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From two-state jump to Gaussian stochastic processes
Physica 149A (1988) 447-471 North-Holland, Amsterdam
FROM TWO-STATE JUMP TO GAUSSIAN STOCHASTIC PROCESSES Kishiko MARUYAMA
and Fumiaki SHIBATA
Department of Physics, Faculty of Science, Ochanomiru
University, Bunkyo-ku,
Tokyo 112, Japan
Received 10 August 1987
A stochastic process, which bridges the two-state jump Markoff process and the Gaussian process, is introduced and fundamental properties of the process are fully analyzed. The resulting expressions are combined with the “partial cumulant” expansion formula to give an exact power spectrum in the form of a continued fraction. A rigorous expression is also obtained for a relaxation function in an analytic form.
1. Introduction In treating various phenomena in statistical and condensed matter physics, we sometimes introduce a stochastic model to simplify a problem. There are two typical stochastic processes which are frequently used’,‘); the two-state jump Markoff and the Gaussian processes. There is a remarkable contrast between the two. The former process is characterized by two values of realization whereas the latter by an infinite number of realizations. In this sense, they are opposite extremes. Moreover, these two processes have been used because of their mathematical simplicity. The higher order “partial cumulants” in the former and the “order cumulants” in the latter vanish identically, yielding simple structures in their expansion formulae (see the main text on the precise definition of the “cumulant”). In actual cases, however, we have to deal with stochastic processes which are neither the two-state jump process nor the Gaussian process. Thus we encounter obstacles inherent in real stochastic processes. We give here a method of treating this difficult problem without losing the simplicity of the two-state jump and the Gaussian processes. The method should be formulated so as to include these two processes as limiting special cases. We introduce a stochastic process composed of a superposition of many two-state jump Markoff processes. Properties of this composed process are fully investigated in the subsequent sections. The partial cumulant expansion 0378-4371/88/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
K. MARUYAMA
448
AND F. SHIBATA
method3) is generally applicable. A simple example is given to serve for illustration of our method; this will clarify the usefulness of our method.
2. Partial cumulant We summarize here general properties of the partial cumulant3) some of which are already known; it may be convenient to give a summary for later development of the present paper. For a certain stochastic process w(t), the jth order partial cumulant is defined as
(4fM4 >. . . w(t,_,)),.,. = g’w(f)g@(t,). . .94lj-,) by using a specific projection 9x=
(x)
operator
(2.1)
9:
)
Q-2)
and 9=1-C??,
(2.3)
where (. . * ) re p resents an average over the process w(t). Sometimes we simplify the stochastic process to be a two-state jump Markoff process or a Gaussian Markoff process. For such processes, multi-time correlation functions can be expressed in terms of double time correlation functions. Moreover for the process of zero mean:
(4))
=o,
(2.4)
the so-called irreducible condition of the partial cumulant is satisfied. Let us illustrate this condition. For this purpose we rewrite the partial cumulant in an alternative form,
Next we note the following: when a single B appears in the correlation o(tj)‘s, for instance,
(4)4ww,)~~~
>>
of
(2.6a)
FROM TWO-STATE JUMP TO GAUSSIAN STOCHASTIC PROCESSES
the correlation
449
is divided into two parts like
which we may call a reducible term, whereas if we have a single ?! instead of 9, the reducible term like (2.6b) is subtracted from the whole correlation and hence the term becomes “irreducible” by definition. We explain that the irreducible condition is satisfied by considering the fourth order partial cumulant of the Gaussian process:
=
(w(t)w(t,)o(t*)o(t,))- (w(t)w(t,))(w(t,)o(t,))
(2.6~)
=
(w(t)o(t*))(w(t,)w(t,))+ (w(t)w(t~))(w(t,)o(t*)).
(2.6d)
The reducible term (the second term) in (2.6~) is cancelled out by a part of the first term and we are left only with the irreducible ones, (2.6d). This means that the irreducible condition is fulfilled for the fourth order partial cumulant of the Gaussian process as stated above. In the same way can prove that the higher order partial cumulants become irreducible by inspecting the right-hand side of (2.5): all the reducible terms are subtracted owing to the presence of 9’s in every intermediate position, We now turn to the two-state jump Markoff process, for which the following theorem of multi-time correlation is proved4): (2.7) fort>t,>t,-*-.
correlation
With the repeated use of (2.7), we find that the j-tuple time function becomes reducible for j 2 3 yielding
according to the irreducible condition. Though this is a known relation, we give here a short proof for later purposes: It may be convenient to rewrite (2.7) in the form
with the projection operator (2.2). The first 2 in the right hand side of (2.5) is expanded into two parts, namely,
K. MARUYAMA
450
=
AND F. SHIBATA
(w(t)w(t,)~2w(t,)~2w(t,). . .22w(t,_,))
- (o(t)~~(t,)~2w(t,)~2w(t,). . * 9w(t,-,))
(2.10)
The second term of (2.10), written in the form
(o(t>)(w(t,)o(t*)w(t,).. .4-AC vanishes owing to (2.4). Furthermore
=
(w(t)o(t*)w(t,)~w(t,). ” -
2
expanding the next 2, we get
'w(tj-l))
(w(t)w(t,)~‘w(t,)~2w(t,). . .2&-,>)
2
(2.11)
where the first term is equal to the second term except for the opposite sign due to (2.9). Thus we have a conclusion that the jth order partial cumulant of two-state jump Markoff process vanishes for i 2 3, (2.8). A short summary of this section is as follows: the partial cumulant defined by (2.1) with the projection (2.2) is irreducible under the condition (2.4). Especially for the two-state jump Markoff process, we have a simple relation (2.8).
3. Structure of the partial cumulant Let us consider the structure of a partial cumulant for a stochastic process I composed of N independent processes each of which is assumed to be a two-state jump Markoff process. Hereafter we assume the following time sequence: t > t, > t2 > . . . . 3.1. Case of N= 2 We first consider a stochastic process given by
W@)(t) = wl(t)
+ W*(t)
)
(3.1)
where each of the constituent processes or(t) and m2(t) is assumed to be a two-state jump Markoff process with zero mean. For each process (n = 1 and
FROM TWO-STATE
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STOCHASTIC
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451
2), we have (49)
=o 7
(w,(t)~~,(t’))
(3.2) = cY,,.A; e-y’r-f”
(3.3)
and (o,(t)On(tl)"'w,(tj-l))p.~.=O
(for
ia3).
(3.4)
The second order partial cumulant for w(t) is given by (W(*)(f)W(*)(fl))P.C. = (4)44)) = 24
+
bJ*w*w
e-Y(~-~,)
(3Sa) (3Sb)
E (+ -)@) )
(3.5c)
where only the signs in front of t and t, in the parenthesis in the exponent are explicitly written in the last member (3.5~). Namely, the form of the exponential function in (3.5b) is suggested by the symbol (+ -), although the multiplicative factor depends on the constituent processes. We can rewrite this partial cumulant as
(W(*)(t)W(*)(tl))p.c. =‘%W by introducing
(3.5d)
t,) ,
the function
ek(t) = eekyr .
(3.6)
A diagrammatic representation of (3.5d) is shown in fig. 1, where the upward arrow denotes the + sign and the downward arrow -. We also assign the horizontal line to (cl,(t - tl). In other words, the contour of this diagram represents the form of the exponential function, (+ -). On the other hand, the
H <
: t
Fig. 1. Second-order
t1 P.C. graph for N = 2.
K. MARUYAMA
452
AND
F. SHIBATA
double arrow indicates two possible choices of wl(t) and u*(t). Once the choice is made at the time t, the corresponding process at t, is uniquely determined, yielding a single arrow at t, . Thus we get the factor of 2 in (3.5d) by calculating the possible number of ways going through the contour from the left to the right in the direction of the arrows. The analysis of the fourth order partial cumulant parallels above: ( 0(2)(t)W(2)(tt)W(2)(t2)~(2)(t~))p,C, =
(wl(t)W,(t*))(w,(t,)o*(t,))
+
+
(W2(t)02(t*))(W,(tl)Wl(t3))
(wl(t>wl(t,))(o,(t,)w,(t,))
+
(W2(t)02(t3))(Wl(tl)Ol(f*))
= 44: e- vV+r,-r,-r3) E
(+
+
-
(3.7a) (3.7b)
_p
(3.7c)
=4A;W- t,)Mt, - t2)+4(t2- td >
(3.7d)
where we have used the same sign convention as (3.5). The graph contributing to (3.7) is shown in fig. 2 where the upward (downward) arrows denote the +(-) sign in (3.7c), that is, the contour of this diagram represents the form of the exponential function, (+ + - -). The first horizontal line of the stairs corresponds to @, whereas the second to &. The double arrow indicates two possibilities of the choice between q(t) and w*(t). The process at time t, (t3) has to be different from the one at time t(t2), otherwise the diagram becomes reducible. Thus we have a single line at t,. Finally we have a front factor of 2* = 4 from fig. 2. Similarly, the sixth order partial cumulant can be written as (0(2)(t)o(2)(t,)W(2)(t2)Wo(t3)00(t~)~~2~(t~))~,~
=8AEe
-y(t+t,-r,+t,-t,-r,)
Fig. 2. Fourth-order
(3.8a)
P.C. graph
for N = 2.
