BROWNIAN DYNAMICS Lecture notes

Göran Wahnströ BROWNIAN DYNAMICS Lecture notes Göteborg, 6 Deceber 6

Brownian dynaics Brownian otion is the observed erratic otion of particles suspended in a fluid (a liquid or a gas) resulting fro their collision with the rapidly oving atos or olecules in the gas or liquid. This transport phenoenon is naed after the botanist Robert Brown. In 87, while exaining grains of pollen in water under a icroscope, Brown observed that inute particles ejected fro the pollen grains executed a continuous erratic otion. He was not able to deterine the echaniss that caused this otion. One suggestion was that it was liferelated, but by observing the sae type of erratic otion also for particles of inorganic atter he was able to rule out that possibility. It took nearly years before Albert Einstein explained in detail how the otion that Brown had observed was a result of the particles being oved by collisions with individual water olecules. The work was published by Einstein 95, in one of his three faous papers fro that year. The paper contained predictions that could be tested experientally and Jean Babtiste Perrin verified Einsteins predictions experientally 98. The sae year Paul Langevin presented an alternative way to study Brownian otion, soewhat ore straightforward copared with Einsteins treatent. The theory by Einstein and its experiental verification was iportant, it served as convincing evidence that atos and olecules exist, a validity not universally recognized at that tie. It also gave considerable insight into the connection between fluctuations and dissipation of energy for systes in theral equilibriu. Brownian dynaics is an exaple of a stochastic process and has served as odel syste for the developent of the theory of noise. Contents Langevin s equation Fokker-Planck equation 8 3 Tie-correlation functions and the power spectru 4 Langevin with external force 4 5 Diffusion equation 7

Langevin s equation Consider a particle with ass oving in a fluid. For siplicity we will consider otion in one diension. To a first approxiation we assue that the interaction with the fluid can be described by a friction or daping force, proportional to the velocity v(t), d v(t) = αv(t) dt This was used by Stokes 85 to describe the otion of a particle in a viscous fluid. The friction force describes the average effect fro the collisions of the olecular fluid particles. Using acroscopic hydrodynaics Stokes showed that for a spherical particle α = 6πνR () where ν is the viscosity of the fluid and R the radius of the particle. By solving the equation of otion an exponential decay for the velocity v(t) = v()e (α/)t is obtained. This can not be true in the case of Brownian otion, because we do not describe the erratic otion of the particle. This is due that Brown considered quite sall particles and fluctuations of the force becoe iportant. We therefore add a fluctuation or stochastic part of the force f(t) = ξ(t) to the equation of otion. This defines the Langevin s equation d v(t) = ηv(t) + ξ(t) () dt where η = α/ will be denoted the friction coefficient. To proceed we need to give soe properties of the stochastic force. It changes in a very erratic and fast way and it depends on the icroscopic otions of the surrounding fluid olecules. We introduce a tie-average A(t) = +τ / A(s)ds (3) τ t τ / where τ is a short tie scale. It is assued to be long copared with an individual olecular collision tie but short copared with the velocity relaxation tie τ = /η, τ /η We assue that the average of the stochastic force is zero ξ(t) = (4) and that for tie differences larger than τ the stochastic force is uncorrelated in tie ξ(t)ξ(t ) = for t t > τ

The Brownian otion odel is not supposed to describe the very rapid otion on the tie scale τ and therefore we ay take the liit τ and write ξ(t)ξ(t ) = qδ(t t ) (5) Furtherore, ξ(t) is assued to be based on the su of a large nuber of independent olecular collisions and ξ(t) can be odelled as a Gaussian rando variable. We have introduced the averaging as a tie-average in Eqn (3). In soe situations it is ore convenient to view the average... as an enseble average, as in the Eqs (4) and (5). We will not ake any clear distinction between these two types of averaging procedures. The Langevin s equation, defined in Eqn (), together with the properties of the stochastic force in Eqs (4) and (5) can be used to solve for the Brownian dynaics. Consider a particle with the initial velocity v oving according to Eqn (). Direct solution leads to v(t) = v e ηt + ds e η(t s) ξ(s) (6) Consider now an enseble of particles all with initial velocity v. If we now ake an average over the enseble we get v(t) = v e ηt i.e. the average velocity decays exponentially to zero. The inforation of the initial velocity v fades away and is essentially lost for ties larger than the relaxation tie τ = /η. The velocity squared is given by and its average by v (t) = ve ηt + v e ηt ds e η(t s) ξ(s) + ds v (t) = ve ηt + ds = v e ηt + q η ds e η(t s) e η(t s ) ξ(s)ξ(s ) e ηt ] ds e η(t s) e η(t s ) qδ(s s ) Here again the initial inforation fades away but the average of v (t) does not approach zero, but the finite value q/η. This is a easure of the erratic otion seen by Brown. We know fro the equipartition theore that if the particle is oving in a fluid at the teperature T v 3 = k BT (7)

at theral equilibriu. This iplies that v (t) should approach k B T/ for large ties and q = η k BT (8) This is a special case of the faous Fluctuation-Dissipation theore. It relates fluctuations, described by q, to dissipation, described by η. Any syste that shows dissipative effects will also show fluctuations and there is a relation between the two. We can now suarize the solution of Langevin s equation for the tie dependence of the ean value µ(t) and the variance σ (t) of an enseble of particles all with the initial velocity v as µ(t) v(t) = v e ηt (9) σ (t) v(t) v(t) ] = k BT e ηt ] () Algorith BD A siple algorith for solving Langevin s equation can be derived by aking the direct discretisation of Eqn () according to v n+ v n = ηv n + ξ n t or v n+ = ( η t)v n + t ξ n The strength of the stochastic force also has to be deterined. Using we have and δ(t t ) t δ n,n ξ n ξ n = ηk BT t δ n,n v n+ = ( η t)v n + (ηk B T t/) G n () where G n is a Gaussian rando nuber with zero ean, unit variance and uncorrelated in tie G n G n = δ n,n. A better algorith can be derived by using the analytical solution of the Langevin s equation. We have with v(t + t) = v()e η(t+ t) + + t ds e η(t+ t s) ξ(s) = v(t)e η t + ζ v (t) () ζ v (t) = + t t ds e η(t+ t s) ξ(s) (3) 4

This iplies that ζ v (t) = and ζ v (t) = + t t = η k BT + t ds e η(t+ t s) ds e η(t+ t s ) ξ(s)ξ(s ) + t t t ds e η(t+ t s) = k BT e η t ] (4) Algorith BD The recoended algorith for solving Langevin s equation is therefore where v n+ = c v n + v th c G n (5) c = e η t and v th = k B T/ and G n is a Gaussian rando nuber with zero ean, unit variance and uncorrelated in tie G n G n = δ n,n. Notice that in the liit η t algorith BD is recovered. Algorith BD is ore accurate for large tie steps t. In Fig. we show the result for 5 trajectories using the algorith BD. The initial velocity is v = 5v th and in the figure we also show the ean value µ(t) and the standarddeviation σ(t) according to Eqs (9) and (). 6 µ(t) µ(t) ± σ(t) 4 Velocity / vth] 4 5 5 Tie / τ] Figure : Five trajectories v(t), all with the sae initial velocity v = v th, together with the analytical result for the ean value µ(t) and the standarddeviation σ(t). 5

We can also obtain an expression for the tie-dependence of the position of the Brownian particle x(t). We still restrict ourselves to otion in one diension. Consider a particle with the initial conditions x() = x and v() = v. By direct integration using the result in Eqn (6) we get x(t) = x + with the average = x + = x + v η = x + v η ds v(s) ds v e ηs + ds s e ηt ] + ds e ηt ] + η x(t) x ] = v η ds e η(s s ) ξ(s ) s ds e η(s s ) ξ(s ) ds e η(t s ) ] ξ(s ) e ηt ] We can also study the fluctuations fro the average value x(t) x ] = v e ηt ] k B T + η ds e η(t s)] η = v e ηt ] k B T + η t η η ( e ηt ) + ] η ( e ηt ) Here, the initial condition for the velocity is fixed at v. We can also consider a syste in equilibriu. The initial condition for the velocity is then v eq = and v eq kb T vth = (6) The ean squared displaceent MSD (t) is then given by MSD (t) x(t) x ] eq k B T ( = ηt e ηt )] η (7) where the superscript eq indicates that we have taken an equilibriu average for the initial velocity. For large ties MSD (t) becoes proportional to tie. This is the diffusive liit. Below we will show that if the Langevin s equation should be consistent with the ordinary diffusion equation in the long tie liit, the diffusion coefficient D has to be equal to D = k BT η (8) We can therefore write the ean squared displaceent as MSD (t) x(t) x ] eq = D t ( e ηt )] (9) η 6

For short ties the particle is oving as a free particle with the theral velocity given by Eqn (6) MSD (t) = (v th t) t η while for large ties it diffuses with the diffusion coefficient D given by Eqn (8) MSD (t) = Dt t η 7

