Lecure 4. Brownian moion. Einsein-Smoluhowski heory of he Brownian moion. Langevin heory of he Brownian moion Approach o equilibrium: Foker-Planck equaion. The flucuaion-dissipaion heorem. The erm Brownian moion derives is name from he boanis Rober Brown whom, in 88, made careful observaions on he iny pollen grains of a plan under a microscope. I was Einsein who, in a number of classic papers (beg. 905) firs provided a sound heoreical analysis of he Brownian moion on he basis of he so called random walk problem and hereby esablished a far-reaching relaionship beween (a) he irreversible naure of his phenomenon (as well as of he phenomena of diffusion and viscosiy) and (b) he mechanism of molecular flucuaions. To illusrae he essenial heme of Einsein s approach, le us consider he problem in one dimension. Le x() denoe he posiion of he Brownian paricle a ime, given ha is posiion coincided wih he poin x0 a ime 0. To simplify maers, we may assume ha each molecular impac (which, on an average, akes place afer a ime τ) causes he paricle o jump a (small) disance l - of consan magniude - in eiher he posiive or negaive direcion of he x-axis. I seems naural o regard he possibiliies Δx+l and Δx-l as equally likely; hough
somewha less naural, we may also regard he successive impacs on, and hence he successive jumps of, he Brownian paricle o be muually uncorreced. The probabiliy ha he paricle is found a he poin x a ime is now equal o he probabiliy ha, in a series of n(/τ * ) successive jumps, he paricle makes m(x/l) more jumps in he posiive direcion of he axis han in he negaive direcion, i.e. i makes ( n+ m) jumps in he posiive direcion and ( n m) jumps in he negaive direcion. The desired probabiliy is given by he binomial expression p n ( m) n n! ( + )! ( )! { n m } { n m } (4.) whence i follows ha m 0 and m n (4.) Thus, for >>τ * we have for he ne displacemen of he paricle x τ ( ) 0 and x ( ) l (4.3)
Accordingly, he roo-mean-square (r.m.s) displacemen of he paricle is proporional o he square roo of he ime elapsed: ( ()) ( / ) x x l rms... * / τ (4.4) I should be noiced ha he proporionaliy of he magniude x r.m.s. of he ne displacemen of he Brownian paricle he square roo of he oal number of elemenary seps is a ypical consequence of he random naure of he seps and i manifess iself in a large variey of physical phenomena occurring in naure. In conras, if he successive seps were fully coheren (or else if he moion were compleely mechanical, predicable and reversible over he ime inerval ), hen he quaniy x would have been proporional o and, accordingly, x r.m.s. would have been proporional o. Smoluchovski s approach o he problem of Browinian moion, which appeared in 906, was essenially he same as ha of Einsein; he difference lay primarily in he mahemaical procedure. Smoluchowski inroduced he probabiliy funcion p n (x 0 x) which denoes he probabiliy ha, in a series of n seps, he Brownian paricle, 3
iniially a he poin x 0, reaches he poin x ; he number x here denoes he disance in erms of he lengh of he elemenary sep. Clearly, p n ( x 0 x) p n ( x 0 z) p( z x) (n ); (4.5) x moreover, since a single sep is equally likely o ake he paricle o he righ or o he lef, while p( zx) δzx, + δ zx, + (4.6) p0 ( zx) δ zx, (4.7) Equaion (4.5) is known as he Smoluchwski equaion. The soluion of his equaion can be wrien in he following way 4
p n n n! ( x0 x) ( + )! ( + )! { n x x } { n x x } 0 0 0 for for x x n 0 x x > n 0 (4.8) Idenifying (x-x 0 ) wih m, we find his resul o be in complee agreemen wi our previous resul (4.). To obain an asympoic form of he probabiliy funcion p n (m), we apply Sirling s formula, / n n! ( πn) ( n / e), o he facorials appearing in he expression (4.) wih he resul ln p n ( m) ( n + - ) ln n ( n m + ) ln{ ( n + m + ) ln{ n[ + ( m / n)]} n[ ( m / n)]} n ln - ln( π ) (4.9) For m<<n (which is generally rue because m r.m.s. n / while n>>) we obain p ( m ) n m n ( n) exp( / π ) (4.0) 5
