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On statistics of the ensemble of oscillators under excitation. I. Classical nonlinear oscillators excited by a weak external force
Chemical Physics 52 (1980) 305-311 © North-Holland Publishing Company
ON STATISTICS OF THE ENSEMBLE OF OSCILLATORS UNDER EXCITATION. I. CLASSICAL NONLINEAR OSCILLATORS EXCITED BY A WEAK E X T E R N A L FORCE V.N. SAZONOV * and S.V. ZATSEPIN
Department of TheoreticalPhysics, P.N. Lebedev PhysicalInstitute, Moscow 11 7924, USSR Received 29 January 1980
The distribution function is searched out of the Fokker-Planck kinetic equation witfi the small parameter at higher derivatives. The analytic solution is produced by means of a double successive application of the averaging method. It is shown that an external harmonic force leads to the appearance of excessive population in some regions of phase space, where the vibrational frequency of oscillator is close to the frequency of the external force. The present model shows that laser radiation can stimulate chemical reactions with high activation energy, whereas reactions with low activation energy will not be stimulated.
1. Introduction We consider the problem of the distribution function of the ensemble o f classical nonlinear onedimensional oscillators, which are excited b y an external harmonic force in a heat bath. This problem is of interest in connection with some laser-chemical experiments e.g. refs. [ 1 - 4 ] . In these experiments a nonthermal action of laser radiation at a gas mixture of resonant and non-resonant molecules was investigated in the deep collisional case %. In the present work the external harmonic force describes the action of laser radiation; the non-resonant molecules and (or) non-resonant degrees o f freedom play the part of the heat bath; the classical non-linear oscillator describes the excited vibrational mode of a resonant molecule (the classical approach is valid because excitation up to large quantum numbers is considered). The mathematical problem is reduced to the solution of the F o k k e r - P l a n c k equation with a small parameter at higher derivatives. With the help of twofold successive applications of the averaging method [6] we get the analytic expression for the distribution function in the case of a weak force This solution shows that the action of the external force leads to * Author to whom correspondence should be addressed.
The multiphoton collisionless case was reviewed in ref. [5].
excessive population in resonant regions o f the phase space, where coy(E) ~ We.
(1)
Here and below ~ v is the self-vibrational frequency of the oscillator, depending on the oscillator energy E, toe is the frequency of the external force.
2. The initial equation
Let q, p, m be the coordinate, the m o m e n t u m and the mass of the oscillator, respectively. V(q) is the potential energy, F(t) is the external force, and W(p, q; t) is the distribution function. The initial equation for the function W(p, q; t) is the kinetic equation ~W+p ~W+[ 3t m 3q
dV+F(t)]~W
-~
-~p = I ,
(2)
where the integral of collisions I describes the interaction of the oscillator with the heat bath. We can take I in the Landau form
I = vmT(3/3p) [3 W/3p + (p/m T) W],
(3)
where v is the collision frequency. T is the temperature of the heat bath. The differential approach (3) o f / i s valid if the oscillator state changes rather little during
KN. Sazonov and S. K Zatsepin /Statistics o f the ensemble o f oscillators
306
where
one act of interaction (other descriptions o f interaction between particles and the heat bath are discussed in refs. [7,8]). The potential V(q) has the following form
W = W(t, u, v),
6co(n) = We - COy(n),
COv(n) = COve + [1/2n(2hmCOen) 1/2]
V(q) = ~moJveq 1 ,2 _2 + V l ( q ) , where V1 (q) is a nonlinear addition including q3 and other higher powers of q. If the external force is absent and t > > u -~ , then any solution of eqs. (2) and (3) has one and the same form and depends neither on time nor on the initial distribution function W(t = 0; q, p); i.e.
W(t > > u -1, p, q) = WG(p, q). The function WG(p, q) is the equilibrium (Gibbs) distribution. Let F(t) = F cos COet. In this case any solution o f ( 2 ) and (3) also has one and the same form, when t > > u -1 irrespective of W(t = 0;p, q); but in this case the function W(t > > v -1 , q, p ) is a periodical function o f time with period 2n/COe. We will use the notation W(t > > v-1 , q, p ) = w(t, q, p); and will refer to w(t, q, p) as a function of the steady-state distribution. The aim of this work is to find the function o f the steady-state distribution.
