chem699.08 lecture #9 calculating time-dependent properties june 28, 2001 mm 1 st ed. chapter 6 --...
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![Page 1: CHEM699.08 Lecture #9 Calculating Time-dependent Properties June 28, 2001 MM 1 st Ed. Chapter 6 -- 333-342 MM 2 nd Ed. Chapter 7.6 -- 374-382 [ 1 ]](https://reader038.vdocuments.net/reader038/viewer/2022110322/56649d225503460f949f87bf/html5/thumbnails/1.jpg)
CHEM699.08
Lecture #9 Calculating Time-dependent Properties
June 28, 2001
MM1stEd. Chapter 6 -- 333-342
MM2ndEd. Chapter 7.6 -- 374-382
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![Page 2: CHEM699.08 Lecture #9 Calculating Time-dependent Properties June 28, 2001 MM 1 st Ed. Chapter 6 -- 333-342 MM 2 nd Ed. Chapter 7.6 -- 374-382 [ 1 ]](https://reader038.vdocuments.net/reader038/viewer/2022110322/56649d225503460f949f87bf/html5/thumbnails/2.jpg)
Calculating Time-dependent Properties
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An advantage of a molecular dynamics (MD) simulation over a Monte Carlo simulation is that each successive iteration of the system is connected to the previous state(s) of the system in time.
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The evolution of a MD simulation over time allows the data, or some property, at one time (t) to be related to the same or different properties at some other time (t+t).
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A time correlation coefficient is a calculated measurement of the degree of correlation for an observed time-dependent property.
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![Page 3: CHEM699.08 Lecture #9 Calculating Time-dependent Properties June 28, 2001 MM 1 st Ed. Chapter 6 -- 333-342 MM 2 nd Ed. Chapter 7.6 -- 374-382 [ 1 ]](https://reader038.vdocuments.net/reader038/viewer/2022110322/56649d225503460f949f87bf/html5/thumbnails/3.jpg)
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Calculating Time-dependent Properties
Our “simple” 2D MD simulation of a single hard sphere moving through an arbitrarily chosen plane.
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![Page 4: CHEM699.08 Lecture #9 Calculating Time-dependent Properties June 28, 2001 MM 1 st Ed. Chapter 6 -- 333-342 MM 2 nd Ed. Chapter 7.6 -- 374-382 [ 1 ]](https://reader038.vdocuments.net/reader038/viewer/2022110322/56649d225503460f949f87bf/html5/thumbnails/4.jpg)
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Calculating Time-dependent Properties
x
yOur “simple” 2D MD simulation of a single hard sphere moving through an arbitrarily chosen plane.
¤
![Page 5: CHEM699.08 Lecture #9 Calculating Time-dependent Properties June 28, 2001 MM 1 st Ed. Chapter 6 -- 333-342 MM 2 nd Ed. Chapter 7.6 -- 374-382 [ 1 ]](https://reader038.vdocuments.net/reader038/viewer/2022110322/56649d225503460f949f87bf/html5/thumbnails/5.jpg)
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Calculating Time-dependent Properties
Is the movement of the sphere in the x direction related to the motion in the y direction?
¤
x
yOur “simple” 2D MD simulation of a single hard sphere moving through an arbitrarily chosen plane.
¤
![Page 6: CHEM699.08 Lecture #9 Calculating Time-dependent Properties June 28, 2001 MM 1 st Ed. Chapter 6 -- 333-342 MM 2 nd Ed. Chapter 7.6 -- 374-382 [ 1 ]](https://reader038.vdocuments.net/reader038/viewer/2022110322/56649d225503460f949f87bf/html5/thumbnails/6.jpg)
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Calculating Time-dependent Properties
t = 0
x
yOur “simple” 2D MD simulation of a single hard sphere moving through an arbitrarily chosen plane.
¤
Is the movement of the sphere in the x direction related to the motion in the y direction?
