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Ergodic behavior in the statistical theory of nuclear reactions
Volume 80B, number 1,2
PHYSICS LETTERS
18 December 1978
ERGODIC BEHAVIOR IN THE STATISTICAL THEORY OF NUCLEAR REACTIONS J.B. FRENCH 1'2 , P.A. MELLO and A. PANDEY 1
lnstituto de Ffsica, UniversidadNationalAutdnoma de Mdxico,Mdxico20, D.F.,Mdxico Received 12 September 1978
A very simple proof is given of the ergodieity of the S-matrix and related quantities, that makes use of the connection between ergodicity and the autocovariance function of the quantity of interest.
Statistical theories of nuclear reactions frequently make use of an ensemble of S-matrices [1,2], ensemble averages of the quantities of physical interest (S-matrix, cross section, etc.) being substituted for the corresponding energy averages. The equality of the two averages has been tested numerically in some specific cases [2] ; more recently, a proof was given [3], within the formalism of ref. [4], by expanding the scattering matrix in powers o f the interaction matrix elements. In this proof the equality of the ensemble and energy averages of all possible combinations of interaction matrix elements and level energies has been taken for granted, as derived from random-matrix theories; however, proofs of ergodicity in such cases have been only very recently given [5]. The purpose of this letter is to present a very simple proof of the ergodicity of the S-matrix and related quantities, that makes use of the connection between ergodicity and the autocovariance function of the quantity of interest [5,6]. We shall see that for F/D >> 1 (where P and D are the mean decay width and the average level spacing, respectively), Agassi et al. [4] give, in their sect. 9, all of the ingredients that are necessary to prove the ergodic behavior of the quantities of interest.
Let F be a complex stationary random function of the energy. We shall use a bar to denote an ensemble average and a bracket for an energy average over an energy interval A or over a corresponding average number of levels p (= A/D):
Supported in part by the U.S. Department of Energy under Contract No. EY-76-S-02-2171.000 and the Instituto Nacional de Energia Nuclear of Mexico. 1 On leave of absence from the Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627, USA. 2 John Simon Guggenheim Memorial Fellow during 1977-78.
does not depend on E, due to stationarity. It follows from eq. (2) that [6]
E+A
(F(E))A
= i- f
F(E')dE'
E
(1)
p
=P 1/
F(E + rD)dr -~ (F)p.
The ensemble variance of this quantity, var (F> = [0012, where f = F - if, can be easily seen to be 1
var(F>p=~
A
f
(A-[el)CF(e)de
-A
(2) =~
P
p2 f (p- Irl)CF(rD)dr' -p
where the autocovariance function
CF(e)
CF(e) -- f*(f)f(E + e)
lim
CF(rD) = 0 ~ lim var(F)p = 0.
