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Exactly soluble multidimensional Fokker-Planck equations with nonlinear drift
Volume 81A, number 6
PHYSICS LETTERS
2 February 1981
EXACTLY SOLUBLE MULTIDIMENSIONAL FOKKER—PLANCK EQUATIONS WITH NONLINEAR DRIFT H. BRAND and A. SCHENZLE Fachbereich Physik, Universitât Essen, 4300 Essen, West Germany Received 3 October 1980 Revised manuscript received 18 November 1980
The time-dependent analytic solutions of three classes of multidimensional Fokker—Planck equations with nonlinear drift are presented together with eigenvalues which are complex and depend essentially on the correlation functions of the fluctuations.
During the last few years the study of multiplicative stochastic processes has become one of the most exciting fields of statistical physics. Experiments in this area deal with very different systems such as chemical reaction dynamics [1] coupled parametric oscifiators in electronic devices [2—4] analogue computers [5] and electrohydrodynamic instabilities in nematic liquid crystals [6]. From the theoretical point of view, multiplicative stochastic processes with a nonlinear drift have been considered in the framework of the Fokker—Planck equation. Stationary solutions of the Fokker—Planck equation for multiplicative stochastic processes have been studied e.g. for the absorptive optical bistability [7] parametric three-wave mixing and Raman scattering [8] and for subharmonic instability [9] in nonlinear optics, for electronic devices [8] and for the electrohydrodynamic instability in nematic liquid crystals [10]. In refs. [8,111 the present authors have solved in detail by analytic methods a large class of time-dependent Fokker—Planck equations. This investigation has shown that multiplicative stochastic processes have exciting properties like an essential dependence of the eigenvalues on the correlation function of the fluctuations and an eigenvalue spectrum which is partly discrete and partly continuous. In the present letter we generalize our results which have been presented in refs. [8,11] to several dimensions and, in addition, we examine a class of stochastic processes which occurs in narrow-band or Rayleigh processes [12]. The Langevin equations for the three classes of stochastic processes to be considered in the following read: type (I): *ax— bx3 +x~0, j~c+dx2 +~ +x~2+x2~3, (1,2) ,
,
,
type (II):
2+e/x2,
(3,4)
i~=—ax+b/x+~0, j—c+(l/x)~1+E2+x~3+dx and type (III): ~=—a/x+b/x3+(1/x)~
2)~
2+~ 0,j’=c+d/x
1+(1/x
2.
(5,6)
A generalization of the processes I, II, III to higher dimensions can be obtained by including additional equations of the structure of eqs. (2), (4) and (6). The fluctuation forces ~ are assumed to form gaussian random processes whose time correlation functions may 0 031—9163/81 /0000—0000/$ 02.50 © North-Holland Publishing Company
321
Volume 81A, number 6
PHYSICS LETTERS
2 February 1981
be approximated by a 6-function: <~)=O, (~j~r)=Qj6jt6(r).
(7)
The Fokker—Planck equation corresponding to the Langevin equations of type (I) reads 3]P} + ~Q 2/ax2)(x2P) = —(a/ax){ [(a + 4Q 0)x bx 0(a + -~-(a2/ay2)[(Q 2 + Q 4)P] + (a/ay)[—(c + dx2)P], 1 + Q2x 3x where we have assumed natural boundary conditions for x and periodic boundary conditions for x, (x
(8)
—
1 can be e.g. the position of a particle). Ify ~ we can interpret eqs. (1), (2) as Langevin equations for the amplitude and phase of a complex stochastic variable. As is well known the general solution of the Fokker—Planck equation can be expanded as follows: P(x,y,t)~j~ CnmPnm(x,y)e_Xflmt,
(9)
where ~nm and Xnm are the corresponding eigenfunctions and eigenvalues. As is easily checked we obtain as eigenfunctions confluent hypergeometric functions multiplied by an exponential and a power law. For the discrete part of the spectrum we obtain in the most general case the following eigenvalues: 2 a +imc+ 2Q0 and eigenfunctions ?~nm~_~_
~nm
=
eim~~~ exp
—
b(l -~Q / 0t,~2n+l ~
a/Q 2Q 0) +m 2 +m2Q 112 (b 0Q3)
+
—
\2
—
(10)
(_ [~- (_-~ m2Q2)hI’2]) x~Qo+1+2~2/’4~Xnm/2Q0+i?fld/2Q0+m2Q1I4Qo)”2 +
~jX2
X
2),
+
—
(11)
0
1F1(—n,2p+ l,V’&x
where ~
ía2
?~
\Q~ ~
2imc
~o
____
~o
,
/
_I
b \2 a=~—j
+m 2
-~—.
