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Elementary statistical models for collision-sequence interference effects with arbitrary persistence of velocity

This content has been downloaded from IOPscience. Please scroll down to see the full text. 2012 J. Phys.: Conf. Ser. 397 012010 (http://iopscience.iop.org/1742-6596/397/1/012010) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 170.130.62.109 This content was downloaded on 31/03/2017 at 01:51 Please note that terms and conditions apply.

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XXI International Conference on Spectral Line Shapes (ICSLS 2012) Journal of Physics: Conference Series 397 (2012) 012010

IOP Publishing doi:10.1088/1742-6596/397/1/012010

Elementary statistical models for collision-sequence interference effects with arbitrary persistence of velocity H T Wheeler1 and J C Lewis2 1

605 - 160 Third Street West, North Vancouver BC Canada V7M 0A9 Department of Physics and Physical Oceanography, Memorial University of Newfoundland, St. John’s NL Canada A1B 3X7 2

E-mail: [email protected], [email protected] Abstract. Elementary statistical models for collision-sequence interference effects usually assume Gaussian-distributed velocities with zero persistence of velocity. Here we extend the treatment to arbitrary persistences of velocity in the range zero to just less than unity. For vector collision-sequence interference the results show the interference dip narrowing with increasing persistence of velocity as expected. Only partial agreement is obtained with some observations in a molecular dynamics simulation in Lennard-Jonesium.

1. Introduction 1.1. Background The elementary statistical models developed by Lewis et al. [1-3] to describe collision-sequence interference effects have mostly used Gaussian-distributed velocities with zero persistence of velocity. Persistences of velocity which run from zero to just less than unity can be modelled by an AR(1) (Box-Jenkins) process [4] allowing the effect of varying this parameter on vector and scalar interference in simple models to be determined. Such processes lead to persistence-of-velocity terms hvi · vi+1 i, hvi · vi+2 i . . . hvi · vi+k i, which form a geometrical progression where vi is the velocity before the ith collision in a collision sequence. More complicated behaviour, however, can be modeled. 1.2. The General Model We assume an infinite set of statistically independent basis vectors ui with identical normal distributions and variance σ 2

 hui · uj i = δi,j σ 2 (1) and a series ai where the squared series

∞ X

a2i = S

i=0

Published under licence by IOP Publishing Ltd

1

(2)

XXI International Conference on Spectral Line Shapes (ICSLS 2012) Journal of Physics: Conference Series 397 (2012) 012010

IOP Publishing doi:10.1088/1742-6596/397/1/012010

is convergent. Correlated velocities vk for k = 0, 1, 2, 3 ... can be defined through ∞ 1 X vk = √ ai uk+i S i=0

The velocities in eq.(3) will have a normal distribution with variance σ 2

2 vk = σ 2

(3)

(4)

The value of hvk · vk+j i for j > 1 will depend on the choice of the series ai . The vectors ui could have other distributions but normal is commonly used. 2. The AR Model A case of particular interest is to take the series ai as a geometric series. Define velocities vk in terms of the ui and a ratio γ with |γ| < 1 as vk =

p

since eq.(3) becomes S=

(1 − γ 2 )

∞ X

∞ X

γ 2i =

i=0

γ i uk+i

(5)

i=0

1 1 − γ2

Then the variance of the vk will be the same as that of the uk

 hvi · vi i = v 2 = σ 2 We have upon assuming a stationary and reversible process

 hvi · vj i = γ |i−j| v 2

where the ratio γ is seen to be the persistence of velocity. The definition used here is * +  hvi · vi+j i (1/j) γ= hv 2 i

3. Vector Collision Sequences 3.1. Equal Collision Intervals As in [1] define impulses fk = vk+1 − vk . From eq. (8)

2



 fk = (vk+1 − vk )2 = 2 v 2 (1 − γ)

 hfk · fj i = h(vk+1 − vk ) · (vj+1 − vj )i = − v 2 γ |k−j|−1(1 − γ)2

(6)

(7)

(8)

(9)

(10) (11)

Upon assuming equal time intervals between collisions, which can be scaled so that tk = k for k ∈ [0, N − 1] for N collisions, we have f (t) =

X k

fk δ(t − k)

2

(12)

XXI International Conference on Spectral Line Shapes (ICSLS 2012) Journal of Physics: Conference Series 397 (2012) 012010

IOP Publishing doi:10.1088/1742-6596/397/1/012010

and the corresponding Fourier transform, with ω = 2π/N f˜(t) =

N −1 X

fk eιωk

(13)

k=0

With the use of eqs.(10) and (11) we have   D E X

 1 ˜ ˜∗ γ j−1 cos ωj  f · f = 2 v 2 (1 − γ) 1 − (1 − γ) N

(14)

j=1

Following [1], after some manipulation, the power spectrum is derived as  

2 1 − cos ω 2 S(ω) = 2 v (1 − γ ) 1 − 2γ cos ω + γ 2

(15)

showing the line shape is dependent on γ.

