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1 Time-Correlation Functions Charusita Chakravarty Indian Institute of Technology Delhi

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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

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3 Time Correlation Functions Time-dependent trajectory of a classical system: Since the classical system is deterministic, a time-dependent quantity can be written as : Correlation function as a time average over a trajectory:

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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.

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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

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6 Typical velocity autocorrelation function Zero slope at origin Rebound from Solvent cage D. Chandler

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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-equilibrium state by applying perturbing field f System allowed to relax freely Equilibrium established time=0 B(t) D. Chandler

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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? Integrate over initial conditions of perturbed system at t=0 Time-dependent value of B for a given set of initial conditions

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9 Linear Response Theory (contd.) Consider the effect of perturbations only upto first-order: D. Chandler, Introduction to Modern Statistical Mechanics

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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

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11 Numerator: Time-dependent behaviour of B:

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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 Macroscopic relaxation Equilibrium Time-correlation function

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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:

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14 Provides an alternative route to evaluate the time-dependent response as an integral over a time-correlation function

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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 FluxGradientLinear Laws DiffusivityMassConcentrationFick’s Law of Diffusion Ionic Conductivity ChargeElectric potentialOhm’s Law ViscosityMomentumVelocityNewton’s Law of Viscosity Thermal Conductivity EnergyTemperatureFourier’s Law of Heat Conduction

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16 Self-diffusivity Consider 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:

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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

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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

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19 Ionic Conductivity Consider an external electric field E x 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. Charge on particle i velocity particle i Rate of change of net dipole moment

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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:

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21 Collective Transport Properties Silica (6000K, 3.0 g/cc) Ionic Conductivity Viscosity

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22 Linear Response Theory and Spectroscopy Let f(t) be a periodic, monochromatic disturbance: Time-dependent energy: Rate of absorption of energy:

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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:

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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=

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25 Linear Response Theory and Spectroscopy (contd.) Using linear response theory Absorption spectrum

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26 Simple Harmonic Oscillator Let the quantity A coupled to the periodic perturbation obey SHO dynamics: Time-dependence of A: Absorption spectrum

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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

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28 Spectroscopic Techniques TechniqueTime-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-InfraredAngular velocity about molecular centre of mass NMRX-component of the magnectisation of the system

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29 Space-Time Correlation Functions: Neutron Scattering Experiments Number density at a point r at a time t: Conservation of particle number: Van Hove Correlation Function for a homogeneous fluid:

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30 Space-Time Correlation Functions (contd) Can divide the double summation into two parts: Self contribution Distinct contribution Fourier transform of the number density

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31 Space-Time Correlation Functions (contd.) Intermediate scattering function Static structure factor Dynamic structure factor Sum rule

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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)

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