- Email: [email protected]
On determining the number of stochastic degrees of freedom in polyatomic molecules
Volume 100. number 3
ON DETERMINING IN POLYATOMIC J.R.
STINE
CHEMICAL
THE NUMBER
9 September 1983
PHYSICS LETTERS
OF STOCHASTIC
DEGREES
OF FREEDOM
MOLECULES
*
Los_~Inntos~~~atiot~al
Laboratory_
Los Alamos. New Mexico
S7.545.
USA
and D-W. NOID * Chemistry Division. Oak Ridge XatiotzaI Laboraroty, Oak Ridge, Tennessee 37830. USA and Deparrntet~t of Chemistry. Universip of Tennessee. Aho_wille. Tennessee 37916, USA
Received 13 hiay 1983; in final form 13 June 1983
Construction of dimrnsionality plots have proved useful for determining the number of integrals of the motion in dynamical systems x\ith t\\o and three degrees of freedom. The slow comergence in the method is caused by the trajectory not accessing phase space uniformly. Here an information dimension is shown to have better convergence properties.
only systems with three or more degrees of freedom can display Arnol’d diffusion [5.6]. The most basic property for characterizing systems with three or more degrees of freedom is simply the number of integrals of the motion. Stine and Noid [7] have described a method for determining this number based on the construction of a dimensionality plot. The main idea of this method is to partition phase space into hyperdimensional cubes (bins) and count the number of bins the trajectory accesses. This number of bins eventually becomes constant and is, of course, dependent upon the size of the bins. From the change in this number with bin size, the dimensionality of the hypersurface in phase space to which the trajectory is restricted can be calculated (dimensionality plot)_ Hence the number of integrals of motion is easily determined. This method has been applied successfully to mod-
1_ Introduction It is well known for dynamical systems with two degrees of freedom, as evidenced by the vast amount of work in the area. that even a relatively simple potential energy surface can produce various types of complicated classical motion [l or] _ Because of this comple.xity in systems with two degrees of freedom, polyatomic systems have received relatively little consideration- Whereas a trajectory for a system with two degrees of freedom can display either quasiperiodic or stochastic motion, a single trajectory for a polyatomic system can display quasiperiodic and stochastic motion simultaneously_ Thus it is seen that just as systems with two degrees of freedom display fundamental differences compared with systems with one degree of freedom so do systems with three degrees of freedom compared to systems with two. In addition to the mixed quasiperiodic- stochastic behavior of a single trajectory, apparently
* Research sponsored by the US Department of Energy under contract W-7405eng26 with the Union Carbide Corporation and contract W-7405+ng-36 with the University of California.
282
el systems with two and three degrees of freedom_ However, this method can suffer from a slow ionvergence of the number of unique bins accessed by the trajectory_ This slow convergence is caused by the highly non-uniform way the trajectory accesses phase space. A similar method is used to determine a different type of dimensionality termed the information di-
0 009~2614/83/0000-0000/S
03.00
0 1983 North-Holland
Volume 100, number3
CHEhfICALPHYSICS LETTERS
mension, which possesses better convergence properties. In the next section, we discuss the non-uniform distribution of phase space for trajectories of the H&on-Heiles system_ The information dimension is then calculated for these trajectories to determine the number of integrals of the motion.
