Correlation in the velocity of a Brownian particle induced by frictional anisotropy and magnetic field
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] J u l Correlation in the velo ity of a Brownianparti le indu ed by fri tional anisotropy andmagneti (cid:28)eldN. Voropajeva and T. Örd ∗ Institute of Theoreti al Physi s, University of Tartu,4 Tähe Str., 51010 Tartu, EstoniaAbstra tWe study the motion of harged Brownian parti les in an external magneti (cid:28)eld. Itis found that a orrelation appears between the omponents of parti le velo ity in the ase of anisotropi fri tion, approa hing asymptoti ally zero in the stationary limit.If magneti (cid:28)eld is smaller ompared to the riti al value, determined by fri tionalanisotropy, the relaxation of the orrelation is non-os illating in time. However, in alarger magneti (cid:28)eld this relaxation be omes os illating. The phenomenon is relatedto the statisti al dependen e of the omponents of transformed random for e ausedby the simultaneous in(cid:29)uen e of magneti (cid:28)eld and anisotropi dissipation.Key words: Charged parti le Brownian motion, CorrelationsPACS: 05.40-a, 05.10.Gg, 02.50.-r1 Introdu tionThe problem of Brownian motion in an external magneti (cid:28)eld was (cid:28)rst in-vestigated in Refs. [1,2,3℄ in onne tion with the di(cid:27)usive pro esses in plasma.About forty years later ertain developments [4,5,6,7,8℄ appeared in this areaagain. In parti ular, anisotropi di(cid:27)usion a ross an external magneti (cid:28)eld was onsidered in Refs. [6,7℄.In the present letter we demonstrate the appearan e of the orrelations in thevelo ity of harged Brownian parti les aused by magneti (cid:28)eld. The e(cid:27)e t ∗ orresponding authorEmail address: teet.ordut.ee (T. Örd).Preprint submitted to Elsevier 30 O tober 2018s possible only in environment hara terized by anisotropi fri tion. In whatfollows we will use a s heme, where the deterministi parts of the sto hasti equations of motion are transformed into independent equations. It simpli(cid:28)essubstantially the derivation of Fokker-Plan k equation in velo ity spa e as wellas makes more easy the understanding of the relevant physi al ba kground.The approa h is di(cid:27)erent from the method of Ref. [8℄, where rotating time-dependent basis was used to transform the Langevin equation in an externalmagneti (cid:28)eld.2 Probability distribution in velo ity spa eThe equations of motion of a Brownian parti le in an external magneti (cid:28)eld −→ B = (0 , , B z ) reads dv i ( t ) dt = − X j =1 λ ij v j + ξ i ( t ) , i = 1 , , x, y, z , (1)where λ ij = β x − ω z ω z β y
00 0 β z , (2) ω z = eB z /mc is the y lotron frequen y and β x,y,z are the fri tion oe(cid:30) ientsfor a parti le moving in the orresponding dire tion. Statisti al properties ofthe random for e are given by the onditions h ξ i ( t ) i = 0 , h ξ i ( t ) ξ j ( t ′ ) i = a i δ ij δ ( t − t ′ ) , (3)where a i = 2 k B T β i /m are the omponents of the intensity of the Langevinsour e.The system of equations (1) an be transformed into new equations of motion du i ( t ) dt = − Λ i u i ( t ) + ζ i ( t ) , (4)where Λ , = 12 (cid:16) β x + β y ± i Ω (cid:17) , Λ = β z , (5) Ω = q ω z − ( β x − β y ) , (6) ζ i ( t ) = X j =1 α ij ξ j ( t ) , (7)2y introdu ing, in general, omplex velo ities u i = X j =1 α ij v j . (8)Here the matrix of the velo ity transformation (8) an be hosen as α ij = q | b , | − b , q | b , | − b , q | b , | q | b , |
00 0 1 , (9)where b , = − Λ , − β x ω z . (10)In Eq. (9) one has to take b if β x > β y , and b if β x < β y . Thereby it isguaranteed that the limit ω z → leads to the unity transformation, α ij = δ ij . In the ase of isotropi fri tion β x,y,z = β the matrix (9) redu es to thefollowing unitary matrix α ij = √ ± i | ω z |√ ω z ± i | ω z |√ ω z √ , (11)being independent of magneti (cid:28)eld strength and fri tion oe(cid:30) ient β . Thelatter transform is in ertain sense lose to the approa h used in Ref. [9℄ in theexamination of the deterministi motion of a harged parti le in a magneti (cid:28)eld. The hoi e of sign in Eq. (11) is arbitrary.On the basis of the Langevin equations (4) one obtains the Fokker-Plan kequation dW ( ~u, t | ~u ) dt = X i =1 ∂∂u i [Λ i u i W ( ~u, t | ~u )] + X i,j =1 A ij ∂ W ( ~u, t | ~u ) ∂u i ∂u j (12)with the initial ondition W ( ~u, | ~u ) = δ ( ~u − ~u ) . In Eq. (12) A ij = X k =1 α ik α jk a k . (13)3he solution of Eq. (12) in terms of the velo ity −→ v is W ( ~v, t | ~v ) = s π ) det [ h ij ] exp − X i,j =1 (cid:16) h − (cid:17) ij ( v i − h v i i )( v j − h v j i ) , (14)where h ij = K K K − K − K K K − K − K K K − K K K K − K
