Stochastic Boundary conditions

Langevin equation
i
i
i
i
r
U
R
v
m
dt
dv
m
∂
∂
−
+
β
−
=
The approach described above is used by M. Berkowitz and J.A. McCammon, Chem. Phys.
Lett.
90
, 215 (1982).
More complex and rigorous descriptions of Langevin particles has been
discussed in literature, e.g. J.C. Tully,
J. Chem. Phys.
73
, 1975 (1980).
δ
(t)
β
m
kT
2
(0)
(t)R
R
i
i
=
R
i
– random white noise forces with Gaussian distribution centered at zero. The width of the
distribution is defined by temperature and should obey the second fluctuationdissipation
theorem [R. Kubo,
Rep. Prog. Theor. Phys.
33
, 425, 1965]:
β
– friction coefficient of an atom that, within the Debye model, can be determined from the
relation
β
= 1/6
π ω
D
, where
ω
D
is the Debye frequency,
ω
D
= k
B
Θ
D
/
ħ
, and
Θ
D
is the Debye T
R
n
is taken from Gaussian random number generator.
Methods for generating random
numbers with Gaussian distribution from evenly distributed random numbers can be found in
[M. Abramovitz,
Handbook of Mathematical Functions
, 9
th
edition, 1970, p. 952]
(
)
(
)
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
−
2
2
2
1
2
2
exp
2
i
i
i
i
R
R
R
R
W
π
0
R
n
=
Δ
t
2kTm
β
R
2
n
=
The second fluctuationdissipation theorem takes care of balancing the increase in energy due
to the random fluctuating force and the decrease in energy due to the friction force.
where
<…>
denotes
average
over
an
equilibrium
ensemble
and
W(R
i
)
is
the
probability distribution of the random force.
To implement in MD we have to average over a timestep
Δ
t:
( )
dt
t
R
Δ
t
1
R
1
n
n
t
t
n
∫
+
=
0
R
R
1
n
n
=
+
• thermal motion of particles is driven by random force
• a friction force and a random force R
i
are added to the equation of motion
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 Fall '11
 Zhigilei
 Thermodynamics, Friction, Heat, Heat Transfer, Fundamental physics concepts, Atomistic Simulations

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