2. Information dimension Stine and Noid [7) have described a method to determine the fractal dimension of the hypersurface in phase space to which the classical trajectory is restricted- This fractal dimension is equal to 212 - k where rz is the number of degrees of freedom and k the number of integrals of the motion. It is the value of k that is important for determining the extent of stochasticity; k = 1 corresponds to a totally stochastic trajectory and k = n corresponds to a totally quasiperiodic trajectory. Intermediate values of k correspond to a single trajectory displaying both stochastic and quasiperiodic motion. In this method the multidimensional phase space, of dimension 2n, is partitioned into 2Lndimensional hypercubes or bins with N of these bins per dimension_ The trajectory is then monitored after each time step to determine its present bin location_ After Ntotal time steps the trajectory will have accessed a number of bins, some of which had been accessed previously_ The number of unique bins, Nunis (those bins that were accessed at least once), is then determined_ At first NUnis grows rapidly as the trajectory accesses new bins, but eventually becomes constant as the trajectory returns to previously accessed bins. The construction of a dimensionality plot requires this process to be repeated for different bii sizes (different values of N) and then ln(NutirJ is plotted as a function of In(N) and the resulting straight line should have a slope of 2n - k. This method has been applied successfully to systems with two and three degrees of freedom. In practrce Ntoti must usually be taken to be quite large before Nutis can be considered constant_ The reason for this behavior is that the bins are not being accessed uniformly as will be demonstrated below. If we define gj as the number of times the ith bin has been accessed, where 1 =Zi
9 September 1983
ability gi of accessing bin i is gi/‘Ntew NOWwe define Gj as the number of bins that have been accessed j times where 1 < j
Wj/Ntcd3
(1)
where Wj =jGj _
(2)
The distributions of Wj are shown for the quasiperiodic and stochastic trajectories in figs. lc and Id, respectively_ The total probability of fmdiig those bins with just a few accesses is extremely small in both cases. This makes the fractal dimensionality plot difficult - a very large value for Ntotal must be chosen to ensure that all accessible bins have been counted. This behavior is not unlike that found for mappings [9-l l] and has been recognized as a drawback to determining the fractal dimension for multidimensional systems [ 121. Another type of dimension, the information dimension [ 10,131, incorporates the relative probability that each bin is accessed_ An entropy-like quantity S is defined as
283
CHEMICAL PHYSICS LETTERS
Volume 100, number 3
This quantity
may be rewritten
s = (A$,,)-1
ci sj -
ln(Nto~)
chastic trajectories are shown in figs. le and 1 f_ These distributions are similar to the distributions of Wfi hence the major contribution to S comes from those bins that are most probable_ If now the entropy is calculated for different bii sizes (N) and its negative is plotted as a function of In(N), then an information plot results and the slope of the resulting line should also be 2n - k_These plots are shown in fig_ 2 for the quasiperiodic and stochastic trajectories. The least-squares values for the slopes are 2-04 and 2.93 for the quasiperiodic and stochastic trajectories. respectively_ These values agree quite well
as .
where Sj=GjiIn~)
(5)
The quantity s,- represents the contribution that bins, accessed exactly j times. make to the total entropy S. The distributions of si for the quasiperiodic and sto-
6
9 September 1983
‘% 20
600
2000
looOa
wi
wi 1000
SO00
10000
3oOo0
si
% 20000 6000 10000 0
0
200
tmo
0
0
60
‘00
160
j
Fig. 1. Bin distributions for a quasiperiodic (plots a, c and e) and stochastic (plots b, d and f) trajectory for system. HereNtotal = 500000. and&,, = 565 for the quasiperiodic case and 160 for the stochastic case.
284
the Henon-Heiles
CHEMICALPHYSICSLETTERS
Volume100, number3
9 September1983
with the values of 2 and 3 that would result with the quasiperiodic trajectory having two integrals of the motion and the stochastic trajectory having one.
&
24 1
26 I
26 1
3.0 1
32 0
3.4 1
36 I
In(N)
Fig. 2. Informationplot for the quasiperiodic(solidLine)and stochastic(dashedline) trajectories.The straightlineshavethe integerslopesof 2 and 3 respectively.
21
3. Discussion
20
16
15
Fig. 3 shows the convergence of the slopes of-the lines produced for the fra&l and information dimensionality plots as a function of Ntotal_These data are for the precessing quasiperiodic trajectory for the H&on-Heilesiystem. The s!opes produced in the information plot have essentially converged to the limiting value of 2 whereas the slopes from the fractal dimensionality plot have not_ Clearly, the fractal dimension requires only a metric whereas the information dimension requires a metric and a measure; that is, the information dimension takes into account the probability of a bin being accessed_ It is expected that the calculation of any diirension that requires only a metric will be identical to the fractal dimension, and calculation of those requiring a metric and a measure will be identical to the information dimension_ However, it can be proved that the information dimension must be greater than or equal to the fractal dimension [14,15]. For dynamical systems it is expected that the fractal and information dimensions will be the same and will also be an integer.
