Diffusion in Equilibrium System

I presented this talk in our discussion group (https://vinayphys.github.io/discussion/) at IMSc.

Transcript

1. DIFFUSION Vinay Vaibhav The Institute of Mathematical Sciences October 07, 2017 Diffusion in Equilibrium System
2. OUTLINE Phenomenological Description of Diffusion Random Walk and Diffusion Langevin Description of Diffusion Fokker-Planck Equation and Diffusion Numerical Calculation of Diffusion Constant
3. Phenomenological Description of Diffusion
4. Fick’s law: Conservation equation: Combining above two, In 1-d: Initial condition: What is the solution of this equation with given initial condition? ~ j = Drc @c(~ r,t) @t + r ·~ j(~ r, t) = 0 @c(~ r,t) @t Dr2c(~ r, t) = 0 @c ( x,t ) @t = D @ 2 c ( x,t ) @x 2 c ( t = 0) = co ( x ) 1 < x < 1 Diffusion equation
5. Solution by Fourier Transform B.C. : ˜ c(t = 0) = ˜ c o (k) ˜ c(k, t) = ˜ c o (k) e k 2 Dt c ( x, t ) = 1 p 4⇡Dt e x 2 4 Dt c ( x, t ) = 1 p 4⇡Dt e ( x xo )2 4 Dt c ( x, t ) = 1 2 ⇡ R 1 1 dk ˜ co ( k ) e Dk 2 t e◆k · x c ( x, t ) = 1 2 ⇡ R 1 1 dk ˜ co ( k ) e◆k · x c ( t = 0) = ( x xo ) c ( t = 0) = ( x ) =) ˜ c ( t = 0) = 1 Define, Diffusion equation becomes, Solution of differential equation: Solution of diffusion equation in real space: If If : What, if initial condition is more complex? @˜ c(k,t) @t + D k2˜ c(k, t) = 0
6. Solution by Green’s Function c ( t = 0) = co 1 ( x x1) + co 2 ( x x2) c ( x, t ) = c o 1 p 4⇡Dt e ( x x1)2 4 Dt + c o 2 p 4⇡Dt e ( x x2)2 4 Dt co ( x ) = R 1 1 co ( q ) 1 p 4 ⇡Dt e ( x q )2 4 Dt dq co ( x ) = R 1 1 co ( q ) ( x q ) dq ( x q ) 1 p 4⇡Dt e ( x q )2 4 Dt Diffusion equation is linear - Principle of linear superposition is applicable If there are two initial point sources of diffusion at two different locations, Solution: Similar idea can be used for any initial condition : Each evolves as So,
7. Consider the diffusion equation in 1-d, Multiply by x2 both sides and integrate over x, Left hand side can be written as: Assuming c(x,t) vanishes at large distances, right hand side becomes: If is c(x,t) normalised then Further integration gives: In 3-d: @c ( x,t ) @t = D @ 2 c ( x,t ) @x 2 R 1 1 dx x 2 @c ( x,t ) @t = R 1 1 dx x 2 D@ 2 c ( x,t ) @x 2 @ h x 2( t )i @t 2 D R 1 1 dx c ( x, t ) @ h x 2( t )i @t = 2D h x 2( t )i = 2 Dt h ~ r 2( t )i = h x 2( t )i + h y 2( t )i + h z 2( t )i = 6 Dt
8. Random Walk and Diffusion
9. Consider random walk in 1-d - Particle can move left/right with constant step length What is the probability of reaching m in N steps? Stirling approximation: Using this approximation, - Divide by 2: m can be even or odd depending on N Introduce dimensions, - Each step is of length a : - Also: W(m, N) = 1 2N N! ( N+m 2 )!( N m 2 )! N! = p 2⇡ NN+ 1 2 e N (1 + O(1/N)) W(m, N) ⇡ 2 p 2⇡N e m2 2N x = am N t = t
10. Probability of reaching x in time t: Consider following limits, So, Average displacement and mean squared displacement after time t : W ( x, t ) = 1 p 2⇡(a2/ t)t e x 2 2( a 2 / t ) t a ! 0 and t ! 0 s.t. a2 t = 2D W ( x, t ) = 1 p 4⇡Dt e x 2 4 Dt h x i = 0 and h x 2i = 2 Dt
11. Langevin Description of Diffusion
