Brownian motion model for collision phenomena: Translational energy distribution

Volume 47, number 3

BROWNIAN

CHEMICAL PHYSICS LETTERS

MOTION MODEL FOR COLLISION

TRANSLATIONAL

ENERGY

1 hfay 197%

PHENOMENA:

DISTRIBUTION*

Byung Chan EU Department of Chemistry. McGill University, Montreal, Quebec, Canada

Reccivcd 12 November 1976 Revised manuscript received 17 January

1977

A brownian motion model is assumed for the relative translational motion of collidmg moiecules with complex interna! structures and the distribution function (probability) of the final relative velocity is obtained from a Fokker-Planck equation for conditional probability in this paper. Under certain further assumptions ae are able to obtdin a distribution. function similar in form to the information theoretic result. A formula for angular distributions isalso obtained, which shows a forward-backward or a side-way peaking. depending on the ratio of “transversal and Iorrgitudhzal temperatures” (friction tensors). The “friction tensors” are given in terms of interaction potentials and can be in principle calculated from their explicit formulas given in the paper.

1. Introduction An important problem which both theory and experiment face as theoretical and experimental information increases in molecular beam experiments and chemical dynamics is the task of “digesting and systematizing” the available data. Bernstein and Levine [l] showed in their series of papers that the information theoretic approach can be useful for such a purpose_ It can give energy dependences of various dynamical processes in a rather compact manner. HOWever, it is a phenomenological theory and as such contains phenomenological coefficients which do not reveal their relationship to the characteristics of the system of interest, e.g., molecular interactions, etc. No physically reasonable correlation of such a kind can be readily made, unless one starts from a dynamical theory of some sort and derives the distribution functions deduced from experiment in their approach. This communication represents a preliminary report on our study in the direction in which we wish to formulate a dynamical statistical theory that retains dynamics in a more simplified way than the “exact” * Supported by grants from the National of Canada.

Research Council

theory, yet provides us with a means to make the afore-mentioned correlations and understand data in a qualitative fashion and also existing phenomenological ?heories like the information theoretic approach. One of the most important implications of the information theoretic approach appears to the writer to be the notion that molecuiar scattering processes can be essentially described, based on a probability theoretic (statistical) concept. The so-called surprisd [l] might be regarded as a correction to the “prior probability” which is one level lower in sophistication as a theory than that statistical theory that should give the probability distribution itself obtained by the surprisal analysis_ It is then logical and legitimate to ask if it is possibIe to construct a statistical theory and just when, where and in what form statistical concepts should be introduced in the theory_ We shall try to answer these questions at [east partially in the subsequent discussions. We wi!I follow up with a fuller account in a later paper. We do this in terms of a model. The model we discuss may be one of possible models. Whether one model is superior to another appears to be of considerabIe interest and should be a subject of discussion_ For now we shall concentrate on one model. In order to

VoIume 47, number 3

make it simple, we shall limit the discussion to nonreactive translational motion only, but it is not difficult to generalize so as to include vibrational and rotational degrees of freedom. In the model we are considering, the relative translational motion of colliding particles with complex internal structures is treated as a stochastic process, which can be properly described only by the probability distribution of the associated :tochastic variable. This stochastic variable is assumed to be subject to a Langevin equation for brownian particIes. Admittedly, a brownian motion model for molecular coIIision is not conceptuaIIy as clear as for heavy particles suspended in a medium. However, if the molecules involved have sufficiently large internal degrees of freedom, a stochastic idealization of the relative motion as we11 as the evolution of some internai motions does not appear unreasonable, since in the course of collision the translational motion may experience almost random fluctuating forces due to complicated interactions between the particles in the molecules invoIved. Even if such an idealization were a little removed from the true picture, statistical (stochastic) formuIation might be the only practicable and mathematically tractable approach open to us for studies of complex scattering systems. At Ieast it will be able to show us how far we are removed from the reality when we compare its result with experiment. We present a formulation of stochastic (brownian model) theory of collision phenomena of complex systems in the subsequent sections.