FROM TWO-STATE JUMP TO GAUSSIAN E
(+
+
-
+
STOCHASTIC
PROCESSES
_)(2)
-
453
(3.8b)
The graph shown in fig. 3 corresponds to (3.8~). Conversely, we can easily read fig. 3 to obtain (3.8~): Assigning I,/+to each of the first step and JI, to the second steps and multiplying by 23, we can reproduce (3.8~) from fig. 3. The above procedure can be extended to calculate higher order partial cumulants with the aid of the diagrammatic representation. As is shown in fig. 4, an arbitrary higher order diagram is obtained as a simple extension of figs. 2 and 3. The diagram is composed of the first horizontal lines representing ~/+((t~~ - t2j+I)‘s and the second ones corresponding to J12(t2j_l - t,)‘s. For 2jth order, the number of paths is 2’. Thus we get ( 0J(2)(t)W(2)(tl) * * . o’2’(t2j_l))p.c.
3.2. Case of N= 3 Next we will consider the following process: W(3)(t) = q(t)
<
(3.10)
+ W2(t) + w,(t) )
; t
t1
f2
f3
tc
f5
Fig. 3. Sixth-order P.C. graph for N = 2.
----__---. t1
f2
f3
t4
f,-3
Fig. 4. Arbitrary order P.C. graph for N = 2.
fj-2
t,-1
K. MARUYAMA
454
AND F. SHIBATA
where, for each of the constituent processes, the relations (3.4) are satisfied. The second order partial cumulant is calculated as (~(~)(t)~(~)(t,))r~~,
(3.2), (3.3) and
= 34: e-y(‘-rlt
(3.11a)
s (+
(3.11b)
_)(3)
(3.11c)
=3A;$,(t-t,),
where the symbols and the functions are the same or simple extensions of the previous ones. The graph representing the second order contribution is drawn in fig. 5 where the triple arrow appears owing to the possible choice of the three independent processes. This gives the front factor in (3.11~). The fourth order partial cumulant becomes ( W(3)(t)o(3)(fl)W(3)(t*)0(3)(f3))P,C, = 124; eP Y@+rl-~z-q
(3.12a)
s (+ + -
(3.12b)
_)(3)
= 12A;llr,(r - f,)k(rl
- &)+,(t, -
~3)
(3.12~)
>
corresponding to fig. 6. As before we can read fig. 6 as representing the form of the exponential function and the front factor: that is, the first horizontal line denotes $r and the second horizontal line q$ while the number of possible ways for the vertical arrows gives the coefficient. The upward arrow at time t is triple owing to three possibilities of choice among W, , w2 and w3, whereas at time t, we have a double arrow because of two remaining possible processes except the one at t. The next downward arrow at time t, becomes double because the two processes which appeared at time t or t, must be paired at t,; the arrow at time t3 is single. As a result, this diagram gives the factor of 3 x 2 X 2 X 1 = 12. Similarly the sixth order partial cumulant is given by
<
: t
Fig. 5. Graphical representation
t1 of the second-order
P.C. for N = 3.
FROM TWO-STATE JUMP TO GAUSSIAN
STOCHASTIC
PROCESSES
455
,
L
\,
<
: t
t1
f3
f2
Fig. 6. Graphical representation
of the fourth-order
P.C. for N = 3.
+ < : t
t1
f2
< :
t
ts
f4
f3
Fig. 7. Graphical representation
t1
f3
f2
f5
f4
of the sixth-order P.C. for N = 3.
( 0(3)(t)W(3)(t,)W(3)(t*)W(3)(t3)0(3)(t4)0(3)(t5))P.C. = 484: e -y(t+tl-r,+tj-t~-r,) + 36A; e-Y(t+r*+r,-l,-l,-r*) s
(+
+
-
+
= 48A:W +
-
_)(3)
- t,)W*
36GM
+
(+
+
+
-
-
- ~*)Jll(~*-
(3.13a)
_)(3)
(3.13b)
t3)W3
- fl)+*(tl - f2)&(‘3(t*-
-
t3)W3
OJll(f4
-
-
b)v+(4
G)
-
fs)
7
(3.13c)
with two diagrams corresponding to (+ + - + - -)(3) and (+ + + - - -)(3) in fig. 7. The first diagram corresponding to (+ + - + - -)(3) gives the front factor of 3 X 2 X 2 X 2 X 2 X 1 = 48 and the second (+ + + - - -)(3) 3 x 2 x 1 x 3 x 2 X 1 = 36. Analogously we can proceed to write down higher order partial cumulants. This will be done in a more systematic way in the next section.
4. General structure
of the partial cumulant
for arbitrary
The foregoing discussions are now generalized P(t)
= $ n=l
w,(t)
)
iV
to the stochastic process (4.1)
K. MARUYAMA
456
AND F. SHIBATA
where each constituent process is assumed to be the two-state jump Markoff process. Namely, the relations (3.2), (3.3) and (3.4) are satisfied. Let us calculate a first few partial cumulants. For the second order, we obtain s
(+
-j(N)
(4.2a)
= N&,(t
(4.2b)
- tl) >
with the diagrammatic representation shown in fig. 8. The notations in (4.2) are the same as the ones in the previous sections. The factor N comes from N possibilities of the choice of a single stochastic process among w1(t), q(t). . . and q,,(t). This is represented by the thick arrow at time t. The fourth order partial cumulant is given by ( 0(N)(t)W(N)(tl)W(N)(t2)W(N)(tj))p,C, F
(+
+
_
=
=‘(N -
-j(N)
(4.3a)
W$W - tlh(fl
- t,)h(t, - td
2
(4.3b)
corresponding to fig. 9. The thick arow at time t is N-fold, and the process at time t, must be different from the one at time t according to the irreducible condition. Thus the thick arrow at time t, is (N - 1)-fold. The double arrow at time t, shows two pairing possibilities with the two processes at time t or t,. There remains only a single process at time t, giving a single arrow. Thus we get the coefficient 2N(N - 1) by calculating the possible numbers of choice and pairing of the stochastic processes from the diagram. There arise no essential difficulties as we proceed beyond the lower orders, and thus we obtain the following general prescriptions of the general term for the 2jth order partial cumulant: (i) Draw all possible 2jth order irreducible diagrams, starting from the left bottom and ending at the right bottom, composed of 2j vertical lines and (2j - 1) horizontal lines. Attach directed arrows one for each to the vertical
t < Fig. 8. Second-order
: t
t1
P.C. graph for arbitrary N.
