqs. (A.1) & (A.2) can be rewritten by

 Solve Eqs. (A.5) & (A.6) under the following conditions[*] by the kinetic Monte Carlo method.

Densities of I-135 & Xe-135

  

^{3 -1}

 

^{4 -2 -1}



5

0.056 3 10 cm 10 cm sec 1.68 / cc sec 2.9 10 / sec

C

I I f

D

I I

k

g 

 





       

  

(A.5)

C D ,

I

I I I

dN k N

dt  



C D D

X

X I I X X

dN k N N

dt  







(A.6)

     

   

3 -1 4 -2 -1 2

5 1 18 2 4 -2 -1

5

0.003 3 10 cm 10 cm sec 9 10 / cc sec 2.1 10 sec 3.5 10 cm 10 cm sec

2.1000000035 10 / sec

C

X X f

D

X X X

k

g 

   

 

  



        

      

 

[*] D. L. Hetrick, Dynamics of Nuclear Reactors, American Nuclear Society, Inc., La Grange Park, IL (1993).

CASE #1: Density of I-135 – Algorithm

Begin: t=0, N_I=0

Calculate k_tot: k_tot k_I^C N_I _I^D

135

135 5

construction rate of I=1.68 / cc sec, destruction rate of I=2.9 10 / sec

C I

D I

k

k ^

 

 

Sample Dt: D  t ln₁ k_tot , t  Dt

Select a transition type:

2 C

I tot

 k k

Simulate a construction:

N_I+= 1

Simulate a destruction:

N_I－= 1

t<t_end

Yes No

No Yes

CASE #1: Density of I-135 – Implementation

Algorithm for Xe-135

Begin with t=0, N_I=0, N_X=0

Calculate k:

4

1 2 3 4

1

; ^C, , ^C, ^D

i I I I X X X

i

k k k k k  N k k k  N







   

Sample Dt: D  t ln₁ k t,   Dt

Select a transition event isatisfying:

1

1 2 1

i i

i i

i^_ k k_   i_ k k_

 

Creation of ¹³⁵I by a fis.:

N_I+= 1

Beta decay of ¹³⁵I:

N_I－= 1, N_X+= 1

t<t_end No Yes

Creation of ¹³⁵Xe by a fis.:

N_X+= 1

Destruction of ¹³⁵Xe:

N_X－= 1

When i=1, When i=2, When i=3, When i=4,

Numerical Results

0 50000 100000 150000 200000 250000 300000

0 10000 20000 30000 40000 50000 60000 70000 80000 90000

Number Density

Time (sec)

135I (KMC)

135Xe (KMC)

135I (Analytic)

135Xe (Analytic)

Mathematical Foundation

of Kinetic Monte Carlo Method

 K. A. Fichthorn and W. H. Weinberg [1] presented the theoretical basis for a dynamical Monte Carlo method in terms of the theory of Poisson process.

 They showed that if

1. A “dynamical hierarchy” of transition probabilities is created which also satisfy the detailed-balance criterion;

2. Time increments upon successful events are calculated appropriately;

3. The effective independence of various events comprising the system can be achieved,

then Monte Carlo methods may be utilized to simulate the Poisson process.

[1] K. A. Fichthorn and W.H Weinberg, “Theoretical Foundations of Dynamical Monte Carlo Simulations,” J. Chem. Phys., 95(2), 1090 (1991).

Kinetic Monte Carlo vs. Poisson Process

 They stated that under a dynamical interpretation, the Monte Carlo method provides a numerical solution to the Master equation:

where X and X′ are successive states of the system.

PX,t) is the probability that the system is in state X at time t.

K(X′→ X) is the probability per unit time that the system will undergo a transition from state X′ to state X.

 The kinetic Monte Carlo method is explained as

• The solution of the Master equation is achieved computationally by choosing randomly among various possible transitions with appropriate probabilities.

• Upon each successful transition, time is typically incremented in integral units of Monte Carlo steps.

Kinetic Monte Carlo vs. Poisson Process (Contd.)

( , )

( ) ( , ) ( ) ( , )

P t

K P t K P t

t  

      



 

X X

X X X X X X X (1)

 We are going to derive a mathematical formulation corresponding to the kinetic Monte Carlo method from the general kinetic equation as

and the initial condition at t=0 given by

 For simplicity, we rewrite Eq. (2) as

Mathematical Foundation of KMC

( , )

( , ) ( , ),

P t

k P t k P t

t  ^  ^{ }

   





^{X X}



^{X X}

X X

X X X (2)

( , )

( , ) ( , );

P t

k P t S t

t

  

 ^X

X X X

,

k k _{ }







X X X

X

( , ) ( , ).

S t k _ P t







^{X X} 

X

X X

(4) (5) (6) ( ) ( , 0)

Q X  P X (3)

 Equation (4) with the initial condition of Eq. (3) is seen to be a first-order linear partial differential equation which has a unique solution. By introducing an integrating factor, Eq. (4) becomes

 Then an integration of Eq. (7) from t=0 yields

 Eq. (8) implies that the probability of state X is made up of the state appeared in the previous time multiplied by the attenuation factor of .

