Brownian Motion: Langevin Equation

Transcription

1 Chapter 6 Brownian Motion: Langevin Equation The theory of Brownian otion is perhaps the siplest approxiate way to treat the dynaics of nonequilibriu systes. The fundaental equation is called the Langevin equation; it contain both frictional forces and rando forces. The fluctuation-dissipation theore relates these forces to each other. The rando otion of a sall particle (about one icron in diaeter) iersed in a fluid with the sae density as the particle is called Brownian otion. Early investigations of this phenoenon were ade by the biologist Robert Brown on pollen grains and also dust particles or other object of colloidal size. The odern era in the theory of Brownian otion began with Albert Einstein. He obtained a relation between the acroscopic diffusion constant D and the atoic properties of atter. The relation is D = RT N A 6πηa = k BT 6πηa where R is the gas constant, N A = /ol is Avogadros nuber, T is the teperature, η is the viscosity of the liquid and a is the radius of the Brownan particle. Also k B = R/N A is Boltzanns constant. The theory of Brownian otion has been extended to situations where the fluctuating object is not a real particle at all, but instead soe collective porperty of a acroscopic syste. This ight be, for exaple, the instantaneous concentration of any coponent of a cheically reacting syste near theral equilibriu. Here the irregular fluctuation in tie of this concentration corresponds to the irregular otion of the dust particle. 6.1 Langevin equation Consider a large particle (the Brownian particle) iersed in a fluid of uch saller particles (atos). Here the radius of the Brownian particle is typically 1 9 < a < The agitated otion of the large particle is uch slower than that of the atos and is the result of rando and rapid collisions due to density fluctuations in the fluid. There are in general three vastly different tiescales in a colloidal syste τ s, τ B, and τ r. Here τ s is the short atoic scale τ s 1 12 s, τ B is the Brownian tiescale for 75

2 76 Chapter 6 Brownian Motion: Langevin Equation Figure 6.1: A large Brownian particle with ass M iersed in a fluid of uch saller and lighter particles. the relaxation of the particle velocity τ B γ 1 3 s and τ r is the relaxation tie for the Brownian particle, i.e. the tie the particle have diffused its own radius τ r = a2 D In general τ s τ B τ r. In dense colloidal suspensions τ r can becoe very long of the order of inutes or hours. While the otion of a dust particle perforing Brownian otion appears to be quite rando, it ust nevertheless be describable by the sae equation of otion as is any other dynaical syste. In classical echanics these are Newton s or Hailtons equations. For siplicity we will consider otion in one diension. The results can easily be generalised to three diensions. Newtons equation of otion for the particle (radius a, ass, position x(t), velocity v(t)) in a fluid ediu (viscosity η) is dv(t) = F (t) (6.1) where F (t) is the total instantaneous force on the particle at tie t. This force is due to the interaction of the Brownian particle with the surrounding ediu. If the pssitions of the oelcules in the surrounding ediu are known as a function of tie, then in principle this force is a known function of tie. In this sense it is not a rando force at all. It is usually not practical or even desirable to look for an exact expression for F (t). Experience tells us that in typical cases this force is doinated by a friction force γv(t),

3 6.1 Langevin equation 77 proportional to the velocity of the Brownian particle. The friction coefficient is given by Stokes law γ = 6πηa (6.2) We also expect a rando force ξ(t) due to rando density fluctuations in the fluid. The equations of otion of the Brownian particle are: dx(t) dv(t) = v(t) = γ v(t) + 1 ξ(t) (6.3) This is the Langevin equations of otion for the Brownian particle. The rando force ξ(t) is a stochastic variable giving the effect of background noise due to the fluid on the Brownian particle. If we would neglect this force (6.3) becoes which has the failiar solution dv(t) = γ v(t) (6.4) v(t) = e t/τ B v(), τ B = γ (6.5) According to this, the velocity of the Brownian particle is predicted to decay to zero at long ties. This cannot be true since in equilibriu we ust have the equipartion theore v 2 (t) eq = k bt (6.6) while (6.5) gives v 2 (t) eq = e 2t/τ B v 2 () eq (6.7) The rando force in (6.3) is therfore necessary to obtain the correct equilibriu. In the conventional view of the fluctuation force it is supposed to coe fro occasional ipacts of the Brownian particle with olecules of the surrounding ediu. The force during an ipact is supposed to vary extreely rapidly over the tie of any observation. The effect of the fluctuating force can be suarized by giving its first and second oents ξ(t) ξ =, ξ(t 1 )ξ(t 2 ) ξ = gδ(t 1 t 2 ) (6.8) The average ξ is an average with respect to the distribution of the realizations of the stochastic variable ξ(t). Since we have extracted the average force γv(t) in the Langevin equation the average of the fluctuating force ust by definition be zero. g is a easure of the strength of the fluctuation force. The delta function in tie indicates that there is no correlation between ipacts in any distinct tie intervals 1 and 2. This loss of correlation is a consequence of the separation of tie scales discussed above. During a short tie interval on scale τ B = 1 3 s, say = 1 5 s, there are still roughly /τ s 1 7 collisons with the atos in the liquid. Therefore any eory between forces at different ties will be lost due to these frequent collisions.

