1 I. BROWNIAN MOTION The dynamics of small particles whose size is roughly 1 µmt or smaller, in a fluid at room temperature, is extremely erratic, and is called Brownian motion. The velocity of such particles is governed by thermal fluctuations. The particle will interact with the bath, i.e. the fluid it is embedded in, gaining and losing kinetic energy from the bath, and thus the velocity of the particle and its position are stochastic variables. Usually there is no hope of solving the problem from a microscopical point of view, since the interactions of the Brownian particle with the surrounding particles are extremely complicated, hence stochastic theories are very useful. Brownian type of dynamics describes dynamics of large molecules in solution, and hence is important in many applications of Chemistry, Biology, and Physics. One original motivation of investigation of Brownian motion by Einstein was to prove the existence of atoms. Today theory of Brownian motion is used as an example of non-equilibrium dynamics which is still close to thermal equilibrium. Many aspects of theory of Brownian motion can be generalized to other types of stochastic dynamics, for example the so called Einstein relations, linear response theory or fluctuation dissipation relation, we study here in the context of Brownian motion have general importance in non-equilibrium systems. We now construct a phenomenological theory for the velocity of a Brownian motion. Our aim is to write an equation of motion for P (V, t)
2 the velocity PDF, given that the velocity of the particle at time t = 0 is V 0. We use four main assumptions: The velocity distribution of a Brownian particle with mass M, in equilibrium is the Maxwell distribution P eq (V ) = M 1/2 ( ) 2πkb T exp v2, (1) 2Mk b T where T is the temperature. This equation does not give us any information of the dynamics, though it does state that the variance of velocity fluctuations is given by the thermal velocity v th = k b T/M. The fact that the velocity distribution is Gaussian is strongly related to the central limit theorem as we show later in the context of kinetic theory of Brownian motion. The second assumption is that the average velocity satisfies the relaxation law v = γ v (2) where γ is the relaxation or damping coefficient which has units of 1/Sec. For example for a spherical particle whose radius is a we know from hydrodynamics the γ = 6πaη/M where η is the viscosity of the fluid. The hydrodynamic Eq. (2) neglects fluctuations and it gives us simple relaxation v(t) = v 0 e γt. From hydrodynamics we know that for Eq. (2) to hold the velocity must not be too high, e.g. if compared to sound velocity. The third assumption is that the velocity of the Brownian particle is continuous. Mathematically this means the a partial differential
3 equation describes the dynamics. The final assumption is that the dynamics is Markovian, namely to determine the velocity distribution in time t + δt, we need to know only the velocity distribution at time t. Physically this means that we neglect the influence of the Brownian particle on the bath. To construct an equation of motion we consider P (v, t) t = B 2 2 P (v, t) + A vp (v, t). (3) v 2 v The coefficients A and B, which seem independent of each other, must be determined from our assumptions. The first term described a diffusion process in velocity space, it is called a fluctuation term. It is clear that B > 0. The role of the second term is to restore the velocity to equilibrium. To see this note that if A = 0 then clearly the variance of velocity is going to increase with time, which is in conflict with Boltzmann s equilibrium statistical mechanics and nature. Multiplying Eq. (3) from the left with v and integrating with respect to v, using the boundary conditions that P (v, t) = 0 when v we have v(t) = A v hence we find A = γ. 7.1 Assume the more general equation P (v, t) t = B m 2 2 v m P (v, t) v 2 + A n v vn P (v, t), (4) show that the law v = γ v implies n = 1 but does not teach us anything on m. Then show that for the equilibrium velocity distribution to be Gaussian we must demand m = 0.