\V 0/
The discrete part of the spectrum consists only of a finite number of eigenvalues subject to the condition 2Q b(l +a/Q0)— -~-m 2 2 + m2Q 2 >2n+l. (12) (b 3Q0)~’ Inequality (12) obviously generalizes condition (6.12) in ref. [8]. In addition there exists a continuous spectrum like in the one-dimensional case [8,11]. The eigenfunctions of the continuous part of the spectrum may be normalized onto the 6-function. Because the general result (10) looks somewhat complicated we examine some special cases. First it is straightforward to check that the results presented in refs. [8,11] are reobtained for m 1 = 0. Ford = 0, Q2 = 0, Q3 = 0 eqs. (11), (12) present the solution to a nonlinear stochastic process considered by Stratonovich [13, eqs. (10.93)] in connection with a parametric amplifier and we obtain in that case for the discrete part of the spectrum 322
Volume 81A, number 6
Xnm
=
2n(a
—
nQ0)
PHYSICS LETTERS
2 February 1981
+ imc.
+
(13)
It is important to notice, that the discrete spectrum becomes complex with the following properties: (i) Re Xnm >0, (ii) Re Xnm > ~ ~ (ill) Im Xnm m. Another special case which emerges in a natural way from eqs. (10), (11) is an equation that appears in the context of non-equilibrium phase transitions with a complex order parameter:
2x~+x~F,
(14)
~=ax~—j3IxI
where a and 13 are complex, or in the present notation
i~ax—bx3+x~
2+~ 0, ~a=c+dx
1,
(15)
and where a = a + ic, j3 = b id. This process has been solved by Graham [17] using an analytic continuation method [18] including correlations between the fluctuating forces ~ and ~i. The discrete part of the spectrum reads in that case 2d2Q 2 + im(c + 2ndQ Xnm = 2n(a nQ0) + + m 0/2b 0/b ad/b). (16) —
—
—
If we set Q2 = 0, Q3 = 0, we obtain an equation which has been derived by Kuznetsov et a!. during tl~eirstudy of electrical fluctuations of a valve oscillator [14] For the eigenvalue spectrum we obtain: 2 + n(2a + 2imdQ 2(Q 2Q 2) + im(c ad/b). (17) ~~nm= —2Q0n 0/b) ÷~m 1 + dcase0/b As is easily checked eq. (17) contains as a special the result of the Fokker—Planck equation for multiplicative stochastic processes that has been considered by Tikhonov and Amiantov for the effect of slow fluctuations on anoscifiator [151. Next we consider class (II) obeying the Fokker—Planck equation .
—
aP/at = —(a/ax) [(—ax+ b/x)P]
2/ax2)P 0(a (18) (a/ay)[(c + dx2 + e/x2)P] + (a2/ay2)[(Q 2 + + Q 2)P]. 1 /x 3x Assuming the same boundary conditions as for class (I) we obtain an exact solution for the corresponding eigenvalue problem. For the discrete part of the spectrum we fmd the following eigenvalues: +
.~
Q
—
ab ~~nm
~
b
—
~a+ +m
1
2
Q
1 2ime 2 +imc+(2imdQ0+a2+m2Q0Q3)hI’2[2n+1+[(~—_—~)+jm2÷~]
2Q
1/2
}
(19) and eigenfunctions —
x
where
e imy exp
—
~
1 2[a ~x
~
2
/a
m2 Q ~
3
÷2imd
1/2
—~—
2),
2÷Qlm2/4Qo+imeI2Qo]1/2
1F1(—n, 2i
~Ib/Qo+1/2+2[(_1/4+bI2Qo)
2IL÷lsel÷[(_~+~)+im2÷~-_]~
+
(20)
1, ~/-1-ax
~
*1 Property (ii) has been predicted by Risken [16].
323
Volume 81A, number 6
PHYSICS LETTERS
2 February 1981
where we have to distinguish the following two cases: (i) for Q1 = 0, Q3 = Owe find Re Xnm > Xno >0, provided that b >0; (ii) for arbitrary values of Q1 and Q3 we find Re Xnm > Xno >0, provided that b/Q0> It seems worth noticing that under the above conditions the probability current /~(x= 0) = 0. We find that an equation derived by Stratonovich for Rayleigh processes [12] emerges as a special case of eqs. (19) and (20). Setting Q2 = 0, Q3 = 0, c = 0 we obtain the analytic solution of this process in the form ~-.
ab
[1 b
1
Xnm=~~
1
\2
Ta~ 2n+[~)
Q1 21 +~mj
1/2
(21)
,
and =
2(2a/Q eim~~~ exp[— 4x 0)] x rb 12 Q
2+Q /QO+h/2+2 12 [(—1/4+b/2Qo)
2/4Qol 1/2 1m
(22)
1 2j~‘ ,~—x2 a X iFi(_n~1 +~(~— +~— m We arrive at the conclusion that the influence of the equation of motion for the phase is very drastic. Ifthe latter equation is neglected, as has been done by Stratonovich, the eigenvalues do nor depend on the correlation function of the fluctuations whereas exciting the motion of the phase leads to an explicit though rather intricate dependence of the eigenvalues on the strength of the amplitude fluctuations. This shows that the eigenvalues of additive stochastic processes will in most cases depend on the correlation function of the fluctuations and only in special cases as e.g. the Ornstein—Uhlenbeck process the eigenvalues are independent of Q. Finally we consider the processes of type (III). The Fokker—Planck equation which is stochastically equivalent to eqs. (5), (6) reads ~-)
aP/ar=—(a/ax){[—a/x
2/ax2)[(1/x2)P] 0(a (a/ay)[(c +d/x2)P] +(a2/ay2)f[~Q 4)Q 1 +(l/2x 2]P}. +
(b
—
~Q
3]P}+ ~Q
0)/x
—
(23)
As has been shown in ref. [8] this type of stochastic processes possesses in one dimension a point of accumulation in its spectrum. In the present multidimensional case we get for the discrete part of the spectrum the result 2 1 2 i (ab/Q 2 xnm = —21 Q (24) a +—Q m + imc — 0 — imd) 2/Q] 1/2}2 0 2 1 2Q0 {2n + 1 + [(b/Q0 — 1)2 + Qm —
D
1nm
—
e
imy
,
1
exp —
2
2x
a ,~
~0
j a
2X
2
+I~~fl~
—
L~I
2+Q2m2/Qol
—
\2
L\1’~ 0
/
~o
+
~o
+
2-~1/2
~o 2),
1F’1(—n, 2p
Q
+~-m 2 21j ~
~i
1/2
X IxIb/Qo+2+[(~’Qo1) where [lb 2p+l’l+II~—lJ
()
,-~
2imc
nm
,
+
a=~-) f a \2 ~ 0
1,
x
s,/~
27~nm 2imc
~0
~o
+
Q
2 1m
~o
The probabifity current j~(x = 0) vanishes if b/Q0 > 1 thus insuring positivity of the eigenvalues. An application of the special cases of the last class may occur in two-particle bound-state systems with rotational degrees of freedom. To summarize: We have solved analytically the Fokker—Planck equation for three classes of stochastic processes 324
Volume 81A, number 6
PHYSICS LETFERS
2 February 1981
with nonlinear drift and we have found that the discrete parts of the eigenvalue spectra can depend in a nonanalytic fashion on the correlation function of the fluctuations. Depending on the special case under investigation the discrete part of the spectrum may either be real giving rise to a monotonous relaxation behaviour, or it may be complex leading to relaxation oscillations. Of course it would be extremely interesting to investigate some of the presented results, e.g. in an electronic experiment. References [1] P. de Kepper andW. Horsthemke, C. R. Acad. Sci. C287 (1978) 251. [2] S. Kabashima, Ann. Isr. Phys. Soc. 2 (1978) 710. [3] S. Kabashima and T. Kawakubo, Phys. Lett. 70A (1979) 375. [41 S. Kabashima, S. Kogure, T. Kawakubo and T. Okada, J. App!. Phys. 50 (1979) 6296. [5] S. Kabashima, preprint (1979). [6] S. Kai, T. Kai, M. Takata and K. Hirakawa, J. Phys. Soc. Japan 47 (1979) 1379. [7] A. Schenz!e and H. Brand, Opt. Commun. 27 (1978) 485. [8] A. Schenz!e and H. Brand, Phys. Rev. A20 (1979) 1628. [9] H. Brand, R. Graham and A. Schenzle, Opt. Commun. 32 (1980) 359. [10] H. Brand and A. Schenzle, J. Phys. Soc. Japan 48(1980)1382. [ill A. Schenz!e and H. Brand, Phys. Lett. 69A (1979) 313. [12] R.L. Stratonovich, Topics in the theory of random noise, Vol. !(Gordon and Breach, New York, 1963). [13] R.L. Stratonovich, Topics in the theory of random noise, Vol.11 (Gordon and Breach, New York, 1967). [14] P.1. Kuznetsov, R.L. Stratonovich and V.!. Tikhonov, in: Nonlinear transformations of stochastic processes, eds. P.!. Kuznetsov, R.L. Stratonovich and V.1. Tikhonov (Pergamon, New York, 1965) p. 252. [15] V.1. Tikhonov and I.N. Amiantov, in: Nonlinear transformations of stochastic processes, eds. P.1. Kuznetsov, R.L. Stratonovich and V.1. Tikhonov (Pergamon, New York, 1965) p. 223. [16]H. Risken, Z. Phys. 251 (1972) 231. [17]R. Graham, Phys. Lett. 80A (1980) 351. [18] R. Graham, Z. Phys. B40 (1980), to be published.
325
PHYSICS LETTERS
2 February 1981
EXACTLY SOLUBLE MULTIDIMENSIONAL FOKKER—PLANCK EQUATIONS WITH NONLINEAR DRIFT H. BRAND and A. SCHENZLE Fachbereich Physik, Universitât Essen, 4300 Essen, West Germany Received 3 October 1980 Revised manuscript received 18 November 1980
The time-dependent analytic solutions of three classes of multidimensional Fokker—Planck equations with nonlinear drift are presented together with eigenvalues which are complex and depend essentially on the correlation functions of the fluctuations.
During the last few years the study of multiplicative stochastic processes has become one of the most exciting fields of statistical physics. Experiments in this area deal with very different systems such as chemical reaction dynamics [1] coupled parametric oscifiators in electronic devices [2—4] analogue computers [5] and electrohydrodynamic instabilities in nematic liquid crystals [6]. From the theoretical point of view, multiplicative stochastic processes with a nonlinear drift have been considered in the framework of the Fokker—Planck equation. Stationary solutions of the Fokker—Planck equation for multiplicative stochastic processes have been studied e.g. for the absorptive optical bistability [7] parametric three-wave mixing and Raman scattering [8] and for subharmonic instability [9] in nonlinear optics, for electronic devices [8] and for the electrohydrodynamic instability in nematic liquid crystals [10]. In refs. [8,111 the present authors have solved in detail by analytic methods a large class of time-dependent Fokker—Planck equations. This investigation has shown that multiplicative stochastic processes have exciting properties like an essential dependence of the eigenvalues on the correlation function of the fluctuations and an eigenvalue spectrum which is partly discrete and partly continuous. In the present letter we generalize our results which have been presented in refs. [8,11] to several dimensions and, in addition, we examine a class of stochastic processes which occurs in narrow-band or Rayleigh processes [12]. The Langevin equations for the three classes of stochastic processes to be considered in the following read: type (I): *ax— bx3 +x~0, j~c+dx2 +~ +x~2+x2~3, (1,2) ,
,
,
type (II):
2+e/x2,
(3,4)
i~=—ax+b/x+~0, j—c+(l/x)~1+E2+x~3+dx and type (III): ~=—a/x+b/x3+(1/x)~
2)~
2+~ 0,j’=c+d/x
1+(1/x
2.
