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1
Time-Correlation Functions
Charusita ChakravartyIndian Institute of Technology Delhi
2
Organization• Time Correlation Function: Definitions and Properties• Linear Response Theory :
– Fluctuation-Dissipation Theorem– Onsager’s Regression Hypothesis– Response Functions
• Chemical Kinetics• Transport Properties
– Self-diffusivity– Ionic Conductivity– Viscosity
• Absorption of Electromagnetic Radiation• Space-time Correlation Functions
3
Time Correlation Functions
• Time-dependent trajectory of a classical system:• Since the classical system is deterministic, a time-dependentquantity can be written as :
• Correlation function as a time average over a trajectory:
average Time )()(1
)(
tyStationari )"()'(1
)'"(
average Time )"()'(1
)",'(
0
0
0
dtBALttC
dtBtALtttC
dtBtALtttC
))(),(( tptr
);,());0(),0(())(),(()( tprAtprAtptrAtA
4
• Time-correlation functions can be written as ensemble averages by summing over all possible initial conditions:
Probability of observing a microstate
• Limiting behaviour
• Alternative definition of time-correlation function in terms of deviations of time-dependent properties from mean values.
)),(exp(/)),(exp(),( prHpdrdprHprP
)()0();,()0;,(),()( tBAtprBprAprPpdrdtC
BAtCt
ABtCt
tBAtC
)( ,When
)( ,0When
)()0()(
0)( ,When
1)( ,0When
)0()0(/)()0()('
tCt
tCt
AA(t)-δA(t)
BAtBAtC
5
• Stationarity for systems with continuous interparticle forces, TCFs must be even functions of the time delay:
• Time-derivative with respect to time origins must be zero
• Short-time expansion of autocorrelation functions
)()( tCtC
)0()()()()()0( BtAstBsAtBA
)()()()(
0)()(
stBsAstBsA
tsBsAds
d
....)0()0()2/1()( 2 AAAtC
6
Typical velocity autocorrelation function
Zero slope at origin
Rebound fromSolvent cage
D. Chandler
7
Small Deviations from Equilibrium:Classical Linear Response Theory
• Apply a weak perturbing field f to the system that couples to some physical property of the system– Electric field/ionic motion– Electromagnetic radiation/charges or molecular dipoles
System prepared in non-equilibriumstate by applying perturbing field f
System allowed to relax freely
Equilibrium established
),(),( prfAprH
),( prH
),( prHtime=0
B(t)
D. Chandler
8
Linear Response Theory (contd.)• Let the time-dependent perturbation be such that
• At t=0, the probability of observing a configuration:
• How will the observed value of a quantity B(t) change with time when the perturbation is turned off at time t=0?
)),(exp(/)),(exp(),( fAprHpdrdfAprHprF
)0;,(),()( tprBprFpdrdtB
Integrate over initial conditions of perturbed system at t=0
Time-dependent value of B for a given set of initial conditions
0for 0
0for 1)(
t
ttf
9
Linear Response Theory (contd.)Consider the effect of perturbations only upto first-order:
fAH
fAHfAprH
1)exp(
)exp()exp()),(exp(
D. Chandler, Introduction to Modern Statistical Mechanics
10
• For t>0, the observed value of B will be given by
• Multiply the numerator and denominator by (1/Q) where Q is the partition function of the unperturbed system
• Denominator )exp( HpdrdQ
fAHpdrd
tBfAHpdrd
tprBprFpdrdtB
1)exp(
)(1)exp(
);,(),()(
Af
HpdrdAHpdrdfHpdrd
fAHpdrdQ
1
)exp(/)exp()exp(
1)exp()/1(
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)()0(
)()0()exp()/1()()exp()/1(
)(1)exp()/1(
tBAfB
tBAHpdrdQftBHpdrdQ
tBfAHpdrdQ
Numerator:
BAftBAfB
AftBAfB
fAtBAfBtB
)()0(
....1)()0(
1)()0()( 1
Time-dependent behaviour of B:
12
Onsager’s regression hypothesis