4 We now impose on the dynamics the request that in equilibrium, namely when t the velocity distribution is Maxwellian which is independent of time. Inserting Eq. (1) in Eq. (2) [ ] B v 2 v P eq(v ) + γvp eq (v) = 0 (5) we find that the condition B/2 = γk b T/M must hold. To conclude we find the equation of motion for the velocity of a Brownian particle P (v, t) t = γ [ kb T M 2 P (v, t) + ] vp (v, t). (6) v 2 v This equation was derived by Rayleigh using a kinetic approach. The approach used in the section to derive Rayleigh s equation, based on a macroscopical (or hydro-dynamical) relaxation law to which fluctuations are added is due to Einstein and also Langevin. 7.2 Find solution to Eq. (6) assuming initial condition is V 0. Check that the solution is normalized, non-negative, and approaches thermal equilibrium in the long time limit. 7.3 Show that v 2 = v 2 0 e 2γt + k bt M ( 1 e 2γt ). 7.4 Consider an ensemble of one dimensional Brownian particle, which at time t = 0 are on the origin. Initially the particles are in thermal equilibrium with temperature T. Consider the random position of the Brownian particle x(t) = t 0 v(t )dt where
5 v(t ) satisfies the dynamics in Eq. (6) show that for 0 t 1 t 2 x(t 1 )x(t 2 ) = k bt M [ 2 γ t 1 1 γ + 1 ( e γt 1 + e γt 2 e ) ] γ(t 2 t 1 ). 2 γ 2 (7) Since this equation implies that when t x 2 (t) 2 k bt Mγ t you have proven the Einstein relation between diffusion constant D = lim t x 2 (t) t and damping γ, namely D = k b T/(Mγ). = 2D A. The Fokker Planck Equation The Fokker Planck equation describes dynamics of a continuous stochastic process y(t) whose dynamics is Markovian. It is used to model many processes where a coordinate y(t) has small jumps, the Rayleigh equation for the velocity of a Brownian particle being a special case. The Fokker-Planck equation is P (y, t) t = y A(y)P (y, t) + 1 2 B(y)P (y, t). (8) 2 y2 The only conditions are that A(y), B(y) are real functions and B(y) > 0. Roughly speaking, the first term on the left hand side describes a drift term, the second describes the fluctuations. The drift term describes the dynamics of the average of y namely it is easy to show ẏ = A(y). For the Brownian motion we used the fact that the average velocity obeys a linear law ẏ = γ y to identify the physical meaning of A(y). However when the hydrodynamic law is not
6 linear we encounter a difficulty. Assume that for some fluid the average velocity of a particle obeys the non linear equation v = β v 2. Then naively we might identify A = βv 2. However this is wrong since then it is easy to see that according to the Fokker-Planck equation v = β v 2 β v 2. This implies that if the dynamics of the average y is described by a non-linear equation we cannot use the simple approach we used in the previous section. Instead one must derive the Fokker Planck equation from some underlying dynamics, wick should then be consistent with the non-linear phenomenological relaxation laws. 7.5 van Kampen shows how to find A(y) and B(y) at least in principle. Let P (y, t y 0, t 0 ) be the solution of the Fokker Planck equation for particle starting on y 0 at time t 0. Take t = t 0 + t and y y 0 = y with t and y being small. Show that for t 0 y t = A(y 0 ), ( y) 2 t = B(y 0 ), ( y) n t = 0n 3. (9) 7.6 Find the stationary solution of the Fokker Planck equation (8). 7.7 Show that solution of Eq. (8) is normalizable. It is more difficult to show that the solution is non-negative. Consider the binomial random walk on a lattice, with jumps to nearest neighbors only, and with probability of 1/2 to jump left or right. We approximate the dynamics using a Fokker Planck equation
7 approach. First we forget about the underlying lattice and treat the problem as a continuum. This means that while the jumps in the random walk model have finite size (the size of the lattice spacing) and hence the position of the particle is not a continuous function of time, we still assume that on a coarse grained level a Fokker- Planck equation might work well. The Fokker-Planck approach gives x/ t = 0, since we do not have bias. Also lim x 0, t 0 x2 /2 t D where x 2 is the lattice spacing square, or more generally the variance of the microscopical jump length, t is the time between jumps, and D by definition is the diffusion constant. We immediately find the Fokker-Planck equation, P (x, t) t = D 2 P (x, t) x 2, (10) which is the famous diffusion equation. From central limit theorem we know that this equation works well only in the long time limit. Still in many Physical situations we are interested in intermediate time scales where t t where t is the time between jumps, however the probability packet is still not in equilibrium (e.g., particles did not reach yet the walls of the container) and then diffusion and Fokker-Planck equations are very useful as excellent approximations.