Xi
J --lr
We introduce new slow variables u and
[(hn/mCOe) 1/2 COS 4d
dq
For a linear oscillator COy and 6CO do not depend on n. The slow variables are convenient because the coefficients in eq. (5) do not depend on time. The stationary solution w(u, v) of eq. (5) does not explicitly depend on time either. Eq. (5) is valid only in the resonance region, when lSCO(n)l < < COe •
(6)
By virtue of (6) we replace COy(n) by COe whenever possible. In this approximation the energy of the oscillator is equal to E = hCOen. Further on the non-linearity of the oscillator is displayed only in the dependence of the oscillator frequency COy and the frequency detuning 6CO on the quantum number n (or on the oscillator energy E). Let us consider some properties of eq. (5) in the case when u = 0, and so eq. (5) can be rewritten in the following form: --+-~t au
3. The first averaging
dV1
d~ cos
-
~v
W
+
av \ 3u
W
=0,
where
v
H = H(u, v)
q = (2h/mCOe)l/2(u cos COet + O sin COet),
u2+v 2
p = -(2hmcoe)l/2(u n COet - v cos COet),
(4)
and insert a new notation n = u 2 + v2 ; f = F/(2hmCOe)l/2; N = T/hCOe. The steady-state distribution function does not depend on time if slow variables are used: w(t, q, p) = w(u, v). Although we consider the classical problem, it is convenient to use values of the quantum theory. We admit that slow variables do not change during the period of the external force; after averaging eqs. (2) and (3) during this period we get
=~f
dnl aCO(nl) +
~fu.
The equations of motion of the oscillator using slow variables can be represented in the hamiltonian form [9] with the hamiltonian H:
du/dt = -3Hfi)v,
dv/dt = 3H/~u.
If at t = 0 the function W(t = O, u, v) = a(u - Uo ) a ( v - Vo),
aw/at -
1
aCO(n)va W / a u FOz W
+ ( ½ f + aCO(n)u)3 w / a v
OW
v ~v
(7)
0
then at t > 0 the function
W(t;u, v) = 6(u - u(t))6(o
v(t)),
i.e. the representation point of the oscillator moves
(8)
KN. Sazonov and S. V. Zatsepin / Statistics o f the ensemble o f oscillators ~oot}
307
with a period T = oo. In the present work we assume that eq. (9) determines the only trajectory in the (u, v) plane, i.e. there are no separatrices on this plane.
%
4. The second I
Fig. 1. Dependence of the oscillator self-frequency w v on the quantum number n or on the energy E = t/wen. The external frequency is shown by a d o t t e d line. The region of the quasicontinuum is n > nqc.
along some definite trajectory (u(t), v(t)) in the (u, v) plane. This trajectory is determined by the condition
H(u, v) = const.
(9)
A possible dependence of coy(n) is shown in fig. 1. The trajectories of the oscillator in the (u, v) plane are represented in fig. 2. The time T of returning of the representation point over the trajectory to the initial position is the pulsation period of oscillator. In the general case T depends on the trajectory. For a linear oscillator T = 27r/16co1 does not depend on the trajectory. In some cases eq. (9) determines more than one closed trajectory in the (u, v) plane for the same value of constant. Then a separatrix exists - a trajectory
l
lY
averaging
To account for the influence of the heat bath we shall treat the collision frequency v as the small parameter. In this case the transition time of representation points from one trajectory to another is much larger than the pulsation period. It is convenient to rewrite eq. (9) in the following form:
at
au,, ~-v w+jc" + a u ~
w+ic° =o,
where
]c,, = -¼vN aW/au - ~vuW,
icy = - I v N a W / a o - ½voW. We again introduce new variables (H, S), where S is the length of the arc of a trajectory from an arbitrary point. By definition W(H, S) is equal to
W(H, S) = W(u(H, S), v(H, SD)IJJ, where J = D(u, v)/D(H, S) is the jacobian of transformation (u, v) -+ (H, 5"). We chose the direction of the increase of S so that J > 0. After some necessary calculations with the help of the following relations:
3H/Ou = (l/J) Oo/OS, OH/Oo= - ( 1 / 3 ) 3u/3S, aS/au = - ( l / J ) av/aH,
aSlav = (1/J) au/aH,
we get
aw/at + (a/as)(w/J) + aicH/aH + a/es/aS = O,
(10)
where ]cH = ]cu o r ~ a s - ]cv a u / a s ,
lcS =Jcv Ou/OH- Jcu Or~all. Fig. 2. Trajectories in the (u, v) plane corresponding to the dependence of ~ov on n as shown in fig. 1. In this case ~co(n) = we - wv(n) < 0 and H3 < H2 < Ha < 0. The arrows show the direction of the representation p o i n t m o t i o n and the direction of the increase of S.