FROM TWO-STATE JUMP TO GAUSSIAN
f
: t
t1
Fig. 9. Fourth-order
f2
STOCHASTIC
457
PROCESSES
f3
P.C. graph for arbitrary N.
line from the left bottom to the right, assigning times t, t,, t,, . . * 7 t*j-l' According to the “irreducible condition” the intermediate arrows never lead to the bottom. Otherwise the diagram becomes reducible. Determine a mathematical expression associated with each diagram by the following items, and sum up those expressions coming from all diagrams. (ii) For the first upward vertical arrow at time t, associate a numerical factor N indicating an N-fold arrow. For the second upward arrows which connect the second level with the first level counting from the bottom except the bottom one, associate a factor of (N - 1) and so on. For the last downward arrow at tZj_r, associate a factor of one. For the second downward arrows which connect the first level with the second level, associate a factor of two and so on. Thus the total numerical factor can be read from a given diagram immediately. (iii) For horizontal lines of the first level counting from the bottom except the bottom one, assign the function I,!J,. Similarly, assign I&, I,$, . . . , I),,, to horizontal lines of the second, third, . . . levels. As an example, we give here an expression of the eighth order partial cumulant. This may serve for illustration. The corresponding diagrams are shown in fig. 10. Using the same notation as before, we find the partial cumulant in the form (+ + - + - + - -)(N) + (+ + - + + - - -)(N) + + (+ + + - + - -
-)(N) +
(+ + + + - - -
(+
+ +
-)(N) )
-
-
+ -
_p
(4.4a)
which correspond to the diagrams (a), (b), (c), (d) and (e) of fig. 10, respectively. Next, let us combine the rules (ii) and (iii) by introducing the following quantity: fjk=k(N-k+l)Qk. Thus
we get
(4.5)
K. MARUYAMA
458
<
a
: t
t1
AND F. SHIBATA
f2
f3
fL
ts
fii
t2
t3
tc
t5
t6
t7
t2
f3
t4
fs
f6
f7
f2
t3
t7
+I < : bt
t1
+t
+I f
:
dt
t1
+ (
:
tt
Fig. 10. Eighth-order
t4 P.C. graphs
f5
f7
f6
for arbitrary
N.
459
FROM TWO-STATE JUMP TO GAUSSIAN STOCHASTIC PROCESSES
where we have suppressed the arguments of &‘s for simplicity: These are (t - tr), (r1 - t2), . . . and (t, - f7), respectively.
5. Simple example As an application of the present formalism, consider an equation of motion for a complex variable x(t) (a model of random frequency modulation5’6)): i(t) = iWCN’(t)x(t) ,
(5.1)
where the stochastic process tiCN)(t) is governed by (4.1). An averaged quantity (x(t)) over the stochastic process can be obtained with the use of the time convolution formula3) (see appendix A):
(i(t))
~2 (-1)’ J dtl J dt2 * * * J 0 '
(w(N)(t)w(N)(tl)'
dt2j-1
0
0 *.
o'N'(t2j-l))p.C.(X(f2j-l))
7
(5.2)
with the initial condition
x(O)= MO)) *
(5.3)
The partial cumulant appearing in (5.2) has been fully analyzed in the preceding section. Eq. (5.2) can be solved by the method of Laplace transforms. Thus we can solve (5.2) in the form
(x), = G'N'[s]x(0)
,
(5.4)
where we have introduced dt eWsr(x(t)) 0
(5.5)
K. MARUYAMA
460
AND F. SHIBATA
and G’N’[~] = (s +
Here Z\“‘[s]
Z’;“‘[s])-’
(5.6)
is explicitly given by
with
fk[Sl= 1 dt
e-s’+k(t) = (S + ky))’
(5.8a)
,
and dt e-S’$k(t) = k(N - k + 1) /(s + ky ) .
Now we can reinterpret the diagrams corresponding as X(;N)[s]: Namely, I+!Q is replaced by fk. The equation (5.7) can be rewritten in the form
$N)[s] = A;i
-
A;~X$N'[s]f,
where the quantity Si”‘[s]
Z;“‘[s] = A;L -
(5.8b)
to the partial cumulants
+ A~f$';y'[s]f&""'[~]f~
+. . . ,
does not contain any f,‘s and is represented
A;~2'N'[s]f2
+ A~~*~~N'[s]f,~~N'[s]f,
+. . . ,
(5.9) by (5.10)
Z iN)[s] being defined analogously. For an arbitrary k (
(5.11) for k < N. On the other hand we find (5.12) and Zlf”‘[s] = 0 for k > N.
461
FROM TWO-STATE JUMP TO GAUSSIAN STOCHASTIC PROCESSES
Substituting the above quantities successively into (5.6) and using (5.8a) and (5.8b), we find GCN’[s] in an explicit form: 1
GcN’[s] =
(5.13)
NA;
.S+
2(N - l)A;
s+y+
3(N - 2)A;
s+2y+
s+3y+ ...
s+(N-l)r+p
NA; s+Ny
In a single two-state jump limit, we have only to put N = 1 in (5.13): (5.14)
This is a well-known result’). While in the Gaussian limit, we let N be large keeping NA: constant. Putting this constant to be (5.15)
NA; = A2, we have
1
G’“‘[s] =
(5.16)
A2
s+
2A2
s+y+
2
s+2y+36
...
This reproduces the result of the Gaussian Markoff process exactly’). It is quite interesting to examine behavior of the power spectrum Z(o) when N varies. The power spectrum is given by Z(w) = (1 /m) Re G’N’[im] .
(5.17)
Fig. 11 displays Z(w) when (Y= 0.2 ((Y= A/y) corresponding to the narrowing case. In this case we have essentially no dependence on N. When (Y becomes large, (Y= 5.0 for instance (see fig. 13), the line shape changes rather
K. MARUYAMA
462
-3
-2
AND
-1
0
F. SHIBATA
1
2
3
U/A Fig. 11. The power spectra I(o) in the case of fast modulation, a = 0.2. The solid line and the dotted line correspond to the two-state jump process (N = 1) and the Gaussian process (N = =), respectively.
drastically with N: When N = 1 we have two peaks corresponding to the two realized values of w(t) while in the Gaussian limit (N+ m) we have an infinite number of realizations. As is clearly seen from fig. 13, we observe changing behavior from the single two-state jump process to the Gaussian process with increasing value of N. That is, we can recognize how the discrete nature turns into the continuous one when N becomes large. 2
a=l.O
1
. .... ..’
2 L
-3
-2
-1
0
1
2
3
O/A Fig. 12. The power spectra I(o) for an intermediate rate of modulation, dashed and the dotted lines correspond to N = 1,2 and ~0, respectively.
(Y= 1. The solid,
the
FROM TWO-STATE
JUMP TO GAUSSIAN
STOCHASTIC
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463
2
a=50
I
Fig. 13. The power spectra I(w) in the case of slow modulation, a = 5. The solid, the dashed, the dashed-dotted and the dotted lines correspond to N = 1,2,3 and m, respectively.
For the value of (Y= 1, we find also several interesting behaviors (see fig. 12) which just reflects the intermediate character of modulation. Our model can be solved in the time domain. In order to see the timeevolution of
@‘N’(t)= (x(t)) /(x(O)) we transform
(5.18)
)
(5.13) in the following form: 1
yGCN’[sy] =
(5.19)
NCYi
s+
2(N - l)&
s+l+ s+2+
3(N - 2)cr; ... 2
s+(N-I)+% with q, = A&y. Since GCN’[s] is the Laplace transform of GCN’(t) (see (5.4) and (5.5)), the left-hand side of (5.19) can be written as
(5.20)
K. MARUYAMA
464
AND F. SHIBATA
where we put t’ = yt in the last line. We find that the inverse transform (5.19) gives cDcN’(tly). Here we note the following expansion formula’):
of
a i 0
e-zU{cosh(U) + a sinh(u)}-”
du 1
=
m(l - a’)
z+ma+
2(m + l)(l - a*>
z+(m+2)a+
z+(m+4)a+
3(m + 2)(1 - a”) z + (m + 6)~ + . . .
(5.21)
In the right-hand side of (5.21), we put m=-N
(1-
u2)l(2a)2 = -cz:,
and (z+mu)/2a=s;
then 1 l(2a) NCX:
(5.21) = S+
2(1; - l)(Y:,
s+l+
$+2+
3(N-2)a: ...
Upon this transformation,
I
the right-hand side of (5.21) is rewritten
as
{cash(u) + a sinh(u)}N e-(2s+N)ou du
0
with a = (1 - 4CX;y We further transform
.
the above expression
(l/2@) / {cosh(t/2u) 0
(5.22)
+ a sinh(t/2u)}N
into the form
e-Nf’2 e-“’ dt
,
FROM TWO-STATE JUMP TO GAUSSIAN STOCHASTIC PROCESSES
465
where we put t=2au. Thus we find the expansion formula as follows: m I
{cosh(t/2a)
+ a sinh(t/2a)}N e-N”Z e-“’ dt
0
1
=
(5.23)
2 Na0
ss
2(N - l)a;
s+l+
s+2+
3(N-2)o; ...
with a given by (5.22). Using (5.19), (5.20) and (5.23), we finally obtain QCN’(t) = {cosh(yt/2a)
+ a sinh(yt/2a)}N
e-Nyr’2 ,
(5.24)
where a is rewritten as a = (1 - 4~x~/N)-l’~ = aN
(5.22’)
with (Y= Aly. In a single two-state jump limit (N = l), we get @‘l’(t) = {cosh(yt/2a,)
+ a, sinh(yt/2a,)}
e-“”
,
(5.25)
while in the Gaussian limit (N-+ 00) @@j(t) = exp[-a2(e-Y’
+ yt - l)] .
