Volume 134.number4
CHEMICAL PHYSICSLETTERS
6 March1987
DUAL LANCZOS TRANSFORMATION THEORY: CLOSED SET OF ALGEBRAIC EQUATIONS
CONNECTING LANCZOS PARAMETERS WITH MOMEI{TS IN MOMENT EXPANSIONS
OF TIME.DEPENDENT QUANTITIES
W.A. WASSAM Jr. and Go. TORRES-VEGA
Departamento de Física, Centro de Investigación y de Estudios Avanzados del IPN,
Apdo. Postal l4-740,07000, Mexico, D.F., Mexico
Received l5 September1986;in final form l0 November 1986
We report the derivation of a closed set of algebraic equations connecting the Lanczos parameters (of dual Lanczos transformation theory) with the moments appearing in the moment expansion (Maclaurin series) of time-dependent quantities. The
utility of this set of equations is illustrated by using them with the aid of symbolic manipulation on a computer to construct a
previously unknown exact continued fraction for the spectral density of the incoherent scattering function associated with a
Fokker-Planck particle in a spatially homogeneous,isotropic environment.
l.Introduction
In characterizingthe temporal and spectral properties of a system, one usually needsthe time evolution or
spectral densities of only a few gross observablesor time correlation functions. One would like to determine
thesequantities without having to appeal to the solution of global linear equations of motion. This goal may be
achieved,or approximately achieved,by utilizing the mathematical apparatusof dual Lanczos transformation
theory t 1-3 ], which provides a number of universal formulae and other results for handling such problems.
Unlike more traditional approachesto time evolution and spectraldensity problems, dual Lanczos transformation theory is able to handle any time evolution and spectral density problem (including those that are
inaccessibleby matrix diagonalization techniquesand those that may not be given the usual continued fraction
representation) associatedwith a globally linear system.In dual Lanczos transformation theory, there are no
symmetry restrictions on the transition operator L appearingin the global propagator exp( - Lü for the vector
lp,) correspondingto the density distribution function or density operator for the systemof interest. Also, there
are no restrictions on the type of dynamical variables that may be considered.Moreover, dual Lanczos transformation theory is written in such a way that it may be applied to classical,quantum, and semi-classical(partly
classicaland partly quantum) systems.
In the presentwork, we report a simple and straightforward derivation of a closedset of algebraicequations
connectingthe Lanczos parameters [ 1,2] of dual Lanczos transformation theory with the moments appearing
in the moment (Maclaurin series) expansion of time-dependent quantities. The utility of this set of equations
is illustrated by using them with the aid of symbolic manipulation on a computer to construct a previously
unknown exact continued fraction for the spectraldensity of the incoherent scatteringfunction associatedwith
a Fokker-Planck particle in a spatially homogeneous,isotropic environment.
2. Dual Lanczos vector space
The initial step in the application of dual Lanczos transformation theory to the solution of time evolution
and spectral density problems is the construction of a dual Lanczos vector space lI,2l bearing the relevant
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dynamical information for determining the quantities of interest. In order to build this vector space,we focus
our attention on sometime-dependentquantity Ao,a(/) and its spectralfunction ao,o(z) (Laplacetransform of
A o . o ( t ) )l l , 2 l ,
Ao.o(t) -- (ro I exp( - íü lpo)
(1)
a o ,o Q ) : ( / o 1 1 z i +i ) - t
(2)
and
l p o ),
where the vectors (rol and lp6) are the starting vectors for building the dual Lanczosvector space,iis the
identity operator, and L is the transition operator characterizing the global dynamics. Ao, o(l) may represent
observable,autocorrelationfunction, cross correlation function, or some
some normalized (Ao,e(/-0)-l)
linear combination of suchquantitiesprovided the vectors (ro I and lp) areof sucha characterthat (re lps) - 1
and the time derivativesof As,o(/) exist at time /:0.
T h e c h o i c e o fd u a l L a n c z o s s t a r t i n g v e c t o(rrso l a n d l p s ) s h o u l d b e b i a s e d i n s u c h a w a y t h a t A 6 , s ( / )( o r
ao,oQ)) representssome quantity or linear combination of quantities that one is actually interestedin. If this
is not possible(rs I and lpo) should be selectedon the basisthat the dual Lanczosvector spacebuilt with these
vectors bearsthe relevant dynamical information for the problem at hand [3 ] .