(5,6)
A generalization of the processes I, II, III to higher dimensions can be obtained by including additional equations of the structure of eqs. (2), (4) and (6). The fluctuation forces ~ are assumed to form gaussian random processes whose time correlation functions may 0 031—9163/81 /0000—0000/$ 02.50 © North-Holland Publishing Company
321
Volume 81A, number 6
PHYSICS LETTERS
2 February 1981
be approximated by a 6-function: <~)=O, (~j~r)=Qj6jt6(r).
(7)
The Fokker—Planck equation corresponding to the Langevin equations of type (I) reads 3]P} + ~Q 2/ax2)(x2P) = —(a/ax){ [(a + 4Q 0)x bx 0(a + -~-(a2/ay2)[(Q 2 + Q 4)P] + (a/ay)[—(c + dx2)P], 1 + Q2x 3x where we have assumed natural boundary conditions for x and periodic boundary conditions for x, (x
(8)
—
1 can be e.g. the position of a particle). Ify ~ we can interpret eqs. (1), (2) as Langevin equations for the amplitude and phase of a complex stochastic variable. As is well known the general solution of the Fokker—Planck equation can be expanded as follows: P(x,y,t)~j~ CnmPnm(x,y)e_Xflmt,
(9)
where ~nm and Xnm are the corresponding eigenfunctions and eigenvalues. As is easily checked we obtain as eigenfunctions confluent hypergeometric functions multiplied by an exponential and a power law. For the discrete part of the spectrum we obtain in the most general case the following eigenvalues: 2 a +imc+ 2Q0 and eigenfunctions ?~nm~_~_
~nm
=
eim~~~ exp
—
b(l -~Q / 0t,~2n+l ~
a/Q 2Q 0) +m 2 +m2Q 112 (b 0Q3)
+
—
\2
—
(10)
(_ [~- (_-~ m2Q2)hI’2]) x~Qo+1+2~2/’4~Xnm/2Q0+i?fld/2Q0+m2Q1I4Qo)”2 +
~jX2
X
2),
+
—
(11)
0
1F1(—n,2p+ l,V’&x
where ~
ía2
?~
\Q~ ~
2imc
~o
____
~o
,
/
_I
b \2 a=~—j
+m 2
-~—.
\V 0/
The discrete part of the spectrum consists only of a finite number of eigenvalues subject to the condition 2Q b(l +a/Q0)— -~-m 2 2 + m2Q 2 >2n+l. (12) (b 3Q0)~’ Inequality (12) obviously generalizes condition (6.12) in ref. [8]. In addition there exists a continuous spectrum like in the one-dimensional case [8,11]. The eigenfunctions of the continuous part of the spectrum may be normalized onto the 6-function. Because the general result (10) looks somewhat complicated we examine some special cases. First it is straightforward to check that the results presented in refs. [8,11] are reobtained for m 1 = 0. Ford = 0, Q2 = 0, Q3 = 0 eqs. (11), (12) present the solution to a nonlinear stochastic process considered by Stratonovich [13, eqs. (10.93)] in connection with a parametric amplifier and we obtain in that case for the discrete part of the spectrum 322
Volume 81A, number 6
Xnm
=
2n(a
—
nQ0)
PHYSICS LETTERS
2 February 1981
+ imc.