12. P(v, t|v o , t o ) At t = 0, P(v, t = 0|v o ) = (v v o ) limt !1 P ( v, t|v o) = m 2 ⇡kBT 1 / 2 exp mv 2 2 kBT = peq ( v ) m˙ v(t) = F(t) F(t) = F int (t) + F ext (t) Fint(t) = ⌘(t) + Fsys(t) Fsys(t) = m v(t) ⌘(t):Random force Langevin model: - Colloidal particles in thermal equilibrium with molecules of fluid - Fluid causes random fluctuations in the velocity of tagged particle - Interaction with other dof is modelled in terms of a stochastic force Consider a subsystem of particles, each with velocity v0 at t = 0 - Follow one of these and describe what happens to it : Probability distribution of velocity v at time t How can we describe this approach to equilibrium? Apply Newton’s equation to a tagged particle where
13. Langevin equation (in absence of external force): Solution: Properties of noise: - Average zero: - Delta correlated: Average and mean squared velocity: Averaging: Pick the set of particles which have initial velocity vo, follow the path of one of these particles and average over all such particles Also, m˙ v = m v + ⌘(t) v(t) = v o e t + 1 m R t 0 dt1 e ( t t1)⌘(t1) ⌘(t) = 0 8 t ⌘(t1)⌘(t2) = (t1 t2) v(t) = v o e t v2(t) = v2 o 2 m 2 e 2 t + 2 m 2 v2(t) t!1 ! hv2ieq =) 2m2 = kBT m No memory
14. A fluctuation-dissipation relation: Velocity correlation: - Assuming (t > t’), - Averaging over all possible initial velocities (for any t): Driving noise is delta correlated but driven velocity process is exponentially correlated - Velocity process: Much slower but retains some memory - Gaussian, stationary and markov: Ornstein-Uhlenbeck process = 2m kBT v(t)v(t0) = v2 o e ( t + t 0) + m2 R t 0 dt1 R t0 0 dt2 e (t t1) e (t t2) (t1 t2) v(t)v(t0) = (v2 o 2 m 2 ) e ( t + t 0) + 2 m 2 e ( t t 0) = (v2 o kBT m ) e ( t + t 0) + kBT m e ( t t 0) hv(t)v(t0)i = kBT m e |t t0|
15. Mean squared displacement: - But - So - is some sort of time scale hX2(t)i = R t 0 dt1 R t 0 dt2 hv(t1)v(t2)i X ( t ) = x ( t ) x (0) = R t 0 dt 0 v ( t 0) hv(t)v(t0)i = kBT m e |t t0| hX2(t)i = R t 0 dt1 R t 0 dt2 kBT m e |t1 t2 | hX2(t)i = 2kBT m 2 [ t 1 + e t] 1 t ! 0 =) hX2(t)i ! kBT m t2 t 1 =) hX2(t)i ! 2kBT m t Asymptotic slope: 2kBT m = 2D Ballistic regime Diffusive regime
16. Displacement Autocorrelation Function The velocity of tagged particle is given by, Integrating this equation once, we get displacement - Conditional mean: The conditional autocorrelation: (t > t’) - Explicitly depends on vo - Equilibrium autocorrelation function: Average right hand side over vo using Maxwellian PDF peq(vo) (t > t’) - Displacement is not a stationary process: ACF depends on t and t’ both - In diffusion regime i.e., Wiener process X(t) = v o (1 e t) + 1 m R t 0 dt1(1 e (t t1))⌘(t1) X(t) = v o (1 e t) hX(t)X(t0)ieq = D (2 t0 1 + e t + e t0 e (t t0)) t 1 hX(t)X(t0)ieq ! 2D min(t, t0) v(t) = v o e t + 1 m R t 0 dt1 e ( t t1)⌘(t1) +D (2 t0 2 + 2e t + 2e t0 e (t+t0) e (t t0 ) X(t)X(t0) = v2 o 2 (1 e t e t0 + e (t+t0))
17. The Ornstein-Uhlenbeck Distribution v(t) = v o e t + 1 m R t 0 dt1 e ( t t1)⌘(t1) The formal solution to Langevin equation is - Linear transformation of Gaussian noise - Linear combination of inde. Gaussian random variables is also Gaussian The PDF of velocity should remain Gaussian at all times - PDF is completely determined by the mean and variance of the velocity - In large time limit, OU goes to Maxwellian distribution v(t) = v o e t v2(t) v(t) 2 = kBT m (1 e 2 t) p ( v, t|v o) = h m 2 ⇡kBT (1 e 2 t ) i1 2 exp h m ( v voe t)2 2 kBT (1 e 2 t ) i What is the probability distribution of velocity v at time t?