8 B. Einstein Relations We consider a Brownian motion under the influence of a driving force field F. F does not depend on time or space, it is a constant driving force, for example if the Brownian particle is charged then F is the charge of the particle times a uniform electric field which is applied on the system, or F could be due to the gravitational field of earth F = mg. The particle will experience a net drift and we assume it will reach a finite velocity (the effects of the size of the container are not important now). The linear equation of motion for the average velocity of a Brownian particle is v = γ v + F M (11) and hence in the long time limit v = F/(Mγ). Now starting with this law for the average velocity we wish to model the behavior of the coordinate x of the particle. We clearly have x/ t = F/(M γ). We also assume that the force is weak, in such a way that the diffusion process is not altered by the external force field, besides the global drift of-course. This means that x 2 /δt = D is the diffusion constant of the Brownian particle in the absence of the external force. And hence according to the Fokker-Planck equation approach we have P (x, t) t = D 2 P (x, t) x 2 F Mγ P x. (12) Now Einstein adds the effect of the container to the process, and considers the equilibrium of such process (in the absence of a container
9 the process never reaches equilibrium, it just exhibits a drift and diffusion). Assume the force is positive, hence the drift is from left to right. The particle are driven towards a wall which we put on x = 0. In equilibrium we have according to Boltzmann ( ) F x P eq (x) = Const exp k b T forx < 0, (13) where we used the fact that the potential energy of the particle is U(x) = F x and the canonical ensemble. Inserting Eq. (13) in Eq. (12) we find a relation between the damping or the dissipation and the diffusion constant D which is the Einstein relation D = k bt Mγ. (14) We also define the mobility µ which is a measure of the response of the particle to a weak external field, by definition v = µf. Then clearly µ = 1/(Mγ) and hence we find the second Einstein relation D = k b T µ. (15) We see that the response of the particles to external driving field, which yields a net current, is related to the fluctuations in the absence of the force fields. Thus µ (transport) D (diffusion) and γ (relaxation, or dissipation) are all related. Theories of transport are many times based on these ideas, though usually a more general framework called linear response theory is used. The idea in many theoretical works is to calculate D from some microscopical theory, and then give a prediction on the response of the system to an external weak perturbation. The
10 advantage is that we do not have to consider the external force field in the first place. Though to apply this scheme we obviously must assume that the response to the external field is linear. II. BROWNIAN MOTION: A SIMPLE KINETIC APPROACH We will consider a simple kinetic approach to obtain Maxwell s velocity distribution. Briefly we consider a one dimensional tracer particle of mass M randomly colliding with gas particles of mass m << M. Four main assumptions are used: (i) molecular chaos holds, implying lack of correlations in the collision process (Stoszzahlansantz), (ii) collision are elastic and impulsive, (iii) gas particles maintain their equilibrium during the collision process, and (iv) rate of collisions is independent of the energy of the colliding particles. Let the probability density function (PDF) of velocity of the gas particles be f (ṽ m ). Our goal is to obtain the equilibrium velocity PDF of the tracer particle W eq (V M ). Questions: (i) Does W eq (V M ) depend on m? (ii) Does W eq (V M ) depend on rate of the collisions R? (iii) Does W eq (V M ) depend on the precise shape of the velocity PDF of the gas particles f (ṽ m )? Answers: (i) no, (ii) no, and (iii) no. We will show that the equilibrium PDF of the tracer particle W eq (V M ) is the Maxwell velocity PDF.
