ce 530 molecular simulationkofke/ce530/lectures/lecture12.ppt.pdf5 approaches to evaluating...

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CE 530 Molecular Simulation

Lecture 12 Dynamical Properties

David A. Kofke

Department of Chemical Engineering SUNY Buffalo

[email protected]

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Review

¡ Several equivalent ways to formulate classical mechanics •  Newtonian, Lagrangian, Hamiltonian •  Lagrangian and Hamiltonian independent of coordinates •  Hamiltonian preferred because of central role of phase space to

development ¡ Molecular dynamics

•  numerical integration of equations of motion for multibody system •  Verlet algorithms simple and popular

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Dynamical Properties

¡ How does the system respond collectively when put in a state of non-equilibrium?

¡ Conserved quantities •  mass, momentum, energy •  where does it go, and how quickly? •  relate to macroscopic transport coefficients

¡ Non-conserved quantities •  how quickly do they appear and vanish? •  relate to spectroscopic measurements

¡ What do we compute in simulation to measure the macroscopic property?

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Macroscopic Transport Phenomena ¡ Dynamical behavior of conserved quantities

•  densities change only by redistribution on macroscopic time scale ¡ Differential balance

¡ Constitutive equation

¡ Our aim is to obtain the phenomenological transport coefficients by molecular simulation •  note that the “laws” are (often very good) approximations that

apply to not-too-large gradients •  in principle coefficients depend on c, T, and v

( , ) 0c tt

∂ +∇ ⋅ =∂r j ( , ) 0p

T tct

∂ +∇ ⋅ =∂r q ( , ) 0D t

Dtρ τ+∇ ⋅ =v r

D c= − ∇j k T= − ∇q ( )xy y xτ ν ρ= − ∇ vFick’s law Fourier’s law Newton’s law

Mass Energy Momentum

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Approaches to Evaluating Transport Properties ¡ Need a non-equilibrium condition ¡ Method 1: Establish a non-equilibrium steady state

•  “Non-equilibrium molecular dynamics” NEMD •  Requires continuous addition and removal of conserved quantities •  Usually involves application of work, so must apply thermostat •  Only one transport property measured at a time •  Gives good statistics (high “signal-to-noise ratio”) •  Requires extrapolation to “linear regime”

¡ Method 2: Rely on natural fluctuations •  Any given configuration has natural inhomogeneity of mass,

momentum, energy (have a look) •  Observe how these natural fluctuations dissipate •  All transport properties measurable at once •  Poor signal-to-noise ratio

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Mass Transfer ¡ Self-diffusion

•  diffusion in a pure substance •  consider tagging molecules and watching how they migrate

¡ Diffusion equations •  Combine mass balance with Fick’s law

•  Take as boundary condition a point concentration at the origin

¡ Solution

¡ Second moment

2 ( , ) 0c D c tt

∂ − ∇ =∂

r

( , ) ( )c t δ=r r

2/ 2( , ) (2 ) exp

2d rc t Dt

Dtπ − ⎛ ⎞

= −⎜ ⎟⎝ ⎠

r

2 2( ) ( , )

2

r t r c t d

dDt

=

=∫ r r RMS displacement increases as t1/2

Compare to ballistic r ~ t

For given configuration, each molecule represents a point of high concentration (fluctuation)

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Interpretation ¡ Right-hand side is macroscopic property

•  applicable at macroscopic time scales

¡ For any given configuration, each atom represents a point of high concentration (a weak fluctuation)

¡ View left-hand side of formula as the movement of this atom •  ensemble average over all initial conditions

•  asymptotic linear behavior of mean-square displacement gives diffusion constant

•  independent data can be collected for each molecule

2( ) 2r t dDt=

r2(t) = dpN drNr12(t) δ (r1)π (r

N ,pN )⎡⎣

⎤⎦∫∫ t=0

2r

t

Slope here gives D

2 21( ) ( )iNr t r t= ∑

Einstein equation

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Time Correlation Function ¡ Alternative but equivalent formulation is possible ¡ Write position r at time t as sum of displacements

0 0

( ) ( )t tdt d ddt

τ τ τ= =∫ ∫rr v

dt

dt

dt

r(t) v

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Time Correlation Function ¡ Alternative but equivalent formulation is possible ¡ Write position r at time t as sum of displacements ¡ Then 0 0