When the influence of the heat bath is negligibly small, one can put in (10)]cs =]cH = 0. According to (10) in this case we have the following equations of motion of the representation point: dH/dt = 0
V.N. S a z o n o v a n d S. V. Z a t s e p i n / S t a t i s t i c s o f t h e e n s e m b l e o f oscillators
308
and dS/dt =J-~. It is necessary for us now to make a more precise definition of the variable S. It is possible to measure the length of the arc of a trajectory with the help of such a coordinate system that the velocity of the representation point over the trajectory will depend neither on time nor on S. Moreover, let the interval of changing of S along an arbitrary trajectory be (0, 1). Then
where T(/-/) is determined by the value of H. Now we can take into account the influence of the heat bath. The currentsjcH and ]cs are proportional to u, so according to (10) we can write -
T(H)(a/cHIOH + alcs/as) ~ vT(H).
Here and below w is the stationary solution of (10) i.e. a steady-state distribution function. Thus, if v T ( H ) < < 1,
(12)
the dependence w(H, S) on S is negligibly small. Let us average eq. (10) over the variable S, i.e. we apply the operator flo dS to both sides of this equation. After the averaging we get ajcH/aH = 0, where fct4 = flo JcH dS. Due to the evident relation/~tt(H = ~o) = 0 the equation a-fcH/aH = 0 is equivalent to ]-cn = 0, where t
JcH = ½P(wlT)(OdS - UOS) -- (uN/4T)[(v's) z + (u's) 2 ] (alaH)(w/T) t
t
+ (uN/4T)(vSU H
+
t
t
USUH)(O/OS)(w/T ).
(13)
The last term can be neglected due to relation (12). During the integration of (13) over the trajectory the variable S changes in the same direction as the representation point because dS/dt > 0. Then 1
f
d S f = sign(H)
0
1
--J dS(v aulaS -
u
av/as)= - 2
sign(H)
0
f du dr, H
1
(I/T)
f dS[OulaS)z +
(arias) 2 ]
0
(11)
dS/dt = 1/J = lIT(H),
aw/aS =
the following relations
2 sign(H) f d u dv (d/dn)(naoa(n)). H
The integrals on the right-hand sides of these relations are taken over the region of the (u, v) plane restricted by the trajectory for a given value of H. After the averaging of (13) with the help of the preceding relations we get Jell = 0. So the equation for the steadystate distribution function w(H) has the following form: 1 _d_ ( _w] / ' d u d v - - d [n6co(n)] + [ -w- t [ ' d u d v 2 dH\T]J H dn \ T ] aH
= O. (15)
This equation can be easily solved in quadratures.
5. The approximation of a weak force
The integrals in eq. (15) can be easily evaluated in the weak force approximation. We shall use parametric equations of trajectories in the form of u = u l + r cos O,
v = r sin O,
where 0 is a parameter. According to (7) we get with an accuracy of first order in f r 2 = n(H),
ul = -f/26aa(n(H)),
where n(H) is determined by the relation
fdu fOu/aS)-' 1
tf
dn ~ko(n) = H. 0
= sign(H) f d o f(ao/as) -1 ,
(14)
H
where f is an arbitrary function of (u, U);~fH is the integral over the trajectory with a given value of H; the integral is taken in the positive direction; the function sign(H) equals +1 ( - 1 ) for a positive (negative) direction of motion of the representation point. Using (14), (11), (8)and the Green theorem we get
The relations obtained are valid only in the case of a weak force, i.e. lull < < r . Now we have f du du
nn(If),
H
f du dv (d/dn) [n6 o3(n)] = rrn(H) 6 o3(n(tt)). tt
V.N.
Sazonov
and S. V. Zatsepin
/Statistics
of the
ensemble
of oscillators
309
This gives according to (15), 44 u>= w?yT(H) Wu,u)
1 =WN) exp I-n(H(u, u))/Nl, = A exp -(2/N) j [
~/wMl)
whereA is a constant. By definition is n(H(u,u)) 1s
dn &w(n) = 1 j
0
0
lC*+V* dn SW(H) t ifu,
With the help of this relation we get n(H(u, u)) with an accuracy of first order in f
n(H(u,u))=u* + II* +fu/sw(u* + u*>. Thus w&u)=
(A/N) exp[-n/N-
fu/NGo(n)],
Fig. 3. The dotted line shows the dependence of the population w(n) on the quantum number n in the equilibrium case, when w(n) = we(n), where WG(n) is Gibbs distribution. The pointed line shows w(n), when excitation by an external force is taken into account and 6 w(n) = 6 w(O) = const corresponding to the linear oscillator case. The solid line shows w(n), when both excitation and non-linearity are taken into account. The second maximum of the solid line is due to the sharp decrease of 18 w(n)l in the region n > ngc (see fig. 1).