4 78 Chapter 6 Brownian Motion: Langevin Equation The reaining atheatical specification of this dynaical odel is that the fluctuating force has a Gaussian distribution deterined by the oents in (6.8). The property (6.8) iply that ξ(t) is a wildly fluctuating function, and it is not at all obvious that the differential equation (6.3) has a unique solution for a given initial condition, or even that dv/ exists. There is a standard existence theore for differential equations which guarantee the existence of a local solution if ξ(t) is continous. A local solution is one which exists in soe neighborhood of the point at which the initial value is given. But even if a solution exists it ay be only local, or it ay not be unique, unless soe stronger conditions are iposed on ξ(t). We can obtain an explicit foral solution of (6.3) as v(t) = e t/τ B v() + 1 dse (t s)/τ B ξ(s) (6.9) but this only transfers the proble elsewhere. How do we know that the integral in (6.9) exists, that it is ore than just a foral sybol? To obtain a eaning to (6.3) and (6.9) we write (6.3) as where dv(t) = γ v(t) + 1 du(t) (6.1) du(t) = ξ(t) (6.11) For an arbitrary nonstochastic continous function f(t) we then find In particular f(t) = 1 gives f(s)dv(s) = γ v(t) v() = γ f(s)v(s)ds + 1 v(s)ds + 1 U(t) U() f(s) du(s) (6.12) = γ x(t) x() + 1 U(t) U() (6.13) We now discuss the integral noice U(t) for long ties t. Dividing t into intervals we have n U(t) U() = U(t k ) U(t k 1 ) (6.14) k=1 with = t < t 1 < t 2 < < t n = t. U(t) is a continous Markov process with zero ean. The continuity follows fro (6.1) since U(t) = U() + ξ(s)ds (6.15) and we ust require that the integral be a continous function of its upper liit, as for ordinary integrals. The Markov property follows since a Brownian particle in a liquid solution undergoes soething of the order of 1 12 rando collisions per second with the particles of the environent. Therefore, we can ake each interval t i t i 1 acroscopically

5 6.2 Exaples 79 very sall even though during it very any collisions occur. These nuerous ipacts destroy all correlations between what happens during the tie interval (t i, t i 1 ) and what has happended before t i 1. This iplies that U(t) is a Markov process, i.e. U(t n ) depends only on U(t n 1 ) etc. On account of the randoness of the otion the rando force ξ(t) ust average to zero, which is also iplied by the separation in (6.3). If we now choose our tie origin so that U() = we ust have U(t k ) =. Fro the discussion above about the rando collisions we also argue that the increents U(t 1 ) U(), U(t 2 ) U(t 1 ),, U(t n ) U(t n 1 ) (6.16) are independent. For long ties we suppose that the rando otion of the ediu has attained a steady state. Then the increents (6.16) are also stationary and identically distributed with zero ean. Applying the central liit theore to (6.14) we deduce that U(t) is Gaussian with zero ean. Therefore, it has all the requireents for a Wiener process, i.e. U(t) = W (t) (6.17) We can now write (6.1) as and the solution in (6.9) becoes 6.2 Exaples dv(t) = γ v(t) + 1 dw (t) (6.18) v(t) = e t/τ B v() + 1 Ornstein-Uhlenbeck process e (t s)/τ B dw (s) (6.19) In the Ornstein-Uhlenbeck process we study a Brownian particle where the equation of otion is given by (6.3) or x(t) = x + v(s)ds v(t) = e t/τ B v + 1 e (t s)/τ B dw (s) (6.2) where x = x() and v = v(). Since dw is a Gaussian process this is also the case for v(t) and x(t). We can therefore obtain the distribution functions for these variables knowing the first and second oents. Taking the average of the second eq. in (6.2) and using dw (t) = we find v(t) = v e t/τ B (6.21) Fro (6.2) we can also calculate the correlation function C v (t 2, t 1 ) = v(t 2 ) v(t 2 ) v(t 1 ) v(t 1 ) = e (t 2 s 2 )/τ B e (t 1 s 1 )/τ B dw (s 2 )dw (s 1 ) (6.22)

