Brownian Motion: Langevin Equation

1 Chapter 6 Brownian motion : LangevinEquationThe theory of Brownian motion is perhaps the simplest approximate way to treat thedynamics of nonequilibrium systems. The fundamental Equation is called the Langevinequation; it contain both frictional forces and random forces. The fluctuation-dissipationtheorem relates these forces to each random motion of a small particle (about one micron in diameter) immersed in afluid with the same density as the particle is called Brownian motion . Early investigationsof this phenomenon were made by the biologist Robert Brown on pollen grains and alsodust particles or other object of colloidal modern era in the theory of Brownian motion began with Albert Einstein.

2 He ob-tained a relation between the macroscopic diffusion constantDand the atomic propertiesof matter. The relation isD=RTNA6 a=kBT6 awhereRis the gas constant,NA= 1023/mol is Avogadros number,Tis the tem-perature, is the viscosity of the liquid andais the radius of the Brownan particle. AlsokB=R/NAis Boltzmanns theory of Brownian motion has been extended to situations where the fluctuatingobject is not a real particle at all, but instead some collective porperty of a macroscopicsystem. This might be, for example, the instantaneous concentration of any componentof a chemically reacting system near thermal equilibrium.

3 Here the irregular fluctuationin time of this concentration corresponds to the irregular motion of the dust Langevin equationConsider a large particle (the Brownian particle) immersed in a fluid of much smallerparticles (atoms). Here the radius of the Brownian particle is typically 10 9m< a <5 10 7m. The agitated motion of the large particle is much slower than that of theatoms and is the result of random and rapid collisions due to density fluctuations in thefluid. There are in general three vastly different timescales in a colloidal system s, B,and r.

4 Here sis the short atomic scale s 10 12s, Bis the Brownian timescale for7576 Chapter 6 Brownian motion : Langevin EquationFigure :A large Brownian particle with massMimmersed in a fluid of much smallerand lighter relaxation of the particle velocity B m 10 3sand ris the relaxation time for the Brownian particle, the time the particle havediffused its own radius r=a2 DIn general s B dense colloidal suspensions rcan become very long of the order of minutes or the motion of a dust particle performing Brownian motion appears to be quiterandom, it must nevertheless be describable by the same Equation of motion as is any otherdynamical system.

5 In classical mechanics these are Newton s or Hamiltons equations . Forsimplicity we will consider motion in one dimension. The results can easily be generalisedto three dimensions. Newtons Equation of motion for the particle (radiusa, massm,positionx(t), velocityv(t)) in a fluid medium (viscosity ) ismdv(t)dt=F(t)( )whereF(t) is the total instantaneous force on the particle at timet. This force is due tothe interaction of the Brownian particle with the surrounding medium. If the pssitionsof the moelcules in the surrounding medium are known as a function of time, then inprinciple this force is a known function of time.

6 In this sense it is not a random force is usually not practical or even desirable to look for an exact expression forF(t).Experience tells us that in typical cases this force is dominated by a friction force v(t), Langevin equation77proportional to the velocity of the Brownian particle. The friction coefficient is given byStokes law = 6 a( )We also expect a random force (t) due to random density fluctuations in the fluid. Theequations of motion of the Brownian particle are:dx(t)dt=v(t)dv(t)dt= mv(t) +1m (t)( )This is theLangevin equationsof motion for the Brownian random force (t) is a stochastic variable giving the effect of background noisedue to the fluid on the Brownian particle.

7 If we would neglect this force ( ) becomesdv(t)dt= mv(t)( )which has the familiar solutionv(t) = e t/ Bv(0), B=m ( )According to this, the velocity of the Brownian particle is predicted to decay to zero atlong times. This cannot be true since in equilibrium we must have the equipartion theorem v2(t) eq=kbTm( )while ( ) gives v2(t) eq= e 2t/ B v2(0) eq 0( )The random force in ( ) is therfore necessary to obtain the correct equilibrium. In theconventional view of the fluctuation force it is supposed to come from occasional impactsof the Brownian particle with molecules of the surrounding medium.

8 The force during animpact is supposed to vary extremely rapidly over the time of any observation. The effectof the fluctuating force can be summarized by giving its first and second moments (t) = 0, (t1) (t2) =g (t1 t2)( )The average is an average with respect to the distribution of the realizations of thestochastic variable (t).Since we have extracted the average force v(t) in the Langevin Equation the averageof the fluctuating force must by definition be a measure of the strength of thefluctuation force. The delta function in time indicates that there is no correlation betweenimpacts in any distinct time intervals dt1and dt2.

9 This loss of correlation is a consequenceof the separation of time scales discussed above. During a short time interval dton scale B= 10 3s, say dt= 10 5s, there are still roughly dt/ s 107collisons with the atomsin the liquid. Therefore any memory between forces at different times will be lost due tothese frequent 6 Brownian motion : Langevin EquationThe remaining mathematical specification of this dynamical model is that the fluctu-ating force has a Gaussian distribution determined by the moments in ( ).The property ( ) imply that (t) is a wildly fluctuating function, and it is not atall obvious that the differential Equation ( ) has a unique solution for a given initialcondition, or even that dv/dtexists.

10 There is a standard existence theorem for differentialequations which guarantee the existence of alocalsolution if (t) is continous. A localsolution is one which exists in some neighborhood of the point at which the initial valueis given. But even if a solution exists it may be only local, or it may not be unique, unlesssome stronger conditions are imposed on (t).We can obtain an explicit formal solution of ( ) asv(t) = e t/ Bv(0) +1m t0dse (t s)/ B (s)( )but this only transfers the problem elsewhere. How do we know that the integral in ( )exists, that it is more than just a formal symbol?