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Surprisal analysis derived from a variational principle for mechanical systems
Volu~nc
98.
number
1 July 1983
CHEMICAL PHYSICS LETTERS
4
SURPRlSAL ANALYSiS DERIVED FROM A VA~ATIO~AL
P~C~P~E
FOR ~EC~ICAL
SYSTEMS
N.Z. TISHBY * and R.D. LEVINE
The ezlpansion of the surprisal in one or a fea observabies is derhed from a variational (action) principle for barn~~o~R systems. Wbenerer phenomcnologiwl sum rules exist or “slow” variables can be identified on physical grounds, the variational approxinution \t11 be accurate nnd a hamiltonian-like formalism for the descriptian of the co&ion in terms of a few variables i\ Slli~‘i~i~~t
In this ietrer we justify the empirical procedure of surprisal an& sis f f ] from mechanical considerations ZIIILI &rltif~ the constraints for which the procedure \\ill be accurdfe. The discussion is in two stages. The first one is based on strict mechanical considerations. In II we show that 3.density matrix (or a density in phase sp~e in classicJ mechanics) N hich is a function of the nte.m values of a few observables provides a vari&ional approsimntion to the solution of the exact equarrons of motion. We also show that the time evohnion ctf thr approsinlate density matrix is also hamiltonwn like and derive an explicit form for this harnilronidn. This hamiltonian provides a complete “reduced” description ofthe time evolution. We have thereby justlticd the introduction ofso calied “stochastic, reduced descripltons” f?] of the dynamics. Since the derivaiion IS b,~rd on a variational principle we are able to delme.tte the conditions under which the variational approsimation will be a good one. As expected on pill\ sic.zl grourlds. this wilt be the case whenever rhe density matri?r depends on variables whose expectation v3Iucs dre slowly varying. From a phenomenofogic~l approach, rhe approximation will be accurate whenc%er ‘“sum ruIct(s)” are known: that is. whenever the * Also Dcpsrtment of Theoretical vrrsity , krwalcm. Isnei.
310
Physics, The Hebrew Uni-
final, post-collision, mean value of an observable A, can be espressed as a Iinear combination of the initial, pre-coIIision. mean values of a few observables,
0) Examples of such relationsabound in all fields ofheavyparticle colIisions [I] _More recently such expansions have been re-e~npi~asized under the heading of “scaling” ES]At the end of the first,strictly
mechanical stage, the density matrix is expressed as a yet unspecified function ofa few observables. In the second stage we invoke the matimum entropy formalism f43 to provide an esplicit form for the density matrix or for the surprisal 4s P(f) =
22 As(r s
-
Here the observables A, are those which are necessary in (1). The time evolution of the Lagrange multipliers, Xs, in (3) is determined b-y one of two equivalent procedures: either by deter~~ni~~ p(r) as the density matrix of matimai entropy subject to the mean values at the time f, CA&r) of the observabies A,,
(AsW) = T~E&)A,l , or from hamittonian-like equations of motion (which wIIf be expIicitly given below). Hence, if before the
0 009-X 14~83~0000-~000~s 03.00 0 1983 bosh-~oliand
CHEMICALPHYSICSLETTERS
Volume98, number4
the density matrix can be expressed as one of maximal entropy then it is a variational approximation that it will remain one of maximal entropy throughout [5] and after the collision. We have previously [6] determined the sufficient conditions under which (2) is an exact expression for the surprisal. While these conditions could be checked and verified for simple model systems [6] it was clear that for realistic problems they will not be exactly satisfied. The present results show that even when (2) is not necessarily an exact result, it remains a variational approximation and can be a very accurate approximation, even with few terms in the sum. It is an important aspect of the variational principle that it correctly identifies (2) as the exact result when the conditions specified in ref. [6] obtain. The present results provide therefore the practical person’s version of ref. [6] _ collision
2. Variational principle We use a variational principle [7] of least action which is an extension to density matrices and observables of the Dirac-Fraenkel [S] --Schwinger [9] variational principle for wavefunctions. Just as in the latter, the wavefunction $ and its dual, ti*, are subject to independent variations then in the version we use [7], observable,A(t) and the density matrix p(r), are independently varied *. The lagrangian we use is X = Tr{A(t)
@p/at - i[p,H])]
(4)
with the action fl I= s
X dt + Tr[p(tl)A(tl)]
.
