The evolution to quantum statistical equilibrium for a simple model

Physica 29 653-661

Verboven, E.

Buyst, L. 1963

THE EVOLUTION TO QUANTUM STATISTICAL EQUILIBRIUM FOR A SIMPLE MODEL II. A NUMERICAL SOLUTION

by E. VERBOVEN *) and L. BUYST Studiecentrum voor Kernenergie, Mol-Dank, Belgie.

Synopsis A simple model, int rod uced by L. Van Hove and E . Verboven, to invest igat e the evolution to qu antum statistical equilibrium, is solved numerically for the diagonal part of the transition probability density. For J. = 0.1 the weak coupling approximation and the numerical solutions agree very well . For ). = 10 the strong coupling an d numerical solution practicall y coincide. F or J. = 1 the numerical solution differs considerably from the weak coupling and strong coupling solutions, showing the ex ist ence of an intermediate coupling region.

1. Introduction. A simple model to investigat e the evolution to quantum statistical equilibrium, introduced by L. Van Hove and one of us (E . Verboven) 1) , has been studied recently in the strong coupling limitz). In this limit the behaviour in time of the transition probability density showed to be entirely different from the weak coupling treatment, which is governed by the Pauli equation and characterized by a relaxation time. For strong coupling the evolution in time is no longer an exponential. Another feature is the appearence of damped oscillations, due to memory effects. There remained however a coupling region where the solution could not be worked out analytically, lying in between the strong coupling and the weak coupling approximation. Therefore we proposed to look for an entirely numerical solution both in this region and in the weak coupling and strong coupling regions. The results show the accuracy of the approximations in the strong and weak coupling limit. In the intermediate coupling region , a different behaviour is observed. The oscillations of the strong coupling treatment do not show up but still th e evolution differs appreciably from an exponentiaL We introduce in sect ion 2 the simple model , using the notation of ref. 2, and define the quantities which are calculated numerically. In section 3 we present the numerical results and compare them with the analytical results already obtained previously in refs. 1 and 2. *} Present ad dr ess : Iustit uut voor T hcoretis cho Fy sica de r Ka thclieke Universiteit, Nijmegen.

-

653 -

654

E. VERB OVEN AND L. BUYST

2. A simple model. Using the notation of ref. 2 our purpose is to evaluate the coefficient of the o(ko - k)-function, occurring in the diagonal part of the transition probability density Pt(kok). We write Pt(kok) as an integral of another function PE, t(kok) co

Pt(kok) =JdE PE,t(kok).

(2.1 )

-co

For the special model introduced in ref. 1 PE, t(kok) has the simple form

k may be interpreted as the wave vector of an electron. (2.2) is obtained from the general theory making the ansatz

{]IV for k,.k' :::;;; a

W ,(kk') = u

0 otherwise,

(2.3)

with a and ]IV given positive constants. (k denotes the length of k, and similarly for k'). The quantity we are interested in is the coefficient of the o(ko - k)function in (2.1) and (2.2). Introducing the notation Pt(k) respectively PE, t(k) for these quantities we have co

Pt(k) =

J dE PE, t(k)

(2.4)

-co

PE, t(k) = 2n1 2

I

.

dl exp(2'l.lt) DE+l(k) DE-l(k)

(2.5)

y

In ref. 1, (2.5) has been evaluated in an approximation based on an analytical continuation of the diagonal part of the resolvent. It is essentially a weak coupling approximation. The result was found to be

PE,t(k) =

sin [2t(Bk - E)J exp(-2Ykt ) n(Bk - E)

(2.6)

and

(2.7) with Bk = k 2 -

4:n;WA,2a

a+k + 2nA,2Wk In -a-k --

(2.8)

THE EVOLUTION TO QUANTUM STATISTICAL EQUILIBRIUM. II

655

and Ylc = 2n 2).2Wk.

(2.9)

In the strong coupling limit, the following expression was obtained 1

;:2 {J

= -

P(t)

dl sin(2Alt)[(12 -

+ (l2 + I) E(l)

I) K(l)

- Inl]

+

o 00

+ fd1Sin(2Alt) l[(1

-l2)

K(+) + (I + 1 E(+) -inJ 2

)

1 1

+

f

dlcos(2Alt)(l +l)[21K(

~ +~) -(1 +12)E( ~ +~)J}

(2.10)

o

with

A = ( 16n:a

3 ?2

y.