11 A. Model and Time Dependent Solution We consider a one dimensional tracer particle with the mass M coupled with bath particles of mass m (these are treated as ideal gas particles). The tracer particle velocity is V M. At random times the tracer particle collides with bath particles whose velocity is denoted with ṽ m. Collisions are elastic hence from conservation of momentum and energy V M + = ξ 1VM + ξ 2ṽ m, (16) where ξ 1 = 1 ɛ 1 + ɛ ξ 2 = 2ɛ 1 + ɛ (17) and ɛ m/m is the mass ratio. In Eq. (16) V + M (V M ) is the velocity of the tracer particle after (before) a collision event. The duration of the collision events is much shorter than any other time scale in the problem. The collisions occur at a uniform rate R independent of the velocities of colliding particles. The probability density function (PDF) of the bath particle velocity is f(ṽ m ). This PDF does not change during the collision process, indicating that re-collisions of the bath particles and the tracer particle are neglected. We now consider the equation of motion for the tracer particle velocity PDF W (V M, t) with initial conditions concentrated on V M (0). Kinetic considerations yield W (V M, t) t =
12 RW (V M, t)+r dvm dṽ m W ( VM, t) f (ṽ m ) δ ( V M ξ 1 VM ξ ) 2ṽ m, (18) where the delta function gives the constrain on energy and momentum conservation in collision events. The first (second) term, on the right hand side of Eq. (18), describes a tracer particle leaving (entering) the velocity point V M at time t. Eq. (18) yields the linear Boltzmann equation W (V M, t) t = RW (V, t) + R ξ 1 ( ) VM ξ 2 ṽ m dṽ m W f (ṽ m ). ξ 1 (19) In Eq. (19) the second term on the right hand side is a convolution in the velocity variables, hence we will consider the problem in Fourier space. Let W (k, t) be the Fourier transform of the velocity PDF W (k, t) = W (V M, t) exp (ikv M ) dv M, (20) we call W (k, t) the tracer particle characteristic function. Using Eq. (19), the equation of motion for W (k, t) is a finite difference equation W (k, t) t = R W (k, t) + R W (kξ 1, t) f (kξ 2 ), (21) where f (k) is the Fourier transform of f(ṽ m ). In Appendix A the solution of the equation of motion Eq. (21) is obtained by iterations W (k, t) = n=0 (Rt) n exp ( Rt) ( ) e ikv M (0)ξ1 n Π n n! i=1 f kξ n i 1 ξ 2, (22) with the initial condition W (k, 0) = exp[ikv M (0)]. The solution Eq. (22) has a simple interpretation. The probability that the tracer particle has collided n times with the bath particles is
13 given according to the Poisson law P n (t) = (Rt)n n! exp ( Rt), (23) reflecting the assumption of uniform collision rate. Let W n (V M ) be the PDF of the tracer particle conditioned that the particle experiences n collision events. It can be shown that the Fourier transform of W n (V M ) is W n (k) = e ikv M (0)ξ n 1 Π n i=1 f ( kξ n i 1 ξ 2 ). (24) Thus Eq. (22) is a sum over the probability of having n collision events in time interval (0, t) times the Fourier transform of the velocity PDF after exactly n collision event W (k, t) = P n (t) W n (k). (25) n=0 It follows immediately that the solution of the problem is W (V M, t) = P n (t)w n (V M ), (26) n=0 where W n (V M ) is the inverse Fourier transform of Wn (k) Eq. (24). B. Equilibrium In the long time limit, t the tracer particle characteristic function reaches an equilibrium W eq (k) lim t W (k, t). (27) This equilibrium is obtained from Eq. (22). We notice that when Rt, P n (t) = (Rt) n exp( Rt)/n! is peaked in the vicinity of n = Rt
14 hence it is easy to see that W eq (k) = lim n Π n i=1 f ( kξ n i 1 ξ 2 ). (28) In what follows we investigate properties of this equilibrium. We will consider the weak collision limit ɛ 0. This limit is important since number of collisions needed for the tracer particle to reach an equilibrium is very large. Hence in this case we may expect the emergence of a general equilibrium concept which is not sensitive to the precise details of the velocity PDF f(ṽ m ) of the bath particles. Remark 1 According to Eq. (24), after a single collision event the PDF of the tracer particle in Fourier space is W1 (k) = f (kξ 2 ) provided that V M (0) = 0. After the second collision event W 2 (k) = f (kξ 1 ξ 2 ) f (kξ 2 ) and after n collision events W n (k) = Π n i=1 f ( kξ n i 1 ξ 2 ). (29) This process is described in Fig. 1, where we show W n (k) for n = 1, 3, 10, 100, 1000. In this example we use a uniform distribution for the bath particles velocity Eq. (44), with ɛ = 0.01, and T = 1. After roughly 100 collision events the characteristic function W n (k) reaches a stationary state, which as we will show is well approximated by a Gaussian (i.e., the Maxwell velocity PDF is obtained).
