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The limit diffusion mechanism of relaxation for spin systems
Physica
122A (1983) 413430
North-Holland Publishing Co.
THE LIMIT DIFFUSION
MECHANISM
OF RELAXATION
FOR SPIN SYSTEMS Lech PAPIEk* Dublin Institute for Advanced Studies, 10 Burlington Road, Dublin 4 Ireland
Received
1 June
1983
The diffusion limit theorem for stochastic differential equations is applied to analyse the dynamical evolutions of spin systems. Bloch equations are derived and the stability of asymptotic evolutions is proved. The theory is applied to nuclear magnetic relaxation of two spins.
1. Introduction and motivation
An effective method of statistical physics uses somewhere certain assumptions which are essentially of a stochastic nature. In the theory of nonequilibrium statistical mechanics advocated by Kubolm3) and Van Kampen4s5), where the time evolution of physical systems is postulated on the basis of stochastic dynamical equations (stochastic Liouville equations), these assumptions are expressed more openly. However, derivations of kinetic equations from stochastic Liouville equations”‘) have, in the past, caused much confusion. It was not clear that assumptions to neglect some correlations, necessary for these derivations, were justified”“) (this subtle problem was exhaustively discussed, for example, in ref. 5). By analogy with the deterministic case one may expect, however, that the Van Hove limit”,‘2) is the proper method for the analysis of stochastic Liouville equations, and clearly should lead to kinetic equations. Fortunately, a mathematical theory exists (the so-called asymptotic theory of stochastic multiplicative equations, which started in 1966 with the Stratonovich’3) and Khasminskii14) papers) which is appropriate for such analysis. We are thus prompted to check how such a theory will work for some simple physical systems. Spin systems, with their finite degrees of freedom and well-established relaxation results are specially good candidates. Thus the object of this paper is to show that the asymptotic theory of stochastic, multiplicative equations 13-“) applies to quantum evolution for spin systems in a random environment, and that it allows us to derive rigorously kinetic equations (Bloch equations) and to determine transport coefficients (relaxation times). Moreover, because our considerations are based on *On
leave of absence
from
Silesian
0378-4371/83/0000-0000/$03.00
University,
0
Katowice,
Poland.
1983 North-Holland
L. PAPIE
414
mathematical results, we are able to dig out those assumptions which are of an “essentially stochastic nature”, under which the relaxation in spin systems appears. The paper is constructed in the following way: Firstly, we state the general problem of the spin system evolution in a random environment. Then, in section 3, we describe a particular example of such a system for which we aim to derive explicit results; and, in section 4, we transform its evolution equation into multiplicative, stochastic form. In section 5 we quote the results which are later, in the more theoretical sections 6 and 7, rigorously justified. Section 6 is thus devoted to the statement of the relevant diffusion limit theorem which allows us to obtain the time evolution of spin systems, while in section 7 the general consequences of this evolution are investigated. Eventually, in section 8, we make the final calculations for the system of two spins.
2. Statement of the problem We aim to derive the Bloch equations and relaxation times for nuclear magnetism from stochastic dynamical equations. Thus the definition of the system under consideration is the same as in the Redfield6) and Kubo”) papers. The Hamiltonian of the system of spins and their molecular environment (bath) can be written in the form
(1)
H = f-f(S) + HP, 41,
where H(S) depends only on the spin variables S, and H(S, q) is the energy of interaction of the spin S and the molecular q degrees of freedom. The stochasticity of H(S, q) is the result of the random variation with time t (1 B 0) of coordinates q( if such probabilistic assumptions are made in consideration of the quantum evolution the theory is called “semiclassical N” ‘“). From quantum mechanical laws we have the evolution of our spin-bath system given by
dp(t) __dt
=+W)l,
PW=Po,
t 20,
where p is the density matrix and His the Hamiltonian (1). Eq. (2) is a stochastic one and its solution p(t) is a stochastic process which determines the evolution of the spin-bath system, while the average Ep(t) (over bath) may be interpreted as the density matrix of the spin subsystem. Our investigations of the quantum evolution (2) will proceed in the sense of the weak coupling limit”~‘*~*’ ). This treatment of the dynamical evolution equations, which leads to derivations of kinetic equations, started with Van Hove’s papers”,‘2), and is now one of the most powerful methods of nonequilibrium
LIMIT DIFFUSION
MECHANISM
OF RELAXATION
415
statistical mechanics*‘). In our case it means that instead of eq. (2) we should postulate y
=; [H, p’(t)]
P’(O)=
)
PO>
t
2 0,
where H is the same Hamiltonian as before, and treat as the (macroscopic) physical evolution that matrix p’(t) which is the solution of (3) for 6-0 and 0 G c*t = z < zo, where r. is arbitrary. E should be interpreted here as a scaling parameter, rather than any established physical quantity. To realize this let us express the time-dependent Hamiltonian H(t) given in (1) as H(t) = H’(t/c)
)
(4)
instead of postulating ad hoc the presence of t in (3). Then (2) yields
T =f[F(t),
pytjj
)
p’(0) = p.
)
where p’(t/c) = p(t). If we now demand H’(t) as we have required earlier for H(t), and 0 G c*t = z < z. will be the same as coupling postulate. In this sense the weak scaling.
tao,
(5)
the same stochastic characteristics for then the solution p’(t) of (5) for 6-0 the solution obtained using the weak coupling postulate is equivalent to time
3. The system of two spins in a random environment Here, we restrict our considerations to a simple system of two spins contained in a molecule placed in the random environment and interacting only through dipoledipole interactions. We assume also the absence of the external magnetic field, i.e. H(S) = 0 (let us notice that H(S, q) is the part of Hamiltonian (1) responsible for the relaxation). Using the example of this system, we are able to show how the method works while the explicit expressions are as simple as possible. The Hamiltonian H(t) = H(S, q) of the spin dipole-dipole interaction is in this case6*18-20) H(t) =
i
(6)
Fk(t)vk,
k=-2
and the Fk(t) are equal to Fk(t) =
0 1’2y*h2r-3(g
l)"Y;"(e(t),
q(t)),
k=O,
fl,
f2,
(7)
where the Y;k are second order spherical harmonics dependent on 8 and cpwhich
L.PAPIE
416
r “joining”
two spins in a
molecule, and r is the length of r. The second order tensor operators in (6) are’9,20)
0“*[sj@
vo= - ;
s, -
gs,OS,
+
s_, 0
Vk appearing
S,)]
= -~(2S,os,-s,os,-s2~s2), & v+‘=
(8)
+(s,oS,,+s,,oSJ = + (S, 0 S, + S, 0 S,) + i(S, 0 S2 + S2 0 SJ .
V’2=
-s+,os*,=
-(s,os,-s2os2)~(s2os,+s,o~2~,
where S,, S, and S, denote
the Pauli
spin matrices
s’=(; A),s*=(pii), s3=(;g,
(9)
where S,, = S, + is,. According to the statement of the problem, the Fk(t) are random variables for any t 3 0. Thus, to define our system we must determine the probabilistic properties of stochastic processes Fk(t). Let us mention the most obvious of them. From (7) and properties of spherical harmonics Yt it follows that (IF) F“(t) are bounded random variables for any t B 0. The natural requirement that the molecular environment is homogeneous in time, and isotropic, implies: (2F) Fk(t) are stationary processes in a wide sense, i.e. EFk(t) = C (without making our considerations less general we may choose C = 0), and Rk,k.(f,, t2) = f?Fk(tl)Fk’*(t2) = Rk,k.(tl - t2), and (3F)
the
correlation
matrix
Rkxk’(s) for
J”(t)
processes
is diagonal,
R,,&) = dk,k.Rk(S) where Rk(S) = EF”(s)E”*(O). (2F) was always assumed in physical models of nuclear relaxation6,” ‘O,**).(3F) follows from properties of spherical harmonics
i.e.
magnetic Y: and the
isotropic initial distribution of the orientation of a molecule carrying two spins (because of homogenity in time this isotropic distribution is preserved during the motion). This was shown, for example, by Ford, Lewis and McConnell (ref. 23, appendix) and by Hubbardz4). Further, the investigations of all specific, stochastic models of rotational motions of molecules demonstrate that25) (4F) correlation functions &(s) = EFk(s)F”*(0) are real, i.e. Rk(s) = R:(S), which implies R&s) = R:(s) = R&S) = R_,(s). At last we know that any nondegenerate stochastic process, in contrast with a deterministic one, must forget, to some extent, its previous evolution. Thus
LIMIT DIFFUSION
MECHANISM
OF RELAXATION
417
(5F) Processes Fk(r) acquire, quickly enough, independence of the time evolution from the previous one. The precise formulation of the condition (5F) we postpone to our later discussion of the mathematical theory of stochastic, multiplicative differential equations.
