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Properties of noise correlation functions of langevin-like equations
Physica 137A (1986) 81-95 North-Holland, Amsterdam
P R O P E R T I E S OF N O I S E C O R R E L A T I O N F U N C T I O N S OF L A N G E V I N - L I K E E Q U A T I O N S Peter SCHRAMM
lnstitut fiir Theoretische Physik, Universitiit Stuttgart, D-7000 Stuttgart-80, Fed. Rep. Germany and
Irwin OPPENHEIM Department of Chemistry, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Received 3 February 1986
The stochastic properties of the noise in Langevin type equations are studied. It is demonstrated that the noise is non-Gaussian and that the three point noise correlation function has an
exponential decay on the slow time scale. Explicit calculations for systems containing one slow linear variable are carried out using mode-coupling techniques. The results are easily extended to four and higher order correlation functions and to systems in which there is more than one slow linear variable. In particular, the noise in the hydrodynamic Langevin equations has the properties described above.
I. Introduction A widely used assumption in nonequilibrium statistical mechanics has b e e n to consider the noise variables occurring in Langevin equations to be Gaussian. This idea originated in Brownian motion 1) but has been used for m a n y other problems including hydrodynamics2). O n e of the most convenient features of this assumption is the equivalence of a Langevin equation with Gaussian white noise and a linear F o k k e r - P l a n c k equation. H o w e v e r , the physical reason why the higher cumulants of the noise should vanish is usually unclear. The assumption of gaussian noise can be justified as long as one is interested only in the first two m o m e n t s of the dynamical variables, since it has been shown that every stochastic process can be replaced by a Gaussian process with the same first and second m o m e n t ( K h i n c h i n - C r a m e r theorem3)). But for the higher m o m e n t s information about the corresponding noise correlations is necessary. Further G r a b e r t and Weidlich showed that non-Gaussian fluctuations in a stationary state lead to a renormalization of the t h e r m o d y n a m i c 0378-4371/86/$03.50 (~) Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
82
P. S C H R A M M A N D I. O P P E N H E I M
forces4). Some authors have avoided the Gaussian assumption and studied the effect of poissonian noiseS). It should be noted in this context that the validity of the fluctuationdissipation theorem of the second kind does not depend on the Gaussian assumption6). For its derivation it is necessary only that the noise is not correlated with the slow variables at earlier times. This means the noise has to be really random, a property which can hold only if the set of slow variables is complete. Otherwise the noise would contain mode-coupling effects and thus have slow parts. In this paper, we demonstrate that the noise in Langevin-type equations derived from molecular theory, is generally non-Gaussian. We study the three point noise correlation function in detail using mode-coupling techniques and show that it is not only non-Gaussian but also has long time decays. The calculations are readily extended to correlation functions involving any number of noise variables. The fundamental equation that is used was derived in ref. 7. We write the expression for an arbitrary dynamical variable B(t):
B(t) = ( B ( t ) Q * ) * ( Q ( t ) Q * ) - 1 ,
Q(t) + IB(t) ,
(1.1)
where IB(t ) is defined by this equation. The quantities Q are all the multilinear modes of all of the slow variables in the system; the superscript * denotes the complex conjugate; the brackets denote an equilibrium average; and the * on the line denotes a sum over all modes. The quantity IB(t ) is a noise term which has the property that
(IB(t)Q* ~ = 0
(1.2)
and is therefore assumed to be quickly decaying in the sense that (1.3)
( IB(t)IB(t') ) ~ ~$(t -- t') . Finally,
MB(t ) -~ ( B(t)Q* ) * ( Q ( t ) Q * )-1 = (BQ*) * (QQ*)-1
__i ([B(O')Q~) * (Q(o-)Q*)-1 o
For t > ~'m where ~'m is a molecular relaxation time,
d(r.
(1.4)
NOISE CORRELATION FUNCTIONS OF LANGEVIN-LIKE EQUATIONS
83 (1.5)
M s ( t ) = M s ( ~ ) ==-M s ,
since (IB(cr)Q*) is assumed to decay to 0 on the time scale r m [see eq. (1.3)]. Because of this we can approximate eq. (1.4) by
(1.6) 0
for our present purposes. Thus eq. (1.1) can be written as (1.7)
B(t) = M s * Q(t) + IB(t )
with M s given by eq. (1.6). In section 2, we study the three point noise correlation functions for a system containing one slow linear variable. We demonstrate that the use of our fundamental equations (1.7) and (1.6) yields a non-gaussian form for the noise correlation functions with slow time decays. Finally, we show that use of these equations yields the correct form for the time asymptotic form of the correlation function
(A(t)A(t)A(t))
= (A(O)A(O)A(O))
,
(1.8)
where A is the slow linear variable. In section 3, we extend the considerations of section 2 to the Langevin-like hydrodynamic equations in which there are five slow linear variables: the n u m b e r density, the energy density and the three components of the momentum density. In section 4, some conclusions and a summary of our results is presented. 2. Noise correlations for a diffusion problem
We consider a system in which there is one slow linear variable A ( r , X ) , N
A ( r , X ) = ~, A j S ( r - r j ) ,
(2.1)
j=l
which is the density of a conserved variable. H e r e r is a position in space, X = r N, P n is the phase point of the N particle system, and Aj is the value of the quantity A for the jth particle in the system. The variable A is a scalar such as the temperature field in a heat transport problem or the concentration of a solute in a solvent.
P. SCHRAMM AND I. OPPENHEIM
84
It is convenient to consider the Fourier transform variable
A k ( X ) =- Ak =- ( eik'r A(r, X ) dr
(2.2)
J
which obeys the equation
Ak(t) = ik. Jk(t),
(2.3)
where Ak(t ) =--Ak(X(t)) and Jk(t) is the vector current for the variable Ak(t); the quantity A =- Ak= 0. We introduce the hierarchy of multi-linear modes, Q, [see ref. 7] which are defined by: Q(0) = 1 , Q ~ 1 ) = / ] k =- Ak - ( Z k ) , Q(2) k-k',k'
= A k - k' A k' _ ( A k k , A k , )
(2.4) -
(Ak_k,Ak A ( AkA
'
etc. where the superscript * denotes the complex conjugate. The Q's are defined such that
(Q~,)Q(,')*) = Au ' (Q(t)Qq)*).
(2.5)
Here the ( ) denote an equilibrium average of the system which is assumed to be uniform and isotropic and the superscript (l) on the Q's denotes the multi-linear order of the mode. The Langevin equations for Q(t) are 7) to order k2:
Qq)(t) = Mll,(t )
*
Qq')(t) + lq)(t) ,
(2.6)
where Mu,(t ) = - / (I(')(cr)l ('')*) dcr * [K w ) ] - l ,
(2.7)
0
Kq,)= (Q(,,~Q(t,~*) ,
(2.8)
the fluctuating force l(r)(t) has the property:
(l~")(t)Q ~t)*) = 0
(2.9)
NOISE CORRELATION FUNCTIONS OF LANGEVIN-LIKE EQUATIONS
85
and is assumed to be a fast variable in the sense that
( l ( t ) ( t ) l ~r)*) = 6(t) .