(5)
%I Here His the actual, true hamiltonian. The action is to be made stationary under independent variations 6&t) and 6A(f), with mixed boundary conditions 3, that p(ru) = pin and A(tl) = A,,r are specified. The requirement of stationary action 61= 0 readily f The analogy becomes more evident when one notes that the trace operation is equivalent to the scalar product. $ Since the exact equations of motion for p(r) and A(f) are fust order in time, one cannot specify p(f) [or A(f)] at both end points.
1 July 1983
leads to Tr@A(t)
@p/at - i]p,fl)]
= 0,
Tr{Gp(t) @A/& -i [A ,H])} = 0 _
(6) (7)
Clearly, if 6A(t) and 6p(t) are arbitrary variations we recover the exact equations of motion. The variational approximation is obtained by requiring (6) and (7) to hold only for limited or restricted variations in p(t) and A(t). A key observation, which is already evident at this point is that ifA is a strict time-dependent constant of the motion [lO,ll] , dA(f)/dtG
aA(t
- i[A,H]
= 0,
(8)
then (7) obtains for any variation 8&t). Say A(t) is nearly a constant of motion so that while dA(r)/dt is not zero, it is “small” (in a sense to be made precise in section 3), then limited variations in p(t) will suffice to ensure that Tr[Gp(r) dA(r)ldt]
=0_
(7’)
In other words, the more nearly is dA(t)/dr zero, the lower is the flexibility that need be built into p(t). This observation will turn to be central in choosing the parameterization of p(f), cf. section 3, on either physical or phenomenological grounds. If the variation 6A(f) a A(t) is allowed, it follows from (6) and (4) that the first term in (5) vanishes and hence that the stationary value of the action lis (A(r)) at t = tl. If the variation 6&t) a p(t) is also allowed it follows from (7’) that for the variational solution p(t), A(t) behaves as a time-dependent constant of the motion, (A(t)) = Tr[p(t) A(t)] = const.
(9)
While we have used a quantum mechanical notation, it should be clear that by replacing i[H, ] by the Poisson bracket {H, ) and interpreting Tr as integra-
tion over phase space, the discussion remains valid in classical
mechanics_
We turn now to the essence of our results which derive from the parameterization of p(t) and of A(r) as proposed and discussed in section 3. In particular we shall show that (i) it is possible to obtain a variational approximation where the density mat;ix at the time t is a function of a few relevant expectation values at that time. A special case is that an initial density 311
matrix of rnasimal entropy will remain a variational appro_ximation throughout the time evolution and (ii) the expectation values of the relevant observables and their conjugate Lagrange multipliers satisfy Hamiltonlike equations and hence can be regarded as a set of generalized coordinates and momenta for a reduced description of the time evolution_
3. Parameterization
1 July 1983
CHEMICAL PHYSICS LETTERS
Volume 98, number 4
and reduced hamiltonian
We restrict the range of allowed variations of p(r) and of A(f) by parameterizing their form in a manner suggested by the esact solution [6]. Alternatively, one can motivate our choice on physical grounds. Explicitly. we seek a solution where the density operator p(r) depends only upon the mean values (AJ(f), cf. (3), of a few (say t?z + 1) linearly independent relevant observables (s = 0.1 . ...m.AO s the identity operator). The precise functional dependence of p on the mean values need not concern us immediately_ One possible choice is made in section 5. The mean values (A,)(t) will serve as variational parameters and will be optimized by the variational principle &I= 0. Note that however p depends on the mean values. it must be a homogeneous function of the first degree [ 121 since multiplying all mean values, including the normalization Tr[&r) A”] . by a constant, must leave p the same escept for multiplication by the same constant_ Hence, by Euler’s theorem
U04
or (12b) where the A, are a set of variational parameters conjugate to the set (A,)(t). Note that (10) and (12) suffice for (9) to be valid. Inserting (10) and (12) in (7) and (6) and requiring that &I= 0 for all possible changes in A,(t) and U>(t) leads to a set of canonical equations:
aigat=-aJcjau,>.
