3509
J. Phys. Chem. 1986,90, 3509-3516 elsewhere:' so only neutral pairs are considered here. The reduced variables introduced in eq 21 are used withy and z representing, respectively, the reduced forms of L and K. In reduced variables the integration limits y + in eq 9 are the appropriate solutions to
y3
+ 0,- z ) 2 - e = 0
(Ala)
y3
+ 0,+ z)Z-
(Alb)
and e
=0
where eq 10 has been used. The limit y + is always the largest solution to eq A l a and can be expressed, using standard methods,39 as
y + = s7 cos (*/3) -
y3;
s6
I0
sg = (-s4/2 - S6)1/3
(A3i)
0 = cos-1 ( - s 4 / [ 2 ( - s p ] )
(A38
and
The ex ression for y - depends on the relative values of c and z. For e l Iz Izmax(e),where zm,,(e) is the reduced form of Km,(E) (see below), y- is the smallest positive solution to eq A l a and can be expressed as
P
y- =
s7
cos (@/3
-
y3
644)
with s7 and defined in eq A3. For z Ie'/', y - is the largest solution to eq A l b and is obtained by substituting y - for y+ on the left-hand side of eq A2 and changing the sign of s1in eq A3a. The quantity z,(c), the maximum value of z for which q(e,z) 1 0, is defined by the requirement y- = y + y q , where
y , = [(z where
+ 4x/3)
+ I)'/'
- 1]/3
645)
Substituting eq A5 into eq A l a and solving give z,,,(e) = 3cI2/2
+ c1
('46)
where c1 = ([2(1 + l/cz)
- c3]'I2 - (c2 + 1))/9
c2 = (1
+ C3)I/Z
c3 = 9 ( ~ / 2 ) ~ / ~ [ (1)3/2 ~4 c4 = (108e
- (c4 + 1)3/2]
+ 1)Il2
(A7a) (A7b) (A7c) (A7d)
Solving instead for e at arbitrary z gives an expression for emh(z), the minimum energy at a given total angular momentum for which ?(€,Z) 2 0 (39) See, for example: Abramowitz,
M.;Stegun, I. A. Handbook of
Mathematical Functions; Dover: New York, 1972.
emin(Z)
= y q 3 + (Z - Y q ) 2
('48)
where y , is given in eq AS.
Dynamical Theory of Statistical Unimolecular Decay Randall S. Dumontt and Paul Brumer* Chemical Physics Theory Group, Department of Chemistry, University of Toronto, Toronto, Ontario, Canada MSS 1Al (Received: February 3, 1986; In Final Form: April 18, 1986)
Statistical unimolecular decay is formally derived through explicit use of molecular dynamics assumptions based upon ergodic theory. Both statistical lifetime and product distributions are considered, and effects due to direct dissociation, relaxation of the initial distribution, and similar heuristic notions are rigorously included. The result is a new model which shows exponential decay with a rate which depends upon both phase space volumes and relaxation time T. The typical RRKM functional form for the rate, denoted k,, is obtained in the limit where T is substantially shorter than the mean gap time. In addition to the ergodic conditions for statistical behavior, other enlightening results include the appropriate interpretation of k;I as the mean gap time, recognition that the statistical model requires T < (kSe)-',the fact that the decay rate may be larger than k, even though the system is mixing, and the possibility of extracting relaxation rates from the rate of exponential decay. Furthermore, the statistical behavior of any product distribution is shown to require that the mean gap times associated with products j must be equal for all j . This requirement is also shown, for the case of multiple chemical product channels, to be implicit in the traditional rate equation approach to unimolecular dissociation.
with RRKM theory being the most widely applied. Although some may regard statistical approximations as interim measures pre+NSERC 1967 Science Scholar (1981-1985); Gulf Canada Ltd. Graduate Fellow (1 985-1 986).
0022-3654/86/2090-3509$01 .50/0
(1) For an introduction, see: Levine, R. D.; Bernstein, R. B. Molecular Reaction Dynamics; Oxford University Press: New York, 1974. (2) For example: Marcus, R. A.; Rice, 0. K.J. Phys. Colloid Chem. 1951, 55, 894. Light, J. C. Discuss. Faraday SOC.1967,44, 14. Klotz, C. J. Phys. Chem. 1971, 75, 1526. Quack, M.; Troe, J. In Gas Kinetics and Energy Transfer; Chemical Society: London, 1977; Vol. 2.