10
t..
11
‘.
1..
12
.
.
I
13
tiN0t3
Fig. 3. Convergenceof the fractal (solidline)and information (dashedline) dimensional&sfor the precessingquasiperiodic trajectory.
We have demonstrated that calculation of the information dimension can have better convergence properties than that of the fractal dimension for dynamical systems. The reason for this better convergence is related to the fact that not all portions of phase space allowed by energy conservation are equally probable_ Calculation of the fractal dimension has those bins in the improbable portions of the phase space contributing just as much as those in the probable portions_ On the other hand, calculation of the information dimension has those bins in the improbable portions of the phase space contributing much less than those in the probable portion. We found that, although calculation of the fractal dimension is not without its numerical difficulties, we believe it is not as hopeless as previously thought [ 121, at least for dynamical systems. One reason for this is that the present method keeps track of only those bii that have been accessed; this number is usually a small 285
Volume 100, number 3
CHEMICAL PHYSICS LETTERS
fraction o f the N ~ bins in the total phase space and is a consequence o f the fact that conservation o f total energy is always at least one integral o f the m o t i o n . Also it has been observed that N need n o t be t o o large to achieve a fractal dimension that is within a b o u t 0.1 o f the true value. Finally, efficient m e t h o d s can be devised such that o n l y a small a m o u n t o f c o m p u t e r m e m o r y is necessary or that the sorting can be done out o f core. It wo u l d be desirable, t h o u g h , to make the calculation o f the fractal and i n f o r m a t i o n dimensions as efficient as possible so that these m e t h o d s can be used as tools in applications to real systems o f chemical interest. O f particular interest is t h e quasiperiodic/stochastic threshold in p o l y a t o m i c systems. These m e t h o d s could be used to investigate the conjecture that, for systems with greater than t w o degrees o f freedom, stochastieity may be m o r e evident (occurring at lower energy) because o f the increased n u m b e r o f low order resonances between pairs o f frequencies. Such calculations could confirm the use o f statistical theories to describe intramolecular energy transfer and unimolecular dissociation.
Acknowledgement Portions o f this manuscript were written while we visited the Theoretical Chemistry D e p a r t m e n t at the University o f Oxford. We w o u ld like to acknowledge
286
9 September 1983
t h e helpful discussions and hospitality With Dr. M.S. Child and partial su p p o r t f r o m N A T O grant 229.81. J R S w o u l d also like t o acknowledge helpful discussions with D o y n e F a r m e r and Erica J e n at Los Alamos.
References [1] D.W. Noid, M.L Koszykowski and R.A. Marcus, Ann. Rev. Phys. Chem. 32 (1981) 267. [2] M. Tabor, Advan. Chem. Phys. 46 (1981) 73. [3] S.A. Rice, Advan. Chem. Phys. 47 (1981) 117. [4] P. Bmmer, Advan. Chem. Phys. 47 (1981) 201. [5] L. Galgani, Nuovo Cimento 62 (1981) 306. [6] B.V. Chirikov. Phys. Rept. 52 (1979) 263. [7] J.R. Stine and D.W. Noid, J. Phys. Chem., to be publJshed. [ 8] M. H6non and C. Heiles, Astron. J. 69 (1964) 73. [9] J.D. Farmer, in: Evolution of order and chaos in physics, chemistry, and biology, ed. H. Haken (Springer, Berlin. 1982) pp. 228-246. [10] J.D. Farmer, Ph.D. Thesis, University of California. Santa Cruz (1981). [ 11 ] B.B. Mandclbrot, Fractals: form, chance, and dimension (Freeman. San Francisco. 1977). [ 12] H.S. Grcenside. A.Wolf, J. Swift and T. Pignataro. Phys. Rev. A25 (1982) 3453. [ 13] J.D. Farmer, E. Ott and J.A. Yorke, The Dimension of Chaotic Attractors [to be published in a special issue of Physica D, Order in Chaos[. [14] J.D. Farmer, Z. Naturforsch. 37a (1982) 1304. [ 15] A. Renyi, Acta Mathematica (Budapest) 10 (1959) 193.