(16) 2n. This gives
wheren=u*+v*.
6. Discussion
0
In the caseof a linear oscillator the exact solution of (5) is known [lo]. In this casethe steady-state distribution function has the form of (16) irrespective of the value off, if v << 1&w/. The restriction v<< /Sol coincides with (12) because the period T in the case of linear oscillator,equals T = 277/164. Thus the non-linearity of oscillator both in the initial equation (5) and in the steady-state distribution function can be taken into account by replacement of 60 by 6w(n). It is interesting to note that formula (16) in the linear oscillator case can be rewritten in the form of wh 4 = V/N) exp
L-Q@,u)P”l ,
where T’ is the temperature of the heat bath, Q(u, u) is the oscillator quasi-energy averaged over the period of the external force (see ref. [ Ill; in this work the concept of “quasi-energy” was extended from the quantum theory [ 121 to the classical one). Let us obtain the distribution as a function of 12 with the help of (16). We put u = nl’* cos cpand u = +“2 sin cpand integrate w over dq from 0 up to
where
Z(n) =AI,Cfi”*/Nl6w(n)l).
(17)
The function w&r) is’the equilibrium normalized Gibbs distribution. The population at energy E = Aw,n distinguishes from the equilibrium population by a factor I(n). The modified Bessel function 1, increases rapidly, if the argument increases [ 131. Thus, the better condition (1) is fulfilled, the more the excessive population is. Fig. 3 shows schematically the dependence w(n) corresponding to fig. 1. The present calculations confirm our conclusion about the appearance of excessive population in the resonant regions of phase space, which was drawn earlier [ 111 with the help of qualitative reasoning. The statistics of the ensemble of quantum oscillators (under some special assumptions) was investigated in ref. [ 141. It was shown that excessive population appears at levels n > n, when
@‘nr+l - EnrP = 0, .
310
V.N. Sazonov and S. V. Zatsepin / Statistics o f the ensemble o f oscillators
This formula corresponds to the classical condition (1). The dependence coy(n), corresponding to the decrease of 15w(n)l as n increases, is shown in fig. 1. This dependence describes the quasicontinuum in high levels of the energy spectrum of polyatomic molecules. There is a resonant pair of levels for an arbitrary laser frequency in the region of the quasicontinuum (i.e. when n ~> nqc ). According to (17) the quantity of the additional population [factor l(n)] increases and can raech a significant value at a large quantum number n. At the same time l(n) ~ 1 for small n. This shows that tile external force can exert a significant stimulating influence on the chemical reactions with a high activation energy, whereas the reactions with a low activation energy will not be stimulated. Specific features of polyatomic molecules are described in our model by decreasing tSw(n)l, when n becomes larger than nqc. Another way is to take into account the stochasticity of intramolecular motion (see e.g. refs. [ 1 5 - 1 9 ] ) . Eqs. (2), (3) and (5) in our work can describe stochasticity too. It arises if a separatrix exists. Indeed, condition (12) does not surely hold if the trajectory approaches the separatrix, because in this case T ~ oo. Thus, even for an arbitrary small collision frequency v the stochastic layer in the (u, v) plane exists. In this layer the motion has an irregular character. Such a motion is described by a rapid diffusion of the distribution function in the neighbourhood of the separatrix. The validity of our model is restricted, mainly, by two assumptions: (1) Assumption (12) can be rewritten as r > > 15col-1, where 6 co is a characteristic frequency detuning value, r is a characteristic relaxation time of resonant molecules. Even if 16601 = 10 -3 cm -1 , 7"> > 3 × 10 -a s. This condition is carried out for V - T relaxation time in many gas mixtures. (2) The external force F ( t ) in (2) was assumed to be a harmonic one: F ( t ) = F cos Wet. So we may formally consider only the time interval 0 ~< t < < (Acoe) -1 , where Awe is the spectral width of the laser radiation. This restriction is formally very strong: if Awe ~ 3 × 10 -2 cm -1 , then t < < 10 -9 s. At the same time the characteristic length of the laser radiation action in refs. [ 1 - 4 ] was 10 - 2 - 1 0 3 s. But actually formula (17), which expresses the main result of our
work, remains correct, if substitution 16~)(n)f-* max{lfico(n)i, r -1 , Awe} is made. Thus, we can conclude: (1) Laser radiation exerts a considerable excessive population at high levels of the energy spectrum of resonant polyatomic molecules. (2) The excessive population factor I [see (17)] increases when the quantum number n increases and (or) the temperature of the heat bath T = ]Vhcoe decreases. We hope these qualitative conclusions are model independent.