Fokker-Planck equation By solving the Langevin s equation we get inforation on the trajectory of the particle. It is a stochastic differential equation so if we repeat the calculation with the sae initial conditions we will get another trajectory. We ay then ask for the ore detailed inforation what the probability is to obtain a certain velocity v at tie t. Therefore, we introduce the probability distribution function f(v, t)dv, which is equal to the probability to find the particle at tie t with velocity between v and v + dv. We will now derive the corresponding partial differential equation for probability distribution function f(v, t). The probability can increase in tie if the particle velocity is changed to v or decrease in tie if the particle velocity is changed fro v. This is described by the Master equation f(v, t) = t dv W (v, v )f(v, t) W ] (v, v)f(v, t) () where W (v, v ) is the probability per unit tie for a transition fro v to v. The critical assuption ade in the Master equation is that the transition probability W (v, v ) is independent of tie, i.e. eory effects are neglected. We introduce the notation for the change of velocity and write y = v v W (v, v ) = W (v v, v ) = W (y, v ) We will later assue that y is sall, but first we rewrite the Master equation as f(v, t) = t = = dv W (v v, v )f(v, t) W (v v, v)f(v, t) ] dy W (y, v )f(v, t) W ( y, v)f(v, t) ] dy W (y, v )f(v, t) W (y, v)f(v, t) ] where we in the last line as ade the change y y in the second ter. We then expand f(v, t) and W (y, v ) in ters of v around v, i.e. for sall values of y and f(v, t) = f(v, t) + (v f(v, t) v) + v (v v) f(v, t) v +... W (y, v ) = W (y, v) + (v W (y, v) v) + v (v v) W (y, v) v +... 8

which iplies to lowest order in y that W (y, v )f(v, t) = W (y, v)f(v, t) y v W (y, v)f(v, t)] + y W (y, v)f(v, t)] v If we insert this into the Master equation we obtain y f(v, t) = dy { y W (y, v)f(v, t)] + t v } W (y, v)f(v, t)] v = { ] } dy yw (y, v) f(v, t) + { ] } v v dy y W (y, v) f(v, t) or where t f(v, t) = v A(v)f(v, t)] + B(v)f(v, t)] () v A(v) = B(v) = dy yw (y, v) dy y W (y, v) This is the Fokker-Planck equation. It is based on the assuption that the velocity only changes by sall aounts, y is sall. This is justified in the case of Brownian dynaics. In other cases where one can not assue that the change of the velocity is sall, as for instance in collisions of neutral atos with the sae ass, a ore appropriate transport equation is the Boltzann equation. In the present case we can use the solution to Langevin s equation in Eqs ()-(4) to obtain explicit expressions for the two coefficients A(v) and B(v). We get A(v) li t B(v) li t v(t + t) v(t) = ηv t v(t + t) v(t)] = η k BT t and the Fokker-Planck equation can now be written as t f(v, t) = η v + k ] BT f(v, t) () v v This is a partial differential equation. The stationary solution is given by the Maxwellian velocity distribution ] f(v, t ) = f eq (v) = πk B T exp v (3) k B T 9

where we have introduced the proper noralisation. If we consider the initial condition with a precisely specified velocity, i.e. f(v, t = ) = δ(v v ), the Fokker-Planck equation has the analytical solution f(v, t) = πk B T ( e ηt ) exp (v v e ηt ) ] k B T ( e ηt (4) ) It shows how the initial sharp distribution spreads out in tie until the final Maxwellian distribution in Eqn (3) is reached at large ties. The Fokker-Planck equation can also be solved nuerically using soe standard grid based ethod. An interesting alternative ethod is to use the connection with the Langevin s equation. By generating a lot of trajectories, all with the initial condition v = v, and then aking a histogra, f(v, t) in Eqn (4) can be obtained. This is not a copetitive nuerical ethod for low diensional probles. However, for high diensional probles, where grid based ethods becoe coputationally too deanding, trajectory-based ethods can be used. They are essentially insensitive to the diensionality of the proble. In Fig. we copare the analytical solution in Eqn (4) with the nuerical result obtained by solving Langevin s equation using the algorith BD. We have generated trajectories and then ade an histogra for five different tie extensions. The agreeent is excellent. Probalitiy density f (v, t) / /vth].5.5 Theoretical τ.5 τ τ τ τ. -4-4 6 Velocity / vth] Figure : The probability distribution function f(v, t) at 5 different ties obtained by generating trajectories. The nuerical solution is copared with the analytical result.