The relaxation of macroscopic non-equilibrium disturbances is governed by the same dynamics as the regression of spontaneous microscopic fluctuations in the equilibrium system
BAtBAfBtB )()0()(
)()0()( tBAftB
Macroscopic relaxation
Equilibrium Time-correlation function
13
Response Functions
For a weak perturbation , we can define a response function:
The response of the system to an impulsive perturbation:
To correspond to the linear response situation studied earlier:
BAtABfAtAtA )()0()()(
)'()',(')( tfttdttA AB
)'()',(')(),( 00 ttttdttAtt ABAB
)'( )',( :tyStationari
'for 0)',( :Causality
tttt
tttt
ABAB
AB
otherwise 0)'( and 0for )'( tftftf
t
ABAB dfttdtftA )()'(')(0
14
• Provides an alternative route to evaluate the time-dependent response as an integral over a time-correlation function
)()0()(
0for 0
0for )()0()(
)()()0()(
ABdftA
t
ttAB
dfBAtABftA
t
AB
t
AB
15
Transport Properties• Flux=-transport coefficient X gradient• Non-equilibrium MD: create a perturbation and watch the
time-dependent response• Equilibrium MD: measure the time-correlation function
Flux Gradient Linear Laws
Diffusivity Mass Concentration Fick’s Law of Diffusion
Ionic Conductivity
Charge Electric potential Ohm’s Law
Viscosity Momentum Velocity Newton’s Law of Viscosity
ThermalConductivity
Energy Temperature Fourier’s Law of Heat Conduction
16
Self-diffusivityConsider an external field that couples to the position of a
tagged particle such that
The steady state velocity of the tagged particle will then be:xxFHH 0'
0
0
0
)()0(
)()0(
)(
)'('
mobility theis where)(
xxx
xx
xvx
xvx
xx
vvdF
vxdF
dF
ttdtF
Ftv
x
x
17
Can one identify the mobility, as defined below, with the macroscopic velocity:
Fick’s Law: Flux of diffusing species= Diffusivity X Concentration gradient
Combining with the equation of continuity derived by imposing conservation of mass of tagged particles, gives :
Diffusion Equation
0
)()0( xxx vvd
),( trcDj
0),(),(
0),(),(
2
trcDt
trc
trjt
trc
18
If the original concentration profile is a delta-function, then the concentration profile at a later time (t) will be a d-dimensional Gaussian:
Consider the second-moment of one-dimensional distribution:
Einstein relation
By writing the displacement as
one can show that
This definition of self-diffusivity will be the same as that of the mobility derived from linear reponse theory
))4/(exp()4/(1),(
)()0,(22/ DtrDttrc
rrcd
Dtxtx 2)0()(2
Dtrtr 6)0()(2
t
iii dttvrtr0
')'()0()(
0
2)()0(6)0()( xx vvdtrtr
19
Ionic ConductivityConsider an external electric field Ex applied to an ionic melt. Under
steady state conditions, the system will develop a net current:
The ionic conductivity per unit volume, will be defined by:
Effect of the external field on the Hamiltonian:
When the field is switched off at t=0, the current will decay towards the zero value characteristic of the unperturbed system.
)()()( tMtreztj xii
ix
Charge on particle i
velocity particle i
Rate of change of netdipole moment
xx VEtj )(
xxEMHH 0'
20
To apply the relation:
to compute the time-dependent decay of the current, we set:
to obtain:
Conductivity per unit volume will then be given by:
)()0()( tABdftAt
xx MBjA and
)()0(
)()0()(
0
0
MMdE
MMdEtj
x
xx
)()0(1)(
0
xxBx
x jjdTVkVE
tj
21
Collective Transport Properties Silica (6000K, 3.0 g/cc)
Ionic Conductivity Viscosity
22
Linear Response Theory and Spectroscopy• Let f(t) be a periodic, monochromatic disturbance:
• Time-dependent energy:
• Rate of absorption of energy:
23
Linear Response Theory and Spectroscopy (contd.)