6 8 Chapter 6 Brownian Motion: Langevin Equation But W (s) is a Wiener process and we have dw (s 2 )dw (s 1 ) = g(ds 2 ds 1 ) (6.23) where g denotes the variance of W (t). Therefore the only contribution to the integral in (6.22) coes when s 1 = s 2, and C v (t 2, t 1 ) = g in(t2,t 1 ) 2 ds 1 e (t 2+t 1 2s 1 )/τ B = gτ B 2 2 e (t 2+t 1 2 in(t 2,t 1 ))/τ B e (t 2+t 1 )/τ B = gτ B 2 2 e ( t 2 t 1 ))/τ B e (t 2+t 1 )/τ B (6.24) Fro the correlation function we also directly get the second oent of v(t) v(t 2 )v(t 1 ) ξ = v 2 gτ B 2 2 e (t 2+t 1 )/τ B + gτ B 2 2 e ( t 2 t 1 ))/τ B (6.25) Here we can also average over the initial velocity distribution where in equilibriu v 2 eq = k B T/. For such a syste the second oent can only depend on the tie difference t 2 t 1 and the first ter has to vanish. We can also see this when t 2 = t 1 = t, then v 2 (t) ξ eq = v 2 eq gτ B 2 2 e 2t/τ B + gτ B 2 2 (6.26) The condition for equilibriu is that v 2 (t) ξ eq = k B T/. This requires g = 2k B T/τ B = 2γk B T (6.27) This iportant result is known as the Fluctuation dissipation theore. It relates the strength g of the rando noise or fluctuating force to the agnitude γ of the friction or dissipation. It expresses the balance between friction which tends to drive any syste to a copletely dead state and noise which tends to keep the syste alive. This balance is required to have a theral equilibriu state at long ties. The variance of v(t) is obtained fro (6.24) for t 2 = t 1 = t and with g = 2γk B T as 2 ξ σv(t) 2 = v(t) v e t/τ B eq = k BT 1 e 2t/τ B (6.28) We can now obtain the conditional probability distribution function for the velocity ( ) 1 1/2 ϱ v (vt v ) = 2πσv(t) 2 e (v µv(t))2 2σv 2(t) (6.29) where µ v (t) = v e t/τ B. Fro (6.2) we can also get an expression for the displaceent of the particle x(t) = x + dsv e s/τ B + 1 = x + v τ B 1 e t/τ B + 1 = x + v τ B 1 e t/τ B + τ B ds dw (u) s e (s u)/τ B dw (u) dse (s u)/τ B u 1 e (t u)/τ B dw (u) (6.3)

7 6.2 Exaples 81 where we in step two changed the order of integration. The average displaceent is then µ x (t) = x(t) ξ = x + v τ B 1 e t/τ B (6.31) An iportant quantity is the ean squared displaceent of the particle fro the starting point. This is obtained as + τ 2 B e t/τ B 1 e (t u)/τ B 1 e (t v)/τ B dw (u)dw (v) ξ x(t) x 2 ξ = v 2 τ 2 B = v 2 τ 2 B = τ 2 B 1 e t/τ B 2 + gτ 2 B 2 1 e t/τ B v 2 2 g 2γ 2 1 e (t u)/τ B + gγ ( t 2 τ B 1 e t/τ B ) (6.32) In equilibriu the first ter vanishes as before. Fro the reaining second ter we get for short and long ties (x(t) x ) 2 = { kb T t2 t 2k B T γ t t (6.33) The result for short ties is the free particle ter, where for short ties x(t) x = v t. The result for long ties can be copared with the diffusion result which gives the Stokes-Einstein result (x(t) x ) 2 = 2Dt (6.34) D = k BT γ = k BT 6πηa (6.35) The second oent of the displacenet becoes σ 2 x(t) = (x(t) µ x (t)) 2 = 2k BT τ 2 B The conditional distribution function for the displaceent is then ( ϱ x (xt x ) = t/τ B e t/τ B 1 2 e 2t/τ B. (6.36) 1 2πσ 2 x(t) ) 1/2 e (x µx(t))2 2σ 2 x (t) (6.37) Fro the relation between the position and velocity we can also get a general Kubo relation x(t) x() = ds v(s) (6.38)

