Statistical error in time correlation functions from ...

Statistical error in time correlation functions from computer experiments on approximately twostate processes Nobuhiro G and Fumiaki Kanô Citation: The Journal of Chemical Physics 75, 4166 (1981); doi: 10.1063/1.442511 View online: http://dx.doi.org/10.1063/1.442511 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/75/8?ver=pdfcov Published by the AIP Publishing

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Letters to the Editor

4166

most familiar one. In the following, by starting with Eq. (ld), we show that kre1 can be expressed in terms of the so-called HF stability matrix. First of all, let us approximate ~o by the single Slater determinant ~F consisting of the HF orbitals. In this case the first and second derivatives of ~F are given by 8fF _ " " HF aQ - L.J L.J C",a a- '" , '"

(2)

a

a2HF

ac

~=LL ~ ~!''''+LL m

a

The elements of matrices A and 8 in Eq. (4) correspond to the configuration interactions between singly excited states (:: '" and ~n) and between the ground and a doubly excited states (~F and ::",.&-n), respectively; they are nothing more then elements of the HF stability matrix. 6.7 At this stage one cannot immediately conclude that whenever the HF solution is unstable, the pseudo-JT distortion occurs. It is interesting, however, that as Eq. (4) shows, there is a definite relation between the HF stability matrix and the force constant formula. The authors would like to thank Dr. A. Toyota for stimulating discussions on the subject. One of them (KD) would also like to acknowledge the financial support of the Japan Society for the Promotion of Science.

m,ft a

( ..~n)

where C lj =

(~II a:~).

Now, substituting these results into Eq. (ld), making use of the Brillouin's theorem and taking some algebra, we obtain the desired formula for kre1 in the HF approximation (4)

IA. Toyota, T. Tanaka. and T. Nakajima, Int. J. Quantum Chern. 10, 917 (1976). 2A. Toyota, M. Saito, and T. Nakajima, Theor. Chim. Acta 56, 231 (1980). ~. F. W. Bader, Mol. Phys. 3, 137 (1960). 4L. Salem, Chern. Phys. Lett. 3, 99 (1969). G. Pearson, Theor. Chim. Acta. 16, 107 (1970). 6J. D. Thouless, The Quantum Mechanics of Many-Body Systems (Academic, New York 1961). 7J. Cizek and J. Paldus, J. Chern. Phys. 47, 3976 (1967).

sa.

Statistical error in time correlation functions from computer experiments on approximately two-state processes Nobuhiro Go Department of Physics. Faculty of Science. Kyushu University. Fukuoka. 812 Japan

Fumiaki KanO Department of Physics. College of Arts and Sciences, Showa University. Fuji- Yoshida, 403 Japan (Received 3 June 1981; accepted 18 June 1981)

Computer experiments are a powerful method for studying dynamics of transitions between states which are locally stable in the sense of statistical phYSics. In this method, time correlation functions are calculated bY replacing an equilibrium ensemble average by a time average over a finite time interval. This replacement causes a statistical error. When two distinct locally stable states are involved, a system stays most of the time in either of the stable states and the transitions between them would take place infrequently. Therefore, records of computer experiments on such systems appear approximately as a Poisson process. The purpose of this note is to estimate the error mentioned above for processes which are apprOximately two-state Poisson processes. Similar analysis has been done by Zwanzig and Ailawadi1 for a case of Gaussian processes. The analysis in this note follows their method. J. Chern. Phys. 75(8), 15 Oct. 1981

We will be interested in dynamical variables whose time-dependent value A(t) is more-or-less constant, when the system stays in either of the locally stable states. Thus A(t) is approximately a Poisson process which takes two more-or-less constant values. We normalize these two values to ± 1. We assume that the two states have the same stability (most computer experiments are done under such a situation). The probability of transition + 1 - - 1 or - 1- + 1 during an infinitesimal time interval ~t is assumed to be given by K~t. Then the probability of an occurrence of l transitions during a time interval t is given by (1)

The conditional probability q(a, a'; t), i. e., the probability of A(t) = a' under the condition of A(O) = a, is given

© 1981 American Institute of Physics 0021-9606/81/204166-02$01.00 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.54.110.73 On: Mon, 24 Feb 2014 21:21:44