(134
acqlat= ax/ax,.
Wb)
where the “hamiltonian”
-X is given by
3~ = i go $Tr(p[H,A,I).
(14)
The “reduced” hamiltonian 3C is a function of the 2m + 2 variables IQ(~) and (A,>(f)_ It is, in fact, linear in the A, and homogeneous, of degree one, in the (A,>, .Ic = C (a3cjax,pr I=
= h, (ii,),
C(aJcla~~~))c~,)= -ipr)_ r
As usual, the dot in (15) denotes the time derivative. The equivalence between the two lines in (15) follows from (9) Tr[p(t)A(t)]
= c r
Xr(Ar> = const.
(16)
In the special case [6] that the set of relevant operators is closed under commutation with the hamiltonian
or 6p = Csc_4~)(r)ap/acA)(r)
_
s
(lob) il+Kl
Note also thar the operators ap/a(A,)(t) and the relevant observables form a biorthogonal set
= c LyrsA,, s
(17)
we have that
Tr[Arap(r)/a(As)(f)]=a(~?(f)laGls>(r)=6,. (ii) For A(f) we choose the parameterization mentary to (10)
compieand (13) reduce to the exact [6] results
(1%
(12a) when (17) obtains, exact. 312
the variational
solution
is, in fact,
1 July 1983
CHEMICAL PHYSICS LETTERS
Volume 98. number 4
4. Sum rules and slow variables
5. Maximal entropy
The boundary conditions on the variational principle are p(tu) and A(tl). Say we take f. to be long before the collision and tl to be a long time after and A(tl) = A,. It follows from (12) and (16) that
We have thus far not specified the functional dependence of p(t) on the mean values of the relevant observables. For all the usual reasons [4.10] and for additional reasons, inherent to the present results but which will not be discussed here, we do so by the variational principle of maximal entropy. In other words, we choose p(t) to be that density matrix which reproduces the mean values, (A,>(f), of the 11~+ 1 relevant observables and is otherwise of maximal entropy. The result is
Di,)out
= c
XF’As’ti
_
c-33
S
The validity of (20) is equivalent to the accuracy of the variational property, 81(f) = 0. Comparing (20) and (1) we conclude that whenever (1) is observed to be (nearly) valid, the variational approximation for p(t), using the set of m + 1 relevant observables A,, will be accurate. Another common situation where one can judge the validity of the approximation is when one can write the full hamiltonian Has H = Ho + HI such that the set of relevant variables is closed under commutation with Ho _It is then possible to explicitly determine ITI+ 1 constants of motion ofHo ,A,(f), some of which may be time dependent, i.e.
ad,.(t)/&
- i[A,(f),Ho]
= 0 _
(21)
We then take as our Ansatz A(t) = c r
X,(t)AJt)
-
cm
Repeating the steps that led to (13), we now find that H is replaced by X1, apt
=
-asc,/aq(t)),
au,(tylat
= a3cl/axr, (139
where [cf. (14)] SC, = i F0 AT(t) Tr{p [Hr,
A,(t)1 I -
(149
The smaller isHI, the more slowly varying are the variables A,(t) and the better is the approximation_ Of course, even with a finite HI, (131) is a variational approximation, superior however to (13) in that the dynamics under Ho is solved for exactly and only HI is treated via a variational approximation_ As in the corresponding case for wavefunctions (e.g. ref. [ 13]), it pays to treat Ho exactly even if Ho is only that part of the hamiltonian that is responsible for elastic scattering.
and surprisal
m
p(r) = exp
I-p.
