CHEM699.08 Lecture #9 Calculating Time-dependent Properties June 28, 2001 MM 1 st Ed. Chapter 6 -- 333-342 MM

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CHEM699.08

Lecture #9 Calculating Time-dependent Properties

June 28, 2001

MM1stEd. Chapter 6 -- 333-342

MM2ndEd. Chapter 7.6 -- 374-382

[ 1 ]

Calculating Time-dependent Properties

¤

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+dt).

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A time correlation coefficient is a calculated measurement of the degree of correlation for an observed time-dependent property.

[ 2 ]

Calculating Time-dependent Properties

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Our “simple” 2D MD simulation of a single hard sphere moving through an arbitrarily chosen plane.

[ 3 ]

Calculating Time-dependent Properties

y

¤

Our “simple” 2D MD simulation of a single hard sphere moving through an arbitrarily chosen plane.

x

[ 3 ]

Calculating Time-dependent Properties

y

¤

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?

x

[ 3 ]

Calculating Time-dependent Properties

y

¤

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?

x

[ 3 ]

t = 0

Calculating Time-dependent Properties

y

¤

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?

x

[ 3 ]

t = 1

t = 0

Calculating Time-dependent Properties

y

¤

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?

x

[ 3 ]

t = 1

t = 2

t = 0

Calculating Time-dependent Properties

y

¤

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?

x

[ 3 ]

t = 1

t = 2

t = 3

t = 0

Calculating Time-dependent Properties

y

¤

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?

x

[ 3 ]

t = 4

t = 1

t = 2

t = 3

t = 0

Calculating Time-dependent Properties

y

¤

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?

x

[ 3 ]

t = 4

t = 5

t = 1

t = 2

t = 3

t = 0

Calculating Time-dependent Properties

¤

If there are two sets of data, x and y, the correlation between them (Cxy) can be defined as:

(1)

[ 4 ]

Calculating Time-dependent Properties

¤

If there are two sets of data, x and y, the correlation between them (Cxy) can be defined as:

(1)

¤

This can also be normalized to a value between -1 and +1 by dividing by the rms of x and y:

(2)

[ 4 ]

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.

[ 5 ]

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.

¤

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)

[ 5 ]

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.

[ 6 ]

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?

¤

Well there’s a way around this.

[ 6 ]

Calculating Time-dependent Properties

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Equation 3 can be re-written without the mean values of x and y:

(4)

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This expression allows for the calculation of cxy on the fly, as the MD simulation progresses!

[ 7 ]

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)

[ 8 ]

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.

[ 8 ]

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)

¤

We can normalize the velocity autocorrelation coefficient thusly:

(7)

[ 9 ]

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.

[ 10 ]

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

[ 10 ]

Calculating Time-dependent Properties

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Shown here are the velocity autocorrelation functions for the MD simulations of argon at two different densities.

¤

At time t = 0 the velocity autocorrelation function is highly correlated as expected, and begins to decrease toward 0.

cvv(t)

Fig.2

[ 11 ]

Time (ps)

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

[ 12 ]

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.

[ 13 ]

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)

[ 14 ]

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)

[ 14 ]

Calculating Time-dependent Properties

¤

Transport Properties

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A mass or concentration gradient will give rise to a flow of material from one region to another until the concentration is even throughout.

¤

The word “transport” suggests the system is at non-equilibrium.

¤

Here we will deal with calculating non-equilibrium properties by considering local fluctuations in a system already at equilibrium.

¤

[ 15 ]

Calculating Time-dependent Properties

¤

The flux (transport of some quantity) can be expressed by Fick’s first law of diffusion thusly:

Jz = -D (dN / dz)

(10)

[ 16 ]

Calculating Time-dependent Properties

¤

The flux (transport of some quantity) can be expressed by Fick’s first law of diffusion thusly:

Jz = -D (dN / dz)

(10)

¤

The time dependence (time-evolution of some distribution) is expressed by Fick’s second law:

¶N (z,t)

¶2N (z,t)

(11)

= D

¶ t

¶ z2

[ 16 ]

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)

3D =

[ 17 ]

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)

3D =

¤

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.

[ 17 ]