8 82 Chapter 6 Brownian Motion: Langevin Equation which for a stationary process gives = = = 2 = 2 (x(t) x()) 2 = s2 ds 2 ds 1 v(s 2 )v(s 1 ) + ds 1 ds 2 v(s 2 )v(s 1 ) s2 ds 2 ds 1 v(s 2 s 1 )v() + ds For t we then find s which gives the Kubo relation du v(u)v() = 2 du(t u) v(u)v() (x(t) x()) 2 = 2t D = Brownian otion in a haronic potential ds 2 s 2 ds 1 v(s 2 )v(s 1 ) s1 ds 1 ds 2 v(s 1 s 2 )v() du v(u)v() u ds du v(u)v() = 2Dt du v(u)v() (6.39) Consider a Brownian particle of ass which is constrained to ove in one diension in a haronic potential V (x) = kx 2 /2. The corresponding force on the particle is F h = kx. The Langevin equations are dx dv = v = γ v ω2 x + 1 ξ(t) where ω 2 = k/. Here the rando force is a Gaussian rando process with oents ξ(t 1 ) = ξ(t 1 )ξ(t 2 ) = 2γk B T δ(t 1 t 2 ) The spectral density is the Fourier transfor of the correlation function S ξ (ω) = e iωt ξ(t)ξ() = 2γk B T which is white noise. We can Fourier transfor the Langevin equations iωx(ω) = v(ω) iωv(ω) = γ v(ω) ω2 x(ω) + 1 ξ(ω)

9 6.2 Exaples 83 5 (a) 1 (b) 4.5 S x (ω) 3 C x (t) ω t Figure 6.2: (a) The spectru S x (ω)/ x 2 for a Brownian particle in a haronic potential plotted versus ω/ω for γ/ =.5, 1. and 1.5ω (b) The correspondning results for the noralized correlation function C x (t)/c x (t = ) plotted versus tω. Then we can solve for x(ω) in ters of the fluctuating force to obtain x(ω) = 1 ξ(ω) ω 2 ω2 γ iω The spectral density is proportional to x(ω) 2 and so S x (ω) = 1 S ξ (ω) 2 ω 2 ω2 γ iω 2 = 2γk BT 1 2 (ω 2 ω2 ) 2 + γ2 ω 2 2 For the corresponding tie dependent correlation function we have C x (t) = 1 2π dωe iωt S x (ω) = 2γk BT 2π 2 dωe iωt 1 (ω 2 ω2 ) 2 + γ2 ω 2 2 The integral can be calculated as a contour integral in the coplex plane by taking the residues at the poles. These are located at ω 2 ω 2 = ± γ γ iω, ω = ±i 2 ± ω 2 γ2 4 2 = ±i γ 2 ± ω 1 with ω 1 = ω 2 γ2 /4 2. For t > and ω = x + iy we see that the exponent iωt = ixt + yt, and the contour in the coplex ω-plane have to be closed by a seicircle in the lower half plane. The poles are then at ω = ±ω 1 iγ/2. By working out the residues one find C x (t) = 2γk BT 2π 2 = 2πi 2γk BT 2π 2 dω e iωt (ω ω 1 i γ 2 )(ω ω 1 + i γ 2 )(ω + ω 1 i γ e i(ω 1 i γ 2 )t ( i γ )(2ω 1 i γ )(2ω 1) + = k { BT ω 2 e γ 2 t cos ω 1 t + γ sin ω 1 t 2ω 1 } e i( ω1 i γ 2 )t 2 )(ω + ω 1 + i γ ( 2ω 1 i γ )( 2ω 1)( i γ ) 2 )