h,(f)A,
1-
analysis
(23)
The procedure of maximal entropy specifies that the + 1 Lagrange multipliers i’+(t) are to be determined from the m + 1 mean values UJ(r), The reader may object that the h(t) have already been defined in this paper as the solutions of eq. (13a). But that is the whole point. As we will argue shortly, the ‘h,(r) which are defined via (23) also satisfy the very same equation. Of course, in (23) it is the X,(ru) which are specified while in (12), the X,. are specified for t = tl _ Having determined p(t) one can obtain the explicit form of the hamiltonian X, eq. (14), and hence, via (13b), determine XA,>(t)~~t or oi,)(t + 6r). Given the mean value of A,. at the time r + 6t one can apply the procedure of maximal entropy to determine p at the time t + 6t. Alternatively, given Jc at the time t, one can, via (13a), determine a set of Lagrange multipliers at ,the time t + 6t. The two routes are equivalent or, in other words, if the density matrix was initially of maximal entropy then it is a variational approximation that it will remain one of maximal entropy (subject to the specified set of relevant observables) throughout its time evolution_ To prove that a density matrix of maximal entropy remains one under time evolution of least action compute apI& directly from (23) and verify that it satisfies (6) whenever the ar satisfy (13a). The details require several steps and so will not be given here. In particular, we have shown that a variational approximation for the post-collision density matrix is given by
m
313
Volume 98, number 4
1 JuIy 1983
CHEMICAL PHYSICS LElTERS References
The accuracy of (24) depends on the choice of relevant set of m + 1 observables A,_ Two useful guidelines are that (i) observables which (approximately) satisfy sum rules are relevant and (ii)observableswhich are slowly varying are relevant.
[l] R-D. Levine, Ann. Rev. Phys. Chem. 29 (1978) 59. [2] S.D. Augustin and H. Rabitz. J. Chem. Phys. 64 (1976)
1223; G.C. SchaU,F.L.McLafferty and J. Rou,J. Chem. Phys. 66 (1977) 3609. B.C. Eu. Chem. Phys. 27 (1978) 301_
I31 A-E. DePristo. J. Chem. Phys. 75 (1981) 3384. [41 R.D. Levine and M. Tribus, eds.. The maximum entropy formalism (hilT Press, Cambridge,
6. Concluding remarks We have been able to apply the proposed formalism both as an analytical and as a numerical (i.e. computational) tool. Space limitations require that the detaiIs be +en elsewhere. The results thus far lead us to believe that the approach is useful not only in providing insight and guidance for surprisal analysis but also in specifying a hamiitonian-like formalism, eqs. (13) for the dynamics using a few relevant variables.
Acknowledgement
1979).
PI G-L. Hofacker and R.D. Levine, Chem. Phys. Letters 33
(1975) 404; RX. Nesbet, Chem. Phys. Letters 42 (1976) 197; J_N_L_Connor, IV- Jakubeta and J_ Mans, Chem. Phys. Letters 44 (1976) 516. l61 Y. Aihassid and R.D. Levine, J. Chem. Phys. 67 (1977) 4321;Phys.Rev.AlS (1978)89:Phys.Rev.C20(1979) 1775. VI R. Balian and hl. Veneroni, Phys. Rev. Letters 47 (1981) 1353. 181P. Kramer and hi. Saraceno, Geometry of the time.dependent variational principle in quantum mechanics (Springer, Berlin, 1981). [PI J. Schwinger, Quantum kinematics and dynamics (Beniamin. New York. 1970) ch. 3 _ IlO1 RD_Levine. Advan. Chem. Phys. 47 (1981) 239.
1111R.D. Levine, Chem, Phys. Letters 79 (1981) 205: We thank Professors S. Alexander and Y. Tikochinsky for discussion. The Fritz Haber Research Center is supPorte-d by the Minerva Gesellschaft fiir die Forschung, mbH. Munich. FRG. This work was supported by the Air Force Office of Scientific Research under Grant AFOSR S ~-0030.
314
C.E. Wulfman and R.D. Levine, Chem. Phys. Letters 84 (1981) 13_ 1121 B. Robertson, Phys. Rev. 144 (1966) 151;in: The maximum entropy formalism, eds. R-D. Levine and M.Tribus (hiiT Press, Cambridge, f979)_ [I31 R-D. Levine, Chem. Phys. Letters 2 (1968) 76.