This in turn gives the
(3)
(4)
ergodic property that 17
Volume 80B, n u m b e r 1,2
lim
p-+~
PHYSICS LETTERS
(F)p = F
(5)
18 December 1978
var(Sat))p
2vcorr It _ an_ 1 pc-o~ ,5
_.l_SabI~- = A
for "almost all" members of the ensemble, i.e., for all except for a set of zero measure. We have written things in eqs. (4) and (5) in terms of the level number rather than the energy parameter to emphasize that it is the number of levels p which is all-important for ergodicity. In the case of energy-level fluctuations in randommatrix theories [5] one can (by taking an appropriately large matrix dimensionality d) have an arbitrarily small energy interval A containing an arbitrarily large number of levels p. One finds then when eq. (4) is satisfied, as it is for the standard ensembles, a very strong "locally generated" ergodicity, which, since the energy interval may be taken small, obtains not only for stationary but also for quasistationary ("locally stationary") processes. For nuclei it is not straightforward to define an appropriate d value but we may take for granted nonetheless that there is local stationarity which is adequate for ergodic behavior. In the randommatrix cases the variances ~ p - 2 1 n p so that strong ergodicity is generated with perhaps 100 levels; in the present case the characteristic form is seen below (eqs. (7) and (9)) to be ~ p - 1 so that 1000 levels or so should be adequate; in regions of high level density, encountered in reactions, there would then be negligible secular variation, over the ergodicity-generating interval, in the density, penetrabilities, etc. We consider first the element Sab of the scattering matrix. Writing Sat) = Sab + sfal' the autocovariance function C(e) can be found from ref. [4], eq. (9.4), to be
c(e) = Sab fl* (E)~b(L n . + ~) (6) = [+irc°rr/(e + ipc°rr)] C(0),
1 F c°rr
A 2 + (Fc°rr)21 °fl ab
(7)
ab rr [ , c o r r
o ab fl
A
0 ab
fl ~.pr Oab
dir
P
O dir '
ab
fl) =~ o,Sab[ f l 2 and odor ~ [Sab [2 are the fluctuaOat tion and the direct cross sections, respectively, for the process (a,b), and F r = Fc°rr/D is the correlation length expressed in units of local spacing. We have ignored terms of order p - 2 lnp to obtain the second form of eq. (7). As a second example, we consider F = SabScd. Following the rules given in ref. [4], the corresponding autocovariance function is found to be where
_ipcorr
iF c°rr I~b f2 PccPdd +Pc'~Pd~cc
C(e)-
e-
Tr P
i~ccll2PaaPbt) +PabPt)a
iF c°rr ~'
Wr P
c + ipc°rr
(rcor,)2
+
(8)
e P t) + e t)vt), PcPdd + P aPae
e2 + (pcorr)2
Tr P
Tr P
which goes to zero as e -+ ~ , so that SabS*d is ergodic. The variance for finite A can again be found explicitly. In units of [SabS*d[ 2, it is given, in the approximation used for the second form of eq. (7), by
var(Sab S~d)p -+ rrP c°rr [SabS*d - Sab S*d ]2 [Sab S,dP2
A
[SabS.cdl2 (9)
w h e r e F c°rr = T r P/(21rp) is the correlation length,
P = 1 - SS-? being Satchler's penetration matrix and P the level density; C(0) is given by C(0) = (PaaPbb + PabPba)/Tr P. Since C(e) -+ e--,~ 0, we conclude that Sab is ergodic. The variance of (Sat)) for a finite value of A can be found explicitly. In units of the square of the corresponding ensemble average Sab we have
, -- S b X * d [2 rr pr ISao. S *ca
p
j%sv2
For a --- c, b = d, this result proves the ergodicit3/of the cross section, the variance being
var (Oab)p -+ n17'c°rr var Oab (~ab)2 where 18
A
Oab = ISab 12.
(~ab)2 '
(10)
Volume 80B, number 1,2
PHYSICS LETTERS
For higher moments o f S, the procedure o f ref. [4], sect. 9, shows that S is a normal (matrix) process so that the autocovariance function can be expressed as sums over products of quantities like (6), its vanishing for large e establishing again the corresponding ergodic property. It can also be easily shown that the variance of the corresponding spectral averages is, like (7) and (9), o f order ( n p c ° r r / A ) = (7r pr/p).
18 December 1978
References [1] P.A. Moldauer, Phys. Rev. 135 (1964) 642. [2] H.M. Hofmann, J. Richert, J.W. Tepel and H.A. Weidenm/Jllcr, Ann. Phys. (NY) 90 (1975) 403. [3] J. Richert and H.A. Weidenmfiller, Phys. Rev. C16 (1977) 1309. [4] D. Agassi, H.A. Weidenmiiller and G. Mantzoura~is, Phys. Rep. 22 (1975) 145. [5] A. Pandey, to be published; J.B. French and A. Pandey, to be published. [6] A.M. Yaglom, An introduction to the theory of stationary random functions, transl, by R.A. Silverman (PrenticeHall, Englewood Cliffs, NJ, 1962).