which is consisen wih he mean values (4.). Taking x o be a coninues variable (and remembering ha p n (m) 0 eiher for even values of m or odd values of m, i.e. in he disribuion ( 4.0), Δm and no ) we can wrie his resul in he Gaussian form: p( x) dx dx x exp (4 D) 4D π (4.) where l * x m / n τ D (4.) The quaniy D inroduced here is idenical wih macroscopic diffusion coefficien of he given sysem. The equaion (4.) connecs i wih he microscopic quaniies l and τ *. 6
To appreciae his connecion, one have o noe ha he problem of Brownian moion can also be looked upon as a problem of Diffusion of he Brownian paricles hrough he medium of he fluid. This poin of view is again due o Einsein. An experimen value for diffusion coefficien can be wrien as following ( Δx) / D (4.3) Le us now o sudy he Brownian moion from he poin of view of diffusion. We denoe he number densiy of he Brownian paricles in he fluid by he symbol n( r, ) and heir curren densiy by he symbol j ( r, ) { n( r, ) v( r, ) }; hen according o Fick s law, j( r, ) D n( r, ) (4.4) where D sands for diffusion coefficien. We also have he equaion of coninuiy, viz. 7
8 0 ), ( ), ( + n r r j (4.5) Subsiuing (4.4) ino (4.5) we obain he diffusion equaion 0 ), ( ), ( n D n r r (4.6) Of he various possible soluions of his equaion, he one relevan o he presen problem is ( ) D r D N n 4 exp 4 ), ( 3/ π r (4.7)
This is a spherically symmeric soluion and is already normalized: 0 n ( r, )4πr dr N (4.8) N being he oal number of (Brownian paricles immersed in he fluid. A comparison of he (3d dimensional) formula (4.7) wih he (one-dimensional) (4.) brings ou mos vividly he relaionship beween he random walk problem on one hand and he phenomenon of diffusion on he oher. I is clear ha in he las approach we have considered he moion of an ensemble of N Brownian paricles placed under equivalen physical condiions, raher han considering he moion of a single paricle over a lengh of ime (as was done in he random walk approach). Accordingly, he averages of various physical quaniies obained here will be in he naure of ensemble averages ; hey mus of course agree wih averages of he same quaniies obained earlier. Now by virue of he disribuion funcion (4.7) we obain 9
4π 4 ( ) > 0; < r ( ) > n( r, ) r dr 6 D < r N 0 (4.9) in complee agreemen wih our earlier resuls, namely x( ) * 0; x ( ) l / τ D (4.0) Thus he ensemble of he Brownian paricles, iniially concenraed a he origin, diffuses ou as ime increases, he naure and he exen of is spread a any ime being given by (4.7) and (4.9) respecively. The diffusion process, which is clearly irreversible, gives us a fairly good picure of he saisical behavior of a single paricle in ensemble. 0
Langevin heory of he Brownian moion. We consider he simples case of a free Brownian paricle, surrounded by a fluid environmen: he paricle is assumed o be free in he sense ha i is no aced upon by any oher force excep he one arising from he molecular bombardmen. The equaion of moion of he paricle will be M dv d F() (4.) where M is he paricle mass, v() he paricle velociy and F() he force acing upon he paricle by virue of he incessan impacs from he fluid molecules. Langevin suggesed ha he sochasic force F() may be wrien as a sum of wo pars: a) an averaged-ou par which represens he viscous drag, -v/b, experienced by he paricle ( accordingly, B is he mobiliy of he sysem, i.e. he drif velociy acquired by he paricle by virue of a uni exernal force. If Sokes law is applicable, hen B/6πηa; η- viscosiy, a-radius of he spherical paricle).
b) A rapidly flucuaing par F(), which over long inervals of ime (as compared wih characerisic ime τ * ), averages ou oo zero; hus we may wrie M dv d v + ( ); F ( ) B 0; F (4.) Taking he ensemble average of (4.) we obain M d d < v > < v B > (4.3) whence i follows ha < v ( ) > v(0) exp( / τ ); τ MB (4.4) Thus, he drif velociy of he paricle decays, a a rae deermined by he relaxaion ime τ, o he ulimae value zero. Dividing Eq. (4.) by he mass of he paricle, we obain an equaion for he insananeous acceleraion. viz.
dv d v + A ( ); A() 0 τ ; (4.5) We consruc he scalar produc of (4.5) wih he insananeous posiion r of he paricle and ake he ensemble average of he produc. In doing so, we make use of he facs ha a) r v ( dr / d) b) d r r ( dv / d) v d, and c) < r A > 0 We obain d d < r d > + τ d < r > < v > (4.6) 3
If he Brownian paricle has already obained hermal equilibrium wih he molecules of he fluid, he quaniy <v >in his equaion may be replaced by is equipariion value 3kT/M. The equaion is hen readily inegraed wih he resul < r > τ τ ( e ) τ 6kT / M (4.7) where he consans of inegraion have been so chosen ha 0 boh <r > and is firs ime-derivaive vanish. We observe ha, for <<τ < 3kT r > M v (4.8) which in consisen wih he reversible equaions of moion (of Newon), whereby one should simply have r v (4.9) 4
On he oher hand, if >>τ, hen < 6kTτ r > (6BkT) M (4.30) which is essenially he same as he Einsein-Smoluchowski resul (4.9); incidenally we obain here a simple, bu imporan, relaionship beween he coefficien of diffusion D and he mobiliy B or for Sokes paricle D BkT (4.3) D kt 6πηa (4.3) The relaionship (4.3) is generally referred o as he Einsein relaion. The irreversible characer of his resul is self-eviden; i is also clear ha i arises essenially from he viscosiy of he sysem. The Einsein relaion (4.3), which connecs he coefficien of diffusion D wih he mobiliy B of he 5
sysem, ells us ha he ulimae source of he viscosiy of he medium (as well as ha of diffusion) lies in he random, flucuaing forces arising from he incessan moion of he fluid molecules. In his conex, if we consider a paricle of charge e and mass M moving in a viscous fluid under he influence of an exernal elecric filed of inensiy E hen he course-grained moion of he paricle will be deermined by he equaion M d < v > < v > ee d B + (4.33) The erminal drif velociy of he paricle would be given by he expression (eb)e, which promps one o define (eb) as he mobiliy of he sysem and denoe i by he symbol ξ. Consequenly, one obains, insead of (4.3) 6
D kt e ξ (4.34) which is, in fac, he original version of he Einsein relaion; some imes i is also referred o as he Nerns relaion. Fokker-Plank equaion. In our analysis of he Brownian moion we have considered he behavior of a dynamical variable, such as he posiion r() of he velociy v() of a Brownian paricle from he poin of view of saisical flucuaions in he value of he variable. A more generalized way of looking a he manner in which and he rae a which a given arbirary disribuion of Brownian paricles approaches a sae of hermal equilibrium is provided by so-called Maser Equaion, a simplified version of which goes afer he names of Fokker an Planck. 7
Le us examine he displacemen x() of he given se of paricles along he x-axis. A any ime, le f(x,)ds be he probabiliy ha an arbirary paricle in he ensemble may have a displacemen beween xδ and x+dx. The funcion f(x,) mus saisfy he normalizaion condiion: f ( x, ) dx (4.35) The maser equaion hen reads: f ( x, ) { f ( x, ) W ( x, x') + f ( x', ) W ( x', x) } dx' (4.36) where W(x,x )dx δ denoes he probabiliy ha in a shor inerval of ime δ a paricle having displacemen x makes a ransiion beween x and x+dx. The firs par of his inegral hus corresponds o all hose ransiions ha remove paricles from displacemen x and, hence represen a ne loss o he funcion f(x,). Similarly, he second par of he inegral corresponds o all hose ransiions ha bring paricles from some oher displacemen x a ime o he displacemen x and, hence, represen a ne gain o he funcion f(x,). The srucure of he Maser Equaion is 8
herefore founded on very simple and sraighforward premises. Under a cerain approximae condiions, his equaion, or any generalizaion hereof (such as he one including velociy, or momenum, coordinaes in he argumen of f), can be reduced o he simpler form f f τ f 0 (4.37) which has proved o be a very useful firs approximaion for sudying problems relaed o he so-called ranspor phenomena. Here, f 0 denoes he equilibrium disribuion funcion (for f 0, when ff 0 ) while τ denoes he relaxaion ime ha characerizes he average rae a which he flucuaions in he sysem end o resore a sae of equilibrium. In sudying Brownian moion on he basis of eqn.(4.36) we can safely assume ha i is only he ransiions beween closely neighboring saes x and x ha have a appreciable probabiliy of occurring in oher words he 9