0 1986 American Chemical Society
3510 The Journal of Physical Chemistry, Vol. 90, No. 16, 1986
, Figure 1. Schematic of unstable molecular region R , transition states S+ and S-, and their 6 broadenings.
themselves in some similarly disconcerting fashion in quantum comp~tations.~Statistical approximations can therefore be expected to be a permanent part of dynamics studies, and the dynamicists’ goal must be to establish conditions under which particular statistical theories provide accurate approximations to the true dynamics5 The vast majority of statistical theories based upon classical mechanics (which is the sole emphasis of this paper) adopt simplifying assumptions from statistical mechanics, such as rapid and complete energy randomization on an energy hypersurface prior to dissociation. As such, they are intimately linked to developments in modern ergodic theory6 and nonlinear mechanics’ which deal with relaxation phenomena and the role of intramolecular couplings in achieving relaxatioms Despite this qualitative link, the vast majority of statistical unimolecular decay theories have not been formulated with sufficient precision to allow input from these areas or to allow an analysis of the assumptions upon which these theories are based. In this paper we formalize many of the concepts in statistical theories of unimolecular decay within the framework of modern ergodic theory. Doing so establishes a new model (the delayed lifetime gap model (DLGM)) which shows exponential decay with a traditional statistical lifetime in a particular limit and which properly incorporates effects due to direct unimolecular decay, trapped trajectory regions on the energy hypersurface, and nonzero relaxation times. The condition of “relaxation prior to dissociation” is formalized, and the relaxation time is precisely defined and shown to influence the rate constant for unimolecular decay, These ideas are also extended to examine the nature of the product distribution in unimolecular decay and conditions for their statistical outcome. The paper is organized as follows: section I1 comments on traditional arguments for exponential unimolecular decay lifetimes and emphasizes the conceptual difficulties associated with rationalizing the statistical vs. dynamical view of unimolecular decay. In section 111 we re-formulate unimolecular decay within a framework wherein definitions of ergodicity, mixing, etc., may be invoked to produce the resultant lifetime distribution. The result is the DLGM which is discussed at length in section IV. Extensions to product distributions are provided in section V, and section VI contains a condensed summary of results of this paper.
-
11. The Exponential Decay Assumption products, treated within Consider unimolecular decay, A the framework of classical mechanics. The system phase space, (3) For reviews, see: Rice, S. A. Adu. Chem. Phys. 1981,47,117.Brumer, P. Ibid. 1981,47, 201. Noid, D. W.; Koszykowski, M. L.; Marcus, R. A. Annu. Rev. Phys. Chem. 1981,32, 261. (4) Although formal relaxation does not occur in the quantum dynamics of conservative systems, quantum dynamics in chaotic systems can be well approximated by classical chaotic motion for reasonable time scales. See, e.g.: Christoffel, K.; Brumer, P. Phys. Rev. A 1986,A33, 1309. (5) (a) Hamilton, I.; Brumer, P. J. Chem. Phys. 198582,1937-1946. (b) For applications to scattering see: Duff, J. W.; Brumer, P. J . Chem. Phys. 1979, 71, 3895-3896. Duff, J. W.;Brumer, P. J . Chem. Phys. 1979, 71, 2693-2702. Duff, J. W.;Brumer, P. J . Chem. Phys. 1977,67,4898-491 1. (6) (a) Arnold, V. I.; Avez, A. Ergodic Problems of Classical Mechanics; Benjamin: New York, 1968. (b) Cornfeld, J. P.; Fomin, S. F.; Sinai, Ya. G. Ergodic Theory; Springer-Verlag: New York, 1980. (c) For applications to molecular processes, see: Dumont, R. S. Ph.D. Dissertation, University of Toronto, 1986. (7) Berry, M. V. In Topics in Nonlinear Dynamics; Jorna, S . , Ed.; American Institute of Physics: New York, 1978. (8) For an introduction, see: Brumer, P. In Encyclopedia of Physical Sciences and Technology, in press.
