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Statistical Mechanics and Multi-Scale Simulation Methods ChBE 591-009

Statistical Mechanics and Multi-Scale Simulation Methods ChBE 591-009. Prof. C. Heath Turner Lecture 16. Some materials adapted from Prof. Keith E. Gubbins: http://gubbins.ncsu.edu Some materials adapted from Prof. David Kofke: http://www.cbe.buffalo.edu/kofke.htm. Dynamical Properties.

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Statistical Mechanics and Multi-Scale Simulation Methods ChBE 591-009

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  1. Statistical Mechanics and Multi-Scale Simulation MethodsChBE 591-009 Prof. C. Heath Turner Lecture 16 • Some materials adapted from Prof. Keith E. Gubbins: http://gubbins.ncsu.edu • Some materials adapted from Prof. David Kofke: http://www.cbe.buffalo.edu/kofke.htm

  2. 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?

  3. 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 Mass Energy Momentum Fick’s law Fourier’s law Newton’s law

  4. 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 anisotropy of mass, momentum, energy • Observe how these natural fluctuations dissipate • All transport properties measurable at once • Poor signal-to-noise ratio

  5. 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 For given configuration, each molecule represents a point of high concentration (fluctuation) RMS displacement increases as t1/2 Compare to ballistic r ~ t

  6. Interpretation Einstein equation • 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 Slope here gives D

  7. Time Correlation Function • Alternative but equivalent formulation is possible • Write position r at time t as sum of displacements dt v r(t) dt dt

  8. Time Correlation Function • Alternative but equivalent formulation is possible • Write position r at time t as sum of displacements • Then r2 in terms of displacement integrals rearrange order of averages correlation depends only on time difference, not time origin Green-Kubo equation

  9. Velocity Autocorrelation Function • Definition • Typical behavior Zero slope (soft potentials) Diffusion constant is area under the curve Asymptotic behavior (t) is nontrivial (depends on d) Backscattering

  10. Other Transport Properties • Diffusivity • Shear viscosity • Thermal conductivity

  11. A(t0) A(t0) A(t0) A(t0) A(t1) A(t1) A(t1) A(t1) A(t2) A(t2) A(t2) A(t2) A(t3) A(t3) A(t3) A(t3) A(t4) A(t4) A(t4) A(t4) A(t5) A(t5) A(t5) A(t5) A(t6) A(t6) A(t6) A(t6) A(t7) A(t7) A(t7) A(t7) A(t8) A(t8) A(t8) A(t8) A(t9) A(t9) A(t9) A(t9) Evaluating Time Correlation Functions Dt Dt • 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 +... A0A1 + A1A2 + A2A3 + A3A4 + A4A5 + A5A6 + A6A7 + A7A8 + A8A9 +... A0A2 + A1A3 + A2A4 + A3A5 + A4A6 + A5A7 + A6A8 + A7A9 A0A5 + A1A6 + A2A7 + A3A8 + A4A9 +...

  12. k-n k 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

  13. Fourier Transform Approach to TCF • Fourier transform • Convolution • Fourier convolution theorem • Application to TCF Sum over time origins With FFT, operation scales as n ln(n)

  14. Coarse-Graining Approach to TCF 1. • Evaluating long-time behavior can be expensive • for time interval T, need to store T/dt 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

  15. 0 n Coarse-Graining Approach to TCF 2.

  16. 0 n Coarse-Graining Approach to TCF 2.

  17. 0 n 0 n Coarse-Graining Approach to TCF 2.

  18. 0 n Coarse-Graining Approach to TCF 2.

  19. 0 n Coarse-Graining Approach to TCF 2.

  20. 0 n 0 n Coarse-Graining Approach to TCF 2.

  21. 0 n Coarse-Graining Approach to TCF 2.

  22. 0 n Coarse-Graining Approach to TCF 2.

  23. 0 n 0 n Coarse-Graining Approach to TCF 2. 0 n

  24. 0 n Coarse-Graining Approach to TCF 2. et cetera

  25. Coarse-Graining Approach to TCF 2. Approximate TCF This term gives the net velocityover this interval 0 n3

  26. Coarse-Graining Approach: Resource Requirements • Memory • for each level, store n sub-blocks • for simulation of length T = nkDt 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

  27. 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

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