Derivation of KMC Algorithm

( , ) ^k ( , )

k t t

e P t S t

t e

     

 ^X X ^X X (7)

( , ) ^{k t} ( ) 0^{t k t} ( , ) P ^X t e ^X Q ^X 



e ^{X } S ^X t dt 

( )

( , ) ( ) ^{k t} 0^{t k t t} ( , )

P ^X t  Q ^X e^{ X} 



e^{ X  } S ^X t dt  ⁽⁸⁾

( )

k t t

e^{ X  }

 The transition probability can be defined by

 Then by multiplying k_X on the both sides of Eq. (8), it can be expressed as

 By introducing the time-flight kernel, T defined by Eq. (10) becomes

is named the first transition source.

Transition Probability Equation

( , ) t k P ( , ) t



X



_X X

( )

( , ) t k e

^{k t}

Q ( )

0^t

k e

^{k t t }

S ( , ) t dt  



^X



^{X X X}

 

^{X X}



^X

(9)

(10)

(12)

( ) (11)

( | )

^{k t t}

T t   t

X

 k e

_X ^{ X  }

0

( , ) t Q ˆ ( , ) t

^t

T t (  t | ) S ( , ) ; t dt  



^X



^X

  

^X



^X

ˆ ( , ) (0 | ) ( )

Q

X

t  T  t

X

Q

X ₍₁₃₎

ˆ ( , )

Q

X

t

 And we define another kernel named the event kernel as

 Using the event kernel C and the transition probability , S(X,t) of Eq. (6) can be expressed as

 The introduction of Eq. (15) into Eq. (12) yields

Transition Density Equation (Contd.)

( , ) ( ) ( , )

( ) ( , )

S t k C P t

C t







 

 

 

  





X X

X

X X X X

X X X (15)

( ) k

C k





   ^{X X}

X

X X (14)

0

( , ) t Q ˆ ( , ) t

^t

K ( , t , ) ( , ) t t dt



    

      

X

X X X X X

( , , ) ( | ) ( )

K

X

  t 

X

t  T t   t

X

 C

X

 

X

(16) (17)

 Let’s consider the solution of Eq. (16) obtained by iteration; thus

Clearly

y

₀ is the first-transition source.

y

₁ means the transition probability from the second-transition state. Similarly,

y

₂ indicates the contribution of the third-

transition state, and so on. If the series converges, it represents a solution to Eq. (16).

Series Solution

(16)

0

1 0 0

0 1

( , ) ˆ ( , )

( , ) ( , , ) ( , )

( , ) ( , , ) ( , )

t

t

n n

t Q t

t K t t t dt

t K t t t dt

y

y y

y y



 



    

 

    

 

 

 

X

X

X X

X X X X

X X X X

 

0

( , )

j j

y t





 ^X 0

( , ) t Q ˆ ( , ) t

^t

K ( , t , ) ( , ) t t dt



    

      

X

X X X X X

 The solution of Eq. (16) can be expressed by the Neumann series:

From Eq. (18), we can find that the transition probability is the sum of the contributions from transition at (X,t) first and after a transition or more.

Neumann Series Solution

(16)

0

( , )

_j

( , );

j

t

^

y t





^X

 

^X

0 1

1 1

0 0 0 0 0

1 0 0 0 0

2 0 0 1 1 1 0 0 1 1

0 0 1 1 1 1 2 2 1

( , , ) ( ) ( ),

( , , ) ( , , ),

( , , ) ( , , ) ( , , ),

( , , ) ( , , ) ( , ,

j

t t

j j j j j j j

K t t t t

K t t K t t

K t t dt K t t K t t

K t t dt dt K t t K t

 



     

   

  

   

   

 

 

 

X

X X

X X X X

X X X X

X X X X X X

X X X X X X



 t_j1)K(X0,t0  X1, )t1

(18)

(20)

0

( , ) t Q ˆ ( , ) t

^t

K ( , t , ) ( , ) t t dt



    

      

X

X X X X X

0

0 0 0 0 0

0

( , )

^t

( , , ) ( ˆ , ),

j

t dt K

j

t t Q t

y    

X

X X X X ⁽¹⁹⁾

Interpretation of KMC by MC Sim. of Series Sol.

Begin: t=0, N_I=0

Calculate k_tot: k_tot k_I^C N_I _I^D

Sample Dt: D  t ln₁ k_tot , t  Dt

Select a transition type:

2 C

I tot

  k k

Simulate a construction:

N_I+= 1

Simulate a destruction:

N_I－= 1

t<t_end

Yes No

No Yes

0 1

1

( )

ln

tot tot

k k

tot tot

tot

T k e k e d

k

  

  

 

   

  

  



( ) k

C k





   ^{X X}

X

X X