Letters to the Editor for t?':. 0 by

,

q(l, 1; t) =q(-l. -1; t) = e-Kt coshKt, q(l.-l;t)=q(-l,l;t)=e- Kt sinhKt,

(2)

Then, the ensemble-averaged moments are given by St

~ S2 ,

(A(St)A(S2)A(S3)A(S4» = exp( - 2K(S4 - S3

+ S2 -

(A(St)A(S2» = exp[ - 2K(S2 - SI)],

for

for

sl

(3) St)] ,

~ S2 ~ S3 ~ S4'

(4)

The deviations ~J(t) and ~II(t) depend on the initial point in the phase space from which the computer experiment was started. For the purpose of determining the distribution of ~l(t) and ~1l(t) we will calculate the first and second moments of these quantities for different choices of the initial point with the weight of the equilibrium distribution. In this calculation we need such quantities as (AA(t»

All moments of odd power vanish. The time correlation function of a classical dynamical variable A(t) is defined as C(t) = (A(s)A(s

+ t»

.

(5)

0\2) =

tlr) ,

where relaxation time

1 =T

IT

:fr {T

0

ds (A(s)A(s

dS t

+ t» = C(t)

(13)

,

~T ds 2(A(S1)A(S2»::; 2(f) -2(fY

(14)

In the calculation of other necessary quantities, Eq. (4) is also used.

Here, (... ) is an ensemble average for the equilibrium distribution. Note that (A(s)A(s + t» is independent of time s. Therefore, we will write (A(s)A(s + t» = (A(O)A(t» = (AA(t». From Eq. (3), we have C(t) = exp( -

4167

After some calculations we have (15)

(6) T

for t« T,

is given by (2K)-1.

(16)

In computer experiments, (AA(t» is replaced by an

for t» T

average AA{t) over a finite time interval T for a single initial point in phase space:

-

AA(t)

lIT

=T

0

dsA(s)A(s

+ t)

(7)

(~II(t)

is identical with C(t), because (A) =0 and (A2)=1. We want to discuss the deviation of C(t) from C(t), ~I1(t)

=C(t) -

C(t) .

(9)

For this purpose we introduce an auxilliary quantity

C(t) defined as C(t)

-A? - - 2 = AA(t) (A2) _ (A)2 =AA(t) -A .

(10)

The deviation of C(t) from the ensemble average is defined as (11) By using Eqs. (10) and (11), we can expand the quantity in Eq. (9) as follows:

~

II

(t)

= ~(t)

C(O)

- C(t)

2(t/T) for

t« T

,

(17)

for t« T

,

for t »T

•

(18)

(8)

(AA(t» - (Ai (A 2) _ (A)2

1-

-2(T/T) for t» T ,

The normalized correlation function C(t) calculated from a record of a,computer experiment for a finite time duration is defined as

A corresponding quantity defined by ensemble-average quantities

~ =

,

Equation (18) is the main result of this note. The error due to finite time averaging in the normalized time correlation function is found to be given by (T/T)1/2 for time t longer than the relaxation time T. When t is shorter than T, the error is smaller. The result of Eq. (18) for t» T is smaller than that obtained previously for Gaussian processes1 by a factor of 2. This result assures that computer time in units of main relaxation time can be shorter by a factor of 2 in cases when a record of a computer experiment is approximately a Poisson process than in cases when a record is approximately a Gaussian process. The result of Eq. (18) for t« T is the same as for the Gaussian process. It is interesting that E:r(t) for t« T is proportional to (rlt)2 in the Poisson process, while it is proportional to (TIT) in Gaussian processes. 1 Based on the analysis in this note, computer experiments of folding and unfolding transition in globular proteins are carried out. 2 This work was supported in part by a grant-in-aid from the Ministry of Education, Japan.

_ ~J(t) - C(t)~J(O)

-

1 +~I(O)

=[~I(t)-C(t)~I(O)][l-~I(O)+ ••• ] .

(12)

IR. Zwanzig and N. K. Ailawadi, Phys. Rev. 182, 280 (1969). • Kana and N. GO, Biopolymers (submitted).

2F

J. Chern. Phys., Vol. 75, No.8, 15 October 1981 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.54.110.73 On: Mon, 24 Feb 2014 21:21:44