Computer Simulation of Thermally Activated Processes: Theory, Algorithms, and a Few Case Applications to Surface Chemistry

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1 , t

2 , x(t)]

**% 0 !&% +"1% %*% "

S[t

1 , t

2 , x(t)] =

Z

t

2 t

1 | ˙x(t) + ∇V (x(t))|

2 dt,

S)ET -3;! ", "- !&% *!&

x(t)

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2

/%!

x(t)

t

_{∈ [t}

1 , t

2 ]

+% + !' *!& ! -' 0 !&% ;"! !

x(t

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9

x(t

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products

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∗

₎

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S[t

1 , t

2 , x(t)]

≥ 2 [V (x(t

∗

))

− V (x(0))]

S)T

(21)

5 -"

t

1 → −∞

9

t

2 → +∞

&% /0%1 ;! " ;".*3!%9 " !&% D:a 0 1% +'Q

Z

t

2 t

1 | ˙x(t) + ∇V (x(t))|

2 dt = 4

Z

t

∗

t

1 |∇V (x(t))|

2 dt

S ))T

= 4

Z

t

∗

t

1 ˙x(t)

· ∇V (x(t))dt

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∗

))

_{− V (x(t}

1 ))] ,

" !&! ! !&% ,"W%! "9%

β

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w (x(t

1 )

|x(t

2 ))

∼ e

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∗

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1 ))]

_{= e}

−βE

A

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! % !&% +"39 %

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W% &1%Q

S[x(t)] =

Z

t

2 t

1 ˙

x + V

0 (x)

2 dt

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t

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˙

∗

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√

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0 _(x

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2 dt

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∗

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η

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η

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∗

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S )=T A !&% , . ! "

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9

t

2 → +∞

!&% ! ! " *"++ , !' !&3 0 1% +'Q

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1 )

|x(t

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1 ,t

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T

0 1% +'Q

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t

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∗

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(22)

(23)

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∇

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_{− τ (s) (τ (s)|∇V (x(s)) = 0.}

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s

−→ i · δs

S ))T

x(s)

−→ x

i

τ

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x

i+1

− x

i

−1

|x

i+1

− x

i

−1

|

.

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∇

_{V (x}

_i

₎

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F (x

i

)

⊥

=

− [∇V (x

i

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− τ

i

(τ

i

|∇V (x

i

))]

⇓

S )(T

x

i

(t + δt) = x

i

(t) + δt F (x

i

(t))

_⊥

.

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F(x

i

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kF(x

i

)

⊥

k

∞

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(25)

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0 ∆x

∆x

L

0 =

√

L

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0 _{− L = L}

_q

1 +

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2 L

2 − 1

∼

1

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2

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i

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⊥

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!&% 1 !3, * 0

F(x

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(26)

6

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F(x

i

)

elastic

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=

−k τ

i

(τ

i

|(2x

i

− x

i+1

− x

i

−1

)) .

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x

i

! ! .%

t

S&%%-!% 9 ;!%9

F(x

i

(t)) = F(x

i

(t))

⊥

+ F(x

i

(t))

elastic

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T 3%9Q

x

i

(t + δt) = x

i

(t) + δt F(x

i

(t)).

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A !&% 8:< .%!&"9 !&% 1 !3, * 0 ;! *%,!' -3;! " !&!

(27)

5 6

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k

i

=

1

2 k

max

+ k

min

− (k

max

− k

min

) cos

π

(V (x

i

)

− V

min

)

(V

max

− V

min

)

.

S)?T

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S! %* 0T "39 *" !"-,0%!,";,;31!3%

ω

(s) =

dτ(s)

ds

A9%%9

!&% 3.+% "- .0% %23 %9 !" %*%%! *!& 9%*%9 " !&% 0,"+,

;31!3% "- !&% *!&

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1

0 |ω(s)| ds.

S) BT

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ω

_i

=

x

i+1

−x

i

|x

i+1

−x

i

|

−

x

i

−x

i−1

|x

i

−x

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|

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Computer Simulation of Thermally Activated Processes: Theory, Algorithms, and a Few Case Applications to Surface Chemistry

×

2

2

×

2

2

×

×

×

τ

f ast

τ

slow

∼ τ

f ast

e

EA

KB T

E

A

E

A

×

×

q

3N

V (q

3N

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q

3N

e

−

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KB T

10

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10

−8

10

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m

E

A

τ

f ast

∼

q

m

V

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(q

min

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τ

slow

τ

f ast

τ

slow

τ

slow

∼ τ

f ast

e

EA

KB T

E

A

 K

B

T

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K

K

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