3 Tie-correlation functions and the power spectru Tie-correlation functions are useful in characterising dynaic properties. Consider a syste in equilibriu. The velocity v(t) is then stationary in tie. We can define the tie-correlation function as C(t) = v(t)v() eq = li T T T v(t + s)v(s)ds The superscript eq indicates that the tie average is taken over a syste in equilibriu. We also notice that v(t) eq =. The correlation function is even in tie C( t) = C(t) (5) This follows fro that v(t) is a classical dynaical variable and it coutes with itself at different ties. We can also introduce the corresponding spectral density using the Fourier transfor FC(t)] Ĉ(ω) = with the corresponding inverse relation F Ĉ(ω)] C(t) = The spectral density is also even in frequency dtc(t)e iωt dω π Ĉ(ω)e iωt Ĉ( ω) = Ĉ(ω) (6) It is instructive to consider the frequency coponents of v(t) obtained by Fourier analysis as well as the corresponding power spectru. To avoid convergence probles introduce { v(t) < t < T v T (t) = otherwise The frequency coponents are then obtained fro the Fourier transfor v T (ω) = v T (t)e iωt dt = We then define the power spectru as T v(t)e iωt dt P(ω) = li T T v T(ω) (7)

There is a direct connection between the power spectru and the corresponding tie-correlation function. It is called the Wiener-Khintchin theore. It can be derived by considering the inverse Fourier transfor of the Power spectru F P(ω)] = = li T T = li T = li T = li T = li T = C(t) dω π P (ω)e iωt T T T T dω T π e iωt T T T T T dsv(s) dsv(s) T dsv(t + s)v(s) v(t + s)v(s)ds T dsv(s)e iωs ds v(s )e iωs ds v(s ) dω π e iω(t+s s ) ds v(s ) δ(t + s s ) T ds δ(t + s s ) This is the Wiener-Khintchin theore. The power spectru for a dynaic variable in an equilibriu syste is equal to the Fourier transfor of the corresponding tie-correlation function, The total power spectru is hence given by P(ω) = Ĉ(ω) (8) dωp(ω) = πc(t = ) (9) Consider the velocity v(t). Using the solution to Langevin s equation in Eqn (6) we get the tie-correlation function C(t) = k BT e η t for a syste in equilibriu. The corresponding power spectru P(ω) = dtc(t)e iωt = k BT η ω + η is a Lorentzian, centered at ω = and with the half width at half axiu equal to η. In Fig. 3 we show a typical spectru for a daped oscillator. We also notice that C(t) has a cusp at t =. Using the true icroscopic otions C(t) has no cusp, dc(t)/dt = for t =. The reason is that when

Powerspectru Frequency Figure 3: The power spectru for a daped oscillator with low (green curve) and high (blue curve) daping. deriving Langevin s equation we disregarded otion on the very short tie scale t < τ, and hence the Langevin s equation is only applicable for t > τ. We can also consider the spectru of the stochastic force ξ(t). Its tiecorrelation is a delta function in tie C ξ (t) = η k BT δ(t) and hence its power spectru is given by P ξ (ω) = η k BT This is independent on frequency and is called white noise. 3

4 Langevin with external force We can also add an external force to Langevin s equation of otion d dt v(t) = ηv(t) + F ext + ξ(t) (3) For instance, if the particle carries a charge e and is placed in a unifor electric field E the external force is F ext = ee. If we apply a constant external force on the particle it will achieve a constant drift velocity v drift at equilibriu. The constant of proportionality B is called the obility v drift = BF ext (3) Taking the ean value of Eqn (3) and considering the steady-state situation where d v((t) /dt =, yields the relation ηv drift = F ext. Cobining this with the relation between the friction and the diffusion coefficients in Eqn (8) we obtain a relation, the Einstein relation, between the obility and the diffusion coefficient B = D (3) k B T If we assue that the external force depend on the position of the particle we have to solve the coupled equations d x(t) dt = v(t) (33) d v(t) dt = ηv(t) + a(x(t)) + ξ(t) (34) where a(x(t)) = (/)F ext (x(t)) is the acceleration caused by the external force. As an exaple, consider the otion of a Brownian particle in an onediensional haronic potential well. The external force is then given by F ext (x) = kx(t) Using the relation k = ω where ω is the corresponding haronic frequency, we obtain the equation of otion d dt v(t) = ηv(t) ω x(t) + ξ(t) or d dt x(t) + η d dt x(t) + ω x(t) = ξ(t) (35) By Fourier transforing we obtain ω iηω + ω ] xt (ω) = ξ T (ω) 4