Dumont and Brumer confined to energy E , may be divided into three regions, denoted R, R, and R+,separated by transition states S- and S+ (see Figure 1). Here R denotes the bound molecular region9 associated with the unstable molecule A, R+is the region of separating products, and R- is the region of colliding reactants which give rise to all states of A. These definitions are such that R and R+are related by time reversal and R is time reversal symmetric. Note that this partitioning of phase space is completely general. For example, if several product arrangement channels are accessible, then R3 and S I are unions of disconnected subsets, each associated with a particular arrangement channel. In terms of these sets, traditional unimolecular decay rate constant formulations essentially consider the evolution of an initially uniform distribution on R into R+. Expectations regarding the nature of the lifetime distribution for statistical decay at fixed energy E are well-known. Specifically, systems displaying rapid energy redistribution prior to decay are presumedlo to display exponential decay at energy E with a statistical lifetime T @ ) = k;l(E). That is, the probability density P ( t ) for decay of the reactant A at time t is given as P ( t ) = P(0) exp[-k,(E)t] = P ( 0 ) exp[-t/~,(E)]
(1)
where the essential statistical feature of k,(E) is that it is directly computable from phase space volumes at energy E. Traditional arguments in favor of this behavior l o rely upon vague notions of “randomness”. Effectively, exponential decay is assumed rather than derived from underlying dynamical approximations. This is true as well for routes which are apparently more dynamical, e.g., derivations via the “random gap” assumption.” Here one considers the distribution of times associated with bound molecular motion between S..and S+,the so-called gap distribution P g ( t ) . Qualitative arguments (substantiated below) relate P ( t ) and Pg(t) by P(t) = k,JmPg(t’) dt’ Assuming P&t) to decay exponentially (the so-called random gap assumption) with rate k, gives analogous behavior for P ( t ) . Thus, such formulations also rely upon the assumption of “random”, Le., exponentially distributed, gaps. None of these arguments can be satisfying to the molecular dynamicist who merely initiates a set of independent deterministic trajectories in some region R and often observes lifetime distributions which can be fit to exponentials.12 Similarly arguments regarding the effects of slow relaxation or partial relaxation, in phase space, in influencing the initial decay rate and magnitude of the decay constant are only qualitatively understood.13 In the following section we re-formulate unimolecular decay so as to permit utilization of modern concepts in ergodic theory to derive the relationship between P ( t ) and P,(t) and to provide insight into conditions for statistical lifetime and product distributions. 111. Dynamical Formulation
(a) Symbolic Dynamics. Our primary interest is in formalizing unimolecular theories in order to forge a link between established, but vague, statistical assumptions and precise concepts in ergodic (9)The definition of R explicitly excludes “trapped” bound molecular regions, i t . , phase space regions of nonzero measure, above the dissociation energy, from which the bound species does not decay. Simple statistical theories assume that R is the entire energy hypersurface at energy E . Note also that the discussion throughout this paper disregards sets of zero measure, e.g., isolated periodic orbits. (10) See, e.g.: Forst, W. Theory of Unimolecular Reactions; Academic: New York, 1973. (1 1) Slater, N. B. Theory of Unimolecular Reacrions; Cornell University Press: Ithaca, NY, 1959. Thiele, E. J. Chem. Phys. 1962,36, 1466. (12)Note that the observation of ”random” distributions need not necessarily imply an underlying random dynamics. See, e.g.: Kac, M. Am. Sci. 1984,72, 282. (13) Bunker, D. L.; Hase, W. L. J . Chem. Phys. 1973,59,4621.Deleon, N.;Berne, B. J. J . Chem. Phys. 1981,75,3495.