00 0 1 K , (15)and the averaged values of the omponents of velo ity are given by h v x i = K K − K K K K − K ) , h v y i = K K − K K K K − K ) , h v z i = − K K . (16)The following notations have been used in Eqs. (15) and (16): K ij = 1 g " ( α ) a + ( α ) a α i α j ϕ + ( α ) a + ( α ) a α i α j ϕ − α α a + α α a Λ + Λ ( α i α j + α j α i ) ϕ ϕ ϕ ; i, j = 1 , ,K = 2Λ a ϕ − , (17) K i = − g ((cid:20) α f + α f (cid:21)" ( α ) a + ( α ) a α i ϕ − α α a + α α a Λ + Λ ϕ ϕ ϕ α i + (cid:20) α f + α f (cid:21)" ( α ) a + ( α ) a α i ϕ − α α a + α α a Λ + Λ ϕ ϕ ϕ α i ; i = 1 , ,K = − a f ϕ − , (18)4 = ( α a + α a ) ( α a + α a )4Λ Λ − ( α α a + α α a ) (Λ + Λ ) ϕ ϕ ϕ , (19) ϕ ij = 1 − e − (Λ i +Λ j ) t , f ij = v j e − Λ i t . (20)3 Correlation between the omponents of velo ityNow we on entrate attention to the orrelation between the omponents ofthe velo ity of a parti le, perpendi ular to a magneti (cid:28)eld, v x and v y . The orrelation fun tion k ( t ) = h v x ( t ) v y ( t ) i − h v x ( t ) ih v y ( t ) i (21)is determined by Eqs. (14) and (15), whi h yield k ( t ) = h . As a result wehave on the basis of Eqs. (5), (9), (15) and (17) k ( t ) = − k B Tm ω z ( β x − β y ) sin (cid:16) Ω t (cid:17) Ω e − ( β x + β y ) t . (22)Consequently, the orrelation fun tion (21) is nonzero only if ω z = 0 and β x = β y , i.e. in the presen e of external magneti (cid:28)eld and fri tional anisotropy. The Fig. 1. Dependen e of the orrelation fun tion k ( t ) on time at k B T /m = 1 and β x = 10 , β y = 2 for various values of y lotron frequen y ω z ≤ ω crz = 4 . Curve 1: ω z = 1 , urve 2: ω z = 2 , urve 3: ω z = 3 , urve 4: ω z = ω crz .5 Fig. 2. Dependen e of the orrelation fun tion k ( t ) on time at k B T /m = 1 and β x = 10 , β y = 2 for various values of y lotron frequen y ω z ≥ ω crz = 4 . Curve 1: ω z = ω crz , urve 2: ω z = 16 , urve 3: ω z = 32 . orrelation approa hes asymptoti ally zero in the stationary limit ( t → ∞ ). Inthis relaxation pro ess one an distinguish two regimes. The time dependen eof k ( t ) is non-os illating (see Fig. 1) if | ω z | < ω crz . The os illating behavior(see Fig. 2) appears if | ω z | > ω crz . Here ω crz = | β x − β y | (23)is the riti al value of the y lotron frequen y separating the regions where Ω in Eq. (22) is imaginary or real quantity orrespondingly.4 Dis ussionIn on lusion, we have found in the short time-s ale the orrelation betweenthe omponents of the velo ity of a harged Brownian parti le aused by ex-ternal magneti (cid:28)eld and fri tional anisotropy. The e(cid:27)e t arises due to thestatisti al dependen e of the omponents of transformed random for e ζ , ( t ) ,determined by Eq. (7). On the basis of Eqs. (7) and (3) we have h ζ ( t ) ζ ∗ ( t ′ ) i = X k α k α ∗ k a k δ ( t − t ′ ) . (24)Whereas in the general ase of anisotropi fri tion and non-zero magneti (cid:28)eld the sum P k α k α ∗ k a k is not equal to zero, it is impossible to transform Note also, that the ne essary ondition for the appearan e of the e(cid:27)e t is non-zerotemperature (see Eq. (22)), whi h unambiguously indi ates to the entirely sto hasti nature of the phenomenon. 6he system (1) into entirely independent equations. Although the deterministi part of the equations of motion (1) an be de oupled, the hannel of orrelationappears in Eqs. (4) for the omponents of sto hasti for e in presen e of anexternal magneti (cid:28)eld and anisotropi dissipation. As a result the omponentsof the velo ity of a Brownian parti le, perpendi ular to the magneti (cid:28)eld, turnout to be orrelated.However, in the ase of isotropi fri tion the expression h ζ i ( t ) ζ ∗ j ( t ′ ) i = 2 k B T βm δ ij δ ( t − t ′ ) (25)is valid for the arbitrary omponents of random for e ζ i ( t ) be ause the on-dition of unitary transformation, P k α ik α ∗ jk = δ ij , holds. In this situation thesystem of equations (1) an be entirely de oupled and the orrelation betweenthe omponents of velo ity is absent. We obtain the same result for anisotropi fri tion if magneti (cid:28)eld equals to zero, due to α ij = δ ijij