(2.10)
The * in eqs. (2.6) and (2.7) denotes a sum over l' and over all intermediate wave vectors. Finally, note that for times greater than a molecular relaxation time
M u, (t) = M u, (oo) =_ Mu, "
(2.11)
Eq. (2.6) can be solved formally to obtain
Q~o(t)
Mt Q~r) iaM<'-~) ~,l~r)(r) d r
:
~11' ~
~-
~11'
(2.12)
--
0
where terms of G(k 2) have been discarded. We now use eq. (2.12) to write: lim (Akl(t)Ak~(t)Ak3(t) ) = ( A ~ A k z A k 3 )
t~
t
0
l
0
t
0
kl
k2
~"113 k3
~ \~
\~1. I~
\~21 ~
\°31!
(2.13) where k I + k 2 + k 3 = O.
(2.14)
Eq. (2.13) is valid since e~t---~ 0 as t - - ~ for all k > 0 . Now, since the left-hand side of eq. (2.13) is nonzero, the right-hand side is nonzero, and I cannot be a Gaussian variable of zero mean. F u r t h e r m o r e , since (I('q)I(z2)I("r3)) is at least of order k 4 which follows from isotropy, it cannot be of the f o r m 6(~ 1 - z2)60" 1 - %) which will yield a term of order k 2 after integration. The left-hand side is of order k °. Thus the three point correlation function of I must have a slow part which on integration yields a term of order k °. The L H S of eq. (2.13) is of order N. We shall consider only those terms on the R H S of eq. (2.13) which are also of order N. In order to do this, we use the facts that M~; is at most of order N 1 - l and t h a t (I(tl)I(t2)I (t3)) is at most of order (l~ + 12 + 1 3 ) / 2 if l 1 + l 2 + l 3 is even and of order (l I + l 2 + 13 - 1 ) / 2 if l I + l 2 + 13 is odd. T h e only terms on the R H S which can be of order N are
86
P. S C H R A M M
A N D I. O P P E N H E I M
those for which l 1 = 12 = 13 = 1 and those for which one of the l's is 2 and the other 2 / ' s are 1. We write the matrix M with elements M u, as
M= F + 0 ,
(2.15)
where F is diagonal both in the mode order, l, and in wave vectors and O is off diagonal either in l or in wave vectors or both. Thus the elements of F are of order N O and the elements of O are at most of order N -1. Thus, e~' -~ e r(k')' ,
(2.16a)
kl t
e~2t = f er(ko(t-~')M1 2 kl
0
k!
er22(k1-k', k,)~
do" ,
(2.16b)
=--- k2D
(2.17a)
k 1- k ' k'
where
F(k) =- el,(k ) =
dO -k 2 .o
and r 2(k, - k', k ' ) :
(2.17b)
-[(k, - k'y +
In order to obtain eq. (2.17a), we have used eq. (2.3), the isotropy of the equilibrium system, and the fact that l(kl)(t)= ik" Jk(t) to Navier-Stokes order (i.e. to order k2). To this order the coefficient of k 2 in (2.17a) is independent of k. To obtain eq. (2.17b) we have used the fact that ~22(kl - k', k') = Ul(kl
- k ' ) --[- ~ l l ( k ' )
(2.18)
--[- ~7(N-1).
Using these facts, we can write eq. (2.13) as:
0
e -k2D(l-'r3)
0
×
+ 3 TL f tirldo-e-kZ~O(t-rl-'OMld~'a 0
0
2
e-tk~+k~lD~i d72 f d~'3
kl; - k 2 ' - k 3
× e -k~°(`-~2) e-k~n(t-'3)(I(_2)k2,_k3(Zl)Ik:(Z2)Ik~(r3)] } ,
0
0
(2.19)
NOISE C O R R E L A T I O N FUNCTIONS OF LANGEVIN-LIKE EQUATIONS
87
where ffkl(7.1) =-Ik(1)1 (7.1) and there are 3 terms like (TL) the last term in which 1(7.1) is replaced by i(2), 1(7.2) is replaced by I(2) and 1(7.3) is replaced by 1 (2). The wave vectors in the last term have been chosen to yield a term of order N. The quantity M 1 2
kl;kl-k',k' f:
is equal to: (2)*
(Ikl(O)Ikl_k,,k,) dO
(Akl_k,Akl_k,)(AklAkl)*
M12k,;k,-k',k'
•
(2.20)
We shall now examine the correlation function wish to retain terms of order N and k 4. We write /kl(T1)Ik2(7.2) = [A k1(7.1) --
Mll (7.1) * Q(/)(7.1 ) ] [ A kI
(Ik~('r1)Ikz(7.2)Ik3(Z3))
k1(7.2) --
and
Mll (7"2) * Q(l')(7.2)1 k2
(2.21)
which follows from eq. (2.6). We define B(7.z) as B(T2) = [Akl(T1 -- 7.2 + 7"2) -Mlt(7.1)*Q(t)(T1 kl
- T2 + T2)]
x [Ak2(7.z) -- Mw(7.2) • Q(r)(7.2) ] ,
(2.22)
k2
Using the general formula given in eq. (1.7), we can now write:
( Ik,(7.1)lkz(7.2)Ik30"3) ) = MBt(7.2)* ( Q(O(rZ)Ik3(r3) ) + ( IB(r2)lk3(r3) ) , (2.23) where MB/(T2)
=
([Akl('rl
-
'/'2) -
Mll ( z l ) * Qq)(rl - 7.2)] kl
× [~lk z _
M l r (T2) , k2
- f (1B(o')l "(j))
Q(r)]Q(D*),
dcr* K (j)-I
K(])-I
(2.24)
0
to Navier-Stokes order and the term (IB(7.Z)Ik3(%)) is assumed to decay quickly and is proportional to k48(7.2 - 7.3)8(rl - 7.2)N. Use of eq. (2.12) in eq. (2.23) results in
88
P. S C H R A M M
A N D I. O P P E N H E I M
r2 AM(r2-a) * ( I(l')(0-)Ik3(7-3) ) d o ( Ik~(7-1 ) Ik2(7-2 ) Ik3('r3 ) ) = MBI(7-2) * f -u' 0
+ (IB(%llk,(r3)} r2
= Mm (7-2)f kl+k2
e-D(kl +kz)2(~z-~) ( Ikl +k2(O')Ik3('r3) ) do" + ( In(r2)Ik3(r3) ) ,
0
(2.25) where in the second equality we have retained only those terms which are of order N. We use the fact that (Ikl+k~(O')Ik~(7-3)) = + 2 k 2 D ( Ak~A T,)6( ~ - 7-3)
(2.26)
to obtain
( I~, (7-1)I< (7-0I~,(7-~) > =
+ M B1
(7-z) e
- Dk2(r2-r3)
2
^
^,
2 k 3 D ( A k 3 A k3)O(7-2 - 7-3)
kl+k 2
+ (Is
(%)Ik3(%)> '
(2.27)
kl+k 2
where 0(7-2 -7-3) is a step function equal to 1 for 7-2 > 7-3 and equal to 0 for 7-2 < 7-3. Note the slow time decay in the first term on the R H S of eq. (2.27). Finally, for 7-1 ~> 7-2 > 7-3, eq. (2.27) becomes
(I<(7-1)/<(7-2)tk,(7-~) > = = + 2 k 2 D e-Dk230"2-'~3)[(Ik1(.q -- 7-2)Ik2Ak3)+
C(Nk4t$(7-1
- 7"2))]
(2.28)
+ {7(Nk46(% - r3)6(7-~ - 7-2))
(2.29)
~-- + 2k~D e Dk]''2--~3) ( Ik,(7-1 - 7-z)Ik/k3)
which is the leading order (in k and N ) contribution to the integral in eq. (2.19). The term
( J~,,ox(~)Jk2xAk3) d~,
(/k1(7-1 - 7-0/k2A k3> %2- 2kl. k2~(7-1 - 7-2) 0
(2.30)
NOISE C O R R E L A T I O N F U N C T I O N S O F L A N G E V I N - L I K E E Q U A T I O N S
89
where J m is the x component of the dissipative current, i.e., I k l = ikl.JDk ~. The k's inside the integral can be set equal to zero. The form of (Ik~(rl)Ik~(r2)Ig3(r3)) depends on the time ordering of %, ~'2, r3. For ~'2, r3 > "l'l we obtain
( Ik,(rl)Ik2(rZ)Ik3(r3) ) = + 2k~D
e -°k~l'2-''l
( Ak lk2(r2 -- r3)Ik3 )
(2.31a)
and for % , z 3 > r 2
= +2k~D e-°k21"l-'~21(Ik,('q -- z3)AkJk3 )
.