4. Reduction of the density matrix equation to a linear differential equation in Euclidean space Let us construct the auxiliary, real, Euclidean space R, in which the quantum evolution equation (3) assumes, in the case of the system of two spins, the form of a linear, differential equation. To this end let us define the scalar product ( , ) for the basis set of vectors {Si 8 S,}:,, , by i,j,k,
(Si @ Sj, Sk 0 SJ = iTr((Si @ Si>* (Sk @ SJ) ,
I = 1,2,3,4,
(10)
where S,, S, and S, are spin Pauli matrices (9), S, = lis the identity 2 x 2 matrix, and * denotes the matrix multiplication. From the above definition it follows that
(si 0
sj,
Sk0 sJ =
s&sj,
9
i,j, k, 1 = 1,2,3,4,
(11)
and in the standard notation for basis vectors, i.e. when S, @OS, = e2, . . . ,
S, 0 S, = e, , we
&@
&=
el6,
(12)
may write instead of (11) (%
ejl
=
6,)
i,j=l,...,16.
(13)
The set {e,>ia, may serve therefore as the orthonormal basis of the real, sixteen-dimensional Euclidean space R6, and any vector z cRi6, z = X:i’“,z,ei, where zi are real, may be thought of as a 4 x 4 hermitian matrix which is a linear combination of basis matrices {St @ S,>&=,. Since vectors from Rf6 are identical with 4 x 4 hermitian matrices, the scalar product (z, y) of z and y may be written as (z,.v) = iTr(z ‘Y) ;
(14)
hence immediately zi = (z, e,) = iTr(z
* eJ
,
(15)
in particular - ITT(z), 4
z16 -
(15’)
418
L. PAPIE
and 1~1’= $Tr(z2) .
(16)
Thus we recognize that any density matrix p for the system of two spins is a vector z ER” for which the last component zr6 is equal to f and /.z/Q 1. The other comdonents z,, i = 1, . . . , 15 of a vector z have by (15) the interpretation of quantum averages of observables e, defined in (12). Moreover, the Hamiltonian H(t) of our problem, defined by (6) (7), (8) may be considered as a stochastic process with values in Ri6. Therefore, since the commutator [ , ] in (3) is the relevant algebraic operation in Rf’, the evolution equation (3) with Hamiltonian (6) is the linear (multiplicative) stochastic differential equation in Ri6. More specifically we may write
%(~,z~ei)=:[H(t),~,z;ej]=~[k~2Fk(t)Vk,Sz = j$,k=:_2 Fk(t)
f, =
z;v
6
z:(O)e,
z
E
Rb6
(17)
Taking the scalar product of the left- and the right-hand side of (17) with basic vector e,, we obtain ; z; = 6 5 $ Fk(t)A;.zj i=lk=-?
,
GO) = z/ >
1=1,...,16,
(18)
where Ai are defined by A i = f (e,, [Vk.e,l),
l,j=1,...,16;
k=O,
+l,
_+2.
(19)
The hermiticity of Vk, ei and e, and the definition of the scalar product in RJ6 imply that matrices (Ak}:= _2 are skew-symmetric, i.e. (Ak)T= -Ak.
(20)
In a vector notation (18) takes the form dz’(t) PZZ dt
’ 6 ,=I_* Fk(t)A kz’(t) 7
Finally, we find by substitution (Ak)* = (-
The combination
z’(O) = z E R:‘.
(21)
of (8) and (12) into (19) that
l)kA -k.
(22)
of (22) with the analogous relation for spherical harmonics, i.e.
y’;*(f) = ( - l>kYF~(~)
or
Fk*(t) = (- l)kFpk(r),
(23)
LIMIT DIFFUSION
MECHANISM
OF RELAXATION
419
ensures that the operator B(t) = X:5 _2 Fk(t)Ak is a real, linear operator from Ri6 into Ri6. This concludes the transformation of the time evolution of our physical system into the problem of linear, multiplicative differential equation in the real, Euclidean space Rf6.
5. Results
To complete the discussion of our example, we now quote results which we will justify rigorously later. All these results follow from the solution p’(t) which we are able to obtain (for c-+0 and 0 < c2t = r < zo, p’(t)+p(r)) for the problem (3) when H is the Hamiltonian of the system given by (6), (7) and (8). This solution determines completely the evolution of the spin-bath system on the macroscopic time-scale. Let p,(r) denote the evolution of the density matrix of the spin subsystem, i.e. p,(z) = I@(z), and let (G) denote the global average (over the spin subsystem and its random environment-bath) of the spin observable G, i.e. (G ) = Tr E(p *G) = Tr(p, *G). The results are (i) The conditions Tr p,(r) = 1 and Tr pi(r) < 1 are fulfilled for any r E[O, ro] if they are true for r = 0; (ii) The observable x0(2$ @ S, + S, @ S2 + S, @IS,), with x0 a real parameter, is invariant during the evolution of the system. The parameter x0 is determined only by the initial conditions of the spin subsystem; (iii) The density matrix of the spin subsystem p,(z) for r+cc relaxes to the limit (equilibrium) density matrix pes given by
a+xo Pe4= i
0
0
a-x0
0 0
0
0
2x0
0
2x0
a-- x0
0
0
0
a + x0
r
)
(24)
where x0 is as in (ii); (iv) The evolutions of (S’) and (S*‘), where S’ and S*’ denote, respectively, z and + 1 components of the total spin operator of two spins S’ = and
gs,0 s, + s, 8 S,)
(25)
420
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are given by
(S’)(T) = (S')(O) * exp - $ (
(27)
11
and (s*‘(r)
= (S”)(O).
(28)
exp
T, and T2 denote, respectively, longitudinal and perpendicular relaxation times, which will be derived in section 8. Because we assumed H(S) = 0 we should compare T, and T2 with the relaxation times for the case of extreme narrowing which were obtained earlier6~‘8~20~22). It IS easy to verify that our results are identical with those known earlier20,22). 6. The solutions of stochastic dynamical equations for spin systems Let us consider the stochastic multiplicative dz’(t) = t i dt k=
Franz’,
equation in R” of the kind
z’(O) = z E R” ,
t30,
-I
where Fk(t), t 3 0, k = 0, + 1, . . , +I are components of a given stochastic process F(t) on a probability space (0, 9, P) (we shall denote integration over 52relative to P, i.e. the bath average, by E), and {A “1: = _, are given constant n x n matrices. This equation is analogous to (21) except that n and I are now arbitrary natural numbers (in (21) n = 16 and I = 2); it describes spin systems which are more complex than those considered in our example. They must, however, be such as to satisfy conditions (20), (22), (23) and (IF)-(SF) for any k = 0, + 1, . . , + 1. In order to obtain the solution for (29) in a rigorous manner we must make the condition (5F) more exact. We assume that processes Fk(t) acquire the independence of the present evolution from the previous one in the sense, that: (5F’) correlation functions Rk(s) converge to zero for S-+CC fast enough to ensure that
s
lSRk(S)I
ds <
c,
(30)
0
and (SF”) process F(t) satisfies the mixing condition’“‘7,2”28), i.e.
(31)
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MECHANISM
421
OF RELAXATION
where 9: c 9,O < s 6 t < co denotes an o-algebra of events generated on s2 by F(u), s < u < t and a(s) fulfills 41 a”‘(s) ds < 00 .
(32)
s 0
As we said earlier, the property of “losing the memory” by process F(t) is obvious. However, the questions in what sense, and how fast it is lost, are open ones. In this context (5F’) and (5F”) should be seen mainly as sufficient conditions for P(t) under which the relaxation in the considered systems appears. Remark 1. It is known26,27)that for Gaussian stationary processes the-mixing condition is equivalent to the complete regularity condition’7,26,27),which allows us to estimate the correlation function. Therefore, by restricting ourselves to Gaussian processes, we may get rid of the condition (5F’). Remark 2. Ergodic Markov processes P(t)
are examples of processes for which
(5F”) is satisfied16). Let us state now some simple consequences of (20) (IF)-(SF’),
(5F”):
Lemma 1. The quantities Jk defined as 4,
R&) ds
Jk =
(33)
,
s 0
are equal to T
s
Rk(s -s’)ds’ds,
(34)
00 and (34) converges to Jk for T-CD as l/T. Proof. Lemma I is implied by (5F’) because, thanks to the partial integration of (34) we have Ts
lim 1 T-m
T ss 0 0
T
R,(s -s’)ds’ds
(T-s)R,(s)ds 0 T
=J+im_;
sR,(s)ds. -
s 0
We notice that by (4F) Jk are positive, and Jk = J-k, k = 0, k 1, . . . , _+1.
(35)
422
L. PAPIE
Corrofary.