98.
number
1 July 1983
CHEMICAL PHYSICS LETTERS
4
SURPRlSAL ANALYSiS DERIVED FROM A VA~ATIO~AL
P~C~P~E
FOR ~EC~ICAL
SYSTEMS
N.Z. TISHBY * and R.D. LEVINE
The ezlpansion of the surprisal in one or a fea observabies is derhed from a variational (action) principle for barn~~o~R systems. Wbenerer phenomcnologiwl sum rules exist or “slow” variables can be identified on physical grounds, the variational approxinution \t11 be accurate nnd a hamiltonian-like formalism for the descriptian of the co&ion in terms of a few variables i\ Slli~‘i~i~~t
In this ietrer we justify the empirical procedure of surprisal an& sis f f ] from mechanical considerations ZIIILI &rltif~ the constraints for which the procedure \\ill be accurdfe. The discussion is in two stages. The first one is based on strict mechanical considerations. In II we show that 3.density matrix (or a density in phase sp~e in classicJ mechanics) N hich is a function of the nte.m values of a few observables provides a vari&ional approsimntion to the solution of the exact equarrons of motion. We also show that the time evohnion ctf thr approsinlate density matrix is also hamiltonwn like and derive an explicit form for this harnilronidn. This hamiltonian provides a complete “reduced” description ofthe time evolution. We have thereby justlticd the introduction ofso calied “stochastic, reduced descripltons” f?] of the dynamics. Since the derivaiion IS b,~rd on a variational principle we are able to delme.tte the conditions under which the variational approsimation will be a good one. As expected on pill\ sic.zl grourlds. this wilt be the case whenever rhe density matri?r depends on variables whose expectation v3Iucs dre slowly varying. From a phenomenofogic~l approach, rhe approximation will be accurate whenc%er ‘“sum ruIct(s)” are known: that is. whenever the * Also Dcpsrtment of Theoretical vrrsity , krwalcm. Isnei.
310
Physics, The Hebrew Uni-
final, post-collision, mean value of an observable A, can be espressed as a Iinear combination of the initial, pre-coIIision. mean values of a few observables,
0) Examples of such relationsabound in all fields ofheavyparticle colIisions [I] _More recently such expansions have been re-e~npi~asized under the heading of “scaling” ES]At the end of the first,strictly
mechanical stage, the density matrix is expressed as a yet unspecified function ofa few observables. In the second stage we invoke the matimum entropy formalism f43 to provide an esplicit form for the density matrix or for the surprisal 4s P(f) =
22 As(r s
-
Here the observables A, are those which are necessary in (1). The time evolution of the Lagrange multipliers, Xs, in (3) is determined b-y one of two equivalent procedures: either by deter~~ni~~ p(r) as the density matrix of matimai entropy subject to the mean values at the time f, CA&r) of the observabies A,,
(AsW) = T~E&)A,l , or from hamittonian-like equations of motion (which wIIf be expIicitly given below). Hence, if before the
0 009-X 14~83~0000-~000~s 03.00 0 1983 bosh-~oliand
CHEMICALPHYSICSLETTERS
Volume98, number4
the density matrix can be expressed as one of maximal entropy then it is a variational approximation that it will remain one of maximal entropy throughout [5] and after the collision. We have previously [6] determined the sufficient conditions under which (2) is an exact expression for the surprisal. While these conditions could be checked and verified for simple model systems [6] it was clear that for realistic problems they will not be exactly satisfied. The present results show that even when (2) is not necessarily an exact result, it remains a variational approximation and can be a very accurate approximation, even with few terms in the sum. It is an important aspect of the variational principle that it correctly identifies (2) as the exact result when the conditions specified in ref. [6] obtain. The present results provide therefore the practical person’s version of ref. [6] _ collision
2. Variational principle We use a variational principle [7] of least action which is an extension to density matrices and observables of the Dirac-Fraenkel [S] --Schwinger [9] variational principle for wavefunctions. Just as in the latter, the wavefunction $ and its dual, ti*, are subject to independent variations then in the version we use [7], observable,A(t) and the density matrix p(r), are independently varied *. The lagrangian we use is X = Tr{A(t)
@p/at - i[p,H])]
(4)
with the action fl I= s
X dt + Tr[p(tl)A(tl)]
.