2. Measured

observables

In order to construct a statistical theory of cohisiona1 phenomena, e.g., a brownian motion model for scattering problems, it is necessary to examine experimental measurements. For example, in the usual moIecuIar beam experiments we measure various cross sections as well as relative transiational velocities (by time-of-flight method or any other means), sometimes angular momenta and internal energies of molecuies in the finaI state of a particular collision process along with their probability of occurrence in terms of intensities of fI uxes. UsuaUy, the number of the basic observables measured in an experiment is much less than needed for a complete specification of the 556

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CHEMICAL PHYSICS LETTERS

state of the system under study. We may classify the observables into two basic categories of measured and unmeasured observables. We will caI1 the former reIevant observables (Ro’s) and the Iatter irreIevant observables (IO’s). This classification entirely depends on experimental setup and conditions. For example, if we measure only the final relative velocity of the collision partners, then the final velocity is a RO, while the anguiar momenta, vibrational energies, etc. of the collision partners are IO’s_ With increased experimental sophistication, the number of RO’s increases accordingly. The complete set of observabIes (hermitean operators in quantum mechanics) may be regarded as a complete set of vectors in the HiIbert space spanned by the vectors (hermitean operators), if we suitably define the norm of vectors. The Hilbert space of hermitean operators are given a metric by the following defmition of the norm: Let {Ai;i= 1,2,...,1~} be the complete set of observables (operators)_ Then ]2,3] U&4,)=

Tr[$(AfAi

++Af)p,]

= Sii,

(1)

where pe may be taken pe = fi(H-E)/Tr 6(H-E), H and E being the hamiltonian and the energy. Here Tr means either trace over quantum mechanical states or taking integration in phase space. This choice implicitly assumes that the distribution of states is microcanonical. More detailed discussions on (1) wiII be made elsewhere. Here the vectors are assumed or&~nomtaI (they can be made orthogonal by Schmidt’s method) and are chosen such that Tr(A,p,) = 0. In scattering experiments&t) are measured after they are allowed to evolve from their initial values at the beginning of the experiment and when they finalIy have reached their asymptotic (plateau) values at t > tc, the collision time. The most important feature of the Ai in the present study is that they do not change in time for t > tc. That is, the observables in a collision experiment (e.g., the relative velocity) stay constant as they move out of the sphere of interaction, if the interaction range is ftite. We shall assume that this is the case in the present study. This point wiII bear a significance, when we come to interpret the so-called stationary distribution function shown below. The basic assumption of the present model is that these observables, especially RO’s are stochastic in

Volume 47, number 3 their time evolution

and thus may be treated as if “particles” moving irregularly through the “medium” of IO’s_ We also assume that r, is much longer than the average time needed for transitions between the internal states which we assume happen numerously during t,. In the next section this idea is put into a mathematical formalism and the distribution function of RO’s is obtained_

they are brownian

3. Fokker-Planck The equations

equation of motion

for the observables

Ai

are &Q(t)/&

= i&$(t),

(2)

where JZ is the Liouville mutator, &4#)

= fi-I

operator

defined

by the com-

[HI A$)]

(3)

in the case of quantum

mechanical description and [H, Ai] pi times i in the case of classical description. In the present paper our discussion is limited to classical formalism_ Since we are restricted to only one RO in the present study, i.e. the relative velocity (normalized by its root mean square) which will be denoted by A, the relevant vector subspace is one-dimensional (three-dimensional, if we decompose the velocity into three components)_ Following the standaid procedure in kinetic theory [2,3], let us introduce a projection operator ‘9such that

L?Ai is the Poisson bracket

94t)

= UIA(t)M.

A’(t) = (1 - 9)A(t).

Then it is possible to show that the equation motion for A(t) is in general given by M(t)/&

= ioA(t)

-Jclsp(r-s)A(s) if(t), 0 which is called a generalized Langevin equation_ d(t) = 1 or K > 1. In fig. I, eq. (29) is schematically plotted. Since 4 is intimateiy related to molecular interactions through the “friction tensor”, it is expected that different systems would give different shapes for angular djstriiutions depending on R. This aspect of (29) appears extremely interesting and should be further studied in the future. In any event the angu!ar distribution (29) suggests that it may be possible to classify anguIar distributions according to the iongitudinal and transversal “temperatures” I;: (i = x, y; 2).

1 May 1977

It is often stated in the molecular beam experimental literature that if the angular distribution is peaked in the forward-backward direction, the collision mechanism involves long-lived collision complexes and is thus statistical [I 21. Therefore, the present result (29) for K > I appears to bear such an assertion out. But then it is also possible to have a side-way peakings if K < 1. Since both have the same origin in the statistical model used here, we may conclude within tile vaIidity of the present model that it is not possible to deduce conciusiveiy the collision mechanism from

the mere shape of angular distributions. One needs more information than just the angular distributionsIf we assumethat the collision time t, is infinite, then J/($1 -+ 1

(30)

and we obtain for the conditional probability W(d&I,-)

= (2~)~~‘~ exp(-&A2),

(31)

which then must be equated with p(A) according to (17). Unfortunately, this form of distribution function does not contain any information on the interaction potentials of the system and consequentiy may not be as useful as (29) for examining experimental data. However, we expect (31) to give rise to an angular distribution similar to (291, if W,i I$> f: