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Byeong-Joo Lee www.postech.ac.kr/~calphad
Byeong-Joo LeeByeong-Joo Lee
Dept. of MSEDept. of MSE
Pohang University Pohang University
of Science and Technologyof Science and Technology
(POSTECH)(POSTECH)
[email protected] @postech.ac.kr
Atomistic SimulationsAtomistic Simulations
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Byeong-Joo Lee www.postech.ac.kr/~calphad
More about Molecular Dynamics More about Molecular Dynamics - Lecture Note of Prof. V. Vitek, 2002- Lecture Note of Prof. V. Vitek, 2002
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Byeong-Joo Lee www.postech.ac.kr/~calphad
Physical Interpretation of MD using the concept of Statistical Mechanics Physical Interpretation of MD using the concept of Statistical Mechanics
Fluctuations
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Byeong-Joo Lee www.postech.ac.kr/~calphad
Physical Interpretation of MD using the concept of Statistical Mechanics Physical Interpretation of MD using the concept of Statistical Mechanics
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Byeong-Joo Lee www.postech.ac.kr/~calphad
Physical Interpretation of MD using the concept of Statistical Mechanics Physical Interpretation of MD using the concept of Statistical Mechanics
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Byeong-Joo Lee www.postech.ac.kr/~calphad
Physical Interpretation of MD using the concept of Statistical Mechanics Physical Interpretation of MD using the concept of Statistical Mechanics
J. S. Rowlinson: Liquids and liquid mixtures, Butterworth: London, 1969
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Byeong-Joo Lee www.postech.ac.kr/~calphad
Physical Interpretation of MD using the concept of Statistical Mechanics Physical Interpretation of MD using the concept of Statistical Mechanics
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Byeong-Joo Lee www.postech.ac.kr/~calphad
Physical Interpretation of MD using the concept of Statistical Mechanics Physical Interpretation of MD using the concept of Statistical Mechanics
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Byeong-Joo Lee www.postech.ac.kr/~calphad
Physical Interpretation of MD Physical Interpretation of MD - Generalized Einstein Relations- Generalized Einstein Relations
dxtxNxN
xtx ),(1
)]0()([ 2
0
2
x
NDxN
Dt
x
Dt
NtxN
4exp
2),(
20
Consider One dimensional Diffusion
If N0 atoms were concentrated at the origin x=0 at time t=0
Mean-square displacement of atoms
Dtxtx 2)]0()([ 2
Dt
rtrt
6
)]0()([lim
2
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Byeong-Joo Lee www.postech.ac.kr/~calphad
Physical Interpretation of MD Physical Interpretation of MD - Green-Kubo Relations- Green-Kubo Relations
)"()'('"2)"()'('")]0()(["
0000
2 tAtAdtdttAtAdtdtAtAmsdtttt
dt
dAtA )(
)0()(2
)]0()([lim
0
2
AAdt
AtAt
Shifting the time origin to t’ :
)'(')0()(0
tAdtAtAt
)0()'"()"()'( AttAtAtA
t
AAdtdtAAdAAddtmsdttttt
1)0()(2")0()(2)0()("2
00
"
00
t
AAdt
AtA t 1)0()(2
)]0()([
0
2
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Byeong-Joo Lee www.postech.ac.kr/~calphad
Physical Interpretation of MD Physical Interpretation of MD - Autocorrelations- Autocorrelations
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Byeong-Joo Lee www.postech.ac.kr/~calphad
Physical Interpretation of MD Physical Interpretation of MD - Autocorrelations- Autocorrelations
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Byeong-Joo Lee www.postech.ac.kr/~calphad
Physical Interpretation of MD Physical Interpretation of MD - Autocorrelations- Autocorrelations
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Byeong-Joo Lee www.postech.ac.kr/~calphad
More about MDMore about MD
Additional literature
D. Chandler: Introduction to Modern Statistical Mechanics, 1987, Oxford: Oxford University Press.
M. P. Allen and D. J. Tildesley: Computer Simulation of Liquids, 1987, Oxford: Oxford University Press.
J. P. Boon and S. Yip: Molecular Hydrodynamics, 1980, New York: McGraw-Hill; also Dover Publications, 1991.
J.M. Haile: Molecular Dynamics Simulation Elementary Method, 1992, John Wiley & Sons,
D. Frenkel and B. Smit: Understanding Molecular Simulations, 1996, New York, Academic Press.