(2.11 )

K(l) and E(l) are the Legendre complete elliptic integrals of first and second kind. From (2.10) it is seen that it is natural to introduce a new time scale given by T

(2.12)

= 2At.

The integrals occurring in (2.10) have been evaluated numerically. In ref. 2 it is shown that the strong coupling treatment is valid as soon as 3a

A~ ( - -

)!-

4nW

whereas in the first approximation A

~(2n~W

r

(2.13)

(2.14)

Therefore in general there remains a coupling region where neither the first nor the second approximation is valid. 3. Numerical solutions. We describe in this section how numerical solutions are obtained for three special values of the coupling constant A, chosen in such a way that, with a special choice of the parameters Wand a, they lie in the strong, intermediate and weak coupling region respectively. The solution is based on the equation a

gz = 4nW J (STc' - l - J,,2g Z)-1 k!2 dk',

o

(3.1)

656

E. VERBOVEN AND L. BUYST

where gz occurs in the diagonal part of the resolvent. (3.1) is solved numerically for the following values of the parameters

a

1, W

=

1

= -,

A = 10, 1 and 0.1.

4n

gz is calculated by Newton's method in the complex plane from F(gz)

=

4nW { a

+ !(l + A2gl)! In

(t (l

+ A2gZ)! -a }

+ A2gZ)! + a

- gz = 0

(3.2)

obtained by integrating (3.1) for real values of l, For A = 10 good agreement was obtained with the strong coupling limit of ref. 2, formula (4.10) and (4.11), except in the very neighbourhood of the branch points. In the weak coupling case (A = 0.1) a different picture is obtained, which is shown in fig. 1.

Imgl

...................-.. ,\

91 for A: 0.1

I

s-»

: I

I I

-0.5 -OA

-ea

·QZ -0.1

0

QI

Q2

Q3

0,4

-I

·2

Fig. 1. The gdunction for A = 0.1.

For A = 1, gz resembles qualitatively the strong coupling result with a displaced symmetry axis, which, for A = 10 is situated in the origin. This numerical solution is brought in Dz(k) which makes it possible to obtain PE,t(k) from (2.5). The contour y of (2.5) is a counter-clockwise contour in the complex l-plane encircling the real axis. By this fact (2.5) can be transformed into

P E , t(k) = -

~2

f

exp(2ilt) 1m IE, z(k) dl

-00

with

(3.3)

657

THE EVOLUTION TO QUANTUM STATISTICAL EQ UILI B R IU M. II

Using the fact that Im [e. l(k ) = - Im fE,-l(k) this gives oc

P E , t(k) =

:2 f

Irn [s, l(k) sin

zu ell

(3.5)

o

which is a real oscillating integral. For A = O. I the numerical ev aluation of (3.5) is in very good agreement with the approximate result of ref . I, formula (3.28). There ex ists a pr eferred P.. Ilkl 1=30 k=Q5

10

),,=0.1

09

1

E

-J

-c

Fig. 2. Pe , ,(Il) for t = 30, k = 0.5 a nd ).

=

0.1.

~ ~1

1 =1 k =0,5

E 10

-I

-2

-3

Fig. 3. Pe ,t(k) for t = 1, k = 0.5 and),

=

1.

11

12

658

E. VERBOVEN AND L. BUYST III

~

• 1 t = 5 k = OS

0.05

- 0.25

-0.0

Fig. 4. Ps.t(k) for t

=

5, It

= 0.5

and A = 1.

}, =1 I' =10 ~ =0.5

Fig. 5. P E , t(k) for t

=

10, h

=

0.5 and A = 1.

= 10k = k 2 and the function P E , t(k) approaches for t -r-s- oo to a delta function I5(E - eTc) as shown in fig. 2. For A= 1, the behaviour for long times is different, There is no tendency to a well preferred value of the energy as in the weak coupling case. This is seen in fig. 3, 4 and 5 for the special case k = 0.5. The same behaviour is found for A = 10. As a function of time all the curves iotPn, t(k) look essentially the same: E value, namely E

THE EVOLUTION TO QUANTUM STATISTICAL EQUILIBRIUM. II

659

a damped oscillation. A specimen is shown for A = 10, E = 0.5 and k = 0.5 in fig. 6. P£,t Ckl ~ ~

10

k •

os

E. os

0.8 -I

-2

-s

Fig. 6. P E , t(k) for E = 0.5, k = 0.5 and A = 10.