(5.26)
Note that time-behavior changes drastically according as (Ys D/2 or (Y> D/2. For (Ys m/2, where a,,, is real, GcN’(t) decreases monotonously in time. For (Y> m/2, however, GcN’(t) evolves in time with oscillation since aN becomes pure imaginary. In fig. 14, we show the time-evolution of QscN’(t) for the two-state jump (N = 1) and the Gaussian processes with LY= 0.2 corresponding to the spectra
K. MARUYAMA
466
-1 Fig. 14. Time-evolution (Y= 0.2 corresponding
AND
F. SHIBATA
L
of aCN’(t) for the two-state jump to the spectra shown in fig. 11.
a=0.2 (N = 1) and the Gaussian
processes
with
in fig. 11. In this case of fast modulation, we find that the dependence of N is not essential in the time domain. Figs. 15 and 16 display time behavior for cy = 1.0 and 5.0 which correspond to figs. 12 and 13, respectively. As CYbecomes large, the “N-effect” difference originated from the N dependence is clearly seen in the time domain, too. When the modulation is slow, (Y= 10.0, we have a time-evolution of QCN’(t) as shown in fig. 17. For the single two-state jump process, we have the oscillating curve. As N becomes large the amplitude of the oscillation reduces. Finally, in the Gaussian limit the oscillation vanishes. Thus we can systematically recognize how the oscillations in the time region change into a smooth Gaussian curve as N becomes large.
-1
L
a=l.O
Fig. 15. Time-evolution of QCN)(r) for N = 1,2 and Gaussian to the spectra shown in fig. 12.
processes
with [Y= 1.0 corresponding
FROM TWO-STATE JUMP TO GAUSSIAN
STOCHASTIC
PROCESSES
467
Fig. 16. Time-evolution of @(N’(f) for N = 1,2,3 and Gaussian processes with (Y= 5.0 corresponding to the spectra shown in fig. 13.
a=10.0 Fig. 17. Time-evolution of QcN)(t) when the modulation is slow, a = 10.0, for the same values of N’s as in fig. 16.
6. Summary
and conclusion
In this paper we find fundamental properties of a (new) stochastic process which is composed of a superposition of N two-state jump processes. According to the formula given in section 4, we can calculate arbitrary order moments and partial cumulants systematically. Therefore, if the formula derived in section 4 are combined with the “partial cumulant expansion formula” summarized in the appendix, we find a basic equation which governs a dynamical evolution of the relevant system. Sometimes physical and other (chemical, electrical, optical and/or communicational) systems are discussed on the basis of the two-state jump (“random
468
K. MARUYAMA
AND F. SHIBATA
telegraph”) Markoff process or the Gaussian process. This is partly because of mathematical simplicity. In addition, these two stochastic processes are complementary to each other in the sense that the former process has only two values of realization while the latter process is characterized by an infinite number of realizations. However in the actual case, we have to treat intermediate situations which are neither the two-state jump nor the Gaussian. With the aid of our formalism developed in this paper, we can treat these intermediate situations including the two-state jump and the Gaussian processes as limiting cases. As a simple but non-trivial example, we have displayed the power spectra and the relaxation function for the model of random frequency modulation. Namely we have exactly solved the problem not only in the frequency domain but also in the time domain. To the authors’ knowledge, there is no work on this subject except the one done by Hiraiwa’) who treated a special problem of Mossbauer spectra, introducing a matrix formalism which may have some relevance to the exciton (electron) theory of Ohata’). Our method may be more systematic and flexible when applied to various problems. Indeed we will develop in forthcoming papers a theory of light scattering using the result of the present paper. Also we will examine the theories of exciton migration ‘O-l*) and of low field resonance13) from our new standpoint established in this paper.
Acknowledgement
We are indebted manuscript.
Appendix
to Professor
N. Hashitsume
for careful reading of the
A
We summarize here the time-convolution an equation of the form
$ J@(t)= gL&)T?(t)
.
expansion formula3). We consider
(A.1)
This equation can be reduced to a set of coupled equations for a relevant part 97’@(t) and an irrelevant part 9 6’(t) by introducing a projection operator 9 and 22 = 1 - 9.
FROM TWO-STATE JUMP TO GAUSSIAN STOCHASTIC PROCESSES
Solving the resulting differential the other one, we get
-L$ W@(t)= @‘i&)9@(t)
469
equation for 9 I@(t) and substituting it into
+
dr @(t, r)SI+(r)
+ .%(t) ,
(A.21
'0
where qt,
T) = g29cl(t)qt,
7)2?Ll(T)
(A.3)
)
with ~(f,i)=exp_[glai,(s)ds].
(A.3
T
These quantities can be expanded in powers of g. Then we can write the second order term of (A.2) in the form
(A.6) where we have defined “partial cumulant”
by (A.7)
with the specified projection 9x=(x)
operator
.
(ASS)
It is already known that the general structure of (A.7) is given by
(A.9 where the sum is taken over all possible division of the average chronological order and 4 is the number of averages in each term.
keeping
K. MARUYAMA
470
Appendix
AND F. SHIBATA
B
In section 5, we solved a model of frequency modulation as an example of application of the formalism. If our concern lies in solving the model itself, there is an alternative method. Of course, this does not diminish the importance and usefulness of our method developed in sections 2-4, which will be used extensively in our future work. Nevertheless, it is of some value to give a simple method of the solution: Eq. (5.1) is formally integrated to give
Mt)x*(W
(x(w*m
= (exp[i i d7 wCN’(T)]).
03.1)
0
Now the stochastic process w(~)( t ) is composed of N independent and thus we have
(exp[i / dr oCv)(r)])
= fi, (exp[i [ dr on(T)])
processes
03.2)
.
0
0
The original equation for x(t), (5.1), is solved in the case of a single two-state jump model (n = 1, for instance):
[dtemX’(exp[i/dr-,(T)])= 0 0
IA’
,
(B.3)
S+’ s+Y
where we have used (2.8). By inverting (B.3) we have
(ev[i/ dTd4])
= {cosh(yt/2a,)
+ a, sinh(yt/2a,)}
Here we note that each constituent process identical. Thus we have from (B.2) and (B.4):
(-4t)x*(O)) = w)x*w
{cosh(yt/2a,)
w,(t)
+ a, sinh(yt/2a,)}N
e-yr’2 .
is independent
ePNy”2 ,
(B.4) and
03.5)
which agrees with (5.24). Moreover, referring to (5.21), we have the continued fraction representation (5.13).
FROM TWO-STATE
JUMP TO GAUSSIAN
STOCHASTIC
PROCESSES
471
Note added in proof The following reference, pertaining to the solution of the simple example treated in section 5, should be added: K. Wbdkiewicz, Z. Phys. B 42 (1981) 95.
References 1) R. Kubo, in: Fluctuation, Relaxation and Resonances in Magnetic Systems, D. ter Haar, ed. (Oliver and Boyd, Edinburgh, 1962); in: Stochastic Processes in Chemical Physics, K.E. Shuler, ed. (Wiley, London, 1969). 2) N.G. van Kampen, Physica 74 (1974) 215, 239. 3) F. Shibata and T. Arimitsu, J. Phys. Sot. Jpn. 49 (1980) 891. 4) R.C. Bourret, U. Frisch and A. Pouqet, Physica 65 (1973) 303. 5) P.W. Anderson, J. Phys. Sot. Jpn. 9 (1954) 316. 6) R. Kubo, J. Phys. Sot. Jpn. 9 (1954) 935. 7) H.S. Wall, Analytic Theory of Continued Fractions (Chelsea Publishing Company, Brox, New York, 1973) formula (94.1). 8) A. Hiraiwa, Master Thesis, University of Tokyo (1975). 9) N. Ohata, Progr. Theor. Phys. 51 (1974) 1332. 10) I. Sato and F. Shibata, Physica 128A (1984) 551. 11) I. Sato and F. Shibata, Physica 135A (1986) 139. 12) I. Sato and F. Shibata, Physica 135A (1986) 388. 13) F. Shibata and I. Sato, Physica 143A (1987) 468.
