Major Concepts. Brownian Motion & More. Chemical Kinetics Master equation & Detailed Balance Relaxation rate & Inverse Phenomenological Rate

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1 Major Conceps Brownian Moion & More Langevin Equaion Model for a agged subsysem in a solven Harmonic bah wih emperaure, T Fricion & Correlaed forces (FDR) Markovian/Ohmic vs. Memory Fokker-Planck Equaion Bolzmann Disribuion a Equilibrium Diffusion Consan Chemical Kineics Maser equaion & Deailed Balance Relaxaion rae & Inverse Phenomenological Rae Brownian & Langevin Dynamics 1

2 Nonequilibrium Dynamics Far-from-equilibrium, sysems are differen! Doesn he solven average i all ou??? Cf. Zwanzig s Topics: Brownian Moion & Langevin Equaions Fokker-Planck Equaions Maser Equaions Reacion Raes & Kineics Classical vs. Quanum Dynamics Linear Response Theory Use hermodynamic quaniies o predic Non-Eq Nonlineariy See, e.g., R. Zwanzig, Nonequilibrium Saisical Mechanics (Oxford Universiy Press, 21) Brownian & Langevin Dynamics 2

3 Brownian Moion A heavy paricle is buffeed by small paricles is moion appears random rue for pollen rue for molecules m dv d = ζv +δf () Brown, Rober, A brief accoun of microscopical observaions made in he monhs of June, July and Augus, 1827, on he paricles conained in he pollen of plans; and on he general exisence of acive molecules in organic and inorganic bodies. Phil. Mag. 4, , Einsein, A. (195), Über die von der molekularkineischen Theorie der Wärme gefordere Bewegung von in ruhenden Flüssigkeien suspendieren Teilchen", Annalen der Physik 17: Brownian & Langevin Dynamics 3

4 Langevin Equaion Langevin Equaion: (in mass-weighed x s) x = "# h x " $U(x) $x Flucuaion Dissipaion Relaion/Theorem: Sochasic Forces: Average is zero Momen is deermined by FDT + % h () % h ()% h ( &) = 2k B T# h ' " & Assume ha hey are Gaussian Correlaed Brownian & Langevin Dynamics 5

5 Langevin Equaion Sysem, x, and bah, y, wih all coordinaes explici, leads o a EoM for x, updaed a each y(): V( x,) m x = +δf ( x, y! ) LE: m x = V x ( ) + f b ( ) γ x GLE: m x = V x ( ) +δf β d 'C b ' x ( ' ) Brownian & Langevin Dynamics 6

6 Langevin Dynamics Langevin Equaion: x = "# h x " $U(x) + % h () $x % h ()% h ( &) = 2k B T# h ' " & Idenify a Reacion/Dynamic Variable (Order Parameer?) The bah coordinaes are subsumed by he Fricion and Random Force Kramers Turnover Raes Mel nikov-pollak-graber-hänggi (PGH) Theory & Raes Shepherd and Hernandez; J. Chem. Phys. 117, (22). (variaional MFPT) Brownian & Langevin Dynamics 7

7 Brownian Moion & he Langevin Equaion Sysem, x, and bah, y, wih all coordinaes explici, leads o a EoM for x, updaed a each y(): LE: m x ( ) = U x + f b γ x GLE: m x C vv () = 1 m e ( m ) = D = 1 m m x = U ( x ) + δf x, y! ( ) = V x + δf [ ] Fokker Planck Equaion : ψ = v γvψ + D ψ v a seady sae : ψ(x) e 1 2 βmv 2 β d Brownian & Langevin Dynamics 9 e β ( K.E.) b x ( ' ) ' δf )δf (' Brown, Rober, A brief accoun of microscopical observaions made in he monhs of June, July and Augus, 1827, on he paricles conained in he pollen of plans; and on he general exisence of acive molecules in organic and inorganic bodies. Phil. Mag. 4, , Einsein, A. (195), Über die von der molekularkineischen Theorie der Wärme gefordere Bewegung von in ruhenden Flüssigkeien suspendieren Teilchen", Annalen der Physik 17: d e ( m ) = 1 In velociy

8 D and Correlaion Funcions Recall: A = 1 d A( ) Le: A() A() A The ime correlaion funcion is: C AA () = 1 d A( + ) A( ) = A()A() The diffusion equaion: 2 n(x, ) =D n(x, ) x2 The diffusion consan is: where n(x, ) is he concenraion D = d v()v() = d C vv () Brownian & Langevin Dynamics 1