Applying
(3F), (20), and lemma
I we find that following
limits:
TJ
and 1 c,,(z) = lim T-x>
T
)(
0 0 exist uniformly
in z. In vector notation,
h(z) = [b,(z)] =
Jk(Ak)+Akz=
f: k=
, k,&Fk’,djAk’z)*
ds’ds I
(37)
(36) and (37) may be put into the form -Dz
(38)
-I
and C(z) = [c,(z)] = where (A”)+ denotes Lemma Proof
k JkAkzzT(/tk)+, k= _, the hermitian
2. The matrices
The matrices
tTD{
=
i
(39)
conjugate
of Ak.
D = CL= _, Jk(Ak)+Ak and
D and
C(z) are nonnegative
definite.
C(z) are real, and for any 5 E R” we have
Jk(Akt)+A kt 3 0
(40)
and
CTC(z)< = i
Jk(p‘4kz)(
)
(41)
k=-I
as the Jk are positive. Let us notice here that matrices D and C(z), though nonnegative definite, may be degenerate since the Ak are skew-symmetric. Now, let us recall the limit theorems for stochastic multiplicative differential equations” I’). For our case the relevant versions of these theorems are given in the papers of Papanicolaou and Varadhan”) and Papanicolaou and Kohler16). It is not difficult to recognize that our problem is analogous to the one considered in theorem 3 of 15), and that after similar transformation, the general, abstract theorem 1 of 15) will apply. It would be more convenient, however, to recall the theorem of Papanicolaou and Kohler16) (PK Theorem). Because Fk(t) are
LIMIT DIFFUSION
MECHANISM
OF RELAXATION
423
bounded and since eq. (29) is linear (and not explicitly dependent on the macroscopic time scale) all the boundedness conditions needed for PI; Theorem are satisfied. Moreover, the existence of the drift &(z) and the diffusion c&z) coefficients, in the sense of limits (35), (36) and (37), and their explicit forms (38) and (39), ensure the fulfillment of the further conditions of the PK Theorem. Thus the PK Theorem applies. Its relevance for the formulation of our problem may be expressed as follows. Theorem. Let z’(t) = z”‘(z), 0 < .c2t = z < z,,, 6 ~(0, l] be the process defined by
eq. (29). Let (lF)-(5F’), (5F”), (20), (22) and (23) hold. Then the process z”‘(z) converge weakly as c--+0 (t-m and 0 < c2t = z < zO,z,, arbitrary) to the diffusion Markov process z(z) with infinitesimal generator L defined by L = f:
(42)
Cij(Z)
i,j= I
where b,(z) and co(z) are, respectively, drift and diffusion coefficients given by (38) and (39). Remark 3. Theorems also exist (ref. 17, theorems 2 and 3, chapter 7, section 3) which allow us to determine the behaviour of the solution z”(t) of the above problem for t E [0, q/t]. It turns out that for such time-intervals process z’(t) does not move significantly from its initial position’7*28).Changes in z’(t) for t E[O, z&l are of the order t’j2, and not of the order 1 as expected for t E[O, ~&2] from the above theorem’7’28). Th’1s explains, to some extent, why we have formulated our problem in the Van Hove limit.
7. Consequences of the diffusion limit solution for spin systems
For further discussion it would be more convenient to represent the diffusion Markov process z(z) with drift vector b(z) and diffusion matrix C(z), as the solution of the Ito stochastic equation29v30).We have dz(z) = b(z(z)) dz + a(z(T)) d W,,
Z(O)=ZER”,
z 20,
(43)
where G(Z) is a square root matrix of C(z) (because C(z) is a nonnegative definite matrix it has a unique square root a(~)~‘)), and W, is a standard, n-dimensional Wiener process. Lemma 3. The mean value Ez(z) of the process Z(T), T 2 0 is contained in K,,, = {z ER”: \zI G r} where r = (z(O)\, and z(0) denotes the starting point of the evolution.
L. PAPIE
424 Proof.
Let us consider
the function (44)
Because
of the following
identities: (45)
and (46) we have /z(*@
Af,,A$z,z,,
+ 4p(p - 1)
c
Jk
k=-I
A~A;;.~,,,z,,,,z,z~
c r,j,m,m’
1)
=
I
Iz(*~-*) >
jzp’1.
A;jA;;z,,,z,,,,
(47)
> The first and the third terms in (47) cancel, symmetric. Therefore (47) yields J%(Z) = 4p(p - 1) f: k= =
LU(Z) d 0, By Ito’s formula
Applying
skew-
JkzTAkzzT(Ak)+z/z12@-*)
definiteness for
Ak are
-/
(48)
of C(z) we obtain
Odp,
(49)
we have also
= Lv(z(z))
the average
w(7))
dz + ____az
o(z(z))
to the last expression
Edv(z(r))=ELv(z(7))<0,
Hence,
the matrices
4p(p - l)zTC(z)zlzl*@-Z),
and by nonnegative
dv(z(r))
since
for
O
d IV, .
(50)
we obtain < 1.
(51)
for p = $, v(z) = Iz(
JEz(7)l < E/47)(
< Elz(O)( = [z(O)/ .
Thus lemma 3 is true, and evidently also it implies that the inequality is preserved during the evolution of the system.
(52)
Tr p:(7) < 1
LIMIT DIFFUSION
MECHANISM
OF RELAXATION
425
Actually, the relations (49) and (51) are stronger than are needed to obtain (52). They also give an opportunity for comment on the stability of the process z(7). Such discussion is, however, complicated by the fact that L may be a degenerate elliptic operator30). The degeneracy of L appears if C(z) turns out to be zero for some set N of points of R”. The form of C(z) implies that if such a set N exists it must be a linear subspace of R”. This is true, since N = n:= _, N(A “), where N(Ak) = {z QR”: A k~ = O}. Moreover, we may notice that for z EN b(z) = 0 and a(z) = 0. Therefore, by (43) N is an invariant subspace with respect to the process z(7). The decomposition of the process z(7) for N and N’ subspaces (where NON’ = R”) allows us to see it as the direct sum of two vectors evolving in time. The first vector is a constant vector z, in N, and the second is equal to the relevant diffusion process z’(7) in N’ with strictly elliptic infinitesimal generator L’ (for z # 0) (this is also clear if eq. (29) is linearly transformed so that the components in N and N’ are separated). For the generator L’ it will then be possible, in the same way as above, to obtain the relation (49) (but with strong inequalities), i.e. L%(z)
< 0)
for
O
z#O.
(53)
But this is a sufficient condition for asymptotic stability in the largez9v30)of the process z’(7). Thus all coordinates of the process z’(7) converge to 0 for 7--m. This means that the system for 7+co relaxes to the uniquely defined equilibrium position zcq= zN0 0, where z,+ N is the constant of the motion determined by initial conditions. Let us now turn our attention to the problem of determining the subsystem evolution, i.e. to the behaviour in time of the average Ez(7) of the process z(7): From (43) we have immediately for m(7) = Ez(7)
dm(4 - d7 =
-Dm(z),
m(O)=z,
or m(z)= exp( - D7)z ,
(55)
where D is the matrix D = I$= _, .I,(,4 “)+Ak. The last relation gives us the rate of decay of m(7). Thus (54) may be called the Bloch equation of the stochastic dynamical system given by (29) (IF)-(SF’), (5F”), (20) (22) and (23). The reason we choose this name becomes clear when we apply (54) to the system considered in 2.
8. Calculation for the system considered in 3
1”) TO find the explicit form of the limit diffusion z(7) in the case of the example considered in this paper we must first know all matrices (A “>i= -7, Applying definitions of the basis set (ei}i6=,(12), (9), and operators (V”}:= _2 (8) (9) to (19) we get after elementary calculations the following form of matrices (Ak)i, :2:
426
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lfll-il
_ -I--c-t-+-
4 __I--
1+11 I I I__ __A:_L_L_I_ Ii’ I I I I I I I I 7 I I / I I - 1--r-t-tI-f --T-f I I I I *Ii i i _ -_I---_-+--___I__ -I --+-•f I I I I I I I I ~_,_,__l__L_L_~_~__l__L_L-~-~--~--~~~--’ I I I I 1_1__l____1_I_~__l__L_L_I_I__I__ I I I I I I :+I; _~__I__~~~~_~_~_~__,__~-~-~-~-~l--~_J-__ I
l-ii
I
I
I+11
I
1;
I;2/
/
I
I
I
I
I
I I
I I
I I
I4
iii
(56’)
I I
I
I I
I I ~_+-+--f--,--~-+--
I
I
I Iii I C-&-4-4_-l--_C-_C-I I I I L_L_I_A_J__L_L__ I I I Ii’
I
I
l_il
I I
I
I
I
I
I
I If11 I I ~_+_~_~__,--_C-_C-I I I I I
I
I
;_;-f-;--;--~-;-I
I
IT11
r-+-t--l--l--~-+--
I
I
I
I
I
I
l
;-,;&,r-T--
I
I
I
I
lil
(56”)
I + ‘I
I
I
I
.-+t-_-I--_I---__++-_+-___-_I__-_t+f-_--_--,---_-+-
I-il lfll I I --I-~--l--L-L-1-~__I__L_L_1_~__I__L_L_.