FROM TWO-STATE JUMP TO GAUSSIAN STOCHASTIC PROCESSES Kishiko MARUYAMA
and Fumiaki SHIBATA
Department of Physics, Faculty of Science, Ochanomiru
University, Bunkyo-ku,
Tokyo 112, Japan
Received 10 August 1987
A stochastic process, which bridges the two-state jump Markoff process and the Gaussian process, is introduced and fundamental properties of the process are fully analyzed. The resulting expressions are combined with the “partial cumulant” expansion formula to give an exact power spectrum in the form of a continued fraction. A rigorous expression is also obtained for a relaxation function in an analytic form.
1. Introduction In treating various phenomena in statistical and condensed matter physics, we sometimes introduce a stochastic model to simplify a problem. There are two typical stochastic processes which are frequently used’,‘); the two-state jump Markoff and the Gaussian processes. There is a remarkable contrast between the two. The former process is characterized by two values of realization whereas the latter by an infinite number of realizations. In this sense, they are opposite extremes. Moreover, these two processes have been used because of their mathematical simplicity. The higher order “partial cumulants” in the former and the “order cumulants” in the latter vanish identically, yielding simple structures in their expansion formulae (see the main text on the precise definition of the “cumulant”). In actual cases, however, we have to deal with stochastic processes which are neither the two-state jump process nor the Gaussian process. Thus we encounter obstacles inherent in real stochastic processes. We give here a method of treating this difficult problem without losing the simplicity of the two-state jump and the Gaussian processes. The method should be formulated so as to include these two processes as limiting special cases. We introduce a stochastic process composed of a superposition of many two-state jump Markoff processes. Properties of this composed process are fully investigated in the subsequent sections. The partial cumulant expansion 0378-4371/88/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
K. MARUYAMA
448
AND F. SHIBATA
method3) is generally applicable. A simple example is given to serve for illustration of our method; this will clarify the usefulness of our method.
2. Partial cumulant We summarize here general properties of the partial cumulant3) some of which are already known; it may be convenient to give a summary for later development of the present paper. For a certain stochastic process w(t), the jth order partial cumulant is defined as
(4fM4 >. . . w(t,_,)),.,. = g’w(f)g@(t,). . .94lj-,) by using a specific projection 9x=
(x)
operator
(2.1)
9:
)
Q-2)
and 9=1-C??,
(2.3)
where (. . * ) re p resents an average over the process w(t). Sometimes we simplify the stochastic process to be a two-state jump Markoff process or a Gaussian Markoff process. For such processes, multi-time correlation functions can be expressed in terms of double time correlation functions. Moreover for the process of zero mean:
(4))
=o,
(2.4)
the so-called irreducible condition of the partial cumulant is satisfied. Let us illustrate this condition. For this purpose we rewrite the partial cumulant in an alternative form,
Next we note the following: when a single B appears in the correlation o(tj)‘s, for instance,
(4)4ww,)~~~
>>
of
(2.6a)
FROM TWO-STATE JUMP TO GAUSSIAN STOCHASTIC PROCESSES
the correlation
449
is divided into two parts like
which we may call a reducible term, whereas if we have a single ?! instead of 9, the reducible term like (2.6b) is subtracted from the whole correlation and hence the term becomes “irreducible” by definition. We explain that the irreducible condition is satisfied by considering the fourth order partial cumulant of the Gaussian process:
=
(w(t)w(t,)o(t*)o(t,))- (w(t)w(t,))(w(t,)o(t,))
(2.6~)
=
(w(t)o(t*))(w(t,)w(t,))+ (w(t)w(t~))(w(t,)o(t*)).
(2.6d)
The reducible term (the second term) in (2.6~) is cancelled out by a part of the first term and we are left only with the irreducible ones, (2.6d). This means that the irreducible condition is fulfilled for the fourth order partial cumulant of the Gaussian process as stated above. In the same way can prove that the higher order partial cumulants become irreducible by inspecting the right-hand side of (2.5): all the reducible terms are subtracted owing to the presence of 9’s in every intermediate position, We now turn to the two-state jump Markoff process, for which the following theorem of multi-time correlation is proved4): (2.7) fort>t,>t,-*-.
correlation
With the repeated use of (2.7), we find that the j-tuple time function becomes reducible for j 2 3 yielding
according to the irreducible condition. Though this is a known relation, we give here a short proof for later purposes: It may be convenient to rewrite (2.7) in the form
with the projection operator (2.2). The first 2 in the right hand side of (2.5) is expanded into two parts, namely,
K. MARUYAMA
450
=
AND F. SHIBATA
(w(t)w(t,)~2w(t,)~2w(t,). . .22w(t,_,))
- (o(t)~~(t,)~2w(t,)~2w(t,). . * 9w(t,-,))
(2.10)
The second term of (2.10), written in the form
(o(t>)(w(t,)o(t*)w(t,).. .4-AC vanishes owing to (2.4). Furthermore
=
(w(t)o(t*)w(t,)~w(t,). ” -
2
expanding the next 2, we get
'w(tj-l))
(w(t)w(t,)~‘w(t,)~2w(t,). . .2&-,>)
2
(2.11)
where the first term is equal to the second term except for the opposite sign due to (2.9). Thus we have a conclusion that the jth order partial cumulant of two-state jump Markoff process vanishes for i 2 3, (2.8). A short summary of this section is as follows: the partial cumulant defined by (2.1) with the projection (2.2) is irreducible under the condition (2.4). Especially for the two-state jump Markoff process, we have a simple relation (2.8).
3. Structure of the partial cumulant Let us consider the structure of a partial cumulant for a stochastic process I composed of N independent processes each of which is assumed to be a two-state jump Markoff process. Hereafter we assume the following time sequence: t > t, > t2 > . . . . 3.1. Case of N= 2 We first consider a stochastic process given by
W@)(t) = wl(t)
+ W*(t)
)
(3.1)
where each of the constituent processes or(t) and m2(t) is assumed to be a two-state jump Markoff process with zero mean. For each process (n = 1 and
FROM TWO-STATE
JUMP TO GAUSSIAN
STOCHASTIC
PROCESSES
451
2), we have (49)
=o 7
(w,(t)~~,(t’))
(3.2) = cY,,.A; e-y’r-f”
(3.3)
and (o,(t)On(tl)"'w,(tj-l))p.~.=O
(for
ia3).
(3.4)
The second order partial cumulant for w(t) is given by (W(*)(f)W(*)(fl))P.C. = (4)44)) = 24
+
bJ*w*w
e-Y(~-~,)
(3Sa) (3Sb)
E (+ -)@) )
(3.5c)
where only the signs in front of t and t, in the parenthesis in the exponent are explicitly written in the last member (3.5~). Namely, the form of the exponential function in (3.5b) is suggested by the symbol (+ -), although the multiplicative factor depends on the constituent processes. We can rewrite this partial cumulant as
(W(*)(t)W(*)(tl))p.c. =‘%W by introducing
(3.5d)
t,) ,
the function
ek(t) = eekyr .
(3.6)
A diagrammatic representation of (3.5d) is shown in fig. 1, where the upward arrow denotes the + sign and the downward arrow -. We also assign the horizontal line to (cl,(t - tl). In other words, the contour of this diagram represents the form of the exponential function, (+ -). On the other hand, the
H <
: t
Fig. 1. Second-order
t1 P.C. graph for N = 2.
K. MARUYAMA
452
AND
F. SHIBATA
double arrow indicates two possible choices of wl(t) and u*(t). Once the choice is made at the time t, the corresponding process at t, is uniquely determined, yielding a single arrow at t, . Thus we get the factor of 2 in (3.5d) by calculating the possible number of ways going through the contour from the left to the right in the direction of the arrows. The analysis of the fourth order partial cumulant parallels above: ( 0(2)(t)W(2)(tt)W(2)(t2)~(2)(t~))p,C, =
(wl(t)W,(t*))(w,(t,)o*(t,))
+
+
(W2(t)02(t*))(W,(tl)Wl(t3))
(wl(t>wl(t,))(o,(t,)w,(t,))
+
(W2(t)02(t3))(Wl(tl)Ol(f*))
= 44: e- vV+r,-r,-r3) E
(+
+
-
(3.7a) (3.7b)
_p
(3.7c)
=4A;W- t,)Mt, - t2)+4(t2- td >
(3.7d)
where we have used the same sign convention as (3.5). The graph contributing to (3.7) is shown in fig. 2 where the upward (downward) arrows denote the +(-) sign in (3.7c), that is, the contour of this diagram represents the form of the exponential function, (+ + - -). The first horizontal line of the stairs corresponds to @, whereas the second to &. The double arrow indicates two possibilities of the choice between q(t) and w*(t). The process at time t, (t3) has to be different from the one at time t(t2), otherwise the diagram becomes reducible. Thus we have a single line at t,. Finally we have a front factor of 2* = 4 from fig. 2. Similarly, the sixth order partial cumulant can be written as (0(2)(t)o(2)(t,)W(2)(t2)Wo(t3)00(t~)~~2~(t~))~,~
=8AEe
-y(t+t,-r,+t,-t,-r,)
Fig. 2. Fourth-order
(3.8a)
P.C. graph
for N = 2.