( ) ( )t tdt d ddt

τ τ τ= =∫ ∫rr v

1

1

21 1 2 2

0 0

1 2 2 10 0

1 2 2 10 0

1 2 1 20 0

10 0

0

( ) ( ) ( )

( ) ( )

2 ( ) ( )

2 (0) ( )

2 (0) ( )

2 2 (0) ( )

t t

t t

t

t

t t

t

r t d d

d d

d d

d d

d d

dDt t d

τ

τ

τ τ τ τ

τ τ τ τ

τ τ τ τ

τ τ τ τ

τ τ τ

τ τ

= ⋅

= ⋅

= ⋅

= ⋅ −

= ⋅

= ⋅

∫ ∫

∫ ∫

∫ ∫

∫ ∫

∫ ∫

∫

v v

v v

v v

v v

v v

v v

r2 in terms of displacement integrals

rearrange order of averages

1 2 1 2 1 2( ) ( ) 2 ( )τ τ τ τ τ τ< + > = >∫ ∫ ∫ ∫ ∫ ∫

correlation depends only on time difference, not time origin

1 2substitute τ τ τ= −

0

1 (0) ( )D dd

τ τ∞

= ⋅∫ v v Green-Kubo equation

t→∞

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Velocity Autocorrelation Function

0

1 (0) ( )D dd

τ τ∞

= ⋅∫ v v

( ) (0) ( )C t t≡ ⋅v v

Backscattering

¡ Definition

¡ Typical behavior Zero slope (soft potentials)

Diffusion constant is area under the curve

Asymptotic behavior (t à ∞) is nontrivial (depends on d)

2(0) /C v dkT m= =

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Other Transport Properties

¡ Diffusivity

¡ Shear viscosity

¡ Thermal conductivity

0

1 ( ) (0)xy xydt tVkT

η σ σ∞

= ∫ 12

1( )

Nyxy x

i i ij y ijii i jm v v x f rσ

= ≠

⎡ ⎤= +⎢ ⎥

⎢ ⎥⎣ ⎦∑ ∑

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1 ( ) (0)T dt q t qVkT

λ∞

= ∫ 21 12 2

1( )

N

i i iji i j

dq m v u rdt = ≠

⎡ ⎤= +⎢ ⎥

⎢ ⎥⎣ ⎦∑ ∑

0

1 (0) ( )D dt tVdρ

∞= ⋅∫ v v [ ]

1

N

ii=

=∑v v

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Evaluating Time Correlation Functions ¡ Measure phase-space property A(rN,pN) for a sequence of time

intervals

¡ Tabulate the TCF for the same intervals •  each time in simulation serves as a new time origin for the TCF

A(t0) A(t1) A(t2) A(t3) A(t4) A(t5) A(t6) A(t7) A(t8) A(t9)

A0A1 A1A2 A2A3 A3A4 A4A5 A5A6 A6A7 A7A8 A8A9


+ + + + + + + + +... ( ) (0)A t AΔ

A0A2 A1A3 A2A4 A3A5 A4A6 A5A7 A6A8 A7A9


+ + + + + + + +... (2 ) (0)A t AΔ

A0A5 A1A6 A2A7 A3A8 A4A9


+ + + +(5 ) (0)A t AΔ +...

Dt ∆t

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Direct Approach to TCF

¡ Decide beforehand the time range (0,tmax) for evaluation of C(t) •  let n be the number of time steps in tmax

¡ At each simulation time step, update sums for all times in (0,tmax) •  n sums to update •  store values of A(t) for past n time steps •  at time step k:

¡ Considerations •  trade off range of C(t) against storage of history of A(t) •  requires n2 operations to evaluate TCF

ci+ = Ak−i Ak i = 1,…,n

k k-n

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Fourier Transform Approach to TCF

¡ Fourier transform

¡ Convolution

¡ Fourier convolution theorem

¡ Application to TCF

ˆ ( ) ( ) i tf f t e dtωω+∞

−

−∞

= ∫

( ) ( ) ( )C t A B t dτ τ τ+∞

−∞

= +∫

!C(ω ) = !A(ω ) !B(ω )

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ˆ( ) ( ) i tf t f e dωπ ω ω+∞

+

−∞

= ∫

0

0 0( ) ( ) ( )t

C t t t t= ⋅ +∑v v Sum over time origins

[ ]2ˆ ˆ( ) ( )C ω ω= v With FFT, operation scales as n ln(n)

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Coarse-Graining Approach to TCF 1.