I
I
I
lil
I
I
I
I
I
I
I
IT,1
I
I
I
I
.~_‘I_1--~--~:l-l~--l--~-i-~_~__~__~_~_ I I I I I I I
I-il&-ll l’__l__r_T_~_~_-~--~-~-.
_-~-~--~--~-~-~-
I
)
I
I
III/l//I//I
I
LIMIT DIFFUSION I
I
I
MECHANISM
I
I
I
OF RELAXATION
I
I
.__t__I_-~-_~-f-_:-1’--~--l
I
I
l*il
421 I
Ifi;
T_;_7-7--r-r-T-. I-
___:__I__~__~_~__:_~-_~--I_-~-~-~-_l~-I--_l~_. I I I I I I I .-_+__I-_l__~_+__C_~-_l_-~-+_~-~--~-~-+_.
I
I
-
II
-
I
I
ITil
I
I_.I \+’
I
I
I
I
1+*
I
i*il-II __I_~__~__L_I_1_~_-l__~-~-~_~-_~--~-~_. _-~-i_-l__l_l_:_J__I__X!:U_p;_.
__;_;-_/_-;-;-
I ~__~__~_~-~-~--~--t-~-.
I
I 1 Ikil -_:_~--~--~-~__:-_I--~__~_~_~_~_-I__+-~-_
/4+2=
illiI
I I-lITi _I__1-_l_-L_~_1_~__,__L_1_1_-1__L_I-I_.
(-2)
lTil _i_~__~__,_~_~_~‘_i__I_l_i_i__/__c_c_
I
ifi/ IQ I I _~__I-_l--r_r_T_~-~-,__~_~_~__:_-I__~-~-.
I
I
_l’--[--~ II
I
_A---_I---_
_;_-/--~--~-;-~_
I 1 lfil I ._C_&&-I--C-L-& I l-11 I I _-L-1-f_l__I__L-_L_1__I__l--1-l
I
I
-i-i-c-.
.-I-
I
I
I I
._~~T~--i--~_~_~-~--I__~_~_T-~_-~-~-T-.
11 Ifi1
’ I _f_l__~_-+-+_tI
t
I
I
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I I
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l__l__+_+-t_l__~_+-+-_ I
I
(56”‘) 2”) Using the explicit form of {II’}:=_~ matrices we find that the subspace N = {z: A% = 0, k = 0, + 1, + 21, which is invariant with respect to the process z(z), is equal to N = -+l •t
e6+
ell) +ye16,
(57)
where x and y are arbitrary real parameters. As x and y are fixed by initial conditions (x = x0 and y = yo), we may write, according to our previous statements, the evolution z(r) as z(z)=ZNOZ'(~)=[xO(e,+e6+e,,)+yOe,6]Oz'(z).
(58)
From the above relation we have Tr pS(z) = Tr Ez(r) = 1 if y, = a. Moreover, taking into account that z’(t) converges to 0 for t-+co, we obtain zeq=
[xO(el
+
e6+
ell> +ie161@
O,
(5%
or, identifying zes with the relevant density matrix, :+x0
0
0
I x0
peq=
8 0
Six o”
a?xo 0
0 0 0 : + X0
.
(60)
L. PAPIE
428
3”) Applying the definition of the global average of the spin observable we may express the time evolution of the components S” (see (25)) and Sk’ (see (26)) of the total spin operator of our system as follows: (S’)(7) = Tr -W(z)*&&
0 S, + S, 0 S,)) = 2(Ez12(7) +
Ez,,(z)) (61)
=
Wh2(7)
+
m15(7)),
and (Sf’)(7)=TrE(z(7).~[(S,OSq+S4OSI)+i(S2OS4+S4OSZ)1) = 2Uz,(7)
+ Ez,,(7)) k i(Ez,(7) + .%(7))1
= Wd7)
+ ~(7))
_+ h.(7)
-t- wd7Nl
(62)
,
where the m,(7) on the right-hand sides of (61) and (62) are determined by the Bloch equation (54). Let us find these elements of matrix D which are necessary to determine the explicit form of evolution for (S’) and (Sk’). From the definition (38) of matrix D and the form (56’), (56”) and (56”‘) of matrices {Ak}: = _2 we find that the required elements are equal to D 12.12 --D
IS,15 =
D 12.15 --D
IS.12 =
-$
Jo -
4(4J, + SJz) ,
;JO-4.8.J2, (63)
De.4= D,,, = D 4.13 =
Dl3.4
D l2,m
DIM
D13,,3
=
D,.,,
=
014.14
=
=
-5+6.4.J,-4.4.5,.
16 D14,8= -4.-J,,-4.4.J,, 6
and -
=
D,p,
=
DC,
=
D13.m
=
Dl4.m
=
0
for any other m than considered above. From these values of D,, i,j = 1, , 16, and the Bloch equation (54), it follows that $
(S’)(7)
i
(s*,)(~)
=
-4(451
+ 16J2)(S2)(7),
(S’)(O)
= m;,
(64)
and = -4(6J,,
+ lOJ, + 4J,)(S*‘)(7),
(S*‘)(O)
= m,+’
(65)
LIMIT DIFFUSION
MECHANISM
OF RELAXATION
429
The above two equations give (S’)(r)=m~exp
-G (
>
and (S”)(z)
= rnt'exp( - ;),
(67)
where ; = 4(45, + 16.Q I
and f = 4(6J, + lOJ, + 45,) . 2
Here T, is to be interpreted as the relaxation time of the z-component of the total spin operator (longitudinal relaxation time), and T2 is to be interpreted as the relaxation time of the perpendicular to the z-component of the total spin operator (perpendicular relaxation time).
Acknowledgements
The author wishes to thank Professor J.T. Lewis for valuable suggestions concerning the problem here discussed, Professor J.R. McConnell and Professor N.G. van Kampen for helpful comments and B. Lenoach for enlightening discussions on applications of asymptotic analysis of stochastic equations.
References 1) R. Kubo, Lectures in Theoretical Physics I, W.E. Brittin and LG. Dunham, eds. (Interscience, New York, 1959). 2) R. Kubo, Fluctuation, Relaxation and Resonance in Magnetic Systems, D. ter Haar, ed. (Oliver and Boyd, Edinburgh, 1962). 3) R. Kubo, J. Math. Phys. 4 (1963) 174. 4) N.G. van Kampen, Physica 74 (1974) 215. 5) N.G. van Kampen, Phys. Rep. 24C (1976) 172. 6) A.G. Redfield, IBM J. Research Develop. 1 (1957) 19. 7) H.S. Howe, J. Fluid Mech. 45 (1971) 769. 8) R.C. Bout-ret, Can. J. Phys. 40 (1962) 782. 9) R.C. Bourret, Nuovo Cim. 26 (1962) 1. 10) J.B. Keller, Proc. Symp. Appl. Math. 13 (1962) 227.
430
L. PAPIEZ
11) L. Van Hove, Physica 21 (1955) 517. 12) L. Van Hove, Physica 22 (1956) 342. 13) R.L. Stratonovich, Conditional Markov Processes and their Application to the Theory of Optimal Control (Elsevier, New York, 1968). 14) R.Z. Khasminskii, Theory Prob. Appl. II (1966) 390. 15) G.C. Papanicolaou and S.R.S. Varadhan. Comm. Pure Appt. Math. 26 (1973) 497. 16) G.C. Papanicolaou and W. Kohler, Comm. Pure Appl. Math. 27 (1974) 641. 17) A.D. Ventzel and M.J. Freidlin, Fluktuacii v Dinamitcheskikh Sistiemakh pod Deistviem Malykh Slutchainykh Vozmushtchenii (Nauka, Moscow, 1979). 18) 19) 20) 21) 22) 23) 24) 25) 26) 27) 28) 29) 30)
K. Tomita, Prog. Theor. Phys. (Kyoto) 19 (1958) 541. A. Abragam, The Principles of Nuclear Magnetism (Clarendon, Oxford, 1961). P.S. Hubbard, Rev. Mod. Phys. 33 (1961) 249. M. Spohn, Rev. Mod. Phys. 53 (1980) 569. N. Bloembergen, E.M. Purcell and R.V. Pound, Phys. Rev. 73 (1948) 679. G.W. Ford, J.T. Lewis and J.R. McConnell, Phys. Rev. Al9 (1979) 907. P.S. Hubbard, Phys. Rev. 180 (1969) 319. J.R. McConnell, Rotational Brownian Motion and Dielectric Theory (Academic Press, London, 1980). Y.A. Rozanov, Stationary Random Processes (Freeman, San Francisco, 1967). I.A. Ibragimov and Y.A. Rozanov. Gaussian Random Processes, (Springer. New York, 1978). R.Z. Khasminskii, Theory Prob. Appl. 11 (1966) 211. L. Arnold, Stochastic Differential Equations (Wiley, New York, 1974). A. Friedman, Stochastic Differential Equations and Applications (Academic Press, New York, 1975).