The basisvectors {(r,l} and { lp,)} (dual Lanczosvectors) spanninga dual Lanczosvector spaceform a bio r t h o g o n a l b a s i s s e t s a t i s f y i n g t h e o r t h o n o r m a l i t y r e l a( rt,i lopn, , ) : ( r T l r , , ) : d , , , ' l l , 2 l . F r o m t h e s t a r t i n g
vectors (rol and lp6), the remaining dual Lanczosbasisvectors may be generatedby meansof the recursion
relationsU,2l
F , +t l P , +) : ( L _ a , i ) l p ) _ F , l p , _ ,)
(3)
É,+r(r"+,1:(r,l(i-a,i)-p,(r"-rl,
G)
and
wh e r es ) 0, ( r - r I an d l p -,) a re n u l l v e c to rsa, n d f o= i .
The Lanczosparameters{a,} and {8,*1} appearingin eqs. (3) and (4) are the only non-vanishingmatrix
elementsof the transition operatori in the dual LanczosrepresentationU,21,
0,: (r,lLlp,)
(5)
and
f ,+t:Q,lí11,*r),
: ( r , *, l i l p , ) .
(6a)
( 6b)
3. Lanczosparametersand moments
If the time derivativesof Ao.o(l) exist at time /-0, the time-dependentquantity Ao,o(/) may be written as a
moment (Maclaurin series)expansion,
6
Ao,o(t):L M¡ttllt,
(7)
I:O
wherethe momenf Mtis given by
M t : d t A r , o ( t : O ) l d t t,
(8a)
:(-1),(rolL,lpo)
( 8b)
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It is convenientto introduce a set of polynomial operatorsV6'*')(-i))
Cauchy integral formula
t
-¿) : (2ni)-' a,/6'*')(t) (zi+ L)-',
/6'*',)(
I
6 March1987
that are formally defined by the
(e)
ó
with the contour C enclosingthe singularitiesof the resolventoperator (zi+ ü-t andfr+r) (z: -,1) denoting
the characteristicfunction of the (s+ 1)-dimensional approximant to the transition operator i in the dual
Lanczosrepresentation.
The polynomial operators defined by eq. ( 9 ) satisfy the recursion relation [ 3 ]
( _L),
fr*,)(_i) : _ (L_o¿,i)
i5',{ _f¡ _B:29-')
*fr"r"f6q(-
ü:ianai6*'r(
(10)
i) is a null operator.
Moreover,
thepolynomialoperators
maybewrittenin
theform [3]
s+l
A
/ 6 ' * ' ) ( - L ) - j :Io ( - l ) i d 6 ' * t ' ¡ ) L i,
(11)
where the expansion coefficients {dÁ'* t"') } are given by
d $ ' + r ' i )- ¿ 5 s ' ; - r ) * a , d 6 ' ' i ) - B z ¿ 5 s - t , L t
for 0(/(s-
(r2)
1,
d $ ' + r ' ' t_ d t ' ' ' - r ) * d ,
(13)
d $ ' + t s' + r )- I
(14)
and
The expansioncoefficientd$, it vanishesby definition forT< 0 andT> s.
Employing eqs. (3), (4) and (10), one can show by mathematical induction that the dual Lanczosbasis
vectors{(r,l} and { lp,)} may be generatedfrom the startingvectors (rol and lp6) and the following relations:
(15)
(1 6 )
Making useof eqs.(11), (15) and (16), as well as the formal definitionsgivenby eqs. (5) and (6b) for the
Lanczosparametersand eq. ( 8b) for the moments,we obtain
/
,
4,:( npkl
\z:0
\-'
/
r
I
dg'.itdg''i'tM¡*¡,*t
(17)
¡.¡':o
and
/
\
s
f?*,: -\If'^
\-r
s+l
) 'F:oof
* '' Dd6s+
'' i')Mi*i,,
(18)
where s)0 and F3= - l. (We adopt the convention fhat B,*, is given by the principlesquareroot of f?*r.)
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Eqs. (17) and (18) along with eqs. (12)-(14) constitutea closedset of algebraicequationsfor generatingthe
Lanczosparametersfrom the moments.
4. Incoherentscatteringfunction
In o r der t oillus t r a te th e u ti l i ty o fe q s . (1 2 )-(1 4 ), (17) and (18),w econsi dertheprobl emofconstructi nga
continued fraction representationof the spectral density (Fourier-Laplace transform) of the incoherent scattering function associatedwith a Fokker-Planck particle in a spatially homogeneous,isotropic environment.