18. Stokes-Einstein-Sutherland Relation Diffusion coefficient D is given by, Assuming the tagged (Brownian) particle to be spherical - Stokes formula gives the expression of the viscous drag on the particle - Equating this force with the magnitude of the systematic part of the internal force i.e. on tagged particle - Putting this in the expression of D, - This relates diffusion coefficient which is a microscopic quantity, to macroscopic quantities like viscosity and temperature - Since itself is strongly dependent on temperature, so the temperature dependence of D mainly comes from rather than numerator D = kBT m Fdrag = 6⇡⌘av m v m = 6⇡⌘a D = kBT 6⇡⌘a ⌘ ⌘
19. Diffusion Coefficient in terms of Velocity Autocorrelation Diffusion coefficient D is defined as: - But MSD is defined as: Now D becomes: - The integration on right hand side can be simplified as, - Along an equilibrium trajectory VACF should not depend on time origin i.e., - Making a substitution and reversing the order of integration, - Replacing t - t’ with t in long time limit and assuming velocity autocorrelation decays to zero in such limit, - This is a special case of so called Kubo-Green formula which relates transport coefficients to equilibrium correlation functions D = limt!1 1 2t hX2(t)ieq hX2(t)ieq = R t 0 dt1 R t 0 dt2 hv(t1)v(t2)ieq D = limt!1 1 2t R t 0 dt1 R t 0 dt2 hv(t1)v(t2)ieq R t 0 dt1 R t 0 dt2 hv(t1)v(t2)ieq = 2 R t 0 dt1 R t1 0 dt2 hv(t1)v(t2)ieq hv(t1)v(t2)ieq = hv(t1 t2)v(0)ieq t1 t2 = t0 R t 0 dt1 R t 0 dt2 hv(t1)v(t2)ieq = 2 R t 0 dt0 R t t0 dt1 hv(t0)v(0)ieq This result is independent of any model considered D = R 1 0 dthv(0)v(t)ieq
20. Fokker-Planck Equation and Diffusion
21. ˙ ⇠ = f(⇠, t) + g(⇠, t)⇣(t) ⇣(t) h⇣(t)i = 0 h⇣(t)⇣(t0)i = (t t0) ⇣(t) g g p(⇠, t|⇠ o , t o ) @ @t p(⇠, t|⇠ o , t o ) = @ @⇠ [f(⇠, t)p(⇠, t|⇠ o , t o )] + 1 2 @ 2 @⇠ 2 [g2(⇠, t)p(⇠, t|⇠ o , t o )] f(⇠, t) Consider a stochastic differential equation (SDE), with GWN: and - Such Markovian process is called diffusion process in mathematical sense - = constant: Additive noise; other form of : Multiplicative noise - is called drift term Above SDE implies and implied by a partial differential equation satisfied by the conditional PDF , called Fokker-Planck equation (FPE) This FPE has been written using Ito interpretation of the SDE - Another famous way which leads to FPE is Stratonovich interpretation - Both interpretations to the SDE lead to same FPE if noise is additive ^