122A (1983) 413430
North-Holland Publishing Co.
THE LIMIT DIFFUSION
MECHANISM
OF RELAXATION
FOR SPIN SYSTEMS Lech PAPIEk* Dublin Institute for Advanced Studies, 10 Burlington Road, Dublin 4 Ireland
Received
1 June
1983
The diffusion limit theorem for stochastic differential equations is applied to analyse the dynamical evolutions of spin systems. Bloch equations are derived and the stability of asymptotic evolutions is proved. The theory is applied to nuclear magnetic relaxation of two spins.
1. Introduction and motivation
An effective method of statistical physics uses somewhere certain assumptions which are essentially of a stochastic nature. In the theory of nonequilibrium statistical mechanics advocated by Kubolm3) and Van Kampen4s5), where the time evolution of physical systems is postulated on the basis of stochastic dynamical equations (stochastic Liouville equations), these assumptions are expressed more openly. However, derivations of kinetic equations from stochastic Liouville equations”‘) have, in the past, caused much confusion. It was not clear that assumptions to neglect some correlations, necessary for these derivations, were justified”“) (this subtle problem was exhaustively discussed, for example, in ref. 5). By analogy with the deterministic case one may expect, however, that the Van Hove limit”,‘2) is the proper method for the analysis of stochastic Liouville equations, and clearly should lead to kinetic equations. Fortunately, a mathematical theory exists (the so-called asymptotic theory of stochastic multiplicative equations, which started in 1966 with the Stratonovich’3) and Khasminskii14) papers) which is appropriate for such analysis. We are thus prompted to check how such a theory will work for some simple physical systems. Spin systems, with their finite degrees of freedom and well-established relaxation results are specially good candidates. Thus the object of this paper is to show that the asymptotic theory of stochastic, multiplicative equations 13-“) applies to quantum evolution for spin systems in a random environment, and that it allows us to derive rigorously kinetic equations (Bloch equations) and to determine transport coefficients (relaxation times). Moreover, because our considerations are based on *On
leave of absence
from
Silesian
0378-4371/83/0000-0000/$03.00
University,
0
Katowice,
Poland.
1983 North-Holland
L. PAPIE
414
mathematical results, we are able to dig out those assumptions which are of an “essentially stochastic nature”, under which the relaxation in spin systems appears. The paper is constructed in the following way: Firstly, we state the general problem of the spin system evolution in a random environment. Then, in section 3, we describe a particular example of such a system for which we aim to derive explicit results; and, in section 4, we transform its evolution equation into multiplicative, stochastic form. In section 5 we quote the results which are later, in the more theoretical sections 6 and 7, rigorously justified. Section 6 is thus devoted to the statement of the relevant diffusion limit theorem which allows us to obtain the time evolution of spin systems, while in section 7 the general consequences of this evolution are investigated. Eventually, in section 8, we make the final calculations for the system of two spins.
2. Statement of the problem We aim to derive the Bloch equations and relaxation times for nuclear magnetism from stochastic dynamical equations. Thus the definition of the system under consideration is the same as in the Redfield6) and Kubo”) papers. The Hamiltonian of the system of spins and their molecular environment (bath) can be written in the form
(1)
H = f-f(S) + HP, 41,
where H(S) depends only on the spin variables S, and H(S, q) is the energy of interaction of the spin S and the molecular q degrees of freedom. The stochasticity of H(S, q) is the result of the random variation with time t (1 B 0) of coordinates q( if such probabilistic assumptions are made in consideration of the quantum evolution the theory is called “semiclassical N” ‘“). From quantum mechanical laws we have the evolution of our spin-bath system given by
dp(t) __dt
=+W)l,
PW=Po,
t 20,
where p is the density matrix and His the Hamiltonian (1). Eq. (2) is a stochastic one and its solution p(t) is a stochastic process which determines the evolution of the spin-bath system, while the average Ep(t) (over bath) may be interpreted as the density matrix of the spin subsystem. Our investigations of the quantum evolution (2) will proceed in the sense of the weak coupling limit”~‘*~*’ ). This treatment of the dynamical evolution equations, which leads to derivations of kinetic equations, started with Van Hove’s papers”,‘2), and is now one of the most powerful methods of nonequilibrium
LIMIT DIFFUSION
MECHANISM
OF RELAXATION
415
statistical mechanics*‘). In our case it means that instead of eq. (2) we should postulate y
=; [H, p’(t)]
P’(O)=
)
PO>
t
2 0,
where H is the same Hamiltonian as before, and treat as the (macroscopic) physical evolution that matrix p’(t) which is the solution of (3) for 6-0 and 0 G c*t = z < zo, where r. is arbitrary. E should be interpreted here as a scaling parameter, rather than any established physical quantity. To realize this let us express the time-dependent Hamiltonian H(t) given in (1) as H(t) = H’(t/c)
)
(4)
instead of postulating ad hoc the presence of t in (3). Then (2) yields
T =f[F(t),
pytjj
)
p’(0) = p.
)
where p’(t/c) = p(t). If we now demand H’(t) as we have required earlier for H(t), and 0 G c*t = z < z. will be the same as coupling postulate. In this sense the weak scaling.
tao,
(5)
the same stochastic characteristics for then the solution p’(t) of (5) for 6-0 the solution obtained using the weak coupling postulate is equivalent to time
3. The system of two spins in a random environment Here, we restrict our considerations to a simple system of two spins contained in a molecule placed in the random environment and interacting only through dipoledipole interactions. We assume also the absence of the external magnetic field, i.e. H(S) = 0 (let us notice that H(S, q) is the part of Hamiltonian (1) responsible for the relaxation). Using the example of this system, we are able to show how the method works while the explicit expressions are as simple as possible. The Hamiltonian H(t) = H(S, q) of the spin dipole-dipole interaction is in this case6*18-20) H(t) =
i
(6)
Fk(t)vk,
k=-2
and the Fk(t) are equal to Fk(t) =
0 1’2y*h2r-3(g
l)"Y;"(e(t),
q(t)),
k=O,
fl,
f2,
(7)
where the Y;k are second order spherical harmonics dependent on 8 and cpwhich
L.PAPIE
416
r “joining”
two spins in a
molecule, and r is the length of r. The second order tensor operators in (6) are’9,20)
0“*[sj@
vo= - ;
s, -
gs,OS,
+
s_, 0
Vk appearing
S,)]
= -~(2S,os,-s,os,-s2~s2), & v+‘=
(8)
+(s,oS,,+s,,oSJ = + (S, 0 S, + S, 0 S,) + i(S, 0 S2 + S2 0 SJ .
V’2=
-s+,os*,=
-(s,os,-s2os2)~(s2os,+s,o~2~,
where S,, S, and S, denote
the Pauli
spin matrices
s’=(; A),s*=(pii), s3=(;g,
(9)
where S,, = S, + is,. According to the statement of the problem, the Fk(t) are random variables for any t 3 0. Thus, to define our system we must determine the probabilistic properties of stochastic processes Fk(t). Let us mention the most obvious of them. From (7) and properties of spherical harmonics Yt it follows that (IF) F“(t) are bounded random variables for any t B 0. The natural requirement that the molecular environment is homogeneous in time, and isotropic, implies: (2F) Fk(t) are stationary processes in a wide sense, i.e. EFk(t) = C (without making our considerations less general we may choose C = 0), and Rk,k.(f,, t2) = f?Fk(tl)Fk’*(t2) = Rk,k.(tl - t2), and (3F)
the
correlation
matrix
Rkxk’(s) for
J”(t)
processes
is diagonal,
R,,&) = dk,k.Rk(S) where Rk(S) = EF”(s)E”*(O). (2F) was always assumed in physical models of nuclear relaxation6,” ‘O,**).(3F) follows from properties of spherical harmonics
i.e.
magnetic Y: and the
isotropic initial distribution of the orientation of a molecule carrying two spins (because of homogenity in time this isotropic distribution is preserved during the motion). This was shown, for example, by Ford, Lewis and McConnell (ref. 23, appendix) and by Hubbardz4). Further, the investigations of all specific, stochastic models of rotational motions of molecules demonstrate that25) (4F) correlation functions &(s) = EFk(s)F”*(0) are real, i.e. Rk(s) = R:(S), which implies R&s) = R:(s) = R&S) = R_,(s). At last we know that any nondegenerate stochastic process, in contrast with a deterministic one, must forget, to some extent, its previous evolution. Thus
LIMIT DIFFUSION
MECHANISM
OF RELAXATION
417
(5F) Processes Fk(r) acquire, quickly enough, independence of the time evolution from the previous one. The precise formulation of the condition (5F) we postpone to our later discussion of the mathematical theory of stochastic, multiplicative differential equations.