15 C. Maxwell Velocity Distribution We consider the case where all moments of f (ṽ m ) are finite and that the following behavior holds: f (ṽ m ) = 1 T/m q(ṽ m/ T/m). (30) q(x) is a non-negative normalized function. The second moment of the bath particle velocity is ṽ 2 m = T m x 2 q(x)dx. (31) Without loss of generality we set x2 q(x)dx = 1. The scaling behavior Eq. (30) and the assumption of finiteness of moments of the PDF yields ṽ 2n m = ( T m ) n q 2n, (32) where the moments of q(x) are defined according to q 2n = x 2n q(x)dx, (33) and we assume that odd moments of q(x) are zero. Thus the small k expansion of the characteristic function is f (k) = 1 T ( ) k2 T 2 2m + q k 4 4 m 4! + O(k6 ). (34) For simplicity we consider only the first three terms in the expansion in Eq. (34). We now obtain the velocity distribution of the tracer particle using Eq. (28) ln [ Weq (k) ] = lim n n i=1 ln [ f ( kξ n i 1 ξ 2 )]. (35)
16 Inserting Eq. (34) in Eq. (35) we obtain ln [ Weq (k) ] = T 2m ɛk2 + q ( 4 3 T 2 2ɛ 4! m) 3 1 + ɛ 2 k4 + O(k 6 ). (36) When ɛ is small we find (using ɛ = m/m) ln [ Weq (k) ] = T k2 2M + ( T M ) 2 q 4 3 2ɛk 4 + O ( k 6). (37) 4! It is important to see that the k 4 term approaches zero when ɛ 0. Hence we find lim ln [ Weq (k) ] = T k2 ɛ 0 2M, (38) inverting to velocity space we obtain the Maxwell velocity PDF lim W eq (V M ) = ɛ 0 ( M exp MV 2 ) M. (39) 2πT 2T We see that the parameters q 2n with n > 1 are the irrelevant parameters of the problem, and hence the Maxwell distribution is stable in the sense that it does not depend on the detailed shape of f(ṽ m ). Remark 1 To complete the proof we will show that the k 6, k 8 and higher order terms in Eq. (37) also approach zero when ɛ 0. Let κ m,2n (κ M,2n ) be the 2n th cumulant of bath particle (tracer particle) velocity. The cumulants describing the bath particle are related to the moments q 2n in the usual way κ m,2 = T/m, κ m,4 = (q 4 1)(T/m) 2 etc. Then using Eq. (28) one can show that κ M,2n = g 2n (ɛ) κ m,2n. (40) From the scaling function Eq. (30) we have κ m,2n = c 2n T n /m n, where c 2n are dimensionless parameters which depend on f(ṽ m ), n = 1, 2,,
17 e.g c 2 = 1, c 4 = q 4 1 etc. The parameters c 2n for n > 1 are the irrelevant parameters of the model in the limit of weak collisions. To see this note that when ɛ 0 we have κ M,2n = (T 2 /M)δ n1. (41) Thus, besides the second cumulant, all cumulants of the tracer particle velocity distribution function are zero. As well known the cumulants of the Gaussian PDF with zero mean are all zero besides second. Eq. (41) shows that the tracer particle reached the Maxwell equilibrium. 1. Numerical Examples We now investigate numerically exact solutions of the problem, and compare these solutions to the stable equilibrium which becomes exact when ɛ 0. We investigate three types of bath particle velocity PDFs: (i) The exponential f (ṽ m ) = ( ) 2m 2 2m ṽm exp, (42) T 2 T2 which yields f(k) = 1 1 + T 2k 2 2m. (43) (ii) The uniform PDF f(ṽ m ) = m 12T 2 if ṽ m < 3T2 m (44) 0 otherwise
18 which yields (iii) The Gaussian PDF f(k) = sin ( 3T 2 m k) 3T2 m k. (45) f(k) = exp ( k2 T 2 2m ). (46) The small k expansion of Eqs. (43,45,46) is f(k) 1 k 2 T 2 /(2m)+, indicating that the second moment of velocity of bath particles ṽ 2 m is identical for the three PDFs. To obtain numerically exact solution of the problem we use Eq. (28) with large though finite n. In all our numerical examples we used M = 1 hence m = ɛ. Thus for example for the uniform velocity PDF Eq. (45) we have ( n W eq (k) exp ln ɛ sin i=1 3T 2 k 3T2 m k ( 1 ɛ 1+ɛ ( 1 ɛ 1+ɛ ) n i 2ɛ 1+ɛ ) n i 2ɛ 1+ɛ ). (47) To obtain equilibrium we increase n for a fixed ɛ and temperature until a stationary solution is obtained. According to our analytical results the bath particle velocity PDFs Eqs. (42,44,46), belong to the domain of attraction of the Maxwellian equilibrium. In Fig. 2 we show W eq (k) obtained from numerical solution of the problem. The numerical solution exhibits an excellent agreement with Maxwell s equilibrium. Thus details of the precise shape of velocity PDF of bath particles are unimportant, and as expected the Maxwell distribution is stable. We note that the convergence rate to equilibrium depends on the value of k. To obtain the results in Fig. 2 I used ɛ = 0.01, T 2 = 2, M = 1 and n = 2000.