¤
![Page 7: CHEM699.08 Lecture #9 Calculating Time-dependent Properties June 28, 2001 MM 1 st Ed. Chapter 6 -- 333-342 MM 2 nd Ed. Chapter 7.6 -- 374-382 [ 1 ]](https://reader038.vdocuments.net/reader038/viewer/2022110322/56649d225503460f949f87bf/html5/thumbnails/7.jpg)
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Calculating Time-dependent Properties
x
y
t = 0 t = 1
Our “simple” 2D MD simulation of a single hard sphere moving through an arbitrarily chosen plane.
¤
Is the movement of the sphere in the x direction related to the motion in the y direction?
¤
![Page 8: CHEM699.08 Lecture #9 Calculating Time-dependent Properties June 28, 2001 MM 1 st Ed. Chapter 6 -- 333-342 MM 2 nd Ed. Chapter 7.6 -- 374-382 [ 1 ]](https://reader038.vdocuments.net/reader038/viewer/2022110322/56649d225503460f949f87bf/html5/thumbnails/8.jpg)
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Calculating Time-dependent Properties
x
y
t = 0 t = 1 t = 2
Our “simple” 2D MD simulation of a single hard sphere moving through an arbitrarily chosen plane.
¤
Is the movement of the sphere in the x direction related to the motion in the y direction?
¤
![Page 9: CHEM699.08 Lecture #9 Calculating Time-dependent Properties June 28, 2001 MM 1 st Ed. Chapter 6 -- 333-342 MM 2 nd Ed. Chapter 7.6 -- 374-382 [ 1 ]](https://reader038.vdocuments.net/reader038/viewer/2022110322/56649d225503460f949f87bf/html5/thumbnails/9.jpg)
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Calculating Time-dependent Properties
x
y
t = 0 t = 1 t = 2 t = 3
Our “simple” 2D MD simulation of a single hard sphere moving through an arbitrarily chosen plane.
¤
Is the movement of the sphere in the x direction related to the motion in the y direction?
¤
![Page 10: CHEM699.08 Lecture #9 Calculating Time-dependent Properties June 28, 2001 MM 1 st Ed. Chapter 6 -- 333-342 MM 2 nd Ed. Chapter 7.6 -- 374-382 [ 1 ]](https://reader038.vdocuments.net/reader038/viewer/2022110322/56649d225503460f949f87bf/html5/thumbnails/10.jpg)
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Calculating Time-dependent Properties
x
y
t = 0 t = 1 t = 2 t = 3 t = 4
Our “simple” 2D MD simulation of a single hard sphere moving through an arbitrarily chosen plane.
¤
Is the movement of the sphere in the x direction related to the motion in the y direction?
¤
![Page 11: CHEM699.08 Lecture #9 Calculating Time-dependent Properties June 28, 2001 MM 1 st Ed. Chapter 6 -- 333-342 MM 2 nd Ed. Chapter 7.6 -- 374-382 [ 1 ]](https://reader038.vdocuments.net/reader038/viewer/2022110322/56649d225503460f949f87bf/html5/thumbnails/11.jpg)
[ 3 ]
Calculating Time-dependent Properties
x
y
t = 0 t = 1 t = 2 t = 3 t = 4 t = 5
Our “simple” 2D MD simulation of a single hard sphere moving through an arbitrarily chosen plane.
¤
Is the movement of the sphere in the x direction related to the motion in the y direction?
¤
![Page 12: CHEM699.08 Lecture #9 Calculating Time-dependent Properties June 28, 2001 MM 1 st Ed. Chapter 6 -- 333-342 MM 2 nd Ed. Chapter 7.6 -- 374-382 [ 1 ]](https://reader038.vdocuments.net/reader038/viewer/2022110322/56649d225503460f949f87bf/html5/thumbnails/12.jpg)
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Calculating Time-dependent Properties
If there are two sets of data, x and y, the correlation between them (Cxy) can be defined as:
¤
(1)
![Page 13: CHEM699.08 Lecture #9 Calculating Time-dependent Properties June 28, 2001 MM 1 st Ed. Chapter 6 -- 333-342 MM 2 nd Ed. Chapter 7.6 -- 374-382 [ 1 ]](https://reader038.vdocuments.net/reader038/viewer/2022110322/56649d225503460f949f87bf/html5/thumbnails/13.jpg)
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Calculating Time-dependent Properties
If there are two sets of data, x and y, the correlation between them (Cxy) can be defined as:
¤
This can also be normalized to a value between -1 and +1 by dividing by the rms of x and y:
¤
(1)
(2)
![Page 14: CHEM699.08 Lecture #9 Calculating Time-dependent Properties June 28, 2001 MM 1 st Ed. Chapter 6 -- 333-342 MM 2 nd Ed. Chapter 7.6 -- 374-382 [ 1 ]](https://reader038.vdocuments.net/reader038/viewer/2022110322/56649d225503460f949f87bf/html5/thumbnails/14.jpg)
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Calculating Time-dependent Properties
A value of cxy = 0 would indicate no correlation between the values of x and y, while a value of 1 indicates a high degree of correlation.