ON DETERMINING IN POLYATOMIC J.R.
STINE
CHEMICAL
THE NUMBER
9 September 1983
PHYSICS LETTERS
OF STOCHASTIC
DEGREES
OF FREEDOM
MOLECULES
*
Los_~Inntos~~~atiot~al
Laboratory_
Los Alamos. New Mexico
S7.545.
USA
and D-W. NOID * Chemistry Division. Oak Ridge XatiotzaI Laboraroty, Oak Ridge, Tennessee 37830. USA and Deparrntet~t of Chemistry. Universip of Tennessee. Aho_wille. Tennessee 37916, USA
Received 13 hiay 1983; in final form 13 June 1983
Construction of dimrnsionality plots have proved useful for determining the number of integrals of the motion in dynamical systems x\ith t\\o and three degrees of freedom. The slow comergence in the method is caused by the trajectory not accessing phase space uniformly. Here an information dimension is shown to have better convergence properties.
only systems with three or more degrees of freedom can display Arnol’d diffusion [5.6]. The most basic property for characterizing systems with three or more degrees of freedom is simply the number of integrals of the motion. Stine and Noid [7] have described a method for determining this number based on the construction of a dimensionality plot. The main idea of this method is to partition phase space into hyperdimensional cubes (bins) and count the number of bins the trajectory accesses. This number of bins eventually becomes constant and is, of course, dependent upon the size of the bins. From the change in this number with bin size, the dimensionality of the hypersurface in phase space to which the trajectory is restricted can be calculated (dimensionality plot)_ Hence the number of integrals of motion is easily determined. This method has been applied successfully to mod-
1_ Introduction It is well known for dynamical systems with two degrees of freedom, as evidenced by the vast amount of work in the area. that even a relatively simple potential energy surface can produce various types of complicated classical motion [l or] _ Because of this comple.xity in systems with two degrees of freedom, polyatomic systems have received relatively little consideration- Whereas a trajectory for a system with two degrees of freedom can display either quasiperiodic or stochastic motion, a single trajectory for a polyatomic system can display quasiperiodic and stochastic motion simultaneously_ Thus it is seen that just as systems with two degrees of freedom display fundamental differences compared with systems with one degree of freedom so do systems with three degrees of freedom compared to systems with two. In addition to the mixed quasiperiodic- stochastic behavior of a single trajectory, apparently
* Research sponsored by the US Department of Energy under contract W-7405eng26 with the Union Carbide Corporation and contract W-7405+ng-36 with the University of California.
282
el systems with two and three degrees of freedom_ However, this method can suffer from a slow ionvergence of the number of unique bins accessed by the trajectory_ This slow convergence is caused by the highly non-uniform way the trajectory accesses phase space. A similar method is used to determine a different type of dimensionality termed the information di-
0 009~2614/83/0000-0000/S
03.00
0 1983 North-Holland
Volume 100, number3
CHEhfICALPHYSICS LETTERS
mension, which possesses better convergence properties. In the next section, we discuss the non-uniform distribution of phase space for trajectories of the H&on-Heiles system_ The information dimension is then calculated for these trajectories to determine the number of integrals of the motion.