10 84 Chapter 6 Brownian Motion: Langevin Equation We notice that at t = we have C x (t = ) = x 2 = k BT/ω 2 equipartiton theore 1 2 ω2 x 2 = 1 2 k BT Siilarly we find the spectral density for the velocity correlatons which is just the S v (ω) = ω 2 S x (ω) and fro this we can get the velocity correlation function C v (t). Dipole-dipole correlation function Many tie correlation functions are related to spectroscopic easureents. For exaple the frequency dependence of the optical absorption coefficient of a substance is deterined by the tie correlation of its electric dipole oent. The absorption coefficient α(ω) at frequency ω is α(ω) = 2πω2 β 3nc e iωt M(t) M() eq where c is the speed of light in vacuu, β = 1/k B T and n is the index of refraction. M(t) is the total electric dipole oent of the syste at tie t. Suppose the syste being investigated is a single dipolar olecule. Then M is just its peranent dipole oent. It has a constant agnitude µ and a tie dependent orientation specified by the unit vector u(t) so that M(t) M() eq = µ 2 u(t) u() eq For a planar rotor with θ(t) the instantaneous angle with the x-axis and u(t) = (cos θ(t), sin θ(t)) u(t) u() eq = cos θ(t) cos θ() + sin θ(t) sin θ() eq = cos(θ(t) θ()) = Re e iθ(t) e iθ() eq We can calculate this quantity using the Langevin equation for rotational Brownian otion. The position x is replaced by the angle θ and the velocity v by the angular velocity Ω, and the ass by the oent of inertia I dθ I dω = Ω = γω + ξ(t) where the fluctuating torque has correlation function ξ(t 2 )ξ(t 1 ) ξ = 2γk B T δ(t 2 t 1 )

11 6.2 Exaples 85 Then with τ R = I/γ and Ω(t) = Ω()e t/τ R + 1 I ds e (t s)/τ R ξ(s) θ(t) = θ() + dsω()e s/τ R + 1 ds due (s u)/τ R ξ(u) I ( ) = θ() + τ R Ω() 1 e t/τ R + 1 ( ) ds 1 e (t s)/τ R ξ(s) γ As before in the Ornstein-Uhlenbeck process we now get (θ(t) θ()) 2 = 2k BT I ( ) γ 2 t/τ R 1 e t/τ R The orientational tie correlation function is s C R (t) = Re e i(θ(t) θ() = Re e i θ(t) But θ(t) = θ(t) θ() is linear in the noise and in the initial angular velocity, and both these have a Gaussian distribution. Then θ(t) also has a Gaussian distribution with a zero ean value and a second oent ( θ(t)) 2. For a Gaussian vaiable X with ean value µ X and variance σx 2 we have the characteristic function Therefore e ikx = e ikµ X k 2 σ 2 X /2 ( C R (t) = e ( θ(t))2 /2 = exp 2k BT I ( ) ) γ 2 t/τ R 1 e t/τ R For short and long ties we then have ( C R (t) = exp 1 ) k B T 2 I t2, t τ R ( C R (t) = exp k ) BT γ t, t τ R (6.4) The rotational correlation function is directly related to the dielectric susceptibility which is easured in dielectric relaxation easureents. The relation is ( ɛ(iω) = ɛ + (ɛ ɛ ) e iωt dc ) R(t) = ɛ + (ɛ ɛ ) 1 iωc R (iω) (6.41) Here ɛ is the static dielectric constant for ω = and ɛ the corresponding one for high frequencies. An exponential relaxation for long ties, as obtained above, is seen for single olecules or low densities. For high densities the relaxation is ore coplex and one often observes a so called Kohlrausch- Willias-Watts for for C R (t) C R (t) = e (t/τ)β, < β < 1 with the relaxation tie τ increasing rapidly with decreasing teperature or increasing density. The corresponding iaginary part of the dielectric susceptibility ɛ (ω) has then a uch broader peak than the Lorentzian shape obtained for an exponential function.