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![Page 15: CHEM699.08 Lecture #9 Calculating Time-dependent Properties June 28, 2001 MM 1 st Ed. Chapter 6 -- 333-342 MM 2 nd Ed. Chapter 7.6 -- 374-382 [ 1 ]](https://reader038.vdocuments.net/reader038/viewer/2022110322/56649d225503460f949f87bf/html5/thumbnails/15.jpg)
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Calculating Time-dependent Properties
A value of cxy = 0 would indicate no correlation between the values of x and y, while a value of 1 indicates a high degree of correlation.
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If x and y are found to only fluctuate around some average value as would be the case for bond lengths, for example, Equation 2 is commonly expressed only as the fluctuating part of x and y.
¤
(3)
![Page 16: CHEM699.08 Lecture #9 Calculating Time-dependent Properties June 28, 2001 MM 1 st Ed. Chapter 6 -- 333-342 MM 2 nd Ed. Chapter 7.6 -- 374-382 [ 1 ]](https://reader038.vdocuments.net/reader038/viewer/2022110322/56649d225503460f949f87bf/html5/thumbnails/16.jpg)
Calculating Time-dependent Properties
One drawback to Equation 3 is that the mean values of x and y can’t accurately be known until the MD simulation has completed all M steps.
¤
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Calculating Time-dependent Properties
One drawback to Equation 3 is that the mean values of x and y can’t accurately be known until the MD simulation has completed all M steps.
¤
Tired of waiting for those pesky MD simulations to finish before generating your time-correlation coefficients?
¤
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Well there’s a way around this.¤
![Page 18: CHEM699.08 Lecture #9 Calculating Time-dependent Properties June 28, 2001 MM 1 st Ed. Chapter 6 -- 333-342 MM 2 nd Ed. Chapter 7.6 -- 374-382 [ 1 ]](https://reader038.vdocuments.net/reader038/viewer/2022110322/56649d225503460f949f87bf/html5/thumbnails/18.jpg)
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Calculating Time-dependent Properties
Equation 3 can be re-written without the mean values of x and y:¤
(4)
This expression allows for the calculation of cxy on the fly, as the MD simulation progresses!
¤
![Page 19: CHEM699.08 Lecture #9 Calculating Time-dependent Properties June 28, 2001 MM 1 st Ed. Chapter 6 -- 333-342 MM 2 nd Ed. Chapter 7.6 -- 374-382 [ 1 ]](https://reader038.vdocuments.net/reader038/viewer/2022110322/56649d225503460f949f87bf/html5/thumbnails/19.jpg)
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Calculating Time-dependent Properties
As the MD simulation proceeds the values of one property can be compared to the same, or another property at a later time:
¤
(5)
![Page 20: CHEM699.08 Lecture #9 Calculating Time-dependent Properties June 28, 2001 MM 1 st Ed. Chapter 6 -- 333-342 MM 2 nd Ed. Chapter 7.6 -- 374-382 [ 1 ]](https://reader038.vdocuments.net/reader038/viewer/2022110322/56649d225503460f949f87bf/html5/thumbnails/20.jpg)
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Calculating Time-dependent Properties
As the MD simulation proceeds the values of one property can be compared to the same, or another property at a later time:
¤
(5)
If x and y are different properties, then Cxy is referred to as a cross-correlation function. If x and y are the same property, then this is referred to as an autocorrelation function.