The incoherent scattering function [a] .S(/<;/) and spectral density s(k; ia¡) for this model system may be
written I
S(ft;t): (exp(ift'q)lexp(-tt) lexp(-lk'q)puo@,q))
(1e)
-'
s(k; ia¡)- (exp( ift. q) | (ia i + ü
Iexp( - ik' q)p.q (p, s) ),
(20)
and
where p and 4 respectively are the momentum and coordinate for the particle of interest and Z is an abstract
operator corresponding to the Fokker-Planck transition operator L(p, q) for a damped free particle in a spatiatly homogeneous,isotropic environment [5]. The vectors (exp(i/<'a) | and lexp( -lk'q)peq@, q)) respectively are dynamical vectorscorrespondingto the phasefunctions exp(ift'a) and exp( -1k'q)ptq(p, q), where
peq(p,q) is the equilibrium distribution function for the particle of interest and ft denotes the changein the
wave vector for the incident photon or neutron depending on whether one is characterizing a light or neutron
scatteringexperiment.
and
If we choosethe dual Lanczos starting vectors (rol and lpo) to be given by (rol-(exp(ik'q)l
-Ik.q)peqj,
(1)
(2),
q)),
(19)
(20)
Adopting
respectively.
assumethe form of eqs.
and
€es.
and
lpo) lexp(
thesestartingvectors,we can make useof eqs.(8b), (12)-(14), (17) and (18) to determinethe relevantLanczos
parametersand so construct a continued fraction representationof s(/<;iroo)l2l.
Employinga symbolicmanipulationscheme[6] that implementedeqs.(8b), (12)-(14), (17) and (18) for
the above-describedproblem on a Burroughs 87800 computer using REDUCE [7], we obtained the results
displayed in tables 1 and 2 for the moments and Lanczos parametersfor a dual Lanczosvector spaceof dimensionality 5. From the resultsin table 2 it is evident that the Lanczos parameters a¡and B/ are of the form at:lf r
and Bl: -l(kuTlm) l/cl2 (we assumethe pattern in a¡ and B/ observedin table 2 for l:0-4 holds for all /),
where/, is the friction coefficient, lkl is the magnitude of the changeft in the wave vector, ks is the Boltzmann
constant, Zis the equilibrium temperature, and m is the mass of the particle. With theseresults,we can immediately write down the following exact continued fraction representationof s(/<;ia¡),
s ( k ; iuo)-
(keTlm)lkl'
ia¡ *
2 ( k r T/ m ) l k l '
írtt * B¡*
ía *28¡'f .
l(knTlm)lkl'
( i a ¡ + l B ) +. .
In the zero damping limit (f ,:0), eq. (21) assumesthe form of the well-known result [4]
t Seeref. I for a discussion of the matrix formulation of time
evolution and spectral density problems.
I ]
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Table I
Mo-Ms for incoherent scattering function S(rq ¡) (free particle, Fokker-Planck and Liouville dynamics)
Moment
M,
Moll
Mto0
M2
M3
M4
M5
M6
M7
MB
Ms
Fokker-Planckdynamicsu)
Liouville
dynamicsu)
- 0*l7k"T/m)
f rlklzk"Tlm¡
- p ? | k l 2 k B T / m )+ 3 ( i e l z k u T l m ) z
p ? l k l z k " T t m ¡ - l 0 É . (l k l , k " T l m ) z
- p f | k l 2 k r T / m ¡ + 2 5 p ? | k l 2 k B T t m ) ' -r s ( l k l 2 k s T t m ) 3
p ? l k l 2 k r T t m ) - 5 6 p ? | k l r k u T t m ) 2+ l 0 5 p f (l k l r k " T / m ) 3
- P f ( l k l z k ' T l m ¡ + I r g l ? ( l k l ' k " T / m ) 2- 4 9 0 \ 7 u k l ' k " T l m ) 3+ 1 0 5 (l k l z k B T l m ) 4
p i u k l z k " T t m ) - 2 4 6 p 7 | k l 2 k B T t m ) '+ l g t 8 p i ( l k l r k " T / m ) 3 - t 2 6 0 p f u k l r k " T t m ) a
- 0ftl 2ksTtm)
0
3|kl2kBTtm)2
0
- 1 5 (l f r l2 k " T t m . 1 3
0
r 0 5 (l f t l2 k " T t m ) a
0
u)Here,
B¡is the friction coefficient,l/cl is the magnitudeof the changeft in the wavevectorof the incidentphotonor neutron,fts is the
Boltzmannconstant,Zis the equilibriumtemperatureandm is the massof the particle.
s(/r;ia;) ( k n T/ m ) l k l 2
í';¿*
2 ( k " Tl m ) l k l 2
ict *
ir¿*..
l(kBTlm)lkl2
T
v D -r ' '
.
