CHEM Lecture #9 Calculating Time-dependent Properties June 28, 2001 MM 1 st Ed. Chapter MM 2 nd Ed. Chapter [ 1 ]

Calculating Time-dependent Properties [ 2 ] 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. ¤ 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). ¤ A time correlation coefficient is a calculated measurement of the degree of correlation for an observed time-dependent property. ¤

[ 3 ] Calculating Time-dependent Properties Our “simple” 2D MD simulation of a single hard sphere moving through an arbitrarily chosen plane. ¤

[ 3 ] Calculating Time-dependent Properties x y Our “simple” 2D MD simulation of a single hard sphere moving through an arbitrarily chosen plane. ¤

[ 3 ] Calculating Time-dependent Properties Is the movement of the sphere in the x direction related to the motion in the y direction? ¤ x y Our “simple” 2D MD simulation of a single hard sphere moving through an arbitrarily chosen plane. ¤

[ 3 ] Calculating Time-dependent Properties t = 0 x 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? ¤

[ 3 ] 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? ¤

[ 3 ] Calculating Time-dependent Properties x y t = 0 t = 1t = 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? ¤

[ 3 ] Calculating Time-dependent Properties x y t = 0 t = 1t = 2t = 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? ¤

[ 3 ] Calculating Time-dependent Properties x y t = 0 t = 1t = 2t = 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? ¤

[ 3 ] Calculating Time-dependent Properties x y t = 0 t = 1t = 2t = 3 t = 4t = 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? ¤

[ 4 ] Calculating Time-dependent Properties If there are two sets of data, x and y, the correlation between them (C xy ) can be defined as: ¤ (1)

[ 4 ] Calculating Time-dependent Properties If there are two sets of data, x and y, the correlation between them (C xy ) 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)

[ 5 ] Calculating Time-dependent Properties A value of c xy = 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 c xy = 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)

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? ¤ [ 6 ] Well there’s a way around this. ¤

[ 7 ] 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 c xy on the fly, as the MD simulation progresses! ¤

[ 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)

[ 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 C xy 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. ¤

[ 9 ] 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: ¤

[ 10 ] Calculating Time-dependent Properties For properties like velocities, the value of c vv at time t = 0 would be 1, while at loner times c vv 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. ¤

Calculating Time-dependent Properties For properties like velocities, the value of c vv at time t = 0 would be 1, while at loner times c vv 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 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 [ 11 ]

Calculating Time-dependent Properties The long time tail of c vv (t) has been ascribed to “hydrodynamic vortices” which form around the moving particles, giving a small additive contribution to their velocity. ¤ Fig.3 [ 12 ]

[ 13 ] 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. ¤

[ 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)

[ 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)

[ 15 ] 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. ¤

[ 16 ] Calculating Time-dependent Properties The flux (transport of some quantity) can be expressed by Fick’s first law of diffusion thusly: ¤ (10) J z =  D (d N / dz)

[ 16 ] Calculating Time-dependent Properties The flux (transport of some quantity) can be expressed by Fick’s first law of diffusion thusly: ¤ (10) J z =  D (d N / dz) The time dependence (time-evolution of some distribution) is expressed by Fick’s second law: ¤ (11)  N (z,t)  t  2 N (z,t)  z 2 = D

Calculating Time-dependent Properties [ 17 ] 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 =

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. ¤ [ 17 ] 3D =

Calculating Time-dependent Properties ~ fin ~ [ 18 ]