4. Reduction of the density matrix equation to a linear differential equation in Euclidean space Let us construct the auxiliary, real, Euclidean space R, in which the quantum evolution equation (3) assumes, in the case of the system of two spins, the form of a linear, differential equation. To this end let us define the scalar product ( , ) for the basis set of vectors {Si 8 S,}:,, , by i,j,k,
(Si @ Sj, Sk 0 SJ = iTr((Si @ Si>* (Sk @ SJ) ,
I = 1,2,3,4,
(10)
where S,, S, and S, are spin Pauli matrices (9), S, = lis the identity 2 x 2 matrix, and * denotes the matrix multiplication. From the above definition it follows that
(si 0
sj,
Sk0 sJ =
s&sj,
9
i,j, k, 1 = 1,2,3,4,
(11)
and in the standard notation for basis vectors, i.e. when S, @OS, = e2, . . . ,
S, 0 S, = e, , we
&@
&=
el6,
(12)
may write instead of (11) (%
ejl
=
6,)
i,j=l,...,16.
(13)
The set {e,>ia, may serve therefore as the orthonormal basis of the real, sixteen-dimensional Euclidean space R6, and any vector z cRi6, z = X:i’“,z,ei, where zi are real, may be thought of as a 4 x 4 hermitian matrix which is a linear combination of basis matrices {St @ S,>&=,. Since vectors from Rf6 are identical with 4 x 4 hermitian matrices, the scalar product (z, y) of z and y may be written as (z,.v) = iTr(z ‘Y) ;
(14)
hence immediately zi = (z, e,) = iTr(z
* eJ
,
(15)
in particular - ITT(z), 4
z16 -
(15’)
418
L. PAPIE
and 1~1’= $Tr(z2) .
(16)
Thus we recognize that any density matrix p for the system of two spins is a vector z ER” for which the last component zr6 is equal to f and /.z/Q 1. The other comdonents z,, i = 1, . . . , 15 of a vector z have by (15) the interpretation of quantum averages of observables e, defined in (12). Moreover, the Hamiltonian H(t) of our problem, defined by (6) (7), (8) may be considered as a stochastic process with values in Ri6. Therefore, since the commutator [ , ] in (3) is the relevant algebraic operation in Rf’, the evolution equation (3) with Hamiltonian (6) is the linear (multiplicative) stochastic differential equation in Ri6. More specifically we may write
%(~,z~ei)=:[H(t),~,z;ej]=~[k~2Fk(t)Vk,Sz = j$,k=:_2 Fk(t)
f, =
z;v
6
z:(O)e,
z
E
Rb6
(17)
Taking the scalar product of the left- and the right-hand side of (17) with basic vector e,, we obtain ; z; = 6 5 $ Fk(t)A;.zj i=lk=-?
,
GO) = z/ >
1=1,...,16,
(18)
where Ai are defined by A i = f (e,, [Vk.e,l),
l,j=1,...,16;
k=O,
+l,
_+2.
(19)
The hermiticity of Vk, ei and e, and the definition of the scalar product in RJ6 imply that matrices (Ak}:= _2 are skew-symmetric, i.e. (Ak)T= -Ak.
(20)
In a vector notation (18) takes the form dz’(t) PZZ dt
’ 6 ,=I_* Fk(t)A kz’(t) 7
Finally, we find by substitution (Ak)* = (-
The combination
z’(O) = z E R:‘.
(21)
of (8) and (12) into (19) that
l)kA -k.
(22)
of (22) with the analogous relation for spherical harmonics, i.e.
y’;*(f) = ( - l>kYF~(~)
or
Fk*(t) = (- l)kFpk(r),
(23)
LIMIT DIFFUSION
MECHANISM
OF RELAXATION
419
ensures that the operator B(t) = X:5 _2 Fk(t)Ak is a real, linear operator from Ri6 into Ri6. This concludes the transformation of the time evolution of our physical system into the problem of linear, multiplicative differential equation in the real, Euclidean space Rf6.
5. Results
To complete the discussion of our example, we now quote results which we will justify rigorously later. All these results follow from the solution p’(t) which we are able to obtain (for c-+0 and 0 < c2t = r < zo, p’(t)+p(r)) for the problem (3) when H is the Hamiltonian of the system given by (6), (7) and (8). This solution determines completely the evolution of the spin-bath system on the macroscopic time-scale. Let p,(r) denote the evolution of the density matrix of the spin subsystem, i.e. p,(z) = I@(z), and let (G) denote the global average (over the spin subsystem and its random environment-bath) of the spin observable G, i.e. (G ) = Tr E(p *G) = Tr(p, *G). The results are (i) The conditions Tr p,(r) = 1 and Tr pi(r) < 1 are fulfilled for any r E[O, ro] if they are true for r = 0; (ii) The observable x0(2$ @ S, + S, @ S2 + S, @IS,), with x0 a real parameter, is invariant during the evolution of the system. The parameter x0 is determined only by the initial conditions of the spin subsystem; (iii) The density matrix of the spin subsystem p,(z) for r+cc relaxes to the limit (equilibrium) density matrix pes given by
a+xo Pe4= i
0
0
a-x0
0 0
0
0
2x0
0
2x0
a-- x0
0
0
0
a + x0
r
)
(24)
where x0 is as in (ii); (iv) The evolutions of (S’) and (S*‘), where S’ and S*’ denote, respectively, z and + 1 components of the total spin operator of two spins S’ = and
gs,0 s, + s, 8 S,)
(25)
420
L.PAPIEi!
are given by
(S’)(T) = (S')(O) * exp - $ (
(27)
11
and (s*‘(r)
= (S”)(O).
(28)
exp
T, and T2 denote, respectively, longitudinal and perpendicular relaxation times, which will be derived in section 8. Because we assumed H(S) = 0 we should compare T, and T2 with the relaxation times for the case of extreme narrowing which were obtained earlier6~‘8~20~22). It IS easy to verify that our results are identical with those known earlier20,22). 6. The solutions of stochastic dynamical equations for spin systems Let us consider the stochastic multiplicative dz’(t) = t i dt k=
Franz’,
equation in R” of the kind
z’(O) = z E R” ,
t30,
-I
where Fk(t), t 3 0, k = 0, + 1, . . , +I are components of a given stochastic process F(t) on a probability space (0, 9, P) (we shall denote integration over 52relative to P, i.e. the bath average, by E), and {A “1: = _, are given constant n x n matrices. This equation is analogous to (21) except that n and I are now arbitrary natural numbers (in (21) n = 16 and I = 2); it describes spin systems which are more complex than those considered in our example. They must, however, be such as to satisfy conditions (20), (22), (23) and (IF)-(SF) for any k = 0, + 1, . . , + 1. In order to obtain the solution for (29) in a rigorous manner we must make the condition (5F) more exact. We assume that processes Fk(t) acquire the independence of the present evolution from the previous one in the sense, that: (5F’) correlation functions Rk(s) converge to zero for S-+CC fast enough to ensure that
s
lSRk(S)I
ds <
c,
(30)
0
and (SF”) process F(t) satisfies the mixing condition’“‘7,2”28), i.e.
(31)
LIMIT DIFFUSION
MECHANISM
421
OF RELAXATION
where 9: c 9,O < s 6 t < co denotes an o-algebra of events generated on s2 by F(u), s < u < t and a(s) fulfills 41 a”‘(s) ds < 00 .
(32)
s 0
As we said earlier, the property of “losing the memory” by process F(t) is obvious. However, the questions in what sense, and how fast it is lost, are open ones. In this context (5F’) and (5F”) should be seen mainly as sufficient conditions for P(t) under which the relaxation in the considered systems appears. Remark 1. It is known26,27)that for Gaussian stationary processes the-mixing condition is equivalent to the complete regularity condition’7,26,27),which allows us to estimate the correlation function. Therefore, by restricting ourselves to Gaussian processes, we may get rid of the condition (5F’). Remark 2. Ergodic Markov processes P(t)
are examples of processes for which
(5F”) is satisfied16). Let us state now some simple consequences of (20) (IF)-(SF’),
(5F”):
Lemma 1. The quantities Jk defined as 4,
R&) ds
Jk =
(33)
,
s 0
are equal to T
s
Rk(s -s’)ds’ds,
(34)
00 and (34) converges to Jk for T-CD as l/T. Proof. Lemma I is implied by (5F’) because, thanks to the partial integration of (34) we have Ts
lim 1 T-m
T ss 0 0
T
R,(s -s’)ds’ds
(T-s)R,(s)ds 0 T
=J+im_;
sR,(s)ds. -
s 0
We notice that by (4F) Jk are positive, and Jk = J-k, k = 0, k 1, . . . , _+1.