FROM TWO-STATE JUMP TO GAUSSIAN E
(+
+
-
+
STOCHASTIC
PROCESSES
_)(2)
-
453
(3.8b)
The graph shown in fig. 3 corresponds to (3.8~). Conversely, we can easily read fig. 3 to obtain (3.8~): Assigning I,/+to each of the first step and JI, to the second steps and multiplying by 23, we can reproduce (3.8~) from fig. 3. The above procedure can be extended to calculate higher order partial cumulants with the aid of the diagrammatic representation. As is shown in fig. 4, an arbitrary higher order diagram is obtained as a simple extension of figs. 2 and 3. The diagram is composed of the first horizontal lines representing ~/+((t~~ - t2j+I)‘s and the second ones corresponding to J12(t2j_l - t,)‘s. For 2jth order, the number of paths is 2’. Thus we get ( 0J(2)(t)W(2)(tl) * * . o’2’(t2j_l))p.c.
3.2. Case of N= 3 Next we will consider the following process: W(3)(t) = q(t)
<
(3.10)
+ W2(t) + w,(t) )
; t
t1
f2
f3
tc
f5
Fig. 3. Sixth-order P.C. graph for N = 2.
----__---. t1
f2
f3
t4
f,-3
Fig. 4. Arbitrary order P.C. graph for N = 2.
fj-2
t,-1
K. MARUYAMA
454
AND F. SHIBATA
where, for each of the constituent processes, the relations (3.4) are satisfied. The second order partial cumulant is calculated as (~(~)(t)~(~)(t,))r~~,
(3.2), (3.3) and
= 34: e-y(‘-rlt
(3.11a)
s (+
(3.11b)
_)(3)
(3.11c)
=3A;$,(t-t,),
where the symbols and the functions are the same or simple extensions of the previous ones. The graph representing the second order contribution is drawn in fig. 5 where the triple arrow appears owing to the possible choice of the three independent processes. This gives the front factor in (3.11~). The fourth order partial cumulant becomes ( W(3)(t)o(3)(fl)W(3)(t*)0(3)(f3))P,C, = 124; eP Y@+rl-~z-q
(3.12a)
s (+ + -
(3.12b)
_)(3)
= 12A;llr,(r - f,)k(rl
- &)+,(t, -
~3)
(3.12~)
>
corresponding to fig. 6. As before we can read fig. 6 as representing the form of the exponential function and the front factor: that is, the first horizontal line denotes $r and the second horizontal line q$ while the number of possible ways for the vertical arrows gives the coefficient. The upward arrow at time t is triple owing to three possibilities of choice among W, , w2 and w3, whereas at time t, we have a double arrow because of two remaining possible processes except the one at t. The next downward arrow at time t, becomes double because the two processes which appeared at time t or t, must be paired at t,; the arrow at time t3 is single. As a result, this diagram gives the factor of 3 x 2 X 2 X 1 = 12. Similarly the sixth order partial cumulant is given by
<
: t
Fig. 5. Graphical representation
t1 of the second-order
P.C. for N = 3.
FROM TWO-STATE JUMP TO GAUSSIAN
STOCHASTIC
PROCESSES
455
,
L
\,
<
: t
t1
f3
f2
Fig. 6. Graphical representation
of the fourth-order
P.C. for N = 3.
+ < : t
t1
f2
< :
t
ts
f4
f3
Fig. 7. Graphical representation
t1
f3
f2
f5
f4
of the sixth-order P.C. for N = 3.
( 0(3)(t)W(3)(t,)W(3)(t*)W(3)(t3)0(3)(t4)0(3)(t5))P.C. = 484: e -y(t+tl-r,+tj-t~-r,) + 36A; e-Y(t+r*+r,-l,-l,-r*) s
(+
+
-
+
= 48A:W +
-
_)(3)
- t,)W*
36GM
+
(+
+
+
-
-
- ~*)Jll(~*-
(3.13a)
_)(3)
(3.13b)
t3)W3
- fl)+*(tl - f2)&(‘3(t*-
-
t3)W3
OJll(f4
-
-
b)v+(4
G)
-
fs)
7
(3.13c)
with two diagrams corresponding to (+ + - + - -)(3) and (+ + + - - -)(3) in fig. 7. The first diagram corresponding to (+ + - + - -)(3) gives the front factor of 3 X 2 X 2 X 2 X 2 X 1 = 48 and the second (+ + + - - -)(3) 3 x 2 x 1 x 3 x 2 X 1 = 36. Analogously we can proceed to write down higher order partial cumulants. This will be done in a more systematic way in the next section.
4. General structure
of the partial cumulant
for arbitrary
The foregoing discussions are now generalized P(t)
= $ n=l
w,(t)
)
iV
to the stochastic process (4.1)
K. MARUYAMA
456
AND F. SHIBATA
where each constituent process is assumed to be the two-state jump Markoff process. Namely, the relations (3.2), (3.3) and (3.4) are satisfied. Let us calculate a first few partial cumulants. For the second order, we obtain s
(+
-j(N)
(4.2a)
= N&,(t
(4.2b)
- tl) >
with the diagrammatic representation shown in fig. 8. The notations in (4.2) are the same as the ones in the previous sections. The factor N comes from N possibilities of the choice of a single stochastic process among w1(t), q(t). . . and q,,(t). This is represented by the thick arrow at time t. The fourth order partial cumulant is given by ( 0(N)(t)W(N)(tl)W(N)(t2)W(N)(tj))p,C, F
(+
+
_
=
=‘(N -
-j(N)
(4.3a)
W$W - tlh(fl
- t,)h(t, - td
2
(4.3b)
corresponding to fig. 9. The thick arow at time t is N-fold, and the process at time t, must be different from the one at time t according to the irreducible condition. Thus the thick arrow at time t, is (N - 1)-fold. The double arrow at time t, shows two pairing possibilities with the two processes at time t or t,. There remains only a single process at time t, giving a single arrow. Thus we get the coefficient 2N(N - 1) by calculating the possible numbers of choice and pairing of the stochastic processes from the diagram. There arise no essential difficulties as we proceed beyond the lower orders, and thus we obtain the following general prescriptions of the general term for the 2jth order partial cumulant: (i) Draw all possible 2jth order irreducible diagrams, starting from the left bottom and ending at the right bottom, composed of 2j vertical lines and (2j - 1) horizontal lines. Attach directed arrows one for each to the vertical
t < Fig. 8. Second-order
: t
t1
P.C. graph for arbitrary N.