(5)
%I Here His the actual, true hamiltonian. The action is to be made stationary under independent variations 6&t) and 6A(f), with mixed boundary conditions 3, that p(ru) = pin and A(tl) = A,,r are specified. The requirement of stationary action 61= 0 readily f The analogy becomes more evident when one notes that the trace operation is equivalent to the scalar product. $ Since the exact equations of motion for p(r) and A(f) are fust order in time, one cannot specify p(f) [or A(f)] at both end points.
1 July 1983
leads to Tr@A(t)
@p/at - i]p,fl)]
= 0,
Tr{Gp(t) @A/& -i [A ,H])} = 0 _
(6) (7)
Clearly, if 6A(t) and 6p(t) are arbitrary variations we recover the exact equations of motion. The variational approximation is obtained by requiring (6) and (7) to hold only for limited or restricted variations in p(t) and A(t). A key observation, which is already evident at this point is that ifA is a strict time-dependent constant of the motion [lO,ll] , dA(f)/dtG
aA(t
- i[A,H]
= 0,
(8)
then (7) obtains for any variation 8&t). Say A(t) is nearly a constant of motion so that while dA(r)/dt is not zero, it is “small” (in a sense to be made precise in section 3), then limited variations in p(t) will suffice to ensure that Tr[Gp(r) dA(r)ldt]
=0_
(7’)
In other words, the more nearly is dA(t)/dr zero, the lower is the flexibility that need be built into p(t). This observation will turn to be central in choosing the parameterization of p(f), cf. section 3, on either physical or phenomenological grounds. If the variation 6A(f) a A(t) is allowed, it follows from (6) and (4) that the first term in (5) vanishes and hence that the stationary value of the action lis (A(r)) at t = tl. If the variation 6&t) a p(t) is also allowed it follows from (7’) that for the variational solution p(t), A(t) behaves as a time-dependent constant of the motion, (A(t)) = Tr[p(t) A(t)] = const.
(9)
While we have used a quantum mechanical notation, it should be clear that by replacing i[H, ] by the Poisson bracket {H, ) and interpreting Tr as integra-
tion over phase space, the discussion remains valid in classical
mechanics_
We turn now to the essence of our results which derive from the parameterization of p(t) and of A(r) as proposed and discussed in section 3. In particular we shall show that (i) it is possible to obtain a variational approximation where the density mat;ix at the time t is a function of a few relevant expectation values at that time. A special case is that an initial density 311
matrix of rnasimal entropy will remain a variational appro_ximation throughout the time evolution and (ii) the expectation values of the relevant observables and their conjugate Lagrange multipliers satisfy Hamiltonlike equations and hence can be regarded as a set of generalized coordinates and momenta for a reduced description of the time evolution_
3. Parameterization
1 July 1983
CHEMICAL PHYSICS LETTERS
Volume 98, number 4
and reduced hamiltonian
We restrict the range of allowed variations of p(r) and of A(f) by parameterizing their form in a manner suggested by the esact solution [6]. Alternatively, one can motivate our choice on physical grounds. Explicitly. we seek a solution where the density operator p(r) depends only upon the mean values (AJ(f), cf. (3), of a few (say t?z + 1) linearly independent relevant observables (s = 0.1 . ...m.AO s the identity operator). The precise functional dependence of p on the mean values need not concern us immediately_ One possible choice is made in section 5. The mean values (A,)(t) will serve as variational parameters and will be optimized by the variational principle &I= 0. Note that however p depends on the mean values. it must be a homogeneous function of the first degree [ 121 since multiplying all mean values, including the normalization Tr[&r) A”] . by a constant, must leave p the same escept for multiplication by the same constant_ Hence, by Euler’s theorem
U04
or (12b) where the A, are a set of variational parameters conjugate to the set (A,)(t). Note that (10) and (12) suffice for (9) to be valid. Inserting (10) and (12) in (7) and (6) and requiring that &I= 0 for all possible changes in A,(t) and U>(t) leads to a set of canonical equations:
aigat=-aJcjau,>.