(35)
422
L. PAPIE
Corrofary.
Applying
(3F), (20), and lemma
I we find that following
limits:
TJ
and 1 c,,(z) = lim T-x>
T
)(
0 0 exist uniformly
in z. In vector notation,
h(z) = [b,(z)] =
Jk(Ak)+Akz=
f: k=
, k,&Fk’,djAk’z)*
ds’ds I
(37)
(36) and (37) may be put into the form -Dz
(38)
-I
and C(z) = [c,(z)] = where (A”)+ denotes Lemma Proof
k JkAkzzT(/tk)+, k= _, the hermitian
2. The matrices
The matrices
tTD{
=
i
(39)
conjugate
of Ak.
D = CL= _, Jk(Ak)+Ak and
D and
C(z) are nonnegative
definite.
C(z) are real, and for any 5 E R” we have
Jk(Akt)+A kt 3 0
(40)
and
CTC(z)< = i
Jk(p‘4kz)(
)
(41)
k=-I
as the Jk are positive. Let us notice here that matrices D and C(z), though nonnegative definite, may be degenerate since the Ak are skew-symmetric. Now, let us recall the limit theorems for stochastic multiplicative differential equations” I’). For our case the relevant versions of these theorems are given in the papers of Papanicolaou and Varadhan”) and Papanicolaou and Kohler16). It is not difficult to recognize that our problem is analogous to the one considered in theorem 3 of 15), and that after similar transformation, the general, abstract theorem 1 of 15) will apply. It would be more convenient, however, to recall the theorem of Papanicolaou and Kohler16) (PK Theorem). Because Fk(t) are
LIMIT DIFFUSION
MECHANISM
OF RELAXATION
423
bounded and since eq. (29) is linear (and not explicitly dependent on the macroscopic time scale) all the boundedness conditions needed for PI; Theorem are satisfied. Moreover, the existence of the drift &(z) and the diffusion c&z) coefficients, in the sense of limits (35), (36) and (37), and their explicit forms (38) and (39), ensure the fulfillment of the further conditions of the PK Theorem. Thus the PK Theorem applies. Its relevance for the formulation of our problem may be expressed as follows. Theorem. Let z’(t) = z”‘(z), 0 < .c2t = z < z,,, 6 ~(0, l] be the process defined by
eq. (29). Let (lF)-(5F’), (5F”), (20), (22) and (23) hold. Then the process z”‘(z) converge weakly as c--+0 (t-m and 0 < c2t = z < zO,z,, arbitrary) to the diffusion Markov process z(z) with infinitesimal generator L defined by L = f:
(42)
Cij(Z)
i,j= I
where b,(z) and co(z) are, respectively, drift and diffusion coefficients given by (38) and (39). Remark 3. Theorems also exist (ref. 17, theorems 2 and 3, chapter 7, section 3) which allow us to determine the behaviour of the solution z”(t) of the above problem for t E [0, q/t]. It turns out that for such time-intervals process z’(t) does not move significantly from its initial position’7*28).Changes in z’(t) for t E[O, z&l are of the order t’j2, and not of the order 1 as expected for t E[O, ~&2] from the above theorem’7’28). Th’1s explains, to some extent, why we have formulated our problem in the Van Hove limit.
7. Consequences of the diffusion limit solution for spin systems
For further discussion it would be more convenient to represent the diffusion Markov process z(z) with drift vector b(z) and diffusion matrix C(z), as the solution of the Ito stochastic equation29v30).We have dz(z) = b(z(z)) dz + a(z(T)) d W,,
Z(O)=ZER”,
z 20,
(43)
where G(Z) is a square root matrix of C(z) (because C(z) is a nonnegative definite matrix it has a unique square root a(~)~‘)), and W, is a standard, n-dimensional Wiener process. Lemma 3. The mean value Ez(z) of the process Z(T), T 2 0 is contained in K,,, = {z ER”: \zI G r} where r = (z(O)\, and z(0) denotes the starting point of the evolution.
L. PAPIE
424 Proof.
Let us consider
the function (44)
Because
of the following
identities: (45)
and (46) we have /z(*@
Af,,A$z,z,,
+ 4p(p - 1)
c
Jk
k=-I
A~A;;.~,,,z,,,,z,z~
c r,j,m,m’
1)
=
I
Iz(*~-*) >
jzp’1.
A;jA;;z,,,z,,,,
(47)
> The first and the third terms in (47) cancel, symmetric. Therefore (47) yields J%(Z) = 4p(p - 1) f: k= =
LU(Z) d 0, By Ito’s formula
Applying
skew-
JkzTAkzzT(Ak)+z/z12@-*)
definiteness for
Ak are
-/
(48)
of C(z) we obtain
Odp,
(49)
we have also
= Lv(z(z))
the average
w(7))
dz + ____az
o(z(z))
to the last expression
Edv(z(r))=ELv(z(7))<0,
Hence,
the matrices
4p(p - l)zTC(z)zlzl*@-Z),
and by nonnegative
dv(z(r))
since
for
O
d IV, .
(50)
we obtain < 1.
(51)
for p = $, v(z) = Iz(
JEz(7)l < E/47)(
< Elz(O)( = [z(O)/ .
Thus lemma 3 is true, and evidently also it implies that the inequality is preserved during the evolution of the system.
(52)
Tr p:(7) < 1
LIMIT DIFFUSION
MECHANISM
OF RELAXATION
425
Actually, the relations (49) and (51) are stronger than are needed to obtain (52). They also give an opportunity for comment on the stability of the process z(7). Such discussion is, however, complicated by the fact that L may be a degenerate elliptic operator30). The degeneracy of L appears if C(z) turns out to be zero for some set N of points of R”. The form of C(z) implies that if such a set N exists it must be a linear subspace of R”. This is true, since N = n:= _, N(A “), where N(Ak) = {z QR”: A k~ = O}. Moreover, we may notice that for z EN b(z) = 0 and a(z) = 0. Therefore, by (43) N is an invariant subspace with respect to the process z(7). The decomposition of the process z(7) for N and N’ subspaces (where NON’ = R”) allows us to see it as the direct sum of two vectors evolving in time. The first vector is a constant vector z, in N, and the second is equal to the relevant diffusion process z’(7) in N’ with strictly elliptic infinitesimal generator L’ (for z # 0) (this is also clear if eq. (29) is linearly transformed so that the components in N and N’ are separated). For the generator L’ it will then be possible, in the same way as above, to obtain the relation (49) (but with strong inequalities), i.e. L%(z)
< 0)
for
O
z#O.
(53)
But this is a sufficient condition for asymptotic stability in the largez9v30)of the process z’(7). Thus all coordinates of the process z’(7) converge to 0 for 7--m. This means that the system for 7+co relaxes to the uniquely defined equilibrium position zcq= zN0 0, where z,+ N is the constant of the motion determined by initial conditions. Let us now turn our attention to the problem of determining the subsystem evolution, i.e. to the behaviour in time of the average Ez(7) of the process z(7): From (43) we have immediately for m(7) = Ez(7)
dm(4 - d7 =
-Dm(z),
m(O)=z,
or m(z)= exp( - D7)z ,
(55)
where D is the matrix D = I$= _, .I,(,4 “)+Ak. The last relation gives us the rate of decay of m(7). Thus (54) may be called the Bloch equation of the stochastic dynamical system given by (29) (IF)-(SF’), (5F”), (20) (22) and (23). The reason we choose this name becomes clear when we apply (54) to the system considered in 2.
8. Calculation for the system considered in 3
1”) TO find the explicit form of the limit diffusion z(7) in the case of the example considered in this paper we must first know all matrices (A “>i= -7, Applying definitions of the basis set (ei}i6=,(12), (9), and operators (V”}:= _2 (8) (9) to (19) we get after elementary calculations the following form of matrices (Ak)i, :2:
426
L. PAPIE III IIll I __~_~__~__~_~_~_~__~__~_~_~-~__~--~-~__ I .-+-_1-
I
II -,__
I
I
I
I I I III ~-+-~-~__,--+_+_~-~__,_-+-+--
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I
I
I
I
l-11
I
T-T--’
I
II
I
I I I
1;
I
:
I
I
I
I
I
I
I
l -~-~~-C-C-I-~-~--C-C-~-~-~__~_L_~-I I I I I I I , / / I
I
I
I
I
I
I
I
I
I
I
I /
I ,
I I
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II
Ifll
Ill I I-l I -;-;--~-t-~-;-i--k-t-+-+-+--F-t-+--
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I
II
1--r-t-t-
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t
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lfll-il
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l-ii
I
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I
1;
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I
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I4
iii
(56’)
I I
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I
I
I Iii I C-&-4-4_-l--_C-_C-I I I I L_L_I_A_J__L_L__ I I I Ii’
I
I
l_il
I I
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I If11 I I ~_+_~_~__,--_C-_C-I I I I I
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r-+-t--l--l--~-+--
I
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l
;-,;&,r-T--
I
I
I
I
lil
(56”)
I + ‘I
I
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.-+t-_-I--_I---__++-_+-___-_I__-_t+f-_--_--,---_-+-
I-il lfll I I --I-~--l--L-L-1-~__I__L_L_1_~__I__L_L_.