3.2. Poisson Distributed Intercollisional Intervals Following [2] assume the collisions times tk follow a Poisson distribution with frequency ν¯. The intervals ∆i will be exponentially distributed. The N collisions lie in the interval [0, T ]. For large N the random time T can be taken as N/¯ ν and then 1 ν¯ = (16) N T so the average over N can be replaced by an average over T . For the current model, with fk = µk and ω ˜ = ω − ω0 , as in the development from [2, eq. 6], we have S(˜ ω ) D ˜ ˜∗ E = f ·f (17) ν¯   j   ∞ X

2 ν¯ = f + 2Re hfk · fk+j i (18)  ν¯ + ι˜ ω  j=1

Writing κ = (1 − γ)¯ ν , the power spectrum is found to be  

2 ω ˜2 S(˜ ω) = 2 v κ 2 κ +ω ˜2

(19)

This shows the shape of the interference dip depends on the product κ and not on ν¯ and γ in a separable manner. The line shape of the dip is an inverted Lorentzian for all values of γ. The results of this and the preceding section 3.1 show the interference dip narrowing with increasing persistence of velocity, as predicted earlier [5-8]. 4. Scalar Interference For γ > 0, only hfk fk+1 i and hfk fk−1 i are different from hf i2 (see [1]). We have from numerical calculations, using the Box-Muller method to construct random normal vectors [10], that hfk fk+1 i decreases with increasing γ and hfk fk+1 i . (1 − γ) hfk fk+1 iγ=0

(20)

hfk fk+1 i → hf i as γ → 1

(21)

2

so that cov(fk , fk+1 ) → 0 as γ → 1. This is to be expected from the argument in [1] on the origin of the covariance: the larger the persistence of velocity, the smaller is the variation in speed from one collision to the next. 3

XXI International Conference on Spectral Line Shapes (ICSLS 2012) Journal of Physics: Conference Series 397 (2012) 012010

IOP Publishing doi:10.1088/1742-6596/397/1/012010

4.1. An Analytic Solution for Two Dimensions Given Gaussian velocities vk and vk+1 in two dimensions with unit variance, these will not be independent for γ > 0. The method of [1] can be used by a change of variable since p |vk+1 − vk | = |(1 − γ)vk − 1 − γ 2 uk+1 | (22) where vk and uk+1 are independent. Since uk+1 and vk have Gaussian distributions with unit variance, we have Z p 1 n hf i = 2 d2 vk d2 uk+1 exp[−(1/2)(vk2 + u2k+1 )] |(1 − γ)vk − 1 − γ 2 uk+1 |n (23) 4π Set

r (1 − γ) 1 − γ2 a = √ vk − uk+1 2 2 √ √ 1+γ 1−γ b= vk + uk+1 2 2 Then

(24a) (24b)

vk2 + u2k+1 a2 2 +b = 2(1 − γ) 2

and

|(1 − γ)vk −

The transformation is 

vk uk+1



=



p

√ 1 − γ 2 uk+1 |n = |a|n ( 2)n

√ a/ 2 p p − (1 + γ)a/ 2(1 − γ)

giving the Jacobian J(γ) as ∂vk /∂a ∂vk /∂b J(γ) = ∂uk+1 /∂a ∂uk+1 /∂b

(25) (26)

 √ 1 + γb √ 1 − γb

√ 1/ p2 = √ − 1 + γ/ 2(1 − γ)

√ 1+γ √ 1−γ

(27)

√ =√ 2 1−γ

Then eq. (23) is evaluated √ Z J(γ)2 ( 2)n n hf i = d2 ad2 b exp{−a2 /[2(1 − γ)] − b2 }|a|n 4π 2 Z Z ∞ 2 √ n ∞ 2 = ( 2) db b exp(−b ) da exp{−a2 /[2(1 − γ)]}|a|n+1 1−γ 0 0 n  = 2n (1 − γ)(n/2) Γ +1 2