and hence the power spectru is given by P x (ω) = li T = k BT ξ T (ω T ω iηω + ω η (ω ω ) + η ω (36) which describes a daped haronic oscillator. By noticing that v T (ω) = iωx T (ω) the power spectru for the velocity is given by P v (ω) = k BT ηω (ω ω ) + η ω (37) The Fokker-Planck equation in Eqn () can be generalized to include the effect fro an external force. We have to obtain the equation for the probability distribution function f(x, v, t). It is given by t + v x + F ext v ] f(x, v, t) = η v v + k BT v ] f(x, v, t) (38) We will now consider the foral solution of Eqn (38) using the Liouville forulation and use this to derive an efficient nuerical schee. We assue that the external force can depend on the position of the Brownian particle F ext (x) and we introduce the acceleration caused by the external force, a(x) = F ext (x)/. The Fokker-Planck can now be written as f(x, v, t) = L f(x, v, t) t with the foral solution and where and f(x, v, t) = e Lt f(x, v, t = ) L = L x + L v + L η L x = v x L v = a(x) v L η = η v + k ] BT v v We split the tie evolution into sall tie steps of size t and approxiate the propagator as e tl f(x, v, t) = e tlx+lv+lη] f(x, v, t) = e t Lη+ t t t Lv+ tlx+ Lv+ Lη] f(x, v, t) e t Lη e t Lv e tlx e t Lv e t Lη f(x, v, t) 5

The syetric decoposition with t/ in the end and in the beginning reduces the error fro O( t ) to O( t 3 ). The two operators tl x and tl v was discussed in appendix A in the lecture notes Molecular dynaics. The first of these operators translate the position coordinate with v t and the second the velocity with a t. The third operator tl η corresponds to the Langevin s equation Eqn ()and for that the algorith BD will be used. The solution of Eqs (33)-(34) can now be written as v(t + ) = c v(t) + v th c G v(t + t/) = v(t + ) + a(t) t x(t + t) = x(t) + v(t + t/) t calculate new (external) accelerations/forces v(t + t) = v(t + t/) + a(t + t) t v(t + t) = c v(t + t) + v th c G where the notation for algorith BD has been used. We notice that in the liit η the velocity Verlet algorith is recovered. Finally: Algorith BD3 The Langevin s equation with an external force (acceleration) that depends on the position can be nuerically solved using the following algorith (that reduces to the velocity Verlet in the liit η ): where ṽ n+ = a n t + c v n + v th c G,n x n+ = x n + ṽ n+ t calculate new external accelerations/forces v n+ = c a n+ t + c ṽ n+ + v th c G,n c = e η t and v th = k B T/ and G and G are two independent Gaussian rando nubers with zero ean, unit variance and uncorrelated in tie G,n G,n = δn,n and G,n G,n = δn,n. 6

5 Diffusion equation If the friction coefficient η is large the velocity will theralize quite rapidly and a ore slow diffusive otion in ordinary space will then take over. Consider the Langevin s equation d dt x(t) = η d dt x(t) + F ext (x) + ξ(t) and assue that the friction ter is large copared with the inertial ter η d d x(t) dt dt x(t) This leads to the stochastic differential equation d dt x(t) = η F ext (x) + ξ(t) (39) η Algorith BD4 The stochastic diffusion equation can be solved using the following algorith x n+ = x n + t η F n ext k B T t + G n (4) η where G n is a Gaussian rando nuber with zero ean, unit variance and uncorrelated in tie G n G n = δ n,n. A corresponding Fokker-Planck equation can be derived for the probability distribution ρ(x, t)dx to find the particle at tie t with position between x and x + dx. In this case the coefficients A(x) and B(x) are given by A(x) li t B(x) li t x(t + t) x(t) = t η F n ext x(t + t) x(t)] = k B T t η The corresponding Fokker-Planck equation is called the Soluchowski equation ρ(x, t) = F ext (x) + k B T ] ρ(x, t) (4) t η x x This is a diffusion equation in the presence of an external force F ext (x). The corresponding stationary distribution is given by ρ(x, t ) ρ eq (x) exp V (x) ] (4) k B T 7

where V (x) is the potential defined by the external force F ext dv (x) (x) dx (43) In the absence of an external force we recover the ordinary diffusion equation t ρ(x, t) = D ρ(x, t) (44) x where we have used the relation D = k B T/η between the diffusion coefficient D and the friction coefficient η, introduced in Eqn (8). With the initial condition ρ(x, t = ) = δ(x x ) we have the solution (x ρ(x, t) = 4πDt exp x ) ] 4Dt which shows how an initial localized distribution spreads out in tie. The Soluchowski equation can also be written as ρ(x, t) = D F ext (x) t x k B T + ] ρ(x, t) (45) x Algorith BD5 Algorith for the Soluchowski equation (45) x n+ = x n + D t k B T F ext n + D t G n (46) where G n is a Gaussian rando nuber with zero ean, unit variance and uncorrelated in tie G n G n = δ n,n. 8