2. Information dimension Stine and Noid [7) have described a method to determine the fractal dimension of the hypersurface in phase space to which the classical trajectory is restricted- This fractal dimension is equal to 212 - k where rz is the number of degrees of freedom and k the number of integrals of the motion. It is the value of k that is important for determining the extent of stochasticity; k = 1 corresponds to a totally stochastic trajectory and k = n corresponds to a totally quasiperiodic trajectory. Intermediate values of k correspond to a single trajectory displaying both stochastic and quasiperiodic motion. In this method the multidimensional phase space, of dimension 2n, is partitioned into 2Lndimensional hypercubes or bins with N of these bins per dimension_ The trajectory is then monitored after each time step to determine its present bin location_ After Ntotal time steps the trajectory will have accessed a number of bins, some of which had been accessed previously_ The number of unique bins, Nunis (those bins that were accessed at least once), is then determined_ At first NUnis grows rapidly as the trajectory accesses new bins, but eventually becomes constant as the trajectory returns to previously accessed bins. The construction of a dimensionality plot requires this process to be repeated for different bii sizes (different values of N) and then ln(NutirJ is plotted as a function of In(N) and the resulting straight line should have a slope of 2n - k. This method has been applied successfully to systems with two and three degrees of freedom. In practrce Ntoti must usually be taken to be quite large before Nutis can be considered constant_ The reason for this behavior is that the bins are not being accessed uniformly as will be demonstrated below. If we define gj as the number of times the ith bin has been accessed, where 1 =Zi
9 September 1983
ability gi of accessing bin i is gi/‘Ntew NOWwe define Gj as the number of bins that have been accessed j times where 1 < j
Wj/Ntcd3
(1)
where Wj =jGj _
(2)
The distributions of Wj are shown for the quasiperiodic and stochastic trajectories in figs. lc and Id, respectively_ The total probability of fmdiig those bins with just a few accesses is extremely small in both cases. This makes the fractal dimensionality plot difficult - a very large value for Ntotal must be chosen to ensure that all accessible bins have been counted. This behavior is not unlike that found for mappings [9-l l] and has been recognized as a drawback to determining the fractal dimension for multidimensional systems [ 121. Another type of dimension, the information dimension [ 10,131, incorporates the relative probability that each bin is accessed_ An entropy-like quantity S is defined as
283
CHEMICAL PHYSICS LETTERS
Volume 100, number 3
This quantity
may be rewritten
s = (A$,,)-1
ci sj -
ln(Nto~)
chastic trajectories are shown in figs. le and 1 f_ These distributions are similar to the distributions of Wfi hence the major contribution to S comes from those bins that are most probable_ If now the entropy is calculated for different bii sizes (N) and its negative is plotted as a function of In(N), then an information plot results and the slope of the resulting line should also be 2n - k_These plots are shown in fig_ 2 for the quasiperiodic and stochastic trajectories. The least-squares values for the slopes are 2-04 and 2.93 for the quasiperiodic and stochastic trajectories. respectively_ These values agree quite well
as .
where Sj=GjiIn~)
(5)
The quantity s,- represents the contribution that bins, accessed exactly j times. make to the total entropy S. The distributions of si for the quasiperiodic and sto-
6
9 September 1983
‘% 20
600
2000
looOa
wi
wi 1000
SO00
10000
3oOo0
si
% 20000 6000 10000 0
0
200
tmo
0
0
60
‘00
160
j
Fig. 1. Bin distributions for a quasiperiodic (plots a, c and e) and stochastic (plots b, d and f) trajectory for system. HereNtotal = 500000. and&,, = 565 for the quasiperiodic case and 160 for the stochastic case.
284
the Henon-Heiles
CHEMICALPHYSICSLETTERS
Volume100, number3
9 September1983
with the values of 2 and 3 that would result with the quasiperiodic trajectory having two integrals of the motion and the stochastic trajectory having one.
&
24 1
26 I
26 1
3.0 1
32 0
3.4 1
36 I
In(N)
Fig. 2. Informationplot for the quasiperiodic(solidLine)and stochastic(dashedline) trajectories.The straightlineshavethe integerslopesof 2 and 3 respectively.
21
3. Discussion
20
16
15
Fig. 3 shows the convergence of the slopes of-the lines produced for the fra&l and information dimensionality plots as a function of Ntotal_These data are for the precessing quasiperiodic trajectory for the H&on-Heilesiystem. The s!opes produced in the information plot have essentially converged to the limiting value of 2 whereas the slopes from the fractal dimensionality plot have not_ Clearly, the fractal dimension requires only a metric whereas the information dimension requires a metric and a measure; that is, the information dimension takes into account the probability of a bin being accessed_ It is expected that the calculation of any diirension that requires only a metric will be identical to the fractal dimension, and calculation of those requiring a metric and a measure will be identical to the information dimension_ However, it can be proved that the information dimension must be greater than or equal to the fractal dimension [14,15]. For dynamical systems it is expected that the fractal and information dimensions will be the same and will also be an integer.