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lil
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_-~-~--~--~-~-~-
I
)
I
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III/l//I//I
I
LIMIT DIFFUSION I
I
I
MECHANISM
I
I
I
OF RELAXATION
I
I
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I
I
l*il
421 I
Ifi;
T_;_7-7--r-r-T-. I-
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I
I
-
II
-
I
I
ITil
I
I_.I \+’
I
I
I
I
1+*
I
i*il-II __I_~__~__L_I_1_~_-l__~-~-~_~-_~--~-~_. _-~-i_-l__l_l_:_J__I__X!:U_p;_.
__;_;-_/_-;-;-
I ~__~__~_~-~-~--~--t-~-.
I
I 1 Ikil -_:_~--~--~-~__:-_I--~__~_~_~_~_-I__+-~-_
/4+2=
illiI
I I-lITi _I__1-_l_-L_~_1_~__,__L_1_1_-1__L_I-I_.
(-2)
lTil _i_~__~__,_~_~_~‘_i__I_l_i_i__/__c_c_
I
ifi/ IQ I I _~__I-_l--r_r_T_~-~-,__~_~_~__:_-I__~-~-.
I
I
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I
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_;_-/--~--~-;-~_
I 1 lfil I ._C_&&-I--C-L-& I l-11 I I _-L-1-f_l__I__L-_L_1__I__l--1-l
I
I
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.-I-
I
I
I I
._~~T~--i--~_~_~-~--I__~_~_T-~_-~-~-T-.
11 Ifi1
’ I _f_l__~_-+-+_tI
t
I
I
I
I
I
I
I I
I
I
I
I
I
I
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l__l__+_+-t_l__~_+-+-_ I
I
(56”‘) 2”) Using the explicit form of {II’}:=_~ matrices we find that the subspace N = {z: A% = 0, k = 0, + 1, + 21, which is invariant with respect to the process z(z), is equal to N = -+l •t
e6+
ell) +ye16,
(57)
where x and y are arbitrary real parameters. As x and y are fixed by initial conditions (x = x0 and y = yo), we may write, according to our previous statements, the evolution z(r) as z(z)=ZNOZ'(~)=[xO(e,+e6+e,,)+yOe,6]Oz'(z).
(58)
From the above relation we have Tr pS(z) = Tr Ez(r) = 1 if y, = a. Moreover, taking into account that z’(t) converges to 0 for t-+co, we obtain zeq=
[xO(el
+
e6+
ell> +ie161@
O,
(5%
or, identifying zes with the relevant density matrix, :+x0
0
0
I x0
peq=
8 0
Six o”
a?xo 0
0 0 0 : + X0
.
(60)
L. PAPIE
428
3”) Applying the definition of the global average of the spin observable we may express the time evolution of the components S” (see (25)) and Sk’ (see (26)) of the total spin operator of our system as follows: (S’)(7) = Tr -W(z)*&&
0 S, + S, 0 S,)) = 2(Ez12(7) +
Ez,,(z)) (61)
=
Wh2(7)
+
m15(7)),
and (Sf’)(7)=TrE(z(7).~[(S,OSq+S4OSI)+i(S2OS4+S4OSZ)1) = 2Uz,(7)
+ Ez,,(7)) k i(Ez,(7) + .%(7))1
= Wd7)
+ ~(7))
_+ h.(7)
-t- wd7Nl
(62)
,
where the m,(7) on the right-hand sides of (61) and (62) are determined by the Bloch equation (54). Let us find these elements of matrix D which are necessary to determine the explicit form of evolution for (S’) and (Sk’). From the definition (38) of matrix D and the form (56’), (56”) and (56”‘) of matrices {Ak}: = _2 we find that the required elements are equal to D 12.12 --D
IS,15 =
D 12.15 --D
IS.12 =
-$
Jo -
4(4J, + SJz) ,
;JO-4.8.J2, (63)
De.4= D,,, = D 4.13 =
Dl3.4
D l2,m
DIM
D13,,3
=
D,.,,
=
014.14
=
=
-5+6.4.J,-4.4.5,.
16 D14,8= -4.-J,,-4.4.J,, 6
and -
=
D,p,
=
DC,
=
D13.m
=
Dl4.m
=
0
for any other m than considered above. From these values of D,, i,j = 1, , 16, and the Bloch equation (54), it follows that $
(S’)(7)
i
(s*,)(~)
=
-4(451
+ 16J2)(S2)(7),
(S’)(O)
= m;,
(64)
and = -4(6J,,
+ lOJ, + 4J,)(S*‘)(7),
(S*‘)(O)
= m,+’
(65)
LIMIT DIFFUSION
MECHANISM
OF RELAXATION
429
The above two equations give (S’)(r)=m~exp
-G (
>
and (S”)(z)
= rnt'exp( - ;),
(67)
where ; = 4(45, + 16.Q I
and f = 4(6J, + lOJ, + 45,) . 2
Here T, is to be interpreted as the relaxation time of the z-component of the total spin operator (longitudinal relaxation time), and T2 is to be interpreted as the relaxation time of the perpendicular to the z-component of the total spin operator (perpendicular relaxation time).
Acknowledgements
The author wishes to thank Professor J.T. Lewis for valuable suggestions concerning the problem here discussed, Professor J.R. McConnell and Professor N.G. van Kampen for helpful comments and B. Lenoach for enlightening discussions on applications of asymptotic analysis of stochastic equations.
References 1) R. Kubo, Lectures in Theoretical Physics I, W.E. Brittin and LG. Dunham, eds. (Interscience, New York, 1959). 2) R. Kubo, Fluctuation, Relaxation and Resonance in Magnetic Systems, D. ter Haar, ed. (Oliver and Boyd, Edinburgh, 1962). 3) R. Kubo, J. Math. Phys. 4 (1963) 174. 4) N.G. van Kampen, Physica 74 (1974) 215. 5) N.G. van Kampen, Phys. Rep. 24C (1976) 172. 6) A.G. Redfield, IBM J. Research Develop. 1 (1957) 19. 7) H.S. Howe, J. Fluid Mech. 45 (1971) 769. 8) R.C. Bout-ret, Can. J. Phys. 40 (1962) 782. 9) R.C. Bourret, Nuovo Cim. 26 (1962) 1. 10) J.B. Keller, Proc. Symp. Appl. Math. 13 (1962) 227.
430
L. PAPIEZ
11) L. Van Hove, Physica 21 (1955) 517. 12) L. Van Hove, Physica 22 (1956) 342. 13) R.L. Stratonovich, Conditional Markov Processes and their Application to the Theory of Optimal Control (Elsevier, New York, 1968). 14) R.Z. Khasminskii, Theory Prob. Appl. II (1966) 390. 15) G.C. Papanicolaou and S.R.S. Varadhan. Comm. Pure Appt. Math. 26 (1973) 497. 16) G.C. Papanicolaou and W. Kohler, Comm. Pure Appl. Math. 27 (1974) 641. 17) A.D. Ventzel and M.J. Freidlin, Fluktuacii v Dinamitcheskikh Sistiemakh pod Deistviem Malykh Slutchainykh Vozmushtchenii (Nauka, Moscow, 1979). 18) 19) 20) 21) 22) 23) 24) 25) 26) 27) 28) 29) 30)
K. Tomita, Prog. Theor. Phys. (Kyoto) 19 (1958) 541. A. Abragam, The Principles of Nuclear Magnetism (Clarendon, Oxford, 1961). P.S. Hubbard, Rev. Mod. Phys. 33 (1961) 249. M. Spohn, Rev. Mod. Phys. 53 (1980) 569. N. Bloembergen, E.M. Purcell and R.V. Pound, Phys. Rev. 73 (1948) 679. G.W. Ford, J.T. Lewis and J.R. McConnell, Phys. Rev. Al9 (1979) 907. P.S. Hubbard, Phys. Rev. 180 (1969) 319. J.R. McConnell, Rotational Brownian Motion and Dielectric Theory (Academic Press, London, 1980). Y.A. Rozanov, Stationary Random Processes (Freeman, San Francisco, 1967). I.A. Ibragimov and Y.A. Rozanov. Gaussian Random Processes, (Springer. New York, 1978). R.Z. Khasminskii, Theory Prob. Appl. 11 (1966) 211. L. Arnold, Stochastic Differential Equations (Wiley, New York, 1974). A. Friedman, Stochastic Differential Equations and Applications (Academic Press, New York, 1975).