22. ˙ v(t) = v(t) + p m ⇣(t) h⇣(t)⇣(t0)i = (t t0) h⇣(t)i = 0 @p @t = @ @v (vp) + 2m2 @2p @v2 Consider the following Langevin equation: with and The corresponding FPE : - Using FD relation between and : - Solution with initial condition gives OU distribution - Can be written like equation of continuity: - Above FD relation can be derived from here also Langevin equation in high friction limit ( is so large that all times ): - The corresponding FPE: - In this regime velocity is essentially delta correlated (GWN) - No any steady distribution for x in long time limit - x is not a stationary process (second moment is 2Dt) but Gaussian and Markov @p @t = @ @v (vp) + kBT m @2p @v2 = 2m kBT p(v, 0|v o ) = (v v o ) @p @t + @J @v = 0 t 1 ˙ x = p m ⇣ ( t ) = p 2 D⇣ ( t ) @p ( x,t ) @t = D@ 2 p ( x,t ) @x 2
23. Phase Space Fokker-Planck Equation Langevin equation: Define, and where Diffusion matrix - Phase space FPE: I.C.- B.C.- What is the solution of this FPE with these I.C. and B.C.? ˙ x v = 0 @⇢ @t = Kij @ @xi ( xj⇢ ) + D ij @ 2 ⇢ @xi@xj @⇢ @t = @ @x (v⇢) + @ @v (v⇢) + kBT m @ 2 ⇢ @v 2 X = ✓ x v ◆ K = ✓ 0 1 0 ◆ X = ✓ x v ◆ ⌘ ✓ x1 x2 ◆ F P E ! =) D = ✓ 0 0 0 kBT m ◆ Drift matrix ⇢ ( x, v, 0) = ( v vo ) ( x xo ) ⇢ ( x, v, t ) ! 0 as | x | / | v | ! 1 Bivariate Gaussian VAN KAMPEN 8.6 page 212 ˙ v + v = 1 m ⌘(t) ˙ X + K = ✓ 0 1 m ⌘(t) ◆
24. Solution of phase-space FPE: - can be solved using matrix green function - - Mean where - Covariance matrix: A general multivariate Gaussian PDF in with given mean and covariance is given by - For phase-space FPE: where and ˙ X + K = ✓ 0 1 m ⌘(t) ◆ G ( t ) = exp( Kt ) G(t) = I + K (e t 1) X(t) = G(t)X o (t) = 2 R t 0 G(t0)DGT (t0) dt0 X o ⌘ ( xo, vo ) (t) = D ✓ C(t) B(t) B(t) A(t) ◆ D = kBT m A(t) = (1 e 2 t) C(t) = 1(2 t 3 + 4e t e 2 t) B(t) = (1 e t)2 µ X = ( x1, x2, ..., xn) p ( X ) = 1 p (2⇡)ndet exp[ 1 2 ( X µ ) T 1 ( X µ )] x = x x ( t ) v = v v(t) p(x, v, t) = 1 2 ⇡D p AC B 2 exp h A ( t )( x )2 2 B ( t )( x )( v )+ C ( t )( v )2 4 D ( t 2+4 e t 2 e 2 t te 2 t) i
25. Brownian Motion as Wiener Process In the diffusion regime: - Position x is the integral of a Gaussian white noise - It is non stationary Gaussian Markov process - Known as Brownian motion in Mathematics literature - With 2D = 1, it is called standard Brownian motion Wiener process: - It is the continuum limit of a simple random walk ˙ x = p 2 D⇣ ( t ) dw(t) = ⇣(t)dt
26. Numerical Calculation of Diffusion Constant
27. Long time regime: Asymptotic slope of MSD vs time curve: Measure Asymptotic slope of MSD vs time curve - slope = 2D (for 1-d system) - slope = 6D (for 3-d system) Calculate MSD at those points of time which look uniformly spaced on log scale t 1 =) hX2(t)i ! 2kBT m t 2kBT m = 2D
28. MSD vs time at different temperatures MSD vs time for different waiting times of an aged system