Major Concepts Langevin Equation Model for a tagged subsystem in a solvent Harmonic bath with temperature, T Friction & Correlated forces (FDR) Markovian/Ohmic vs. Memory Chemical Kinetics Master equation & Detailed Balance Relaxation rate & Inverse Phenomenological Rate Brownian Motion Langevin Equation Fokker-Planck Equation Boltzmann Distribution at Equilibrium Diffusion Constant Lecture #23 Brownian & Langevin Dynamics 1
Nonequilibrium Dynamics Far-from-equilibrium, systems are different! Doesn t the solvent average it all out??? Cf. Zwanzig s Topics: Brownian Motion & Langevin Equations Fokker-Planck Equations Master Equations Reaction Rates & Kinetics Classical vs. Quantum Dynamics Linear Response Theory Use thermodynamic quantities to predict Non-Eq Nonlinearity See, e.g., R. Zwanzig, Nonequilibrium Statistical Mechanics (Oxford University Press, 2001) Lecture #23 Brownian & Langevin Dynamics 2
Recall: Correlation Functions The time correlation function is: The diffusion equation: The diffusion constant is: Lecture #23 Brownian & Langevin Dynamics 3
Langevin Equation System, x, and bath, y, with all coordinates explicit, leads to a EoM for x, updated at each y(t): V( x,0) m x = +δf ( x, y ) LE: m x t ( ) = V x ( t ) ( ) + f b ( t) γ x GLE: m x t ( ) = V x ( t ) ( ) +δf t ( ) β dt t 'C b t t' 0 ( ) x ( t' ) Lecture #23 Brownian & Langevin Dynamics 4
Langevin Dynamics Langevin Equation: Identify a Reaction/Dynamic Variable (Order Parameter?) The bath coordinates are subsumed by the Friction and Random Force Kramers Turnover Rates Mel nikov-pollak-grabert-hänggi (PGH) Theory & Rates Shepherd and Hernandez; J. Chem. Phys. 117, 9227-9233 (2002). (variational MFPT) Lecture #23 Brownian & Langevin Dynamics 5
Brownian Motion & the Langevin Equation System, x, and bath, y, with all coordinates explicit, leads to a EoM for x, updated at each y(t): ( ) m x = U x + δf x, y ( ) LE: m x t ( ( )) ( ) = U x t + f b t ( ) γ x GLE: m x t ( ( )) ( ) = V x t + δf t ( ) β dt t 0 ( ) b x ( t' ) ' δf t)δf (t' [ ] Fokker Planck Equation : ψ t = v γvψ + D ψ v at steady state : ψ(x) e 1 2 βmv 2 e β ( K.E.) In velocity Brown, Robert, A brief account of microscopical observations made in the months of June, July and August, 1827, on the particles contained in the pollen of plants; and on the general existence of active molecules in organic and inorganic bodies. Phil. Mag. 4, 161-173, 1828. Einstein, A. (1905), Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen", Annalen der Physik 17: 549-560 Lecture #23 Brownian & Langevin Dynamics 6
Chemical Kinetics Simple Kinetics Phenomenology Master Equation Detailed Balance 1 E.g.: apparent rate for isomerization : τ rxn = kab + k BA Microscopic Rate Formula Relaxation time Plateau time Lecture #23 Brownian & Langevin Dynamics 7
Rates The rate is: 1 E.g., in the apparent rate for isomerization : τ rxn = k AB + k BA k(0) is the transition state theory rate After an initial relaxation, k(t) plateaus (Chandler): the plateau or saddle time: t s k(t s ) is the rate (and it satisfies the TST Variational Principle) After a further relaxation, k(t) relaxes to 0 Other rate formulas: Miller s flux-flux correlation function Langer s Im F Lecture #23 Brownian & Langevin Dynamics 8
Transition State Theory Objective: Calculate reaction rates Obtain insight on reaction mechanism Eyring, Wigner, Others.. 1. Existence of Born-Oppenheimer V(x) 2. Classical nuclear motions 3. No dynamical recrossings of TST Keck,Marcus,Miller,Truhlar, Others... Extend to phase space Variational Transition State Theory Formal reaction rate formulas Pechukas, Pollak... PODS 2-Dimensional non-recrossing DS Full-Dimensional Non-Recrossing Surfaces Miller, Hernandez developed good action-angle variables at the TS using CVPT/Lie PT to construct semiclassical rates Jaffé, Uzer, Wiggins, Berry, Others... extended to NHIM s, etc (Marcus: Science 256 (1992) 1523) Lecture #23 Brownian & Langevin Dynamics 9
Brownian Motion Chandler m x t ( ) = V x ( t ) ( ) + f b ( t) γ x Doi m x = U ξv + f D = kt ξ = 1 βξ Fokker Planck Equation : ψ t = 1 ξ U ψ + D ψ In position at steady state : ψ(x) e V Dξ e βv Brown, Robert, A brief account of microscopical observations made in the months of June, July and August, 1827, on the particles contained in the pollen of plants; and on the general existence of active molecules in organic and inorganic bodies. Phil. Mag. 4, 161-173, 1828. Einstein, A. (1905), Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüsigkeiten suspendierten Teilchen", Annalen der Physik 17: 549-560 Lecture #23 Brownian & Langevin Dynamics 10