¤
The autocorrelation function can be though of as an indication of how long the system retains a “memory” of its previous state.
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![Page 21: CHEM699.08 Lecture #9 Calculating Time-dependent Properties June 28, 2001 MM 1 st Ed. Chapter 6 -- 333-342 MM 2 nd Ed. Chapter 7.6 -- 374-382 [ 1 ]](https://reader038.vdocuments.net/reader038/viewer/2022110322/56649d225503460f949f87bf/html5/thumbnails/21.jpg)
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Calculating Time-dependent Properties
An example is the velocity autocorrelation coefficient which gives an indication of how the velocity at time (t) correlates with the velocity at another time.
¤
(6)
(7)
We can normalize the velocity autocorrelation coefficient thusly:¤
![Page 22: CHEM699.08 Lecture #9 Calculating Time-dependent Properties June 28, 2001 MM 1 st Ed. Chapter 6 -- 333-342 MM 2 nd Ed. Chapter 7.6 -- 374-382 [ 1 ]](https://reader038.vdocuments.net/reader038/viewer/2022110322/56649d225503460f949f87bf/html5/thumbnails/22.jpg)
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Calculating Time-dependent Properties
For properties like velocities, the value of cvv at time t = 0 would be 1, while at loner times cvv would be expected to go to 0.
¤
The time required for the correlation to go to 0 is referred to as the correlation time, or the relaxation time. The MD simulation must be at least long enough to meet the relaxation time, obviously.
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![Page 23: CHEM699.08 Lecture #9 Calculating Time-dependent Properties June 28, 2001 MM 1 st Ed. Chapter 6 -- 333-342 MM 2 nd Ed. Chapter 7.6 -- 374-382 [ 1 ]](https://reader038.vdocuments.net/reader038/viewer/2022110322/56649d225503460f949f87bf/html5/thumbnails/23.jpg)
Calculating Time-dependent Properties
For properties like velocities, the value of cvv at time t = 0 would be 1, while at loner times cvv would be expected to go to 0.
¤
The time required for the correlation to go to 0 is referred to as the correlation time, or the relaxation time. The MD simulation must be at least long enough to meet the relaxation time, obviously.
¤
For long MD simulations the relaxation times can be calculated relative to several starting points in order to reduce the uncertainty.
¤
Fig.1
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![Page 24: CHEM699.08 Lecture #9 Calculating Time-dependent Properties June 28, 2001 MM 1 st Ed. Chapter 6 -- 333-342 MM 2 nd Ed. Chapter 7.6 -- 374-382 [ 1 ]](https://reader038.vdocuments.net/reader038/viewer/2022110322/56649d225503460f949f87bf/html5/thumbnails/24.jpg)
Calculating Time-dependent Properties
Shown here are the velocity autocorrelation functions for the MD simulations of argon at two different densities.¤
Time (ps)
c vv(
t)At time t = 0 the velocity autocorrelation function is highly correlated as expected, and begins to decrease toward 0.¤
Fig.2
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![Page 25: CHEM699.08 Lecture #9 Calculating Time-dependent Properties June 28, 2001 MM 1 st Ed. Chapter 6 -- 333-342 MM 2 nd Ed. Chapter 7.6 -- 374-382 [ 1 ]](https://reader038.vdocuments.net/reader038/viewer/2022110322/56649d225503460f949f87bf/html5/thumbnails/25.jpg)
Calculating Time-dependent Properties
The long time tail of cvv(t) has been ascribed to “hydrodynamic vortices” which form around the moving particles, giving a small additive contribution to their velocity.
¤
Fig.3
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Calculating Time-dependent Properties
This slow decay of the time correlation toward 0 can be problematic when trying to establish a time frame for the MD simulation, and also in the derivation of some properties.
¤
Transport coefficients require the correlation function to be integrated between time t = 0 and t = ¤
In cases where the time correlation has a long time-tail there will be fewer blocks of data over a sufficiently wide time span to reduce the uncertainty in the correlation coefficients.