The Journal of Physical Chemistry, Vol. 90, No. 16. 1986 3511
Dynamical Theory of Statistical Unimolecular Decay theory. To do so necessitates that decay dynamics be cast in the language of ergodic theory and, at least initially, that the continuous time dynamics be replaced by a discrete time mapping. Specifically, we consider dynamics at multiples of a fixed time increment 6, defining 7"x as the propagation of a phase space point x for n time increments (Le,, x(t=n6) = 7"x).I4 Consider then Figure 1; decay consists of the evolution, into R+, of a distribution which is initially uniform9 on R. At this stage we require that the transition states S* be defined so that recrossing from R + to R or R to R- does not occur. Later below it proves necessary to impose additional conditions on R to ensure statistical decay to R;; the relationship between these conditions and statistical behavior are discussed in detail in Appendix A. Within discrete dynamics, S- and S+ are broadened from surfaces to volumes, S, and S+6,comprising all points which enter or have left R during a time interval 6. In what follows, time parameters associated with the discrete dynamics are measured in units of 6. These include t, t', T , and T< which are also used in connection with the flow. In the latter context the conversion to 6 independent units is implicit. Since our concern is with the behavior of trajectories with respect to reaction, it suffices to characterize them in terms of their location in R+,R-, or R as a function of time. Specifically, we may classify each phase space point, through its trajectory, as a member of a set (A03
A I , . * * I Am) = (xlx E Ao, T X E A I , ..', F XE A m )
(3)
where Ai is one of either R+, R-, or R. For example, (R-, R, R, R+) denotes the set of points which are in R- at time zero and which will be in R at times 6 and 26 and in R+ at time 36. For convenience we denote the set of points which are initiated in R and remain there for the following t - 1 time steps before entering R+ as r' and the set of points which are initiated in R and remain ; is in R for t - 1 time steps before passing into R+ as s - ~ ~that
To obtain the desired relationship, consider
T'r' = #*I
u
s-;+1
(9)
which merely states that the set of all trajectories which spend t in R either originate (one step before) in R or R-. Since r'+I and s - ~ ~are + ~disjoint and T i s measure-preserving
p ( T I r f )= p ( r ' ) = p ( r ' + ' )
+~(s-~~+l)
(10)
This result, in conjunction with the definitions in eq 5-8, gives
Pdt+l) - Pdt) = -+s,dR) P,,&)
(1 1)
or in the continuous time case dP(t)/dt = -k,(R) P,(t)
(12)
k,(R) = T,-I(R) = lim k,,(R)/6
(13)
where &-a
k , d R ) = p(S-J/dR) = IL(S+J/P(R) The last equality arises from the measure-preserving property of time reversibility. It is convenient to introduce an additional measure on S+ and S-. Specifically, let B be a Borel set15bin S+ (or S-) so that B6 = U;=,,TfBis a Borel set in R (orR-), Le. the "6 broadening of B". We define the measure p , ( B ) as Pg(B) = lim P(B6)/6
(14)
8-0
If one uses eq 14, eq 13 assumes the form U R ) = 7S-'(R) = pg(s+)/p(R)
(15)
To make contact with the chemical literature, note that eq 15 may be writtedc as
rf = (R, ..., R, R+); where R appears t times s - l = (R-, R,
..., R, R+); where R appears t - 1 times
(4)
The spaces S, and R are thus partitionedlSaaccording to the time required to reach R+. That is m
m
R = U r ' ; S-* = Us-gf r=
I
f=l
This symbolic dynamics approach allows one to focus explicitly upon only those features essential to the dynamics of unimolecular dissociation. Below we utilize this description of the dynamics to obtain the relationship of the gap distribution to the lifetime distribution. this relationship is then used in the derivation of a model for the lifetime distribution in the presence of relaxation. ( b ) The Gap and Lifetime Distributions. Consider first the relationship between the gap distribution Pg,6(t),describing evolution between K6and S+6,and the lifetime distribution P , ( t ) , describing motion from R to R+ through S+6.With p an invariant measure these are, for the discrete time map, defined as
where dx and dx'are volume elements on R and S+,respectively, ii is the unit normal to S+,v is the phase space velocity dxfdt, and H is the Hamiltonian. Thus, eq 15 is of the form k, = flux/volume. Although this is the traditional form, it does include two generalizations. First, it applies to any choice of R, and associated S-, S+,including R equal to the entire energy hypersurface (minus permanently trapped regions). Second, the transition state can be a general phase space dividing surface. This derivation of the relationship between the gap and lifetime distributions, given by eq 12 and 15, replaces the heuristic arguments" leading to eq 2. Equation 12 in conjunction with the fact that P,(t), as a probability distribution, is nonnegative and L' is normalized to unity has a number of direct consequences: (i) P ( t ) must be strictly nonincreasing, and integrating eq 12 over all time gives (ii) Moments of the two distributions are related by X - d t tnPg(t)= n + , ( R ) l0e d t t"lP(t);
n
>0
(iii) For the specific choice of n = 1 we have and for the continuous time case as P g ( t ) = lim P,,,(t/S)/G
(7)
P ( t ) = lim P 8 ( t / 6 ) / 6
(8)
6-0
6-0
(14) Alternatively, one may consider the base automorphism of a special representation of the continuous time flow, e.g., a Poincare surface of section. See ref 6b. (15) (a) A partition of M is a set of disjoint subsets of M , whose union is M . (b) Bore1 sets are a class of sets obtained by taking countable unions and intersections of open or closed sets. All Borel sets are measureable.