(2.31b)
When eqs. (2.29)-(2.31) are substituted into eq. (2.19), the result is
+ f (Jm(o")J~A) do"/D,
(2.32)
0
where we have used the fact that k 1 + k 2 + k 3 = 0. We next turn our attention to the terms in eq. (2.19) which involve i(2). The terms in /(2) which contribute to a n N 2 contribution from the correlation function of the 3 I's are
I_k2('Cl)A_k3(~'1) + I_k3(Z,)A_~:2(rl) . When this is substituted into the correlation function we obtain:
( l(-2)k2,_k3(~'l)Ikz(7"2)Ik3('r3)) = ( I_kz(T1)Ik2(T2) ) ( m _k3(71)Ik3(7"3)) + (I_k3(Ta)Ik3(Z3)) (A_kz(~'l)Ik2(~'2)), (2.33) Again we write:
( Ikz(~'z)l_kz(rl) ) = 2k~D ( Ak A ~) 6(r~ - ~'z)
(2.34)
and r1
= f e--Dk]O'l--°')(lk3(ff)Ik3(T3) ) do" 0
= e-°k~'-~3~21,~o ( AkA L) 0(~', - ~'3).
(2.35)
90
P. SCHRAMM AND I. OPPENHEIM
Thus: ( I~2.-k3(7-1 )Ik2(7-Z)Ik3(Z3)) =4D2(AA,
2 2 ) 2k2k3[c~('rl - 7-2)
e-Ok](rl-'3)O(7-1 7"3)
+ ~ ( r l -- "r3) e-Dk2(~l-~2)0(rl -- 7-2)],
(2.36)
where we have set k = 0 in the correlation functions of A. Again, we note a long time decay of the fluctuating force correlation function. Substitution of eq. (2.36) and its analogs into eq. (2.19) yields
-
fo doD
+
(2.37)
In order to obtain eq. (2.37) we have used the relation
M12 kl 'k2'k3
=-f0 (Ik,(o-)Z(k22),k3)do-(Ak?Z2)-'(mk3flZ3)-1 o ^
^~
( Ak,A k,)
do-(Ak2Ak 2) (AkAk3) (2.38)
which follows from the definition o f Q(2), eq. (2.4). Eq. (2.37) together with eq. (2.32) yields for eq. (2.19)
(Ak,Ak2Ak~) = ( Ak,Akfik3) "
(2.39)
3. Noise correlations for hydrodynamics
It is easy to extend the considerations of section 2 to the case of several slow linear variables. In hydrodynamics, there are five slow linear densities: the number density, the energy density, and the three components of the momentum density. These five densities will be denoted by A~, where a specifies the particular density and its wave vector.
NOISE CORRELATION FUNCTIONS OF LANGEVIN-LIKE EQUATIONS
91
The noise correlation functions of interest are those in eq. (2.19). We shall go through the argument for the linear noise correlation function and see that the procedure is essentially identical to that in section 2. We consider the correlation function (lal(T1)la2(72)/a3(73)) where the (3/i noise has wave vector ki, and again k 1 + k 2 + k 3 = 0. We define B~,~2(72) by
B.,.:(72) =
/~1(71 -- 77-2 + 7 2 ) L 2 ( 7 2 )
= [A,~,(% -- 72 + 72) -- M , ~ d ( % ) * Q ( t ) ( % × [A,~2(72) - M . 2 r ( r 2 )
- 72 + 7211
(3.1)
* Q(r)(72) ] .
We use the arguments following eq. (2.22) to arrive at the analog of eq. (2.25) for z I/> r 2 > 73: 7-2
( ia1(7111,~2(7211,~3(731) =
MB.(72 ) f c,, ~-r(.2-~) (I, (0-11.3(73)) 0
+ (I,(721/~3(73)) ,
(3.2/
where the wave vector associated with T and 6 is k3, and repeated G r e e k subscripts are s u m m e d over. The quantity MBr(~-2) is given by
MRs,(72) =
( [ A ~ , ( r 1 - r2) - M,~,I(T1) * Q(t)(r 1 - r2) ]
× [ A 2 - M~2r (721 * Q"')]A,,).
(/~i*) -1 Y'3'
72
- f (Is(or)I,) do'. (/~{*)~,~ ,
(3.3)
0
where again the wave vector associated with y ' is k 3. The matrix F is the hydrodynamic matrix
r = [(A,i*) - f (l(~)t) d~]. (Li*)-m
(3.4)
0
to order k 2 and has convective (Euler) parts as well as dissipative ( N a v i e r Stokes) parts. The Euler terms are proportional to ik and the N a v i e r - S t o k e s parts to - k 2.
92
P. SCHRAMM AND I. OPPENHEIM T h e term of interest, with long time correlations, is ( I(r,
) 1,~2(T2)1~,3(T3)) T2
= (I(r.
^ ,
-
T2)/,~"Av, _ > (,4,4),,,
1
+1"(r2--7t) /
e,~
~
(I~(o")I~3(r3)) do".
0
(3.5) We use the fact that r2
f (l~(o-)l~3('r3) ) do"= -2r o.