¤
![Page 27: CHEM699.08 Lecture #9 Calculating Time-dependent Properties June 28, 2001 MM 1 st Ed. Chapter 6 -- 333-342 MM 2 nd Ed. Chapter 7.6 -- 374-382 [ 1 ]](https://reader038.vdocuments.net/reader038/viewer/2022110322/56649d225503460f949f87bf/html5/thumbnails/27.jpg)
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Calculating Time-dependent Properties
Another example is the net dipole moment of the system. This requires the summation of the individual dipoles (vector quantities) of each molecule in the system -- which will change over time.
¤
(8)
![Page 28: CHEM699.08 Lecture #9 Calculating Time-dependent Properties June 28, 2001 MM 1 st Ed. Chapter 6 -- 333-342 MM 2 nd Ed. Chapter 7.6 -- 374-382 [ 1 ]](https://reader038.vdocuments.net/reader038/viewer/2022110322/56649d225503460f949f87bf/html5/thumbnails/28.jpg)
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Calculating Time-dependent Properties
Another example is the net dipole moment of the system. This requires the summation of the individual dipoles (vector quantities) of each molecule in the system -- which will change over time.
¤
(8)
The total dipole correlation function is expressed as:¤
(9)
![Page 29: CHEM699.08 Lecture #9 Calculating Time-dependent Properties June 28, 2001 MM 1 st Ed. Chapter 6 -- 333-342 MM 2 nd Ed. Chapter 7.6 -- 374-382 [ 1 ]](https://reader038.vdocuments.net/reader038/viewer/2022110322/56649d225503460f949f87bf/html5/thumbnails/29.jpg)
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Calculating Time-dependent Properties
Transport Properties¤
A mass or concentration gradient will give rise to a flow of material from one region to another until the concentration is even throughout.
¤
Here we will deal with calculating non-equilibrium properties by considering local fluctuations in a system already at equilibrium.¤
The word “transport” suggests the system is at non-equilibrium. ¤
Examples: temperature gradient, mass gradient, velocity gradient, etc.¤
![Page 30: CHEM699.08 Lecture #9 Calculating Time-dependent Properties June 28, 2001 MM 1 st Ed. Chapter 6 -- 333-342 MM 2 nd Ed. Chapter 7.6 -- 374-382 [ 1 ]](https://reader038.vdocuments.net/reader038/viewer/2022110322/56649d225503460f949f87bf/html5/thumbnails/30.jpg)
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Calculating Time-dependent Properties
The flux (transport of some quantity) can be expressed by Fick’s first law of diffusion thusly:¤
(10)Jz = D (dN / dz)
![Page 31: CHEM699.08 Lecture #9 Calculating Time-dependent Properties June 28, 2001 MM 1 st Ed. Chapter 6 -- 333-342 MM 2 nd Ed. Chapter 7.6 -- 374-382 [ 1 ]](https://reader038.vdocuments.net/reader038/viewer/2022110322/56649d225503460f949f87bf/html5/thumbnails/31.jpg)
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Calculating Time-dependent Properties
The flux (transport of some quantity) can be expressed by Fick’s first law of diffusion thusly:¤
(10)Jz = D (dN / dz)
The time dependence (time-evolution of some distribution) is expressed by Fick’s second law:¤
(11)N (z,t)
t
2N (z,t)
z2= D
![Page 32: CHEM699.08 Lecture #9 Calculating Time-dependent Properties June 28, 2001 MM 1 st Ed. Chapter 6 -- 333-342 MM 2 nd Ed. Chapter 7.6 -- 374-382 [ 1 ]](https://reader038.vdocuments.net/reader038/viewer/2022110322/56649d225503460f949f87bf/html5/thumbnails/32.jpg)
Calculating Time-dependent Properties
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Einstein showed that the diffusion coefficient (D) is related to the mean square of the distance, and in 3-dimensions this is given by:¤
(12)3D =
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Calculating Time-dependent Properties
Einstein showed that the diffusion coefficient (D) is related to the mean square of the distance, and in 3-dimensions this is given by:¤
(12)
It is important to point out that Fick’s law only applies at long time durations, such as the case above. To a good approximation some duration where “t” effectively approaches infinity as far as the simulation is concerned will be sufficient.
¤
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3D =
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Calculating Time-dependent Properties
~ fin ~
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