9 Langevin Equaion Field Free Recall wih V= : m d 2 x d 2 Which is a firs-order diff. eq. in v: m dv d This can be readily solved: dx = ζm d + ξ = ζmv + ξ() v() = v() exp( ζ) + 1 m dsexp ζ( s) Of course, we have o average over he noise: v() 2 = v() 2 exp( 2ζ) + 1 d s $ dsexp ζ(2 s $ s) m 2 ξ( $ s )ξ(s) ξ(s) Brownian & Langevin Dynamics 11

10 Langevin Equaion Field Free For arbirary noise: v() = v() exp( ζ) + 1 m dsexp ζ( s) Assuming Markovian Noise.. This gives us he long-ime resul: v() 2 = v() 2 exp( 2ζ) + 1 d s $ dsexp ζ(2 s $ s) m 2 where he firs erm wen o zero, and i s he second erm ha conribues! ξ( $ s )ξ(s) ξ(s) lim v()2 = + 3kTmζ = 3kT / m if he FDR is: ξ()ξ( #) = 2kTmζδ # m 2 ζ Brownian & Langevin Dynamics 12

11 Sysem, x, and bah, y, wih all coordinaes explici, leads o a EoM for x, updaed a each y(): f! y = c i y i mω 2 = GLE: m x Zwanzig Hamilonian = V x ( ) where he Fricion Kernel N H = p2 N * 2m +V (x) + p 2 $ + m i ω 2 2m i y i c ' i, & x) +, i % m i ω i=1 i ( dp d = H i=1 N * 2 1 $ c ' -, & i ) / dp βc, m i % ω b () i=1 i ( / +. +δf = dv (x) dx β d + c i y i 'C b ' x ( ' ) γ() C b () ξ()ξ() = 1 m c i m i ω cos ( ω 2 i ) i Brownian & Langevin Dynamics 13 N i= N i=1 N i=1,. -. d = d V (x) mω2 x 2 dx 1 m i & ( ' f c i ω i ) + *! y /. / / x

12 Chemical Kineics Simple Kineics Phenomenology Maser Equaion Deailed Balance 1 E.g.: apparen rae for isomerizaion : τ rxn = kab + k BA Microscopic Rae Formula Relaxaion ime Plaeau ime Brownian & Langevin Dynamics 14

13 Transiion Sae Theory Objecive: Calculae reacion raes Obain insigh on reacion mechanism Eyring, Wigner, Ohers.. 1. Exisence of Born-Oppenheimer V(x) 2. Classical nuclear moions 3. No dynamical recrossings of TST Keck,Marcus,Miller,Truhlar, Ohers... Exend o phase space Variaional Transiion Sae Theory Formal reacion rae formulas Pechukas, Pollak... PODS 2-Dimensional non-recrossing DS Full-Dimensional Non-Recrossing Surfaces Miller, Hernandez developed good acion-angle variables a he TS using CVPT/Lie PT o consruc semiclassical raes Jaffé, Uzer, Wiggins, Berry, Ohers... exended o NHIM s, ec Brownian & Langevin Dynamics (Marcus: Science 256 (1992) 1523) 15

14 The rae is: Raes C () = (x()) (x()) where ( ) is he heaviside funcion and is a DS k() = d d k() is he ransiion sae heory rae Afer an iniial relaxaion, k() plaeaus (Chandler): he plaeau or saddle ime: s k( s ) is he rae (and i saisfies he TST Variaional Principle) Afer a furher relaxaion, k() relaxes o Oher rae formulas: Miller s flux-flux correlaion funcion Langer s Im F 1 E.g., in he apparen rae for isomerizaion : τ rxn = k AB + k BA ln C () Brownian & Langevin Dynamics 16

15 Brownian Moion Chandler m x = V x ( ) + f b ( ) γ x Doi m x = U ξv + f D = kt ξ = 1 βξ Fokker Planck Equaion : ψ = & 1 ( ' ξ U ψ + D ψ ) + * In posiion a seady sae : ψ(x) e V Dξ e βv Brown, Rober, A brief accoun of microscopical observaions made in he monhs of June, July and Augus, 1827, on he paricles conained in he pollen of plans; and on he general exisence of acive molecules in organic and inorganic bodies. Phil. Mag. 4, , Einsein, A. (195), Über die von der molekularkineischen Theorie der Wärme gefordere Bewegung von in ruhenden Flüsigkeien suspendieren Teilchen", Annalen der Physik 17: Brownian & Langevin Dynamics 17