Note that eq 17 and 18 provide the precise definition of 7,: it is the mean gap time between S- and S+ and the reciprocal of P(0). Further analysis, contained in section c, is necessary to provide the relation of s,(R) to the decay time and to properly define R for statistical applications. ( c ) Assumptions of Chaos. Further characterization of P(t ) and P , ( f ) requires details of the dynamics, which we propose to formulate in terms of ergodic theory based assumptions. However, since ergodic theory6 considers dynamics on a bounded manifold, it is not directly applicable to the unbounded phase space asso-
3512 The Journal of Physical Chemistry, Vol. 90, No, 16, I986
ciated with molecular decay. For this reason we first introduce a related auxiliary bounded system upon which conditions of chaos are imposed and then determine their effect on the molecular decay. This approach is analogous to traditional methods where the full Hamiltonian is partitioned into terms describing the bound molecule, the dissociative continuum, and their mutual coupling. As shown below, a careful treatment yields a new model, the delayed lifetime gap model (DLGM) for P ( t ) and P,(t). The simple statistical theory assumption that P&t) and P ( t ) are exponential with rate k,(E) is shown to arise only as a limiting case. The auxiliary bounded system of interest is constructed by composing the original flow within R with any p,-preserving mapping from S- to S+ which recycles outgoing density back into R. The time evolution operator associated with this bounded flow is denoted Tb. In practice, the pg-preserving map is chosen to be the reversal of the velocity associated with the phase space coordinate normal to the incoming transition state, a transformation which takes R+ R- and S+ S-. Care is taken below to utilize chaotic properties of the bounded flow which are independent of this particular choice of the map. If the bounded flow were required to be mixing in R, then, setting p ( R ) = 1, we obtain
-
-
lim 4 A
I -m
n T,”B)
= ,441 4 B )
(19)
for all Bore1 sets A , B in R. This condition, in its general form, is of little use for two reasons: (a) the long time limit on a bounded system is not relevant to the dissociation problem, and (b) the condition that mixing be achieved for all sets A, B is unnecessary since ultimate interest is only in the issue of reactivity. Thus, we choose to impose a finite time version of eq 19 on the specific choice A = rl and B = S+ That is
Note that eq 20 is required to hold for all times t. Equation 20 has the qualitative interpretation that the number of trajectories initiated in R and reacting in time t , which had their origin in S-$at a prior time t’, is independent of t’for times t’> T. Alternatively, a distribution initially uniform on S- must relax in time T to a distribution which overlaps r‘ in the same fashion as does the distribution uniform on R, Le., only according to p ( I ‘ ) . This equation contains the central feature of a statistical theory,16i.e., the partitioning of the initial to final dynamics into separable terms, one dependent only upon the initial state and one dependent solely on thefinal state. Note that both sets r1 and T{‘S-6 are expected to be highly convoluted and strung out over R due to the complicated dynamics, a feature relied upon to justify eq 20. Equation 20 applies to the auxiliary bounded system, and it is necessary to relate this condition to the unbounded flow associated with dissociation. Doing so leads to specific conditions on 7, as follows. (i) First, note that the unbounded flow possesses a smallest gap time, denoted T )
(53)
Equation 52 is the condition for statistical product distributions obtained above whereas eq 53 is precisely such a product distribution. This argument obviously generalizes to any definition of product state j and any number of such final channels. Two relevant comments emerge from this discussion. First, the equality of the j-specific mean gap times, as a requirement for statistical behavior, is readily tested through measurements of the rate of production of products in statej (see eq 49). Second, the traditional setup for decay, e.g. eq 46, already includes the statistical assumption of equal j-specific gap times by assuming that all available N ( t ) contribute to the decay of each and every product channel. VI. Summary In this paper we have provided an ergodic theory basis for statistical theories of unimolecular decay. The approach provides insight into conditions under which statistical theories obtain and
the dynamical implications of statistical product distributions and exponential decay rates: (1) The decay probability density P ( t ) and the gap distribution P,(t) have been rigorously related through dP(t)/dt = -k,(R) Pg(t). The traditional statistical decay time k,(R)-’ is seen to be properly interpreted as the mean gap time. (2) A specific mixing condition (Fq 22) for molecular relaxation in dissociating systems was formulated. The condition has a number of noteworthy qualtitative features. First, it shows the necessity of finite rate mixing conditions in molecular decay problems. Second, this condition necessarily subdivides the dynamics into direct and indirect components, where only the latter is amenable to statistical treatment. Third, a nonzero relaxation time 7 emerges explicitly within this formulation. (3) Invoking the mixing condition leads to an analytically soluble model for P ( t ) which is asymptotically well-approximated by exponential decay with rate K. The rate of decay is directly affected by the relaxation time T and equals k,(R) in the limit T 0. Further, K is bounded as k,(R) < K < k,(R)e, allowing for larger than statistical decay rates when the system is mixing. (4) The gap distribution for the set of statistical trajectories is not everywhere exponential if T is nonzero. (5) A product distribution is statistical if the mean gap time associated with each product j equals k;I(R). (6) The traditional rate formulation for decay into multiple products implicitly assumes equal mean gap times for each product
-
n(j)
i.