10
t..
11
‘.
1..
12
.
.
I
13
tiN0t3
Fig. 3. Convergenceof the fractal (solidline)and information (dashedline) dimensional&sfor the precessingquasiperiodic trajectory.
We have demonstrated that calculation of the information dimension can have better convergence properties than that of the fractal dimension for dynamical systems. The reason for this better convergence is related to the fact that not all portions of phase space allowed by energy conservation are equally probable_ Calculation of the fractal dimension has those bins in the improbable portions of the phase space contributing just as much as those in the probable portions_ On the other hand, calculation of the information dimension has those bins in the improbable portions of the phase space contributing much less than those in the probable portion. We found that, although calculation of the fractal dimension is not without its numerical difficulties, we believe it is not as hopeless as previously thought [ 121, at least for dynamical systems. One reason for this is that the present method keeps track of only those bii that have been accessed; this number is usually a small 285
Volume 100, number 3
CHEMICAL PHYSICS LETTERS
fraction o f the N ~ bins in the total phase space and is a consequence o f the fact that conservation o f total energy is always at least one integral o f the m o t i o n . Also it has been observed that N need n o t be t o o large to achieve a fractal dimension that is within a b o u t 0.1 o f the true value. Finally, efficient m e t h o d s can be devised such that o n l y a small a m o u n t o f c o m p u t e r m e m o r y is necessary or that the sorting can be done out o f core. It wo u l d be desirable, t h o u g h , to make the calculation o f the fractal and i n f o r m a t i o n dimensions as efficient as possible so that these m e t h o d s can be used as tools in applications to real systems o f chemical interest. O f particular interest is t h e quasiperiodic/stochastic threshold in p o l y a t o m i c systems. These m e t h o d s could be used to investigate the conjecture that, for systems with greater than t w o degrees o f freedom, stochastieity may be m o r e evident (occurring at lower energy) because o f the increased n u m b e r o f low order resonances between pairs o f frequencies. Such calculations could confirm the use o f statistical theories to describe intramolecular energy transfer and unimolecular dissociation.
Acknowledgement Portions o f this manuscript were written while we visited the Theoretical Chemistry D e p a r t m e n t at the University o f Oxford. We w o u ld like to acknowledge
286
9 September 1983
t h e helpful discussions and hospitality With Dr. M.S. Child and partial su p p o r t f r o m N A T O grant 229.81. J R S w o u l d also like t o acknowledge helpful discussions with D o y n e F a r m e r and Erica J e n at Los Alamos.
References [1] D.W. Noid, M.L Koszykowski and R.A. Marcus, Ann. Rev. Phys. Chem. 32 (1981) 267. [2] M. Tabor, Advan. Chem. Phys. 46 (1981) 73. [3] S.A. Rice, Advan. Chem. Phys. 47 (1981) 117. [4] P. Bmmer, Advan. Chem. Phys. 47 (1981) 201. [5] L. Galgani, Nuovo Cimento 62 (1981) 306. [6] B.V. Chirikov. Phys. Rept. 52 (1979) 263. [7] J.R. Stine and D.W. Noid, J. Phys. Chem., to be publJshed. [ 8] M. H6non and C. Heiles, Astron. J. 69 (1964) 73. [9] J.D. Farmer, in: Evolution of order and chaos in physics, chemistry, and biology, ed. H. Haken (Springer, Berlin. 1982) pp. 228-246. [10] J.D. Farmer, Ph.D. Thesis, University of California. Santa Cruz (1981). [ 11 ] B.B. Mandclbrot, Fractals: form, chance, and dimension (Freeman. San Francisco. 1977). [ 12] H.S. Grcenside. A.Wolf, J. Swift and T. Pignataro. Phys. Rev. A25 (1982) 3453. [ 13] J.D. Farmer, E. Ott and J.A. Yorke, The Dimension of Chaotic Attractors [to be published in a special issue of Physica D, Order in Chaos[. [14] J.D. Farmer, Z. Naturforsch. 37a (1982) 1304. [ 15] A. Renyi, Acta Mathematica (Budapest) 10 (1959) 193.