FROM TWO-STATE JUMP TO GAUSSIAN
f
: t
t1
Fig. 9. Fourth-order
f2
STOCHASTIC
457
PROCESSES
f3
P.C. graph for arbitrary N.
line from the left bottom to the right, assigning times t, t,, t,, . . * 7 t*j-l' According to the “irreducible condition” the intermediate arrows never lead to the bottom. Otherwise the diagram becomes reducible. Determine a mathematical expression associated with each diagram by the following items, and sum up those expressions coming from all diagrams. (ii) For the first upward vertical arrow at time t, associate a numerical factor N indicating an N-fold arrow. For the second upward arrows which connect the second level with the first level counting from the bottom except the bottom one, associate a factor of (N - 1) and so on. For the last downward arrow at tZj_r, associate a factor of one. For the second downward arrows which connect the first level with the second level, associate a factor of two and so on. Thus the total numerical factor can be read from a given diagram immediately. (iii) For horizontal lines of the first level counting from the bottom except the bottom one, assign the function I,!J,. Similarly, assign I&, I,$, . . . , I),,, to horizontal lines of the second, third, . . . levels. As an example, we give here an expression of the eighth order partial cumulant. This may serve for illustration. The corresponding diagrams are shown in fig. 10. Using the same notation as before, we find the partial cumulant in the form (+ + - + - + - -)(N) + (+ + - + + - - -)(N) + + (+ + + - + - -
-)(N) +
(+ + + + - - -
(+
+ +
-)(N) )
-
-
+ -
_p
(4.4a)
which correspond to the diagrams (a), (b), (c), (d) and (e) of fig. 10, respectively. Next, let us combine the rules (ii) and (iii) by introducing the following quantity: fjk=k(N-k+l)Qk. Thus
we get
(4.5)
K. MARUYAMA
458
<
a
: t
t1
AND F. SHIBATA
f2
f3
fL
ts
fii
t2
t3
tc
t5
t6
t7
t2
f3
t4
fs
f6
f7
f2
t3
t7
+I < : bt
t1
+t
+I f
:
dt
t1
+ (
:
tt
Fig. 10. Eighth-order
t4 P.C. graphs
f5
f7
f6
for arbitrary
N.
459
FROM TWO-STATE JUMP TO GAUSSIAN STOCHASTIC PROCESSES
where we have suppressed the arguments of &‘s for simplicity: These are (t - tr), (r1 - t2), . . . and (t, - f7), respectively.
5. Simple example As an application of the present formalism, consider an equation of motion for a complex variable x(t) (a model of random frequency modulation5’6)): i(t) = iWCN’(t)x(t) ,
(5.1)
where the stochastic process tiCN)(t) is governed by (4.1). An averaged quantity (x(t)) over the stochastic process can be obtained with the use of the time convolution formula3) (see appendix A):
(i(t))
~2 (-1)’ J dtl J dt2 * * * J 0 '
(w(N)(t)w(N)(tl)'
dt2j-1
0
0 *.
o'N'(t2j-l))p.C.(X(f2j-l))
7
(5.2)
with the initial condition
x(O)= MO)) *
(5.3)
The partial cumulant appearing in (5.2) has been fully analyzed in the preceding section. Eq. (5.2) can be solved by the method of Laplace transforms. Thus we can solve (5.2) in the form
(x), = G'N'[s]x(0)
,
(5.4)
where we have introduced dt eWsr(x(t)) 0
(5.5)
K. MARUYAMA
460
AND F. SHIBATA
and G’N’[~] = (s +
Here Z\“‘[s]
Z’;“‘[s])-’
(5.6)
is explicitly given by
with
fk[Sl= 1 dt
e-s’+k(t) = (S + ky))’
(5.8a)
,
and dt e-S’$k(t) = k(N - k + 1) /(s + ky ) .
Now we can reinterpret the diagrams corresponding as X(;N)[s]: Namely, I+!Q is replaced by fk. The equation (5.7) can be rewritten in the form
$N)[s] = A;i
-
A;~X$N'[s]f,
where the quantity Si”‘[s]
Z;“‘[s] = A;L -
(5.8b)
to the partial cumulants
+ A~f$';y'[s]f&""'[~]f~
+. . . ,
does not contain any f,‘s and is represented
A;~2'N'[s]f2
+ A~~*~~N'[s]f,~~N'[s]f,
+. . . ,
(5.9) by (5.10)
Z iN)[s] being defined analogously. For an arbitrary k (
(5.11) for k < N. On the other hand we find (5.12) and Zlf”‘[s] = 0 for k > N.
461
FROM TWO-STATE JUMP TO GAUSSIAN STOCHASTIC PROCESSES
Substituting the above quantities successively into (5.6) and using (5.8a) and (5.8b), we find GCN’[s] in an explicit form: 1
GcN’[s] =
(5.13)
NA;
.S+
2(N - l)A;
s+y+
3(N - 2)A;
s+2y+
s+3y+ ...
s+(N-l)r+p
NA; s+Ny
In a single two-state jump limit, we have only to put N = 1 in (5.13): (5.14)
This is a well-known result’). While in the Gaussian limit, we let N be large keeping NA: constant. Putting this constant to be (5.15)
NA; = A2, we have
1
G’“‘[s] =
(5.16)
A2
s+
2A2
s+y+
2
s+2y+36
...
This reproduces the result of the Gaussian Markoff process exactly’). It is quite interesting to examine behavior of the power spectrum Z(o) when N varies. The power spectrum is given by Z(w) = (1 /m) Re G’N’[im] .
(5.17)
Fig. 11 displays Z(w) when (Y= 0.2 ((Y= A/y) corresponding to the narrowing case. In this case we have essentially no dependence on N. When (Y becomes large, (Y= 5.0 for instance (see fig. 13), the line shape changes rather
K. MARUYAMA
462
-3
-2
AND
-1
0
F. SHIBATA
1
2
3
U/A Fig. 11. The power spectra I(o) in the case of fast modulation, a = 0.2. The solid line and the dotted line correspond to the two-state jump process (N = 1) and the Gaussian process (N = =), respectively.
drastically with N: When N = 1 we have two peaks corresponding to the two realized values of w(t) while in the Gaussian limit (N+ m) we have an infinite number of realizations. As is clearly seen from fig. 13, we observe changing behavior from the single two-state jump process to the Gaussian process with increasing value of N. That is, we can recognize how the discrete nature turns into the continuous one when N becomes large. 2
a=l.O
1
. .... ..’
2 L
-3
-2
-1
0
1
2
3
O/A Fig. 12. The power spectra I(o) for an intermediate rate of modulation, dashed and the dotted lines correspond to N = 1,2 and ~0, respectively.
(Y= 1. The solid,
the
FROM TWO-STATE
JUMP TO GAUSSIAN
STOCHASTIC
PROCESSES
463
2
a=50
I
Fig. 13. The power spectra I(w) in the case of slow modulation, a = 5. The solid, the dashed, the dashed-dotted and the dotted lines correspond to N = 1,2,3 and m, respectively.
For the value of (Y= 1, we find also several interesting behaviors (see fig. 12) which just reflects the intermediate character of modulation. Our model can be solved in the time domain. In order to see the timeevolution of
@‘N’(t)= (x(t)) /(x(O)) we transform
(5.18)
)
(5.13) in the following form: 1
yGCN’[sy] =
(5.19)
NCYi
s+
2(N - l)&
s+l+ s+2+
3(N - 2)cr; ... 2
s+(N-I)+% with q, = A&y. Since GCN’[s] is the Laplace transform of GCN’(t) (see (5.4) and (5.5)), the left-hand side of (5.19) can be written as
(5.20)
K. MARUYAMA
464
AND F. SHIBATA
where we put t’ = yt in the last line. We find that the inverse transform (5.19) gives cDcN’(tly). Here we note the following expansion formula’):
of
a i 0
e-zU{cosh(U) + a sinh(u)}-”
du 1
=
m(l - a’)
z+ma+
2(m + l)(l - a*>
z+(m+2)a+
z+(m+4)a+
3(m + 2)(1 - a”) z + (m + 6)~ + . . .
(5.21)
In the right-hand side of (5.21), we put m=-N
(1-
u2)l(2a)2 = -cz:,
and (z+mu)/2a=s;
then 1 l(2a) NCX:
(5.21) = S+
2(1; - l)(Y:,
s+l+
$+2+
3(N-2)a: ...
Upon this transformation,
I
the right-hand side of (5.21) is rewritten
as
{cash(u) + a sinh(u)}N e-(2s+N)ou du
0
with a = (1 - 4CX;y We further transform
.
the above expression
(l/2@) / {cosh(t/2u) 0
(5.22)
+ a sinh(t/2u)}N
into the form
e-Nf’2 e-“’ dt
,
FROM TWO-STATE JUMP TO GAUSSIAN STOCHASTIC PROCESSES
465
where we put t=2au. Thus we find the expansion formula as follows: m I
{cosh(t/2a)
+ a sinh(t/2a)}N e-N”Z e-“’ dt
0
1
=
(5.23)
2 Na0
ss
2(N - l)a;
s+l+
s+2+
3(N-2)o; ...
with a given by (5.22). Using (5.19), (5.20) and (5.23), we finally obtain QCN’(t) = {cosh(yt/2a)
+ a sinh(yt/2a)}N
e-Nyr’2 ,
(5.24)
where a is rewritten as a = (1 - 4~x~/N)-l’~ = aN
(5.22’)
with (Y= Aly. In a single two-state jump limit (N = l), we get @‘l’(t) = {cosh(yt/2a,)
+ a, sinh(yt/2a,)}
e-“”
,
(5.25)
while in the Gaussian limit (N-+ 00) @@j(t) = exp[-a2(e-Y’
+ yt - l)] .