References [ 1] G.N. Basov, E.P. Markin, A.N. Oraevsky and A.V. Pankratov, Dokl. Akad. Nauk SSSR 198 (1971) 1043; G.N. Basov, A.N. Oraevsky and A.V. Pankratov, Kvantovaya Elektron. 3 (1976) 814 [Sov. J. Quantum Electron. 6 (1976) 4431. [2] H.R. Bachmann, R. Rink, H. NiSth and K.L. Kompa, Chem. Phys. Letters 33 (1975) 261; 45 (1977) 169. [3] R.N. Zitter and D. Koster, J. Am. Chem. Soc. 99 (1977) 5491. [4] V. Slezak, J. Caballero, A. Burgos and E. Quel, Chem. Phys. Letters 54 (1978) 105. [5] R.V. Ambartzumian and V.S. Letokhov, in: Chemical and biochemical applications of lasers, Vol. 3, ed. C.B. Moore (Academic Press, New York, 1977). [6] R.V. Khokhlov, Dokl. Akad. Nauk SSSR 97 (1954) 411; Usp. Fiz. Nauk 87 (1965) 17. [7] T.K. Melik-Barkhudarov, Zh. Eksp. Teor. Fiz. 75 (1978) 97. [8] S. Mukamel, Chem. Phys. 37 (1979) 33. [9] V.I. Gorchakov and V.N. Sazonov, Zh. Eksp. Teor. Fiz. 70 (1976)467 (Sov. Phys. JETP 43 (1976) 241). [10] B.Ya. Zel'dovich, A.M. Perelomov and V.S. Popov, Zh. Eksp. Teor. Fiz. 55 (1968) 588; 57 (1969) 196. [ 11 ] V.N. Sazonov and S.V. Zatsepin, Teor. Mat. Fiz. 40 (1979) 311. [12] Ya.B. Zel'dovich, Zh. Eksp. Teor. Fiz. 51 (1966) 1492 [Soy. Phys. JETP 26 (1967) 861]; Usp. Fiz. Nauk 110 (1973) 139 [Usp. 16 (1973) 427]; V.I. Ritus, Zh. Eksp. Teor. Fiz. 51 (1966) 1544 [Sov. Phys. JETP 26 (1967) 1041]. [13] I.S. Gradshtein and I.M. Ryzhik, Tablitsy integralov (Moscow, 1971). [14] V.N. Sazonov, Zh. Eksp. Teor. Fiz. 77 (1979) 1767. [15] E.V. Suryak, Zh. Eksp. Teor. Fiz. 71 (1976) 2039 [Sov. Phys. JETP 44 (1976) 1070]. [16] D.W. Oxtoby and S.A. Rice, J. Chem. Phys. 65 (1976) 1676. [17] D.W. Noid, M.L. Koszykowski, R.A. Marcus and J.D.
V.N. Sazonov and S. V. Zatsepin / Statistics o f the ensemble o f oscillators
McDonald, Chem. Phys. Letters 51 (1977) 540; D.W. Noid and J.R. Stine, Chem. Phys. Letters 65 (1979) 153. [18] K.D. H~insel, Chem. Phys. Letters 57 (1978) 619; Chem. Phys. 33 (1978) 35; J. Chem. Phys. 70 (1979) 1830. [19] F.M. Israilev, J. Gazati, J. Ford and B.V. Chirikov, Insti-
311
tute of Nuclear Physics, preprint N 46, Novosibirsk (1978); F.M. Israilev and D.L. Shepelensky, Institute of Nuclear Physics, preprint N 97, Novosibirsk (1978); Dokl. Akad. Nauk SSSR 249 (1979) 1103.