19
PHYSICS LETTERS
18 December 1978
ERGODIC BEHAVIOR IN THE STATISTICAL THEORY OF NUCLEAR REACTIONS J.B. FRENCH 1'2 , P.A. MELLO and A. PANDEY 1
lnstituto de Ffsica, UniversidadNationalAutdnoma de Mdxico,Mdxico20, D.F.,Mdxico Received 12 September 1978
A very simple proof is given of the ergodieity of the S-matrix and related quantities, that makes use of the connection between ergodicity and the autocovariance function of the quantity of interest.
Statistical theories of nuclear reactions frequently make use of an ensemble of S-matrices [1,2], ensemble averages of the quantities of physical interest (S-matrix, cross section, etc.) being substituted for the corresponding energy averages. The equality of the two averages has been tested numerically in some specific cases [2] ; more recently, a proof was given [3], within the formalism of ref. [4], by expanding the scattering matrix in powers o f the interaction matrix elements. In this proof the equality of the ensemble and energy averages of all possible combinations of interaction matrix elements and level energies has been taken for granted, as derived from random-matrix theories; however, proofs of ergodicity in such cases have been only very recently given [5]. The purpose of this letter is to present a very simple proof of the ergodicity of the S-matrix and related quantities, that makes use of the connection between ergodicity and the autocovariance function of the quantity of interest [5,6]. We shall see that for F/D >> 1 (where P and D are the mean decay width and the average level spacing, respectively), Agassi et al. [4] give, in their sect. 9, all of the ingredients that are necessary to prove the ergodic behavior of the quantities of interest.
Let F be a complex stationary random function of the energy. We shall use a bar to denote an ensemble average and a bracket for an energy average over an energy interval A or over a corresponding average number of levels p (= A/D):
Supported in part by the U.S. Department of Energy under Contract No. EY-76-S-02-2171.000 and the Instituto Nacional de Energia Nuclear of Mexico. 1 On leave of absence from the Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627, USA. 2 John Simon Guggenheim Memorial Fellow during 1977-78.
does not depend on E, due to stationarity. It follows from eq. (2) that [6]
E+A
(F(E))A
= i- f
F(E')dE'
E
(1)
p
=P 1/
F(E + rD)dr -~ (F)p.
The ensemble variance of this quantity, var (F> = [0012, where f = F - if, can be easily seen to be 1
var(F>p=~
A
f
(A-[el)CF(e)de
-A
(2) =~
P
p2 f (p- Irl)CF(rD)dr' -p
where the autocovariance function
CF(e)
CF(e) -- f*(f)f(E + e)
lim
CF(rD) = 0 ~ lim var(F)p = 0.
This in turn gives the
(3)
(4)
ergodic property that 17
Volume 80B, n u m b e r 1,2
lim
p-+~
PHYSICS LETTERS
(F)p = F
(5)
18 December 1978
var(Sat))p
2vcorr It _ an_ 1 pc-o~ ,5
_.l_SabI~- = A
for "almost all" members of the ensemble, i.e., for all except for a set of zero measure. We have written things in eqs. (4) and (5) in terms of the level number rather than the energy parameter to emphasize that it is the number of levels p which is all-important for ergodicity. In the case of energy-level fluctuations in randommatrix theories [5] one can (by taking an appropriately large matrix dimensionality d) have an arbitrarily small energy interval A containing an arbitrarily large number of levels p. One finds then when eq. (4) is satisfied, as it is for the standard ensembles, a very strong "locally generated" ergodicity, which, since the energy interval may be taken small, obtains not