(3.6)
,
0
where
F°
is the dissipative part of F, to write
(/~,(r, )Io2(T2)I~3(%)) : - 2 { l,~l(Y,
-- ~-2)I~2A~ , >. < AA* > ~-,~
+l'(z 2 " r ~ ) r D / ~ *
• e~
,~
160k-,~ko~la3 >
(3.7)
for q'l > '7"2 > '7"3" Eq. (3.7) is of o r d e r N. A g a i n , note that 7 ' , Y, (5 and 0 have the wave vector k 3. T h e only difference b e t w e e n eq. (3.7) and the c o r r e s p o n d i n g e q u a t i o n in section 2, eq. (2.29), is the fact that there are several slow m o d e s which have different vectorial and time reversal symmetries. In particular, if any of the 0/'s = N, the noise 14 = 0 and eq. (3.7) yields 0. If we d e n o t e the time reversal signature of A s by G , then g u = E"E = 1, ep = - 1 and their currents have the opposite signature. If e~IG2G3 = - 1 , the left-hand sides of eqs. (3.5) and (3.7) can be of o r d e r k 3 whereas if e~ ea2G3 = 1, the left-hand sides must be at least of o r d e r k 4. T h e right-hand sides of eqs. (3.5) and (3.7) are at least of o r d e r k 4 and can only be the d o m i n a n t c o m b i n a t i o n to the 3 noise correlation functions w h e n the p r o d u c t of e's is 1. In particular, if 0/, = 0/2 = O13 = E, 7 ' and y must be N or E and 6 must be E. If a I = 0/2 ~- P and a 3 = E, then again y ' and y must be N or E and 6 must be E. If 0/, = 0/3 = P and 0/2 = E, then y ' and y must be P and 6 must be P. T h e cases w h e n the p r o d u c t of e's is - 1 are 0/1 = 0/2 = 0/3 = P and those in which o n e of the 0/'s is P and the o t h e r two are E. If 0/1 = 0/2 = °z3 = P, 7 and y ' must be N or E and 6 must be P. If 0/, = P and o/2 ~--- 0/3 = E , y ' and y must be P and 6 must be E. If a, = 0/2 = E and 0/3 = P, Y' and y must be N or E and 6 must be P. Ft T h e a p p r o p r i a t e factors to the used for e ~ can be o b t a i n e d f r o m the matrix
NOISE CORRELATION FUNCTIONS OF LANGEVIN-LIKE EQUATIONS "g"(e~+'+e~-') + 2 - g, e ~hr m
~Ft =
1
t
mn
X"--~-h(e:'+'+e:'-'-2e~'r)
x. --(e~+'+e ~ ' - 2 e ~r)
-c-o( & + ' - e ~-')
m
m
x~h (e~'+'+e~'-')+2 X" eZ'r C°h (e . . . . e ~-')
mn
mn
m
mn
93
0
0
0
0
CoX.(e~÷' - e~-')
Co(e ~÷' - e ~-t)
C2(e v+' + e v-')
0
0
0
0
0
2C2o e - ~'k2'
0
0
0
0
0
2C~ e "~k2'
(3.8) in which the first r o w and c o l u m n are E, the second row and c o l u m n are N, the third r o w and c o l u m n are the longitudinal m o m e n t u m and the fourth and fifth rows and columns are for the two c o m p o n e n t s of the transverse m o m e n t u m . Here: y+_ = + ikC o -
-/-s k2
,
Yr = -Irk
2 ,
(3.9)
vt = r t / m n ,
w h e r e C O is the adiabatic s o u n d speed, ~7 is the shear viscosity, and P s (Fr) is the s o u n d (heat) m o d e a t t e n u a t i o n coefficient; Xe = ( Op / Oe),, , Xn = ( OP / On) e, P is the hydrostatic p r e s s u r e and h is the e n t h a l p y density. T h e correlation functions which contain I t2) are also easy to analyze. 1(2)
"r
( ,~,,~(,)l,,2(z2)I~,3(r3))
~-- (I,~,(z,)I,~2(z2)) ( .,?t
a~('/'l)Ia3(T3))
+ (I,~i(z,)I,~3(z3))(A~q(7"1)I~,2('r2))
(3.10)
which follows f r o m eq. (2.33). H e r e a I has the w a v e vector - k 2 , a~ the w a v e v e c t o r - k 3 , a 2 the w a v e v e c t o r k 2 and a 3 the w a v e v e c t o r k 3. Again, we use the relations [eq. (2.34)]
(
) = -2r
(A
o)
-
(3.11)
,
w h e r e 13 has the w a v e v e c t o r - k 2 , and [eq. (2.35)]
( A ~i(T1)I~3(r3) ) = --2 er('q-r3)/"D
7-3)
w h e r e Y and Y' h a v e the w a v e v e c t o r - k 3. Eq. (3.10) b e c o m e s
(3.12)
94
P. SCHRAMM AND I. OPPENHEIM
X [ ~ ("/'1 _~ ~(T1
'/'2)0(1"1
--
T3)0(T 1 - -
D
D
F('r t -"r3)
\ r , Dc ~ i vrl Df3'~ e F('r . l # ,I 1"2)] 1,
"r2)l
(3.13)
where /3' has the wave v e c t o r - k 2. The left-hand side of eq. (3.13) has the signature -e~le~ie,2e~3 and is of order k 4 when this is - 1 . This is the case, e.g., if a 1 = a ' 1 = a 2 = a 3 = P. In this case, y ' = / 3 = y = / 3 ' = P. For % = a~ = a 2 = O~3 = E , "y' a n d / 3 must be N or E a n d / 3 ' = y = E. The appropriate factors of e rt can be obtained from eq. (3.8). The only differences between the analysis for the hydrodynamic noise correlation functions and those in section 2 are the proliferation of indices and the existence of Euler terms in the hydrodynamic matrix which yield factors of s i n ( k C o t ) and c o s ( k C o t ) in some of the propagators.
4. Summary We have d e m o n s t r a t e d that the noise in Langevin-like equations is not Gaussian by studying three point noise correlation functions. We have assumed that the two point correlation functions are delta-function correlated, i.e. (I(tx)I(t2))
- -
~(tl -- t2)
(4.1)
but that the three (and higher) point correlation functions have slow decays. While we have considered only equilibrium correlation functions, the results can readily be extended to nonequilibrium correlation functions as well.
Acknowledgements One of us (P.S.) would like to thank the chemistry d e p a r t m e n t of M I T where this work was carried out for their w a r m hospitability. A portion of this work was supported by the National Science Foundation under grant CHE84-10682.
References 1) S. Chandrasekhar, Rev. Mod. Phys. 15 (1943) 1. G.E. Uhlenbeck and L.S. Ornstein, Phys. Rev. 36 (1930) 823. M.C. Wang and G.E. Uhlenbeck, Rev. Mod. Phys. 17 (1945) 323.
NOISE CORRELATION FUNCTIONS OF LANGEVIN-LIKE EQUATIONS 2) Reviews: N.G. van Kampen, Adv. Chem. Phys. 34 (1976) 245. R. Graham, in: Springer Tracts in Modern Physics, vol. 66 (Springer, Berlin, 1973). R. Kubo, Rep. Progr. Phys. 29 (1966) 255. And more recently e.g. P. Grigolini, J. Stat Phys. 27 (1982) 283. R. Graham and A. Schenzle, Phys. Rev. A 25 (1982) 1731, 26 (1982) 1676. W. van Saarloos, D. Bedeaux and P. Mazur, Physica I10A (1982) 247. 3) A. Khinchin, Math. Ann. 109 (1933) 604. 4) H. Grabert and W. Weidlich, Phys. Rev. A 21 (1980) 2147. 5) P. H~inggi, Z. Phys. B31 (1978) 407. A. Onuki, J. Stat. Phys. 19 (1978) 325. 6) H. Grabert, J. Stat. Phys. 19 (1978) 479. 7) J. Machta and I. Oppenheim, Physica l12A (1982) 361.
