6、传递性质的统计力学计算,分子动力学初步

i p / 2mi 1 ij ) 2 v (r
2 i j i
Momenta of molecule i.
The quantities F, F and are multi-particle properties, properties of the system as a whole, and so additional averaging over the N particles is not possible. Consequently viscosity and thermal conductivity are subject to much greater statistical imprecision than D.
The non-normalized correlation function is defined as
CAB (t ) A(t ) B(0)
ens
A((t )) B((0))
ens
Time correlation functions and transport coefficients---continue
Diffusion coefficient
We can use time correlation function or Einstein relation to calculate transport coefficients in computer simulations. Or go back to first principles and conducting a suitable non-equilibrium simulation. For using equilibrium MD, we give here the equations in the microcanonical ensemble, for a fluid composed of N identical molecules. The diffusion coefficient is given by Green-Kubo formula or Einstein relationship 1 The center-of-mass velocity D dt v i (t ) v i (0) 3 0 of a single molecules
Einstein relationship
4 V 2t (V ) ( L (t ) L (0) Pt )2 3 kBT
Thermal conductivity
Green-Kubo Formula
V T kBT 2


0
dt j (t ) j (0)
A component of the energy current, i.e., the time derivative of .
2.3.4 Potential truncation and its corrections
The chemical potential may also be related to g(r)
kBT ln( ) 4 d r 2v(r ) g (r, )dr
3 0 0
This is achieved by expanding the partition function in powers of Planck’s constant ħ = h/2.
Efull EC 2 N r 2v(r )dr
rC
Pfull P C (2 / 3)
2


rC
r 3[dv(r ) / dr ]dr
full C 4 r 2v(r )dr
rC

Subscript C refers to the quantities calculated from simulation.
1 F pi pi / mi rij fij V i i j i
Einstein relationship
V 2t ( L (t ) L (0)) 2 kBT
L
1 ri pi V i
Bulk viscosity
Green-Kubo formula
1) They give a clear picture of the dynamics in a fluid
2) tA is related to macroscopic transport coefficients
3) Fourier transform is related to experimental spectra.
1 2 2tD ri (t ) ri (0) 3
3directions
Molecular position at time t.
Velocity autocorrelation function and mean squared displacement
1.0 0.8
Cvv(t)
0.6 0.4 0.2 0.0 -0.2 0 1
cAB A B / ( A) ( B)
A A A ens
2 ( A) A2
ens
A2
ens
A
2 ens
The absolute value of cAB lies between 0 and 1, with values close to 1 indicating a high degree of correlation. By considering A and B to be evaluated at two different times, the resulting quantity is a function of the time difference and called time correlation function cAB(t).

1 ri ( i i ) V i
Make no contribution if ri=0, as in the case in a normal one-component MD simulation. Einstein relationship
How to get i ?
V 2 2tT ( ( t ) (0)) kBT 2
How to get the energy per molecule?
Assuming pairwise potentials, the potential energy of two molecules is taken to be divided equally between them:
(t ) A (0) dt A
0
Transport coefficient
A variable appearing in the perturbation term in the Hamiltonian.
Einstein relation
2t ( A(t ) A(0)) 2
1

Thermal de Broglie wavelenth
Parameter coupling the two atom
In the MD or MC, in order to save time, we introduce a spherical cutoff. The cutoff distance should be sufficient large to ensure that it is a small perturbation, but it must be no greater than BOXL/2 for the consistency with minimum image convention.
So that
cAB (t ) CAB (t ) / ( A) ( B)
cAA (t ) CAA (t ) / ( A) CAA (t ) / CAA (0)
2
Transport coefficients are defined in terms of the response of a system to a perturbation.
Shear viscosity
Green-Kubo Formula:
V kBT


0
dt F (t ) F (0)
1 F pi pi / mi ri fi V i i
xy, yz, zx
An off-diagonal element of the pressure tensor
2.3.5 Time correlation functions and transport coefficients
Correlation between two different quantities A and B are measured in the usual statistical sense, via the correlation coefficient cAB
4 V V 3 kBT


0
dt F (t ) F (0)
F (t ) F (t ) F F (t ) P
In calculation, we use
1 1 F (t ) Tr[ F (t )] F 3 3
and
F (t ) F (t ) P
Long-range corrections
Therefore simulation calculates only the part of integration in energy, pressure and chemical potential equations (from 0 to rC). They should be corrected. Assuming the radial distribution function as unit in the distance r > rC, the energy, pressure and chemical potential can be corrected by
=0.863 g/cm 3 =1.396 g/cm
3
2
3
4
t / ps
Velocity autocorrelation functions for liquid argon
Variation in mean squared displacement during a MD simulation of argon
2.3.6 Quantum corrections
Most of this course will deal with the computer simulation of system within the classical approximation. Even within the limitations of a classical simulation, it is still possible to estimate quantum corrections of thermodynamic functions.
Time correlation functions and transport coefficients---continue
For identical phase functions, cAA(t) is called an autocorrelation function and its time integral (from t =0 to t =) is a correlation time tA. These functions are of great interest in computer simulation because