(134
acqlat= ax/ax,.
Wb)
where the “hamiltonian”
-X is given by
3~ = i go $Tr(p[H,A,I).
(14)
The “reduced” hamiltonian 3C is a function of the 2m + 2 variables IQ(~) and (A,>(f)_ It is, in fact, linear in the A, and homogeneous, of degree one, in the (A,>, .Ic = C (a3cjax,pr I=
= h, (ii,),
C(aJcla~~~))c~,)= -ipr)_ r
As usual, the dot in (15) denotes the time derivative. The equivalence between the two lines in (15) follows from (9) Tr[p(t)A(t)]
= c r
Xr(Ar> = const.
(16)
In the special case [6] that the set of relevant operators is closed under commutation with the hamiltonian
or 6p = Csc_4~)(r)ap/acA)(r)
_
s
(lob) il+Kl
Note also thar the operators ap/a(A,)(t) and the relevant observables form a biorthogonal set
= c LyrsA,, s
(17)
we have that
Tr[Arap(r)/a(As)(f)]=a(~?(f)laGls>(r)=6,. (ii) For A(f) we choose the parameterization mentary to (10)
compieand (13) reduce to the exact [6] results
(1%
(12a) when (17) obtains, exact. 312
the variational
solution
is, in fact,
1 July 1983
CHEMICAL PHYSICS LETTERS
Volume 98. number 4
4. Sum rules and slow variables
5. Maximal entropy
The boundary conditions on the variational principle are p(tu) and A(tl). Say we take f. to be long before the collision and tl to be a long time after and A(tl) = A,. It follows from (12) and (16) that
We have thus far not specified the functional dependence of p(t) on the mean values of the relevant observables. For all the usual reasons [4.10] and for additional reasons, inherent to the present results but which will not be discussed here, we do so by the variational principle of maximal entropy. In other words, we choose p(t) to be that density matrix which reproduces the mean values, (A,>(f), of the 11~+ 1 relevant observables and is otherwise of maximal entropy. The result is
Di,)out
= c
XF’As’ti
_
c-33
S
The validity of (20) is equivalent to the accuracy of the variational property, 81(f) = 0. Comparing (20) and (1) we conclude that whenever (1) is observed to be (nearly) valid, the variational approximation for p(t), using the set of m + 1 relevant observables A,, will be accurate. Another common situation where one can judge the validity of the approximation is when one can write the full hamiltonian Has H = Ho + HI such that the set of relevant variables is closed under commutation with Ho _It is then possible to explicitly determine ITI+ 1 constants of motion ofHo ,A,(f), some of which may be time dependent, i.e.
ad,.(t)/&
- i[A,(f),Ho]
= 0 _
(21)
We then take as our Ansatz A(t) = c r
X,(t)AJt)
-
cm
Repeating the steps that led to (13), we now find that H is replaced by X1, apt
=
-asc,/aq(t)),
au,(tylat
= a3cl/axr, (139
where [cf. (14)] SC, = i F0 AT(t) Tr{p [Hr,
A,(t)1 I -
(149
The smaller isHI, the more slowly varying are the variables A,(t) and the better is the approximation_ Of course, even with a finite HI, (131) is a variational approximation, superior however to (13) in that the dynamics under Ho is solved for exactly and only HI is treated via a variational approximation_ As in the corresponding case for wavefunctions (e.g. ref. [ 13]), it pays to treat Ho exactly even if Ho is only that part of the hamiltonian that is responsible for elastic scattering.
and surprisal
m
p(r) = exp
I-p.