(22)
for a free particle governed by Liouville dynamics.
In the high damping limit where (mf? /ksTl kl')rrz y 1, eq.(21 ) is well approximaiedby the result [4 ]
s(/c;iar)-I/(iat + Dlkl2)
for Smoluchowski dynamics, where D -k"T
(23)
/mB¡ is the diffusion coefficient.
Table2
for incoherentscatteringfunctionS(ft; ¡) (freeparticle,Fokker-Planckand Liouville dynamics)
Lanczosparameters
Lanczos
parameter
ots00
p?
Fokker-Planck
dynamics u)
Liouville
dynamics u)
(ksTtm)lkl2
- (ksTlm)lkl2
ft0
-2(k"Ttm)lkl2
-2(ksT/m)lkl2
pi
d3
pZ
-3(k"r/ñWl2
3F,
-4(k"Tfift12
-3(k"T/m)lhl2
0
-4ft"T/m)lklz
da
48,
0
ot1
pi
ot22fr0
"
') Hcre, is the f¡iction coeffici€nt,
l¡ | is the ma8litude of the ch¡nge É in the wavev€ctorofthe incide[t photod or neutron, kB is th€
Ér
Boltzmain conrla¡l, U is the equilibrium tempe¡ature,and m is the massofthe particle.
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5. Concludingremarks
In the present work we have reported a simpie and straightforward derivation of a closed set of algebraic
equationsconnectingthe Lanczosparametersof dual Lanczostransformationtheory t1-3] with the moments
appearingin the moment (Maclaurin series)expansionof time-dependentquantities.The utility of this set of
algebraic equations was illustrated by using them with the aid of symbolic manipulation on a computer to
construct an exact continued fraction for the spectral density (Fourier-Laplace transform) of the incoherent
scatteringfunction associatedwith a Fokker-Planck particle in a spatially homogeneous,isotropic environment.
An advantageof working with the algebraicresults obtained here for generatingthe Lanczosparametersover
the other methods U,,2) is that the presentresultsenableus to connectthe Lanczosparameters,and hencethe
temporal and spectralpropertiesof a system,to microscopicinteractions (and phenomenologicalparameters
in mesoscopicdescription) in a more transparentway. Moreover, this set of equationsenablesus to handle
more easily few body, many bod.y, and multidimensional/rnultimode (several degreesof freedom) systems
without having to appealto a finite-dimensional initial representationwith a discreteindex. Another advantage
is that they are very easy to employ symbolically on a computer [6] in performing exact computations and
approximate computations involving infinite order summations of certain contributions (defined by the size
of scalingparameters)to a probiem.
The only closed set of algebraic equations that we aware of for connecting the parameters appearing in a
continued fractions representation of a tirne correlation function with moments are the results of Allen and
Diestler [8] for autocorrelation functions involving classicaldynarnical variables with definite time reversal
parity and for systemsgoverned by Liouville dynamics. Since Allen and Diestler worked in the languageof
Hilbert spacetheory employing unnormalized Mori vectors [1] and employeda dynamical embeddingprocedure, it is not possible to make a simple and direct comparison between the results of these investigators and
the results obtained by us, which certainly are more general.
Acknowledgement
We wish to expressour appreciation to IFUNAM at UNAM for use of their computational facilities. Also,
we would like to give special thanks to A. Rosario Guzmán S. for her contributions to the initial stagesof the
development of a collection of symbolic codesfor performing symbolic manipulation on computers. This work
was partially supported through a CONACyT feliowship granted to GTV
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Solution of Time Evolution and Spectral Density Problems, J. Chem. Phys., submitted for publication.
[4] J.P. Boon and S. Yip, Molecular hydrodynamics (McGraw-Hill, New York, 1980).
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Computers: A General Approach for Studying the Temporal and Spectral Properties of ClassicalSystemswith SelectedApplications,
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360