ransiion probabiliy funcion W(x,x ) is sharply peaked around he value x x and falls rapidly o zero elsewhere. Denoing he inerval (x -x) by he symbol ξ, we may wrie W ( x, x') W ( x; ξ ) (4.38) where W ( x; ξ ) is a funcion wih a sharp peak around he value ξ0, falling rapidly o zero elsewhere. This enables us o expand he righ-hand side of eqn.(4.37) as a Taylor series around he value ξ0. Reaining erms up o he second order, we obain f ( x, ) d x x { μ ( x) f ( x, ) } + { μ ( x) f ( x, )} (4.39) where μ ( x) < δx > ξw ( x; ξ ) dξ δ < v > δ x (4.40) and 0
< ( δx) > δ μ ( x) ξ W ( x; ξ ) dξ (4.4) δ Equaion (4.39) is known as he Fokker-Plank equaion, which occupies a classic place in he field of Brownian moion and flucuaions. We now consider a specific sysem of Brownian paricles (of negligible mass), each paricle being aced upon by linear (elasic) resoring force, F x -λx, and having a mobiliy B in he surrounding medium. The mean viscous force -<v x >/B, mus be balanced by he linear resoring force; consequenly < v x B > + F x 0 (4.4) and, hence Nex, in view of eqn. (4.30) we have v μ ( x) λ x B x (4.43)
( δx) δ μ ( x) BkT (4.44) Subsiuing (4.43) and 4.44) ino Fokker-Planck equaion (4.39), we obain f d x f x λ ( xf ) + BkT (4.45) Le us apply his equaion o an ensemble of Brownian paricles, iniially concenraed a he poin xx 0. In his conex, we noe ha, in he absence of he resoring force (λ0), eqn.(4.45) reduces o he one-dimensional diffusion equaion
f D x f ; ( D Bk) (4.46) which confirms o our earlier resuls (4.6) and (4.3). The presen derivaion shows ha he process of diffusion is neiher more nor less hen a random walk a he molecular level. The flucuaion-dissipaion heorem. In he Langevin s heory of Brownian moion we obained a resul of considerable imporance, namely B M M 6kT C M 6kT K A ( s) ds K F ( s) ds 6kT τ (4.47) Here, K A (s) and K F (s) are respecively, he auocorrelaion funcions of he flucuaing acceleraion A() of he Brownian paricle and he flucuaing force F() experienced by i: K A ( s) A(0) A( s) < F(0) F( s) > K ( s) F (4.48) M M 3
Equaion (4.47) esablishes a fundamenal relaionship beween (a) he coefficien /B of he averaged-ou par of he oal force F () experienced by he Brownian paricle due o he impacs of he fluid molecules and (b) he saisical characer of he flucuaing par F() of ha force (see Langevin s equaion (4.)). In oher words, i connecs he coefficien of he viscosiy of he fluid, which represens he dissipae forces operaing in he sysem, wih he emporal characer of he molecular flucuaions; he conen of he equaion (4.48) is herefore referred o as a flucuaion-dissipaion heorem. The mos sriking feaure of his heorem is ha i relaes in a mos fundamenal manner, he flucuaions of a physical quaniy peraining o he Equilibrium Sae of he given sysem o a dissipaive process which in pracice, is realized only when he sysem is subjec o an exernal force ha drives i away from equilibrium. Consequenly, his heorem enables us o deermine he noequilibrium properies of a given saisical sysem are in one of is equilibrium saes! Thus he saisical mechanics of irreversible processes is, in sense, reduced o he saisical mechanics of equilibrium saes. 4
A his sage we recall ha in relaionship beween he coefficien of diffusion D and he mobiliy B viz. DBkT. Combining his relaionship wih (4.47) we obain D K F ( s) ds 6( kt ) (4.49) Now he diffusion coefficien can be direcly relaed o he auocorrelaion funcion K v (s) of he flucuaing variable v(). For his one sars wih he observaion ha by definiion r( ) v( u) du 0 (4.50) whence i follows ha < r ) > < v( u) v 00 ( ( u ) > du du (4.5) Changing he variables as 5
S ( u + u ) and s u u ) (4.5) ( and replacing he elemen (du du ) by he corresponding (dsds) we also will change he limis of inegraion and afer some simplificaions we can wrie < r ( ) > / ( S ) ds Kv ( s) ds + ds 0 S S / K v ( S ) ( s) ds (4.53) In he case of a saionary ensemble he funcion K v (s)would be K v kt s τ ( s) 3 e (4.54) M 6
I will be noed ha he ime scale for he correlaion funcion K v (s) is provided by he relaxaion ime τ of he Brownian moion which is many orders of magniude larger han he characerisic ime τ * ha provides he ime scale for correlaion funcions K A (s) and K F (s). If >>τ < r > 6D (4.55) in he same limi eqn.(4.53) reduces o < r ( ) > ds K ( s) ds K ( s) ds 0 v v (4.56) Comparing he wo resuls, we obain he desired relaionship, viz. D Kv ( s) ds 6 (4.57) 7
We can also noe from (4.49) and (4.57) K ( 6 ) ( s) ds K ( s) ds kt v F (4.58) I is obvious ha he equaions, which conain a flucuaion-dissipaion heorem, can be adaped o any siuaion ha involves a dissipaive mechanism. 8