h,(f)A,
1-
analysis
(23)
The procedure of maximal entropy specifies that the + 1 Lagrange multipliers i’+(t) are to be determined from the m + 1 mean values UJ(r), The reader may object that the h(t) have already been defined in this paper as the solutions of eq. (13a). But that is the whole point. As we will argue shortly, the ‘h,(r) which are defined via (23) also satisfy the very same equation. Of course, in (23) it is the X,(ru) which are specified while in (12), the X,. are specified for t = tl _ Having determined p(t) one can obtain the explicit form of the hamiltonian X, eq. (14), and hence, via (13b), determine XA,>(t)~~t or oi,)(t + 6r). Given the mean value of A,. at the time r + 6t one can apply the procedure of maximal entropy to determine p at the time t + 6t. Alternatively, given Jc at the time t, one can, via (13a), determine a set of Lagrange multipliers at ,the time t + 6t. The two routes are equivalent or, in other words, if the density matrix was initially of maximal entropy then it is a variational approximation that it will remain one of maximal entropy (subject to the specified set of relevant observables) throughout its time evolution_ To prove that a density matrix of maximal entropy remains one under time evolution of least action compute apI& directly from (23) and verify that it satisfies (6) whenever the ar satisfy (13a). The details require several steps and so will not be given here. In particular, we have shown that a variational approximation for the post-collision density matrix is given by
m
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Volume 98, number 4
1 JuIy 1983
CHEMICAL PHYSICS LElTERS References
The accuracy of (24) depends on the choice of relevant set of m + 1 observables A,_ Two useful guidelines are that (i) observables which (approximately) satisfy sum rules are relevant and (ii)observableswhich are slowly varying are relevant.
[l] R-D. Levine, Ann. Rev. Phys. Chem. 29 (1978) 59. [2] S.D. Augustin and H. Rabitz. J. Chem. Phys. 64 (1976)
1223; G.C. SchaU,F.L.McLafferty and J. Rou,J. Chem. Phys. 66 (1977) 3609. B.C. Eu. Chem. Phys. 27 (1978) 301_
I31 A-E. DePristo. J. Chem. Phys. 75 (1981) 3384. [41 R.D. Levine and M. Tribus, eds.. The maximum entropy formalism (hilT Press, Cambridge,
6. Concluding remarks We have been able to apply the proposed formalism both as an analytical and as a numerical (i.e. computational) tool. Space limitations require that the detaiIs be +en elsewhere. The results thus far lead us to believe that the approach is useful not only in providing insight and guidance for surprisal analysis but also in specifying a hamiitonian-like formalism, eqs. (13) for the dynamics using a few relevant variables.
Acknowledgement
1979).
PI G-L. Hofacker and R.D. Levine, Chem. Phys. Letters 33
(1975) 404; RX. Nesbet, Chem. Phys. Letters 42 (1976) 197; J_N_L_Connor, IV- Jakubeta and J_ Mans, Chem. Phys. Letters 44 (1976) 516. l61 Y. Aihassid and R.D. Levine, J. Chem. Phys. 67 (1977) 4321;Phys.Rev.AlS (1978)89:Phys.Rev.C20(1979) 1775. VI R. Balian and hl. Veneroni, Phys. Rev. Letters 47 (1981) 1353. 181P. Kramer and hi. Saraceno, Geometry of the time.dependent variational principle in quantum mechanics (Springer, Berlin, 1981). [PI J. Schwinger, Quantum kinematics and dynamics (Beniamin. New York. 1970) ch. 3 _ IlO1 RD_Levine. Advan. Chem. Phys. 47 (1981) 239.
1111R.D. Levine, Chem, Phys. Letters 79 (1981) 205: We thank Professors S. Alexander and Y. Tikochinsky for discussion. The Fritz Haber Research Center is supPorte-d by the Minerva Gesellschaft fiir die Forschung, mbH. Munich. FRG. This work was supported by the Air Force Office of Scientific Research under Grant AFOSR S ~-0030.
314
C.E. Wulfman and R.D. Levine, Chem. Phys. Letters 84 (1981) 13_ 1121 B. Robertson, Phys. Rev. 144 (1966) 151;in: The maximum entropy formalism, eds. R-D. Levine and M.Tribus (hiiT Press, Cambridge, f979)_ [I31 R-D. Levine, Chem. Phys. Letters 2 (1968) 76.