Lecture 9: Dynamic Properties
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1 Shor Course on Molecular Dynamcs Smulaon Lecure 9: Dynamc Properes Professor A. Marn Purdue Unversy
2 Hgh Level Course Oulne 1. MD Bascs. Poenal Energy Funcons 3. Inegraon Algorhms 4. Temperaure Conrol 5. Boundary Condons 6. eghbor Lss 7. Inalzaon and Equlbrum 8. Exracng Sac Properes 9. Exracng Dynamc Properes 1. on-equlbrum MD Las Updaed 8/9
3 Dynamc Properes Tme correlaon funcons Sngle parcle correlaons Velocy auocorrelaon funcon Collecve correlaons Sress auocorrelaon funcon Transpor coeffcens Ensen relaons Green-Kubo relaons Vscosy Dffuson Las Updaed 8/9
4 Tme Correlaon Funcon Two me dependen sgnals A and B; C quanfes he correlaon beween hem τ 1 C ( lm A( B( + d A( B( + τ τ A correlaon funcon s nvaran under ranslaons of he me orgn A and B dfferen: C s a cross-correlaon A and B he same: C s an auo-correlaon Las Updaed 8/9
5 Tme Correlaon Funcon In he lm of no delay me, C( s he sac correlaon funcon * Ofen normalze by hs value: In he long me lm, behavor of C( depends on perodcy of A and B If non-perodc (relevan o MD, hey wll become uncorrelaed Las Updaed 8/9
6 Tme Correlaon Funcon Characersc feaures of an auocorrelaon funcon A Zero slope a C ( d A( A( + d C( d Tme orgn Delay me A d Las Updaed 8/9
7 Tme Correlaon Funcon Implemenaon max 1 C ( d A( A( + max d max L d Δ ndex A sn ( π ( 1 (.4 A + 4 C d Example.4 max ( d.4 6. C(.4.46 d 4 [ A( A( A(. A( A(.4 A( A(.6 (.6.4] Las Updaed 8/9
8 Tme Correlaon Funcon Example con C ( d d Auocorrelaon Funcon Delay Tme Las Updaed 8/9
9 Tme Correlaon Funcon Sngle parcle correlaons Dynamc quany s a propery of ndvdual parcles Velocy auocorrelaon funcon ψ ( 3kT m 1 v v ψ ( d ( ( + d ψ( d d Las Updaed 8/9
10 Tme Correlaon Funcon Uncerany measuremen Equal Caresan coordnae conrbuons x & ( x& ( + d y& ( y& ( + d z& ( z& ( + d Sascal uncerany ψˆ ( d ε ( ± τ [ 1 ψˆ ( ] d d max τ [ d ] dd ψˆ ( 1 e -1 ( ψ ˆ ( ~ exp /τ d d ψˆ ( τ me for o decrease from 1 o e -1 d τ max d Las Updaed 8/9
11 Tme Correlaon Funcon Collecve correlaons Properes of he whole sysem Easer o compue Less accurae Sress auocorrelaon funcon J J J J xx yx zx J J J xy yy zy J J J xz yz zz J αβ m v v α β + 1 j r jβ F jα ϕ ρ 1 Jαβ ( Jαβ ( + d 3kT ( d for α β Las Updaed 8/9
12 Tme Correlaon Funcon Sress auocorrelaon funcon ϕ ρ 1 Jαβ ( Jαβ ( + d 3kT ( d J αβ m v v α β + 1 j r jβ F jα φ( d Shear modulus d Knec erm: correlaon of momenum ranspor caused by aomc moon Poenal erm: correlaon of momenum ranspor caused by neraomc forces Cross erm: couplng of he aomc moons and forces Las Updaed 8/9
13 Transpor Coeffcens Ensen relaons Transpor coeffcens from dfferenang a correlaon funcon wh respec me Green-Kubo relaons Transpor coeffcens from negrang a correlaon funcon over me General concep Flux -coeffcen x graden Transfer per un area n un me Ressance o flow Drvng force for he flux Las Updaed 8/9
14 Transpor Coeffcens Example: 1D dffuson Fck s law and conservaon of mass x& D x ( x& + x Combne and solve for D x ( x, πd x exp 4D Second momen of he dsrbuon s he mean-square dsplacemen 1 x x ( x, dx [ ( x( ] Las Updaed 8/9
15 Transpor Coeffcens Example: 1D dffuson con. Combne [ x( x( ] D Applcable when he me s large compared o he average me beween aomc collsons Fnally, Ensen s relaon D lm [ x( x( ] Mean-squared dsplacemen Las Updaed 8/9
16 Transpor Coeffcens Example: 1D dffuson con. Tme dervave x &( dx d x( x( x& ( d Square boh sdes and average over me orgns msd [ x( x( ] d d x& ( x& ( Use negrand symmery, shf he me orgn, and change varables msd d dτ x& ( τ x& ( - Las Updaed 8/9
17 Transpor Coeffcens Example: 1D dffuson con. Inegrae and solve for msd [ x( x( ] τ dτ x& ( τ x& ( 1 Take he long-me lm lm [ x( x( ] [ x( x( ] D lm VACF dτ x& ( τ x& ( Recall Ensen s relaon D dτ x& ( τ x& ( Las Updaed 8/9
18 Transpor Coeffcens Green-Kubo relaons Dffuson (3D D ψ ( d 1 v v ψ ( d ( ( + d 3 Vscosy η ϕ( d ϕ ρ 1 Jαβ ( Jαβ ( + d 3kT ( d Las Updaed 8/9
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