12 86 Chapter 6 Brownian Motion: Langevin Equation Langevin dynaics of a Gaussian polyer chain Consider a polyer olecule with N beads or particles of ass connected by springs with force constant k, and oving in a fluid of sall olecules. The interaction potential is N 1 U = r(l, t) r(l 1, t)2 2 l=1 The equation of otion of bead l reads d2 2 r(l, t) = γ d r(l, t) + k r(l + 1, t) 2r(l, t) + r(l 1, t) + ξ(l, t) Here γ is the friction constant. Neglecting inertial effects, i.e. the acceleration ter this gives d r(l, t) = k γ r(l + 1, t) 2r(l, t) + r(l 1, t) + 1 ξ(l, t) γ Let s consider a ring polyer or a chain with periodic boundary conditions. discrete Fourier transfor is N R(q, t) = e iql r(l, t) with the inverse transfor l=1 Then a r(l, t) = 1 N N e iq kl R(q, t) k=1 and where q k = π + 2πk N, k = 1,..., N This follows since periodic boundary conditions iply r(l, t) = r(l+n, t) and so exp(iqn) = 1. We also have the relation N l=1 e i(q q )l = e i(q q ) 1 ei(q q )N 1 e i(q q ) { q q = N q = q and siilarly when suing over q k. For R(q, t) we find the equation The solution is given by d R(q, t) = 2k γ 1 cos q R(q, t) + 1 ξ(q, t) γ R(q, t) = 1 γ with the relaxation tie τ = γ/2k, and so r(l, t) = 1 Nγ q ds e (1 cos q)(t s)/τ ξ(q, s) ds e (1 cos q)(t s)/τ e iql ξ(q, t s)

13 6.2 Exaples 87 We are interested to calculate the correlation function C(l, t) = r(l, t) r(, ) 2 which is a easure of the ean squared displaceent for the different beads. Then C(l, t) = 1 N 2 γ 2 ds ds e (1 cos q)(t s)/τ e (1 cos q )(t s )/τ q q e iql ξ(q, t s)ξ(q, s) e iq l ξ(q, t s )ξ(q, s ) Now, the fluctuating force ξ is uncorrelated on different beads and at different ties and so ξ α (q, t)ξ β (q, t = ξ α (l, t)ξ β (l, t = gδ l,l δ(t t )δ α,β N l=1 N l e iql e iq l ξ α (l, t)ξ β (l, t = g = Ngδ q+q, δ(t t )δ α,β N e i(q+q )l δ(t t )δ α,β Inserting this into the expression for C we can perfor the tie integrals and one suation over q, to obtain C(l, t) = 3g N 2 γ 2 δ q+q, = q q ds l=1 ds e (1 cos q)(t s)/τ e (1 cos q)(t s )/τ δ(s s ) e iql δ(t s + s ) e iq l δ(t s + s ) + δ(s s ) 3g 1 e (1 cos q)t/τ cos ql N2kγ 1 cos q q In the liit N, q becoes a continous variable in the interval ( π, π and C(l, t) = 3g 1 π dq 1 e (1 cos q)t/τ cos ql 2kγ 2π π 1 cos q For t = we have C(l, ) = 3g 1 π 1 cos ql dq 2kγ 2π π 1 cos q = e iq = z = 3g 1 dz 1 (z l 1) 2 2kγ 2πi C z l z 1 = 3g 1 2kγ 2πi C ( dz 1 l 1 ) 2 z l z k k= where C is a unit-circle in the z-plane. Since 1 dz 1 2πi z = δ,1 C

14 88 Chapter 6 Brownian Motion: Langevin Equation 1 C(l,t)/C(1,) t/τ Figure 6.3: The function C(l, t)/c(1, ) versus tτ for l = 4. The asyptotoc behaviour for t/τ 1is shown as dashed curves. we see that there are l ters where z k z p = z l 1 and so C(l, ) = 3g l = C(1, )l 2kγ To find a ore explicit expression for the tiedependence we can take the derivative of C(l, t) 3g 1 π C(l, t) = t γ 2 dq e (1 cos q)t/τ cos ql = 3g 1 π 2π π γ 2 e t/τ dq e iql cos q(t/τ) e 2π π = 3g γ 2 e t/τ I l (t/τ) where I l is the odified Bessel function of order l. Integrating we find C(l, t) = C(l, ) + g S(l, t/τ) 2kγ where S(l, x) = xe x I (x) + I 1 (x) + l e x I (x) 1 l 1 + xe x (l n)i n (x) In figure 6.3 C(l, t) is plotted for l = 4. For t τ we have the asyptotic expansion I n (x) = e x (2πx) 1/2, x 1 which gives a (t) dependence of C(l, t) for large ties. In particular for l = this gives the long tie behaviour C(, t) = g ( ) kt 1/2 γk πγ n=1

15 6.2 Exaples 89 The dashed curves in fig. 6.3 shows this long tie behaviour. For sall l-values this asyptotic behaviour is reached for rather short ties.