only for stationary but also for quasistationary ("locally stationary") processes. For nuclei it is not straightforward to define an appropriate d value but we may take for granted nonetheless that there is local stationarity which is adequate for ergodic behavior. In the randommatrix cases the variances ~ p - 2 1 n p so that strong ergodicity is generated with perhaps 100 levels; in the present case the characteristic form is seen below (eqs. (7) and (9)) to be ~ p - 1 so that 1000 levels or so should be adequate; in regions of high level density, encountered in reactions, there would then be negligible secular variation, over the ergodicity-generating interval, in the density, penetrabilities, etc. We consider first the element Sab of the scattering matrix. Writing Sat) = Sab + sfal' the autocovariance function C(e) can be found from ref. [4], eq. (9.4), to be
c(e) = Sab fl* (E)~b(L n . + ~) (6) = [+irc°rr/(e + ipc°rr)] C(0),
1 F c°rr
A 2 + (Fc°rr)21 °fl ab
(7)
ab rr [ , c o r r
o ab fl
A
0 ab
fl ~.pr Oab
dir
P
O dir '
ab
fl) =~ o,Sab[ f l 2 and odor ~ [Sab [2 are the fluctuaOat tion and the direct cross sections, respectively, for the process (a,b), and F r = Fc°rr/D is the correlation length expressed in units of local spacing. We have ignored terms of order p - 2 lnp to obtain the second form of eq. (7). As a second example, we consider F = SabScd. Following the rules given in ref. [4], the corresponding autocovariance function is found to be where
_ipcorr
iF c°rr I~b f2 PccPdd +Pc'~Pd~cc
C(e)-
e-
Tr P
i~ccll2PaaPbt) +PabPt)a
iF c°rr ~'
Wr P
c + ipc°rr
(rcor,)2
+
(8)
e P t) + e t)vt), PcPdd + P aPae
e2 + (pcorr)2
Tr P
Tr P
which goes to zero as e -+ ~ , so that SabS*d is ergodic. The variance for finite A can again be found explicitly. In units of [SabS*d[ 2, it is given, in the approximation used for the second form of eq. (7), by
var(Sab S~d)p -+ rrP c°rr [SabS*d - Sab S*d ]2 [Sab S,dP2
A
[SabS.cdl2 (9)
w h e r e F c°rr = T r P/(21rp) is the correlation length,
P = 1 - SS-? being Satchler's penetration matrix and P the level density; C(0) is given by C(0) = (PaaPbb + PabPba)/Tr P. Since C(e) -+ e--,~ 0, we conclude that Sab is ergodic. The variance of (Sat)) for a finite value of A can be found explicitly. In units of the square of the corresponding ensemble average Sab we have
, -- S b X * d [2 rr pr ISao. S *ca
p
j%sv2
For a --- c, b = d, this result proves the ergodicit3/of the cross section, the variance being
var (Oab)p -+ n17'c°rr var Oab (~ab)2 where 18
A
Oab = ISab 12.
(~ab)2 '
(10)
Volume 80B, number 1,2
PHYSICS LETTERS
For higher moments o f S, the procedure o f ref. [4], sect. 9, shows that S is a normal (matrix) process so that the autocovariance function can be expressed as sums over products of quantities like (6), its vanishing for large e establishing again the corresponding ergodic property. It can also be easily shown that the variance of the corresponding spectral averages is, like (7) and (9), o f order ( n p c ° r r / A ) = (7r pr/p).
18 December 1978
References [1] P.A. Moldauer, Phys. Rev. 135 (1964) 642. [2] H.M. Hofmann, J. Richert, J.W. Tepel and H.A. Weidenm/Jllcr, Ann. Phys. (NY) 90 (1975) 403. [3] J. Richert and H.A. Weidenmfiller, Phys. Rev. C16 (1977) 1309. [4] D. Agassi, H.A. Weidenmiiller and G. Mantzoura~is, Phys. Rep. 22 (1975) 145. [5] A. Pandey, to be published; J.B. French and A. Pandey, to be published. [6] A.M. Yaglom, An introduction to the theory of stationary random functions, transl, by R.A. Silverman (PrenticeHall, Englewood Cliffs, NJ, 1962).
19