19 III. APPENDIX A In this Appendix the solution of the equation of motion for W (k, t) Eq. (21) is obtained, the initial condition is W (k, 0) = exp[ikv M (0)]. The inverse Fourier transform of this solution yields W (V M, t) with initial condition W (V M, 0) = δ[v M V M (0)]. Introduce the Laplace transform W (k, s) = 0 W (k, t) exp ( st) dt. (48) Using Eq. (21) we have s W (k, s) e ikv M (0) = R W (k, s) + R W (kξ 1, s) f (kξ 2 ), (49) this equation can be rearranged to give W (k, s) = eikv M (0) R + s + R R + s W (kξ 1, s) f (kξ 2 ). (50) This equation is solved using the following procedure. Replace k with kξ 1 in Eq. (50) W (kξ 1, s) = eikξ 1V M (0) R + s + R R + s W ( kξ 2 1, s) f (kξ2 ξ 1 ). (51) Eq. (51) may be used to eliminate W (kξ 1, s) from Eq. (50), yielding Re ikξ 1V M (0) (R + s) 2 f (kξ2 ) + W (k, s) = eikv M (0) R + s + R 2 (R + s) 2 W ( k 2 ξ 1, s ) f (kξ2 ξ 1 ) f (kξ 2 ). (52) Replacing k with kξ 2 1 in Eq. (50) W (kξ 2 1, s) = eikξ2 1 V M (0) R + s + R R + s W ( kξ1 3, ( ) s) f kξ2 ξ1 2. (53)
20 Inserting Eq. (53) in Eq. (52) and rearranging W (k, s) = eikv M (0) R + s + Reikξ1VM (0) (R + s) 2 f (kξ2 ) + R 2 e ikξ2 1 V M (0) f (R + s) 3 (kξ2 ξ 1 ) f ( ) R 3 ( (kξ 2 )+ W kξ 3 1 R + s, ( s) f kξ2 ξ1) 2 f (kξ2 ξ 1 ) f (kξ 2 ). Continuing this procedure yields (54) W (k, s) = eikv M (0) R + s + R n (R + s) n+1 eikξn 1 V M (0) Π n f ( ) i=1 kξ1 n i ξ 2. (55) n=1 Inverting to the time domain, using the inverse Laplace s t transform yields Eq. (22). The solution Eq. (22) may be verified by substitution in Eq. (21).
21 1 0.8 W n (k) 0.6 0.4 0.2 0 0-30 -20-10 0 10 20 30-20 -10 0 10 20 1 1 W n (k) 0.5 0.5 0 0-20 -10 0 10 20 k -4-2 0 2 4 k FIG. 1: We show the dynamics of the collision process: the tracer particle characteristic function, conditioned that exactly n collision events have occurred, W n (k) versus k. The velocity PDF of the bath particle is uniform and ɛ = 0.01. We show n = 1 (top left), n = 3 (top right), n = 10 (bottom left) and n = 100 n = 1000 (bottom right). For the latter case we have W 100 (k) W 1000 (k), hence the process has roughly converged after 100 collision events. The equilibrium is well approximated with a Gaussian characteristic function indicating that a Maxwell Boltzmann equilibrium is obtained.
22 1 W eq (k) 0.5 0-2 0 2 k FIG. 2: The equilibrium characteristic function of the tracer particle, Weq (k) versus k. We consider three types of bath particles velocity PDFs (i) exponential (squares), (ii) uniform (circles), and (iii) Gaussian (diamonds). The velocity distribution of the tracer particle M is well approximated by Maxwell s distribution plotted as the solid curve W eq (k) = exp ( k 2). For the numerical results I used: M = 1, T 2 = 2, n = 2000, and ɛ = 0.01.