¡ Evaluating long-time behavior can be expensive •  for time interval T, need to store T/δt values (perhaps for each atom)

¡ But long-time behavior does not (usually) depend on details of short time motions resolved at simulation time step

¡ Short time behaviors can be coarse-grained to retain information needed to compute long-time properties •  TCF is given approximately •  mean-square displacement can be computed without loss

Method due to Frenkel and Smit

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(1) (1)v

n0 Σ

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(1) (1)v

n0

(1) (2)v

Σ

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(1) (1)v

n0

(1) (2)v (1) (3)v

Σ

(2) (1)v

n0 Σ

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(1) (1)v (1) (2)v (1) (3)v

n0 Σ

(2) (1)v

(1) (4)v

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(1) (1)v (1) (2)v (1) (3)v

n0 Σ

(2) (1)v

(1) (4)v (1) (5)v

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(1) (1)v (1) (2)v (1) (3)v

n0 Σ

(2) (1)v

(1) (4)v (1) (5)v (1) (6)v

n0 Σ

(2) (2)v

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(1) (1)v (1) (2)v (1) (3)v

n0 Σ

(2) (1)v

(1) (4)v (1) (5)v (1) (6)v

(2) (2)v

(1) (7)v

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(1) (1)v (1) (2)v (1) (3)v

n0 Σ

(2) (1)v

(1) (4)v (1) (5)v (1) (6)v

(2) (2)v

(1) (7)v (1) (8)v

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(1) (1)v (1) (2)v (1) (3)v

n0 Σ

(2) (1)v

(1) (4)v (1) (5)v (1) (6)v

(2) (2)v

(1) (7)v (1) (8)v (1) (9)v

n0 Σ

(2) (3)v

n0 Σ

(3) (1)v

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(1) (1)v (1) (2)v (1) (3)v

n0 Σ

(2) (1)v

(1) (4)v (1) (5)v (1) (6)v

(2) (2)v

(1) (7)v (1) (8)v (1) (9)v

(2) (3)v

(3) (1)v

(1) (10)vet cetera

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(1) (1)v (1) (2)v (1) (3)v

(2) (1)v

(1) (4)v (1) (5)v (1) (6)v

(2) (2)v

(1) (7)v (1) (8)v (1) (9)v

(2) (3)v

(3) (1)v

(1) (10)v

n3 0

This term gives the net velocity over this interval

Approximate TCF 3 (3) (3)(0) ( ) (0) (1)n t⋅ Δ ≈ ⋅v v v v

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Coarse-Graining Approach: Resource Requirements

¡ Memory •  for each level, store n sub-blocks •  for simulation of length T = nkΔt requires k × n stored values

compare to nk values for direct method

¡ Computation •  each level j requires update (summing n terms) every 1/nj steps •  total

•  scales linearly with length of total correlation time compare to T2 or T ln(T) for other methods

T × nΔt

1+ 1n+ 1

n2 +…+ 1nk

⎛⎝⎜

⎞⎠⎟= t × n

Δtn− n−k

n−1

≈ T × nΔt

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Summary ¡ Dynamical properties describe the way collective behaviors

cause macroscopic observables to redistribute or decay ¡ Evaluation of transport coefficients requires non-equilibrium

condition •  NEMD imposes macroscopic non-equilibrium steady state •  EMD approach uses natural fluctuations from equilibrium

¡ Two formulations to connect macroscopic to microscopic •  Einstein relation describes long-time asymptotic behavior •  Green-Kubo relation connects to time correlation function

¡ Several approaches to evaluation of correlation functions •  direct: simple but inefficient •  Fourier transform: less simple, more efficient •  coarse graining: least simple, most efficient, approximate

ce 530 molecular simulationkofke/ce530/lectures/lecture12.ppt.pdf5 approaches to evaluating...

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