(28)

(29a) (29b) (29c)

This result is in agreement with previous results for γ = 0 and with numerical calculations for γ > 0. 4.2. Comparison to the γ = 0 Value From numerical calculations in three dimensions and from eq. (29c), we have for a Gaussian velocity distribution with unit variance hf n i = (1 − γ)n/2 hf0n i where hf0n i is the quantity evaluated for γ = 0. 4

(30)

XXI International Conference on Spectral Line Shapes (ICSLS 2012) Journal of Physics: Conference Series 397 (2012) 012010

IOP Publishing doi:10.1088/1742-6596/397/1/012010

5. Comparison to Lennard-Jones Molecular Dynamics Results

Fig. 1 Dip for σµ = σ.

Fig. 2 Dip for σµ = 1.1σ.

Fig. 3 AR-model vs Fig. 2

Figs. 1 and 2 show interference dips from molecular dynamics simulations of Lennard-Jonesium mixtures where the parameters were chosen for Ar and H2 , with the model induced dipole moment also of Lennard-Jones type. These data were first published in [9]. Taking the induced dipole moment proportional to the force, µk ∝ fk = vk+1 − vk , the data of Fig. 1 were well fit using the AR-model eq.(19). The value of κ so obtained was then used in a stochastic simulation to fit Fig. 2 with µk = cµ fk |fk |β . Note that the physical parameters must be kept the same so the two fits are not independent. In this second fit both γ and β were varied by generating random velocities with a Gaussian distribution and the desired persistence of velocity. Fig. 3 shows the best fit overlaid with the cusp from Fig. 2. This best fit is for cµ ≈ 0.34, β ≈ 0.47 and γ ≈ 0. It was not possible to reproduce the cusp with an expected large value of γ as the larger γ is the broader the resulting interference dip: normally increasing γ would narrow the dip but here the fit to the Fig. 2 case is constrained by the value of κ derived from fitting the Fig. 1 case. 6. Conclusions The theory developed in [1, 2] for the zero persistence of velocity case has been extended to the arbitrary persistence of velocity case using the AR-model although certain quantities could only be evaluated numerically. It is fairly straightforward to generate velocities vk using eqs. (3) or (5), and then express the induced dipole moment µk as some function of the force (impulse) fk = vk+1 −vk to do stochastic simulations for cases that are too complex to obtain an analytical solution. The AR-model was successful in reproducing the Lennard-Jones results of [9] for the case σµ = σ but, using consistent physical parameters, here κ, a satisfactory fit to the case σµ = 1.1σ could not be obtained. The reason for this is not clear, but may reflect differing roles for the soft and hard parts of the interaction in the simulation results so that the simple fitting formula used, µk = cµ fk |fk |β , is not adequate. 7. Acknowledgement We thank Terrence Tricco for his assistance in carrying out and analysing the molecular dynamics simulations discussed above and reported in [9]. References [1] [2] [3] [4]

Lewis J C 2008 Phys. Rev. A 77 062702 Lewis J C 2009 International Journal of Spectroscopy 561697 Lewis J C and Herman R M 2011 International Review of Atomic and Molecular Physics 2 25 Box G, Jenkins G M and Reinsel G C 1994 Time Series Analysis: Forecasting and Control (Prentice-Hall) 3rd ed. [5] Lewis J C and van Kranendonk J 1972 Can. J. Phys. 50 352–67

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XXI International Conference on Spectral Line Shapes (ICSLS 2012) Journal of Physics: Conference Series 397 (2012) 012010

IOP Publishing doi:10.1088/1742-6596/397/1/012010

[6] Lewis J C and van Kranendonk J 1972 Can. J. Phys. 50 2902–13 [7] Lewis J C 1985 Phenomena Induced by Intermolecular Interactions ed Birnbaum G ( New York, Plenum Press) 215–57. [8] Lewis J C and R M Herman 2003 Phys. Rev. A 68 032703 [9] Lewis J C 2004 Proc. of the 17th Int. Conf. on Spectral Line Shapes ed Dalimier E (Frontier Group) 358–60. [10] Box G E P and Muller M E 1958 A note on the generation of random normal deviates The Annals of Mathematical Statistics 29 No. 2 610–1

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