+
(13)
It is important to notice, that the discrete spectrum becomes complex with the following properties: (i) Re Xnm >0, (ii) Re Xnm > ~ ~ (ill) Im Xnm m. Another special case which emerges in a natural way from eqs. (10), (11) is an equation that appears in the context of non-equilibrium phase transitions with a complex order parameter:
2x~+x~F,
(14)
~=ax~—j3IxI
where a and 13 are complex, or in the present notation
i~ax—bx3+x~
2+~ 0, ~a=c+dx
1,
(15)
and where a = a + ic, j3 = b id. This process has been solved by Graham [17] using an analytic continuation method [18] including correlations between the fluctuating forces ~ and ~i. The discrete part of the spectrum reads in that case 2d2Q 2 + im(c + 2ndQ Xnm = 2n(a nQ0) + + m 0/2b 0/b ad/b). (16) —
—
—
If we set Q2 = 0, Q3 = 0, we obtain an equation which has been derived by Kuznetsov et a!. during tl~eirstudy of electrical fluctuations of a valve oscillator [14] For the eigenvalue spectrum we obtain: 2 + n(2a + 2imdQ 2(Q 2Q 2) + im(c ad/b). (17) ~~nm= —2Q0n 0/b) ÷~m 1 + dcase0/b As is easily checked eq. (17) contains as a special the result of the Fokker—Planck equation for multiplicative stochastic processes that has been considered by Tikhonov and Amiantov for the effect of slow fluctuations on anoscifiator [151. Next we consider class (II) obeying the Fokker—Planck equation .
—
aP/at = —(a/ax) [(—ax+ b/x)P]
2/ax2)P 0(a (18) (a/ay)[(c + dx2 + e/x2)P] + (a2/ay2)[(Q 2 + + Q 2)P]. 1 /x 3x Assuming the same boundary conditions as for class (I) we obtain an exact solution for the corresponding eigenvalue problem. For the discrete part of the spectrum we fmd the following eigenvalues: +
.~
Q
—
ab ~~nm
~
b
—
~a+ +m
1
2
Q
1 2ime 2 +imc+(2imdQ0+a2+m2Q0Q3)hI’2[2n+1+[(~—_—~)+jm2÷~]
2Q
1/2
}
(19) and eigenfunctions —
x
where
e imy exp
—
~
1 2[a ~x
~
2
/a
m2 Q ~
3
÷2imd
1/2
—~—
2),
2÷Qlm2/4Qo+imeI2Qo]1/2
1F1(—n, 2i
~Ib/Qo+1/2+2[(_1/4+bI2Qo)
2IL÷lsel÷[(_~+~)+im2÷~-_]~
+
(20)
1, ~/-1-ax
~
*1 Property (ii) has been predicted by Risken [16].
323
Volume 81A, number 6
PHYSICS LETTERS
2 February 1981
where we have to distinguish the following two cases: (i) for Q1 = 0, Q3 = Owe find Re Xnm > Xno >0, provided that b >0; (ii) for arbitrary values of Q1 and Q3 we find Re Xnm > Xno >0, provided that b/Q0> It seems worth noticing that under the above conditions the probability current /~(x= 0) = 0. We find that an equation derived by Stratonovich for Rayleigh processes [12] emerges as a special case of eqs. (19) and (20). Setting Q2 = 0, Q3 = 0, c = 0 we obtain the analytic solution of this process in the form ~-.