• Time-dependent value of A reflects response to applied field:
• The average rate of absorption or energy dissipation is given by:
• For a periodic field:
24
Linear Response Theory and Spectroscopy (contd.)• Fourier transform of response function is defined as:
• Compute average rate of absorption of energy over one time period T=
25
Linear Response Theory and Spectroscopy (contd.)
• Using linear response theory
• Absorption spectrum
26
Simple Harmonic Oscillator• Let the quantity A coupled to the periodic perturbation obey
SHO dynamics:
• Time-dependence of A:
• Absorption spectrum
)()( 2
02
2
tAdt
tAd
)cos()0(
)cos()0()0()sin()0()0(
)()0()(
)cos()0()sin()0()(
2 tA
tAAtAA
tAAtA
tAtAtA
o
oo
oo
)()()0()cos(4 0
22
tAAtdtfE
27
Infrared Absorption by a Dilute Gas of Polar Molecules
• X-component of total dipole moment will couple to oscillating electric field .
• Independent dipole approximation:• Perturbed Hamiltonian:
• Change in dipole moment with time:
• Thermal distribution of angular velocities, P(), will be reflected in absorption profile
i
ixM
)()( 0 tEMHtH xx
)cos()3/()()0(3
)()0()()0()()0(
2 tNt)(N/
tttMMi
ixixi j
jxixxx
28
Spectroscopic Techniques
Technique Time-correlation Function
Dynamical quantity
Dielectric relaxation
Unit vector along the molecular permanent dipole moment
Infra-red absorption
Unit vector along the molecular transient dipole, typically due to a normal mode vibration
Raman Scattering
Unit vector along the molecular transient dipole, typically due to a normal mode vibration
Far-Infrared Angular velocity about molecular centre of mass
NMR X-component of the magnectisation of the system
)()0( tMM xx
i
ii t)()0(
)(cos
)()0(
1
P
t
)(cos
2/1)]()0([3
2
P
t
)()0( t
29
Space-Time Correlation Functions:Neutron Scattering Experiments
j
j tt )(),( rrr
mNdt rr ),(
rrrrr dtN
tGm
)0,(),(1
),(
ij
jim
tN
)()0(1
rrr
Number density at a point r at a time t:
Conservation of particle number:
Van Hove Correlation Function for a homogeneous fluid:
30
Space-Time Correlation Functions (contd)
),(),(),( tGtGtG dS rrr
ij
jim
S tN
tG )()0(1
),( rrrr
)()0,( rr SG
)()0,( rr gGd
rrk rk dett i),(),( j
ti je )(rk
Can divide the double summation into two parts:
Self contribution
Distinct contribution
ji
jim
D tN
tG )()0(1
),( rrrr
Fourier transform of the number density
31
Space-Time Correlation Functions (contd.)
rrk rk detGtF ),(),(
)0,(),(1
kk tNm
dtetFS t
),(
2
1),( kk
)(),( kk SdS
)0,()( kk FS
Intermediate scattering function
Static structure factor
Dynamic structure factor
Sum rule
32
References• D. Frenkel and B. Smit, Understanding Molecular
Simulations: From Algorithms to Applications• D. C. Rapaport, The Art of Molecular Dynamics
Simulation (Details of how to implement algorithms for molecular systems)
• M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids (SHAKE, RATTLE, Ewald subroutines)
• Haile, Molecular Dynamics Simulation: Elementary Methods
• D. Chandler, Introduction to Modern Statistical Mechanics (Linear Response Theory)
• D. A. McQuarrie, Statistical Mechanics (Spectroscopic Properties)
• J.-P. Hansen and I. R. McDonald, The Theory of Simple Liquids (Almost everything)