In order to obtain Pt(k) we did not use PE, t(k) and integrate over E, because we then should have to perform an integration from - 0 0 to of an oscillating function PE, t(k) with variable period and which is also a function of time. We used instead the general expression

+=

co

J dE Jdl_2nl_ exp(2ilt) DE+l(k) DE-l(k)

Pt(k) =

(3.6)

2

-0<)

l'

and integrated first over E and afterwards over l. The expression (3.6) can be transformed into Pt(k) = _1_JeXP(2ilt) F(k, l) ell 2n 2

with

(3.7)

u-: (3.8)

F(k, l) = -2iJ DE-l(k) Irn DE+l(k) dE, t.-i

which is a smooth function that can be pre-calculated in function of l independent of time. y- is an integration path just below the real I-axis and lr and 12 are the branch points of gl. This gives 00

Pt(k) =

~2

f

Re F(k, l) cos

o which are real integrals.

00

zu dl - ~2

f 0

Irn F(k, l) sin

zu dl

(3.9)

660

E. VERBOVEN AND L. BUYST

The numerical evaluation of the integrals of the form p

p

J t(x) cos px dx

J t(x) sin px dx

and

is performed by a method based on the expansion of t(x) into a series of Legendre polynomials and repeated integration by parts. Integration is first attempted over the whole range by using a five-point formula. The result is compared with the sum of the results obtained by using the same formula consecutively over two half intervals. If these agree to a preassigned accuracy, the integration is regarded as successful. Otherwise the interval is halved and the same process applied to each half. The results are collected in fig. 7, where we put together the numerical solutions for A = 0.1, A = 1 and A = 10, as well as the analytical solutions in weak coupling and strong coupling approximation. We give each of the

,

a ....... u

e,

1\ I .... I

b_

'~'"

d -

,I

0.1

-

,,=

1

strong coupling (analytical)

":'.

:

'X':,.

I

10

~

>"'\::•••••

\ I

k

=o.s

'-':;"

I

';:""

-c-,

I

I

''':,:''

.~,

I

M

=.

numerlcat solution for

I

0.7

);

= ,,:= ".....' \ : ' - - - - W.ak coupling approximation for =1

\~.'"

1

Numerlca4 solution for

e _ _ NumerIcal setullcn far

-,

~

...

....- Wtak c:oupl1ng apprexlrnatlcn for 1.::1: 0.1

~~

I

'~"

1

'~:~...'::.:..

" \

\

\ \ \ \

".".

...........

\ \

\

\,

, ' ...... \ ' ..........~ 10

Fig. 7. The diagonal part of the transition probability density Pt(h) for k = 0.5 and different J. values,

latter for the three values of A. Because we use the time scale 7' = 2 At the strong coupling solutions remain the same for A = 10, 1 and 0.1. Because A r--J Athe evolution occurs in a time scale of order A-1 . For weak coupling (A = 0.1) the numerical and weak coupling solutions are very close to each

THE EVOLUTION TO QUANTUM STATISTICAL EQUILIBRIUM. II

661

other (curves a and b in fig. 7), but entirely different from the strong coupling approximation (curve d). There exists a relaxation time of order A-2 • For A = 10 the graph shows no difference between the numerical and the strong coupling approximation (curve d). This confirms the results of ref. 2. Oscillations appear as already observed in ref. 2. The analytical solution for weak coupling practically coincides with the vertical axis and is not shown. For A = 1 both the weak coupling (curve e) and strong coupling solution (curve d) differ appreciably from the numerical solution (curve c). No exponential behaviour is observed, nor oscillations. This solution emphasizes the existence of an intermediate coupling region where no oscillations occur, but one cannot describe the evolution with a relaxation time. Because we suppose the general behaviour of the non diagonal part of Pt(kok) to be of the same nature, we did not repeat the lengthy numerical calculations for this quantity. Acknowledgements. The authors are indebted to Dr. M. D'Hont, scientific director of the S. C. K., for permission of publication of the present article. They express also their gratitude to Prof. L. Van Hove for a critical reading of the manuscript and to Mr. J. Vandervee for his assistance in the numerical calculations. Received 31·10·62

REFERENCES 1) Van Have, L. and Vc r b c v e n, E., Physiea 27 (1961) 418. 2) Jauner, A., Van Have, L. and Verboven, E., Physica 28 {I 962) 1341.