ab
[1 b
1
Xnm=~~
1
\2
Ta~ 2n+[~)
Q1 21 +~mj
1/2
(21)
,
and =
2(2a/Q eim~~~ exp[— 4x 0)] x rb 12 Q
2+Q /QO+h/2+2 12 [(—1/4+b/2Qo)
2/4Qol 1/2 1m
(22)
1 2j~‘ ,~—x2 a X iFi(_n~1 +~(~— +~— m We arrive at the conclusion that the influence of the equation of motion for the phase is very drastic. Ifthe latter equation is neglected, as has been done by Stratonovich, the eigenvalues do nor depend on the correlation function of the fluctuations whereas exciting the motion of the phase leads to an explicit though rather intricate dependence of the eigenvalues on the strength of the amplitude fluctuations. This shows that the eigenvalues of additive stochastic processes will in most cases depend on the correlation function of the fluctuations and only in special cases as e.g. the Ornstein—Uhlenbeck process the eigenvalues are independent of Q. Finally we consider the processes of type (III). The Fokker—Planck equation which is stochastically equivalent to eqs. (5), (6) reads ~-)
aP/ar=—(a/ax){[—a/x
2/ax2)[(1/x2)P] 0(a (a/ay)[(c +d/x2)P] +(a2/ay2)f[~Q 4)Q 1 +(l/2x 2]P}. +
(b
—
~Q
3]P}+ ~Q
0)/x
—
(23)
As has been shown in ref. [8] this type of stochastic processes possesses in one dimension a point of accumulation in its spectrum. In the present multidimensional case we get for the discrete part of the spectrum the result 2 1 2 i (ab/Q 2 xnm = —21 Q (24) a +—Q m + imc — 0 — imd) 2/Q] 1/2}2 0 2 1 2Q0 {2n + 1 + [(b/Q0 — 1)2 + Qm —
D
1nm
—
e
imy
,
1
exp —
2
2x
a ,~
~0
j a
2X
2
+I~~fl~
—
L~I
2+Q2m2/Qol
—
\2
L\1’~ 0
/
~o
+
~o
+
2-~1/2
~o 2),
1F’1(—n, 2p
Q
+~-m 2 21j ~
~i
1/2
X IxIb/Qo+2+[(~’Qo1) where [lb 2p+l’l+II~—lJ
()
,-~
2imc
nm
,
+
a=~-) f a \2 ~ 0
1,
x
s,/~
27~nm 2imc
~0
~o
+
Q
2 1m
~o
The probabifity current j~(x = 0) vanishes if b/Q0 > 1 thus insuring positivity of the eigenvalues. An application of the special cases of the last class may occur in two-particle bound-state systems with rotational degrees of freedom. To summarize: We have solved analytically the Fokker—Planck equation for three classes of stochastic processes 324
Volume 81A, number 6
PHYSICS LETFERS
2 February 1981
with nonlinear drift and we have found that the discrete parts of the eigenvalue spectra can depend in a nonanalytic fashion on the correlation function of the fluctuations. Depending on the special case under investigation the discrete part of the spectrum may either be real giving rise to a monotonous relaxation behaviour, or it may be complex leading to relaxation oscillations. Of course it would be extremely interesting to investigate some of the presented results, e.g. in an electronic experiment. References [1] P. de Kepper andW. Horsthemke, C. R. Acad. Sci. C287 (1978) 251. [2] S. Kabashima, Ann. Isr. Phys. Soc. 2 (1978) 710. [3] S. Kabashima and T. Kawakubo, Phys. Lett. 70A (1979) 375. [41 S. Kabashima, S. Kogure, T. Kawakubo and T. Okada, J. App!. Phys. 50 (1979) 6296. [5] S. Kabashima, preprint (1979). [6] S. Kai, T. Kai, M. Takata and K. Hirakawa, J. Phys. Soc. Japan 47 (1979) 1379. [7] A. Schenz!e and H. Brand, Opt. Commun. 27 (1978) 485. [8] A. Schenz!e and H. Brand, Phys. Rev. A20 (1979) 1628. [9] H. Brand, R. Graham and A. Schenzle, Opt. Commun. 32 (1980) 359. [10] H. Brand and A. Schenzle, J. Phys. Soc. Japan 48(1980)1382. [ill A. Schenz!e and H. Brand, Phys. Lett. 69A (1979) 313. [12] R.L. Stratonovich, Topics in the theory of random noise, Vol. !(Gordon and Breach, New York, 1963). [13] R.L. Stratonovich, Topics in the theory of random noise, Vol.11 (Gordon and Breach, New York, 1967). [14] P.1. Kuznetsov, R.L. Stratonovich and V.!. Tikhonov, in: Nonlinear transformations of stochastic processes, eds. P.!. Kuznetsov, R.L. Stratonovich and V.1. Tikhonov (Pergamon, New York, 1965) p. 252. [15] V.1. Tikhonov and I.N. Amiantov, in: Nonlinear transformations of stochastic processes, eds. P.1. Kuznetsov, R.L. Stratonovich and V.1. Tikhonov (Pergamon, New York, 1965) p. 223. [16]H. Risken, Z. Phys. 251 (1972) 231. [17]R. Graham, Phys. Lett. 80A (1980) 351. [18] R. Graham, Z. Phys. B40 (1980), to be published.
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