95
P R O P E R T I E S OF N O I S E C O R R E L A T I O N F U N C T I O N S OF L A N G E V I N - L I K E E Q U A T I O N S Peter SCHRAMM
lnstitut fiir Theoretische Physik, Universitiit Stuttgart, D-7000 Stuttgart-80, Fed. Rep. Germany and
Irwin OPPENHEIM Department of Chemistry, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Received 3 February 1986
The stochastic properties of the noise in Langevin type equations are studied. It is demonstrated that the noise is non-Gaussian and that the three point noise correlation function has an
exponential decay on the slow time scale. Explicit calculations for systems containing one slow linear variable are carried out using mode-coupling techniques. The results are easily extended to four and higher order correlation functions and to systems in which there is more than one slow linear variable. In particular, the noise in the hydrodynamic Langevin equations has the properties described above.
I. Introduction A widely used assumption in nonequilibrium statistical mechanics has b e e n to consider the noise variables occurring in Langevin equations to be Gaussian. This idea originated in Brownian motion 1) but has been used for m a n y other problems including hydrodynamics2). O n e of the most convenient features of this assumption is the equivalence of a Langevin equation with Gaussian white noise and a linear F o k k e r - P l a n c k equation. H o w e v e r , the physical reason why the higher cumulants of the noise should vanish is usually unclear. The assumption of gaussian noise can be justified as long as one is interested only in the first two m o m e n t s of the dynamical variables, since it has been shown that every stochastic process can be replaced by a Gaussian process with the same first and second m o m e n t ( K h i n c h i n - C r a m e r theorem3)). But for the higher m o m e n t s information about the corresponding noise correlations is necessary. Further G r a b e r t and Weidlich showed that non-Gaussian fluctuations in a stationary state lead to a renormalization of the t h e r m o d y n a m i c 0378-4371/86/$03.50 (~) Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
82
P. S C H R A M M A N D I. O P P E N H E I M
forces4). Some authors have avoided the Gaussian assumption and studied the effect of poissonian noiseS). It should be noted in this context that the validity of the fluctuationdissipation theorem of the second kind does not depend on the Gaussian assumption6). For its derivation it is necessary only that the noise is not correlated with the slow variables at earlier times. This means the noise has to be really random, a property which can hold only if the set of slow variables is complete. Otherwise the noise would contain mode-coupling effects and thus have slow parts. In this paper, we demonstrate that the noise in Langevin-type equations derived from molecular theory, is generally non-Gaussian. We study the three point noise correlation function in detail using mode-coupling techniques and show that it is not only non-Gaussian but also has long time decays. The calculations are readily extended to correlation functions involving any number of noise variables. The fundamental equation that is used was derived in ref. 7. We write the expression for an arbitrary dynamical variable B(t):
B(t) = ( B ( t ) Q * ) * ( Q ( t ) Q * ) - 1 ,
Q(t) + IB(t) ,
(1.1)
where IB(t ) is defined by this equation. The quantities Q are all the multilinear modes of all of the slow variables in the system; the superscript * denotes the complex conjugate; the brackets denote an equilibrium average; and the * on the line denotes a sum over all modes. The quantity IB(t ) is a noise term which has the property that
(IB(t)Q* ~ = 0
(1.2)
and is therefore assumed to be quickly decaying in the sense that (1.3)
( IB(t)IB(t') ) ~ ~$(t -- t') . Finally,
MB(t ) -~ ( B(t)Q* ) * ( Q ( t ) Q * )-1 = (BQ*) * (QQ*)-1
__i ([B(O')Q~) * (Q(o-)Q*)-1 o
For t > ~'m where ~'m is a molecular relaxation time,
d(r.
(1.4)
NOISE CORRELATION FUNCTIONS OF LANGEVIN-LIKE EQUATIONS
83 (1.5)
M s ( t ) = M s ( ~ ) ==-M s ,
since (IB(cr)Q*) is assumed to decay to 0 on the time scale r m [see eq. (1.3)]. Because of this we can approximate eq. (1.4) by
(1.6) 0
for our present purposes. Thus eq. (1.1) can be written as (1.7)
B(t) = M s * Q(t) + IB(t )
with M s given by eq. (1.6). In section 2, we study the three point noise correlation functions for a system containing one slow linear variable. We demonstrate that the use of our fundamental equations (1.7) and (1.6) yields a non-gaussian form for the noise correlation functions with slow time decays. Finally, we show that use of these equations yields the correct form for the time asymptotic form of the correlation function
(A(t)A(t)A(t))
= (A(O)A(O)A(O))
,
(1.8)
where A is the slow linear variable. In section 3, we extend the considerations of section 2 to the Langevin-like hydrodynamic equations in which there are five slow linear variables: the n u m b e r density, the energy density and the three components of the momentum density. In section 4, some conclusions and a summary of our results is presented. 2. Noise correlations for a diffusion problem
We consider a system in which there is one slow linear variable A ( r , X ) , N
A ( r , X ) = ~, A j S ( r - r j ) ,
(2.1)
j=l
which is the density of a conserved variable. H e r e r is a position in space, X = r N, P n is the phase point of the N particle system, and Aj is the value of the quantity A for the jth particle in the system. The variable A is a scalar such as the temperature field in a heat transport problem or the concentration of a solute in a solvent.
P. SCHRAMM AND I. OPPENHEIM
84
It is convenient to consider the Fourier transform variable
A k ( X ) =- Ak =- ( eik'r A(r, X ) dr
(2.2)
J
which obeys the equation
Ak(t) = ik. Jk(t),
(2.3)
where Ak(t ) =--Ak(X(t)) and Jk(t) is the vector current for the variable Ak(t); the quantity A =- Ak= 0. We introduce the hierarchy of multi-linear modes, Q, [see ref. 7] which are defined by: Q(0) = 1 , Q ~ 1 ) = / ] k =- Ak - ( Z k ) , Q(2) k-k',k'
= A k - k' A k' _ ( A k k , A k , )
(2.4) -
(Ak_k,Ak A ( AkA
'
etc. where the superscript * denotes the complex conjugate. The Q's are defined such that
(Q~,)Q(,')*) = Au ' (Q(t)Qq)*).
(2.5)
Here the ( ) denote an equilibrium average of the system which is assumed to be uniform and isotropic and the superscript (l) on the Q's denotes the multi-linear order of the mode. The Langevin equations for Q(t) are 7) to order k2:
Qq)(t) = Mll,(t )
*
Qq')(t) + lq)(t) ,
(2.6)
where Mu,(t ) = - / (I(')(cr)l ('')*) dcr * [K w ) ] - l ,
(2.7)
0
Kq,)= (Q(,,~Q(t,~*) ,
(2.8)
the fluctuating force l(r)(t) has the property:
(l~")(t)Q ~t)*) = 0
(2.9)
NOISE CORRELATION FUNCTIONS OF LANGEVIN-LIKE EQUATIONS
85
and is assumed to be a fast variable in the sense that
( l ( t ) ( t ) l ~r)*) = 6(t) .