These results constitute the first major steps in formalizing statistical theories of reaction dynamics and relating statistical molecular behavior to ergodic theory. Further numerical and formal studies are in progress.
Acknowledgment. We thank the donors of the Petroleum Research Fund, administered by the American Chemical Society, for support of this work. R.S.D. thanks P. Nardonne and R. Kapral for introducing him to delay differential equations.
Appendix A. The Choice of Transition State In this paper the phase space is partitioned into regions R, R+, and R-, separated by transition states which allow no recrossing. Further conditions required to satisfy eq 22 were alluded to in section 111. Since various definitions of the dividing surface are often adopted, a number of comments are in order. (a) Although not a necessary requirement, the transition state is often defined in terms of a coordinate space dividing surface. Such a traditional approach might ignore the nonrecrossing requirement and choose a dividing surface which is chemically sensible, i.e., generally “too close in” to the molecular region for the nonrecrossing requirement to hold. The resultant partitioning of phase space into R‘, RL, and R; with transition states SL and S‘+ is equivalent to constructing a modified dynamics where recrossing is neglected. The relationships derived in this paper still hold for this modified system, but the resultant distributions, compared to the exact dynamics, will contain errors due to the incorrect nonrecrossing assumption. (b) Errors associated with the recrossing of S i can be corrected by eliminating, via short time dynamics, those transition-state points corresponding to other than a first crossing. Such an approach involves the replacement of S’* and R’by S(’), C S’* and R ( ’ ) 3 R’, respectively. (R’is augmented by trajectory segments between consecutive crossings of S’+ and SL.) The alternative is to select the transition states sufficiently far into the product channels to ensure no recrossing. In this case direct trajectories, not included when S’* or S ( ’ ) , are utilized, can be eliminated, again, by short time dynamics. The system and SCz)*which results differs from that given by R(’) and S(I),in the location of the transition states and in the size of the molecular phase space. Specifically, is obtained from R(’) by appending additional segments at either end of each trajectory. Since the parameters relevant to the DLGM, k, and T < , differ in the two approaches, it is probable that these approaches are inequivalent with respect to the application of this statistical model. Indeed, in (c) below, it is shown that a transition-state choice for which
3516 The Journal of Physical Chemistry, Vol. 90, No. 16, 1986
the DLGM applies is unique with respect to “time translation”, a transformation which can, at least approximately, relate the choices S(’), and S(2)h. (c) Section IIIc discusses the need to tailor the region R in order to eliminate direct dynamics and to satisfy the mixing condition of eq 22 leading to the DLGM model. In tests of the DLGM” a divergence criterion in phase space5 is used to define this region R, There is, however, a feature of the dynamics, associated with satisfying eq 22 and linked to the definition of R, which leads to our current belief that eq 22 imposes a condition on R which makes its choice unique. Further work is necessary, however, to produce a noncomputational route to determining R. Specifically, consider the effect of time translations on the transition states and the corresponding effect on R. Here we define the time translated transition state as S-” = T’/2S- (with the inverse translation on S+, defining S,”).The modified R is then given by U
Ru = R
+ sgn ( u ) U ( T ‘ S - U FS+) 1=0
(All
and the associated modified gap and lifetime distributions PgU(t) and P’(t) satisfy Pg”(t)= Pg(t-u) P ( t ) = (1
+ k,u)-IP[(t-u)+]
(‘4-2)
where (u)+ = u for u > 0 and 0 for u I 0 and -u < T