(5.26)
Note that time-behavior changes drastically according as (Ys D/2 or (Y> D/2. For (Ys m/2, where a,,, is real, GcN’(t) decreases monotonously in time. For (Y> m/2, however, GcN’(t) evolves in time with oscillation since aN becomes pure imaginary. In fig. 14, we show the time-evolution of QscN’(t) for the two-state jump (N = 1) and the Gaussian processes with LY= 0.2 corresponding to the spectra
K. MARUYAMA
466
-1 Fig. 14. Time-evolution (Y= 0.2 corresponding
AND
F. SHIBATA
L
of aCN’(t) for the two-state jump to the spectra shown in fig. 11.
a=0.2 (N = 1) and the Gaussian
processes
with
in fig. 11. In this case of fast modulation, we find that the dependence of N is not essential in the time domain. Figs. 15 and 16 display time behavior for cy = 1.0 and 5.0 which correspond to figs. 12 and 13, respectively. As CYbecomes large, the “N-effect” difference originated from the N dependence is clearly seen in the time domain, too. When the modulation is slow, (Y= 10.0, we have a time-evolution of QCN’(t) as shown in fig. 17. For the single two-state jump process, we have the oscillating curve. As N becomes large the amplitude of the oscillation reduces. Finally, in the Gaussian limit the oscillation vanishes. Thus we can systematically recognize how the oscillations in the time region change into a smooth Gaussian curve as N becomes large.
-1
L
a=l.O
Fig. 15. Time-evolution of QCN)(r) for N = 1,2 and Gaussian to the spectra shown in fig. 12.
processes
with [Y= 1.0 corresponding
FROM TWO-STATE JUMP TO GAUSSIAN
STOCHASTIC
PROCESSES
467
Fig. 16. Time-evolution of @(N’(f) for N = 1,2,3 and Gaussian processes with (Y= 5.0 corresponding to the spectra shown in fig. 13.
a=10.0 Fig. 17. Time-evolution of QcN)(t) when the modulation is slow, a = 10.0, for the same values of N’s as in fig. 16.
6. Summary
and conclusion
In this paper we find fundamental properties of a (new) stochastic process which is composed of a superposition of N two-state jump processes. According to the formula given in section 4, we can calculate arbitrary order moments and partial cumulants systematically. Therefore, if the formula derived in section 4 are combined with the “partial cumulant expansion formula” summarized in the appendix, we find a basic equation which governs a dynamical evolution of the relevant system. Sometimes physical and other (chemical, electrical, optical and/or communicational) systems are discussed on the basis of the two-state jump (“random
468
K. MARUYAMA
AND F. SHIBATA
telegraph”) Markoff process or the Gaussian process. This is partly because of mathematical simplicity. In addition, these two stochastic processes are complementary to each other in the sense that the former process has only two values of realization while the latter process is characterized by an infinite number of realizations. However in the actual case, we have to treat intermediate situations which are neither the two-state jump nor the Gaussian. With the aid of our formalism developed in this paper, we can treat these intermediate situations including the two-state jump and the Gaussian processes as limiting cases. As a simple but non-trivial example, we have displayed the power spectra and the relaxation function for the model of random frequency modulation. Namely we have exactly solved the problem not only in the frequency domain but also in the time domain. To the authors’ knowledge, there is no work on this subject except the one done by Hiraiwa’) who treated a special problem of Mossbauer spectra, introducing a matrix formalism which may have some relevance to the exciton (electron) theory of Ohata’). Our method may be more systematic and flexible when applied to various problems. Indeed we will develop in forthcoming papers a theory of light scattering using the result of the present paper. Also we will examine the theories of exciton migration ‘O-l*) and of low field resonance13) from our new standpoint established in this paper.
Acknowledgement
We are indebted manuscript.
Appendix
to Professor
N. Hashitsume
for careful reading of the
A
We summarize here the time-convolution an equation of the form
$ J@(t)= gL&)T?(t)
.
expansion formula3). We consider
(A.1)
This equation can be reduced to a set of coupled equations for a relevant part 97’@(t) and an irrelevant part 9 6’(t) by introducing a projection operator 9 and 22 = 1 - 9.
FROM TWO-STATE JUMP TO GAUSSIAN STOCHASTIC PROCESSES
Solving the resulting differential the other one, we get
-L$ W@(t)= @‘i&)9@(t)
469
equation for 9 I@(t) and substituting it into
+
dr @(t, r)SI+(r)
+ .%(t) ,
(A.21
'0
where qt,
T) = g29cl(t)qt,
7)2?Ll(T)
(A.3)
)
with ~(f,i)=exp_[glai,(s)ds].
(A.3
T
These quantities can be expanded in powers of g. Then we can write the second order term of (A.2) in the form
(A.6) where we have defined “partial cumulant”
by (A.7)
with the specified projection 9x=(x)
operator
.
(ASS)
It is already known that the general structure of (A.7) is given by
(A.9 where the sum is taken over all possible division of the average chronological order and 4 is the number of averages in each term.
keeping
K. MARUYAMA
470
Appendix
AND F. SHIBATA
B
In section 5, we solved a model of frequency modulation as an example of application of the formalism. If our concern lies in solving the model itself, there is an alternative method. Of course, this does not diminish the importance and usefulness of our method developed in sections 2-4, which will be used extensively in our future work. Nevertheless, it is of some value to give a simple method of the solution: Eq. (5.1) is formally integrated to give
Mt)x*(W
(x(w*m
= (exp[i i d7 wCN’(T)]).
03.1)
0
Now the stochastic process w(~)( t ) is composed of N independent and thus we have
(exp[i / dr oCv)(r)])
= fi, (exp[i [ dr on(T)])
processes
03.2)
.
0
0
The original equation for x(t), (5.1), is solved in the case of a single two-state jump model (n = 1, for instance):
[dtemX’(exp[i/dr-,(T)])= 0 0
IA’
,
(B.3)
S+’ s+Y
where we have used (2.8). By inverting (B.3) we have
(ev[i/ dTd4])
= {cosh(yt/2a,)
+ a, sinh(yt/2a,)}
Here we note that each constituent process identical. Thus we have from (B.2) and (B.4):
(-4t)x*(O)) = w)x*w
{cosh(yt/2a,)
w,(t)
+ a, sinh(yt/2a,)}N
e-yr’2 .
is independent
ePNy”2 ,
(B.4) and
03.5)
which agrees with (5.24). Moreover, referring to (5.21), we have the continued fraction representation (5.13).
FROM TWO-STATE
JUMP TO GAUSSIAN
STOCHASTIC
PROCESSES
471
Note added in proof The following reference, pertaining to the solution of the simple example treated in section 5, should be added: K. Wbdkiewicz, Z. Phys. B 42 (1981) 95.
References 1) R. Kubo, in: Fluctuation, Relaxation and Resonances in Magnetic Systems, D. ter Haar, ed. (Oliver and Boyd, Edinburgh, 1962); in: Stochastic Processes in Chemical Physics, K.E. Shuler, ed. (Wiley, London, 1969). 2) N.G. van Kampen, Physica 74 (1974) 215, 239. 3) F. Shibata and T. Arimitsu, J. Phys. Sot. Jpn. 49 (1980) 891. 4) R.C. Bourret, U. Frisch and A. Pouqet, Physica 65 (1973) 303. 5) P.W. Anderson, J. Phys. Sot. Jpn. 9 (1954) 316. 6) R. Kubo, J. Phys. Sot. Jpn. 9 (1954) 935. 7) H.S. Wall, Analytic Theory of Continued Fractions (Chelsea Publishing Company, Brox, New York, 1973) formula (94.1). 8) A. Hiraiwa, Master Thesis, University of Tokyo (1975). 9) N. Ohata, Progr. Theor. Phys. 51 (1974) 1332. 10) I. Sato and F. Shibata, Physica 128A (1984) 551. 11) I. Sato and F. Shibata, Physica 135A (1986) 139. 12) I. Sato and F. Shibata, Physica 135A (1986) 388. 13) F. Shibata and I. Sato, Physica 143A (1987) 468.