(2.10)
The * in eqs. (2.6) and (2.7) denotes a sum over l' and over all intermediate wave vectors. Finally, note that for times greater than a molecular relaxation time
M u, (t) = M u, (oo) =_ Mu, "
(2.11)
Eq. (2.6) can be solved formally to obtain
Q~o(t)
Mt Q~r) iaM<'-~) ~,l~r)(r) d r
:
~11' ~
~-
~11'
(2.12)
--
0
where terms of G(k 2) have been discarded. We now use eq. (2.12) to write: lim (Akl(t)Ak~(t)Ak3(t) ) = ( A ~ A k z A k 3 )
t~
t
0
l
0
t
0
kl
k2
~"113 k3
~ \~
\~1. I~
\~21 ~
\°31!
(2.13) where k I + k 2 + k 3 = O.
(2.14)
Eq. (2.13) is valid since e~t---~ 0 as t - - ~ for all k > 0 . Now, since the left-hand side of eq. (2.13) is nonzero, the right-hand side is nonzero, and I cannot be a Gaussian variable of zero mean. F u r t h e r m o r e , since (I('q)I(z2)I("r3)) is at least of order k 4 which follows from isotropy, it cannot be of the f o r m 6(~ 1 - z2)60" 1 - %) which will yield a term of order k 2 after integration. The left-hand side is of order k °. Thus the three point correlation function of I must have a slow part which on integration yields a term of order k °. The L H S of eq. (2.13) is of order N. We shall consider only those terms on the R H S of eq. (2.13) which are also of order N. In order to do this, we use the facts that M~; is at most of order N 1 - l and t h a t (I(tl)I(t2)I (t3)) is at most of order (l~ + 12 + 1 3 ) / 2 if l 1 + l 2 + l 3 is even and of order (l I + l 2 + 13 - 1 ) / 2 if l I + l 2 + 13 is odd. T h e only terms on the R H S which can be of order N are
86
P. S C H R A M M
A N D I. O P P E N H E I M
those for which l 1 = 12 = 13 = 1 and those for which one of the l's is 2 and the other 2 / ' s are 1. We write the matrix M with elements M u, as
M= F + 0 ,
(2.15)
where F is diagonal both in the mode order, l, and in wave vectors and O is off diagonal either in l or in wave vectors or both. Thus the elements of F are of order N O and the elements of O are at most of order N -1. Thus, e~' -~ e r(k')' ,
(2.16a)
kl t
e~2t = f er(ko(t-~')M1 2 kl
0
k!
er22(k1-k', k,)~
do" ,
(2.16b)
=--- k2D
(2.17a)
k 1- k ' k'
where
F(k) =- el,(k ) =
and r 2(k, - k', k ' ) :
(2.17b)
-[(k, - k'y +
In order to obtain eq. (2.17a), we have used eq. (2.3), the isotropy of the equilibrium system, and the fact that l(kl)(t)= ik" Jk(t) to Navier-Stokes order (i.e. to order k2). To this order the coefficient of k 2 in (2.17a) is independent of k. To obtain eq. (2.17b) we have used the fact that ~22(kl - k', k') = Ul(kl
- k ' ) --[- ~ l l ( k ' )
(2.18)
--[- ~7(N-1).
Using these facts, we can write eq. (2.13) as:
0
e -k2D(l-'r3)
0
×
+ 3 TL f tirldo-e-kZ~O(t-rl-'OMld~'a 0
0
2
e-tk~+k~lD~i d72 f d~'3
kl; - k 2 ' - k 3
× e -k~°(`-~2) e-k~n(t-'3)(I(_2)k2,_k3(Zl)Ik:(Z2)Ik~(r3)] } ,
0
0
(2.19)
NOISE C O R R E L A T I O N FUNCTIONS OF LANGEVIN-LIKE EQUATIONS
87
where ffkl(7.1) =-Ik(1)1 (7.1) and there are 3 terms like (TL) the last term in which 1(7.1) is replaced by i(2), 1(7.2) is replaced by I(2) and 1(7.3) is replaced by 1 (2). The wave vectors in the last term have been chosen to yield a term of order N. The quantity M 1 2
kl;kl-k',k' f:
is equal to: (2)*
(Ikl(O)Ikl_k,,k,) dO
(Akl_k,Akl_k,)(AklAkl)*
M12k,;k,-k',k'
•
(2.20)
We shall now examine the correlation function wish to retain terms of order N and k 4. We write /kl(T1)Ik2(7.2) = [A k1(7.1) --
Mll (7.1) * Q(/)(7.1 ) ] [ A kI
(Ik~('r1)Ikz(7.2)Ik3(Z3))
k1(7.2) --
and
Mll (7"2) * Q(l')(7.2)1 k2
(2.21)
which follows from eq. (2.6). We define B(7.z) as B(T2) = [Akl(T1 -- 7.2 + 7"2) -Mlt(7.1)*Q(t)(T1 kl
- T2 + T2)]
x [Ak2(7.z) -- Mw(7.2) • Q(r)(7.2) ] ,
(2.22)
k2
Using the general formula given in eq. (1.7), we can now write:
( Ik,(7.1)lkz(7.2)Ik30"3) ) = MBt(7.2)* ( Q(O(rZ)Ik3(r3) ) + ( IB(r2)lk3(r3) ) , (2.23) where MB/(T2)
=
([Akl('rl
-
'/'2) -
Mll ( z l ) * Qq)(rl - 7.2)] kl
× [~lk z _
M l r (T2) , k2
- f (1B(o')l "(j))
Q(r)]Q(D*),
dcr* K (j)-I
K(])-I
(2.24)
0
to Navier-Stokes order and the term (IB(7.Z)Ik3(%)) is assumed to decay quickly and is proportional to k48(7.2 - 7.3)8(rl - 7.2)N. Use of eq. (2.12) in eq. (2.23) results in
88
P. S C H R A M M
A N D I. O P P E N H E I M
r2 AM(r2-a) * ( I(l')(0-)Ik3(7-3) ) d o ( Ik~(7-1 ) Ik2(7-2 ) Ik3('r3 ) ) = MBI(7-2) * f -u' 0
+ (IB(%llk,(r3)} r2
= Mm (7-2)f kl+k2
e-D(kl +kz)2(~z-~) ( Ikl +k2(O')Ik3('r3) ) do" + ( In(r2)Ik3(r3) ) ,
0
(2.25) where in the second equality we have retained only those terms which are of order N. We use the fact that (Ikl+k~(O')Ik~(7-3)) = + 2 k 2 D ( Ak~A T,)6( ~ - 7-3)
(2.26)
to obtain
( I~, (7-1)I< (7-0I~,(7-~) > =
+ M B1
(7-z) e
- Dk2(r2-r3)
2
^
^,
2 k 3 D ( A k 3 A k3)O(7-2 - 7-3)
kl+k 2
+ (Is
(%)Ik3(%)> '
(2.27)
kl+k 2
where 0(7-2 -7-3) is a step function equal to 1 for 7-2 > 7-3 and equal to 0 for 7-2 < 7-3. Note the slow time decay in the first term on the R H S of eq. (2.27). Finally, for 7-1 ~> 7-2 > 7-3, eq. (2.27) becomes
(I<(7-1)/<(7-2)tk,(7-~) > = = + 2 k 2 D e-Dk230"2-'~3)[(Ik1(.q -- 7-2)Ik2Ak3)+
C(Nk4t$(7-1
- 7"2))]
(2.28)
+ {7(Nk46(% - r3)6(7-~ - 7-2))
(2.29)
~-- + 2k~D e Dk]''2--~3) ( Ik,(7-1 - 7-z)Ik/k3)
which is the leading order (in k and N ) contribution to the integral in eq. (2.19). The term
( J~,,ox(~)Jk2xAk3) d~,
(/k1(7-1 - 7-0/k2A k3> %2- 2kl. k2~(7-1 - 7-2) 0
(2.30)
NOISE C O R R E L A T I O N F U N C T I O N S O F L A N G E V I N - L I K E E Q U A T I O N S
89
where J m is the x component of the dissipative current, i.e., I k l = ikl.JDk ~. The k's inside the integral can be set equal to zero. The form of (Ik~(rl)Ik~(r2)Ig3(r3)) depends on the time ordering of %, ~'2, r3. For ~'2, r3 > "l'l we obtain
( Ik,(rl)Ik2(rZ)Ik3(r3) ) = + 2k~D
e -°k~l'2-''l
( Ak lk2(r2 -- r3)Ik3 )
(2.31a)
and for % , z 3 > r 2
= +2k~D e-°k21"l-'~21(Ik,('q -- z3)AkJk3 )
.
(2.31b)
When eqs. (2.29)-(2.31) are substituted into eq. (2.19), the result is
+ f (Jm(o")J~A) do"/D,
(2.32)
0
where we have used the fact that k 1 + k 2 + k 3 = 0. We next turn our attention to the terms in eq. (2.19) which involve i(2). The terms in /(2) which contribute to a n N 2 contribution from the correlation function of the 3 I's are
I_k2('Cl)A_k3(~'1) + I_k3(Z,)A_~:2(rl) . When this is substituted into the correlation function we obtain:
( l(-2)k2,_k3(~'l)Ikz(7"2)Ik3('r3)) = ( I_kz(T1)Ik2(T2) ) ( m _k3(71)Ik3(7"3)) + (I_k3(Ta)Ik3(Z3)) (A_kz(~'l)Ik2(~'2)), (2.33) Again we write:
( Ikz(~'z)l_kz(rl) ) = 2k~D ( Ak A ~) 6(r~ - ~'z)
(2.34)
and r1
= f e--Dk]O'l--°')(lk3(ff)Ik3(T3) ) do" 0
= e-°k~'-~3~21,~o ( AkA L) 0(~', - ~'3).
(2.35)
90
P. SCHRAMM AND I. OPPENHEIM
Thus: ( I~2.-k3(7-1 )Ik2(7-Z)Ik3(Z3)) =4D2(AA,
2 2 ) 2k2k3[c~('rl - 7-2)
e-Ok](rl-'3)O(7-1 7"3)
+ ~ ( r l -- "r3) e-Dk2(~l-~2)0(rl -- 7-2)],
(2.36)
where we have set k = 0 in the correlation functions of A. Again, we note a long time decay of the fluctuating force correlation function. Substitution of eq. (2.36) and its analogs into eq. (2.19) yields
-
fo
+
(2.37)
In order to obtain eq. (2.37) we have used the relation
M12 kl 'k2'k3
=-f0 (Ik,(o-)Z(k22),k3)do-(Ak?Z2)-'(mk3flZ3)-1 o ^
^~
( Ak,A k,)
do-(Ak2Ak 2) (AkAk3) (2.38)
which follows from the definition o f Q(2), eq. (2.4). Eq. (2.37) together with eq. (2.32) yields for eq. (2.19)
(Ak,Ak2Ak~) = ( Ak,Akfik3) "
(2.39)
3. Noise correlations for hydrodynamics
It is easy to extend the considerations of section 2 to the case of several slow linear variables. In hydrodynamics, there are five slow linear densities: the number density, the energy density, and the three components of the momentum density. These five densities will be denoted by A~, where a specifies the particular density and its wave vector.
NOISE CORRELATION FUNCTIONS OF LANGEVIN-LIKE EQUATIONS
91
The noise correlation functions of interest are those in eq. (2.19). We shall go through the argument for the linear noise correlation function and see that the procedure is essentially identical to that in section 2. We consider the correlation function (lal(T1)la2(72)/a3(73)) where the (3/i noise has wave vector ki, and again k 1 + k 2 + k 3 = 0. We define B~,~2(72) by
B.,.:(72) =
/~1(71 -- 77-2 + 7 2 ) L 2 ( 7 2 )
= [A,~,(% -- 72 + 72) -- M , ~ d ( % ) * Q ( t ) ( % × [A,~2(72) - M . 2 r ( r 2 )
- 72 + 7211
(3.1)
* Q(r)(72) ] .
We use the arguments following eq. (2.22) to arrive at the analog of eq. (2.25) for z I/> r 2 > 73: 7-2
( ia1(7111,~2(7211,~3(731) =
MB.(72 ) f c,, ~-r(.2-~) (I, (0-11.3(73)) 0
+ (I,(721/~3(73)) ,
(3.2/
where the wave vector associated with T and 6 is k3, and repeated G r e e k subscripts are s u m m e d over. The quantity MBr(~-2) is given by
MRs,(72) =
( [ A ~ , ( r 1 - r2) - M,~,I(T1) * Q(t)(r 1 - r2) ]
× [ A 2 - M~2r (721 * Q"')]A,,).
(/~i*) -1 Y'3'
72
- f (Is(or)I,) do'. (/~{*)~,~ ,
(3.3)
0
where again the wave vector associated with y ' is k 3. The matrix F is the hydrodynamic matrix
r = [(A,i*) - f (l(~)t) d~]. (Li*)-m
(3.4)
0
to order k 2 and has convective (Euler) parts as well as dissipative ( N a v i e r Stokes) parts. The Euler terms are proportional to ik and the N a v i e r - S t o k e s parts to - k 2.
92
P. SCHRAMM AND I. OPPENHEIM T h e term of interest, with long time correlations, is ( I(r,
) 1,~2(T2)1~,3(T3)) T2
= (I(r.
^ ,
-
T2)/,~"Av, _ > (,4,4),,,
1
+1"(r2--7t) /
e,~
~
(I~(o")I~3(r3)) do".
0
(3.5) We use the fact that r2
f (l~(o-)l~3('r3) ) do"= -2r o.
(3.6)
,
0
where
F°
is the dissipative part of F, to write
(/~,(r, )Io2(T2)I~3(%)) : - 2 { l,~l(Y,
-- ~-2)I~2A~ , >. < AA* > ~-,~
+l'(z 2 " r ~ ) r D / ~ *
• e~
,~
160k-,~ko~la3 >
(3.7)
for q'l > '7"2 > '7"3" Eq. (3.7) is of o r d e r N. A g a i n , note that 7 ' , Y, (5 and 0 have the wave vector k 3. T h e only difference b e t w e e n eq. (3.7) and the c o r r e s p o n d i n g e q u a t i o n in section 2, eq. (2.29), is the fact that there are several slow m o d e s which have different vectorial and time reversal symmetries. In particular, if any of the 0/'s = N, the noise 14 = 0 and eq. (3.7) yields 0. If we d e n o t e the time reversal signature of A s by G , then g u = E"E = 1, ep = - 1 and their currents have the opposite signature. If e~IG2G3 = - 1 , the left-hand sides of eqs. (3.5) and (3.7) can be of o r d e r k 3 whereas if e~ ea2G3 = 1, the left-hand sides must be at least of o r d e r k 4. T h e right-hand sides of eqs. (3.5) and (3.7) are at least of o r d e r k 4 and can only be the d o m i n a n t c o m b i n a t i o n to the 3 noise correlation functions w h e n the p r o d u c t of e's is 1. In particular, if 0/, = 0/2 = O13 = E, 7 ' and y must be N or E and 6 must be E. If a I = 0/2 ~- P and a 3 = E, then again y ' and y must be N or E and 6 must be E. If 0/, = 0/3 = P and 0/2 = E, then y ' and y must be P and 6 must be P. T h e cases w h e n the p r o d u c t of e's is - 1 are 0/1 = 0/2 = 0/3 = P and those in which o n e of the 0/'s is P and the o t h e r two are E. If 0/1 = 0/2 = °z3 = P, 7 and y ' must be N or E and 6 must be P. If 0/, = P and o/2 ~--- 0/3 = E , y ' and y must be P and 6 must be E. If a, = 0/2 = E and 0/3 = P, Y' and y must be N or E and 6 must be P. Ft T h e a p p r o p r i a t e factors to the used for e ~ can be o b t a i n e d f r o m the matrix
NOISE CORRELATION FUNCTIONS OF LANGEVIN-LIKE EQUATIONS "g"(e~+'+e~-') + 2 - g, e ~hr m
~Ft =
1
t
mn
X"--~-h(e:'+'+e:'-'-2e~'r)
x. --(e~+'+e ~ ' - 2 e ~r)
-c-o( & + ' - e ~-')
m
m
x~h (e~'+'+e~'-')+2 X" eZ'r C°h (e . . . . e ~-')
mn
mn
m
mn
93
0
0
0
0
CoX.(e~÷' - e~-')
Co(e ~÷' - e ~-t)
C2(e v+' + e v-')
0
0
0
0
0
2C2o e - ~'k2'
0
0
0
0
0
2C~ e "~k2'
(3.8) in which the first r o w and c o l u m n are E, the second row and c o l u m n are N, the third r o w and c o l u m n are the longitudinal m o m e n t u m and the fourth and fifth rows and columns are for the two c o m p o n e n t s of the transverse m o m e n t u m . Here: y+_ = + ikC o -
-/-s k2
,
Yr = -Irk
2 ,
(3.9)
vt = r t / m n ,
w h e r e C O is the adiabatic s o u n d speed, ~7 is the shear viscosity, and P s (Fr) is the s o u n d (heat) m o d e a t t e n u a t i o n coefficient; Xe = ( Op / Oe),, , Xn = ( OP / On) e, P is the hydrostatic p r e s s u r e and h is the e n t h a l p y density. T h e correlation functions which contain I t2) are also easy to analyze. 1(2)
"r
( ,~,,~(,)l,,2(z2)I~,3(r3))
~-- (I,~,(z,)I,~2(z2)) ( .,?t
a~('/'l)Ia3(T3))
+ (I,~i(z,)I,~3(z3))(A~q(7"1)I~,2('r2))
(3.10)
which follows f r o m eq. (2.33). H e r e a I has the w a v e vector - k 2 , a~ the w a v e v e c t o r - k 3 , a 2 the w a v e v e c t o r k 2 and a 3 the w a v e v e c t o r k 3. Again, we use the relations [eq. (2.34)]
(
) = -2r
(A
o)
-
(3.11)
,
w h e r e 13 has the w a v e v e c t o r - k 2 , and [eq. (2.35)]
( A ~i(T1)I~3(r3) ) = --2 er('q-r3)/"D
7-3)
w h e r e Y and Y' h a v e the w a v e v e c t o r - k 3. Eq. (3.10) b e c o m e s
(3.12)
94
P. SCHRAMM AND I. OPPENHEIM
X [ ~ ("/'1 _~ ~(T1
'/'2)0(1"1
--
T3)0(T 1 - -
D
D
F('r t -"r3)
\ r , Dc ~ i vrl Df3'~ e F('r . l # ,I 1"2)] 1,
"r2)l
(3.13)
where /3' has the wave v e c t o r - k 2. The left-hand side of eq. (3.13) has the signature -e~le~ie,2e~3 and is of order k 4 when this is - 1 . This is the case, e.g., if a 1 = a ' 1 = a 2 = a 3 = P. In this case, y ' = / 3 = y = / 3 ' = P. For % = a~ = a 2 = O~3 = E , "y' a n d / 3 must be N or E a n d / 3 ' = y = E. The appropriate factors of e rt can be obtained from eq. (3.8). The only differences between the analysis for the hydrodynamic noise correlation functions and those in section 2 are the proliferation of indices and the existence of Euler terms in the hydrodynamic matrix which yield factors of s i n ( k C o t ) and c o s ( k C o t ) in some of the propagators.
4. Summary We have d e m o n s t r a t e d that the noise in Langevin-like equations is not Gaussian by studying three point noise correlation functions. We have assumed that the two point correlation functions are delta-function correlated, i.e. (I(tx)I(t2))
- -
~(tl -- t2)
(4.1)
but that the three (and higher) point correlation functions have slow decays. While we have considered only equilibrium correlation functions, the results can readily be extended to nonequilibrium correlation functions as well.
Acknowledgements One of us (P.S.) would like to thank the chemistry d e p a r t m e n t of M I T where this work was carried out for their w a r m hospitability. A portion of this work was supported by the National Science Foundation under grant CHE84-10682.
References 1) S. Chandrasekhar, Rev. Mod. Phys. 15 (1943) 1. G.E. Uhlenbeck and L.S. Ornstein, Phys. Rev. 36 (1930) 823. M.C. Wang and G.E. Uhlenbeck, Rev. Mod. Phys. 17 (1945) 323.
NOISE CORRELATION FUNCTIONS OF LANGEVIN-LIKE EQUATIONS 2) Reviews: N.G. van Kampen, Adv. Chem. Phys. 34 (1976) 245. R. Graham, in: Springer Tracts in Modern Physics, vol. 66 (Springer, Berlin, 1973). R. Kubo, Rep. Progr. Phys. 29 (1966) 255. And more recently e.g. P. Grigolini, J. Stat Phys. 27 (1982) 283. R. Graham and A. Schenzle, Phys. Rev. A 25 (1982) 1731, 26 (1982) 1676. W. van Saarloos, D. Bedeaux and P. Mazur, Physica I10A (1982) 247. 3) A. Khinchin, Math. Ann. 109 (1933) 604. 4) H. Grabert and W. Weidlich, Phys. Rev. A 21 (1980) 2147. 5) P. H~inggi, Z. Phys. B31 (1978) 407. A. Onuki, J. Stat. Phys. 19 (1978) 325. 6) H. Grabert, J. Stat. Phys. 19 (1978) 479. 7) J. Machta and I. Oppenheim, Physica l12A (1982) 361.
95