ON STATISTICS OF THE ENSEMBLE OF OSCILLATORS UNDER EXCITATION. I. CLASSICAL NONLINEAR OSCILLATORS EXCITED BY A WEAK E X T E R N A L FORCE V.N. SAZONOV * and S.V. ZATSEPIN
Department of TheoreticalPhysics, P.N. Lebedev PhysicalInstitute, Moscow 11 7924, USSR Received 29 January 1980
The distribution function is searched out of the Fokker-Planck kinetic equation witfi the small parameter at higher derivatives. The analytic solution is produced by means of a double successive application of the averaging method. It is shown that an external harmonic force leads to the appearance of excessive population in some regions of phase space, where the vibrational frequency of oscillator is close to the frequency of the external force. The present model shows that laser radiation can stimulate chemical reactions with high activation energy, whereas reactions with low activation energy will not be stimulated.
1. Introduction We consider the problem of the distribution function of the ensemble o f classical nonlinear onedimensional oscillators, which are excited b y an external harmonic force in a heat bath. This problem is of interest in connection with some laser-chemical experiments e.g. refs. [ 1 - 4 ] . In these experiments a nonthermal action of laser radiation at a gas mixture of resonant and non-resonant molecules was investigated in the deep collisional case %. In the present work the external harmonic force describes the action of laser radiation; the non-resonant molecules and (or) non-resonant degrees o f freedom play the part of the heat bath; the classical non-linear oscillator describes the excited vibrational mode of a resonant molecule (the classical approach is valid because excitation up to large quantum numbers is considered). The mathematical problem is reduced to the solution of the F o k k e r - P l a n c k equation with a small parameter at higher derivatives. With the help of twofold successive applications of the averaging method [6] we get the analytic expression for the distribution function in the case of a weak force This solution shows that the action of the external force leads to * Author to whom correspondence should be addressed.
The multiphoton collisionless case was reviewed in ref. [5].
excessive population in resonant regions o f the phase space, where coy(E) ~ We.
(1)
Here and below ~ v is the self-vibrational frequency of the oscillator, depending on the oscillator energy E, toe is the frequency of the external force.
2. The initial equation
Let q, p, m be the coordinate, the m o m e n t u m and the mass of the oscillator, respectively. V(q) is the potential energy, F(t) is the external force, and W(p, q; t) is the distribution function. The initial equation for the function W(p, q; t) is the kinetic equation ~W+p ~W+[ 3t m 3q
dV+F(t)]~W
-~
-~p = I ,
(2)
where the integral of collisions I describes the interaction of the oscillator with the heat bath. We can take I in the Landau form
I = vmT(3/3p) [3 W/3p + (p/m T) W],
(3)
where v is the collision frequency. T is the temperature of the heat bath. The differential approach (3) o f / i s valid if the oscillator state changes rather little during
KN. Sazonov and S. K Zatsepin /Statistics o f the ensemble o f oscillators
306
where
one act of interaction (other descriptions o f interaction between particles and the heat bath are discussed in refs. [7,8]). The potential V(q) has the following form
W = W(t, u, v),
6co(n) = We - COy(n),
COv(n) = COve + [1/2n(2hmCOen) 1/2]
V(q) = ~moJveq 1 ,2 _2 + V l ( q ) , where V1 (q) is a nonlinear addition including q3 and other higher powers of q. If the external force is absent and t > > u -~ , then any solution of eqs. (2) and (3) has one and the same form and depends neither on time nor on the initial distribution function W(t = 0; q, p); i.e.
W(t > > u -1, p, q) = WG(p, q). The function WG(p, q) is the equilibrium (Gibbs) distribution. Let F(t) = F cos COet. In this case any solution o f ( 2 ) and (3) also has one and the same form, when t > > u -1 irrespective of W(t = 0;p, q); but in this case the function W(t > > v -1 , q, p ) is a periodical function o f time with period 2n/COe. We will use the notation W(t > > v-1 , q, p ) = w(t, q, p); and will refer to w(t, q, p) as a function of the steady-state distribution. The aim of this work is to find the function o f the steady-state distribution.
Xi
J --lr
We introduce new slow variables u and
[(hn/mCOe) 1/2 COS 4d
dq
For a linear oscillator COy and 6CO do not depend on n. The slow variables are convenient because the coefficients in eq. (5) do not depend on time. The stationary solution w(u, v) of eq. (5) does not explicitly depend on time either. Eq. (5) is valid only in the resonance region, when lSCO(n)l < < COe •
(6)
By virtue of (6) we replace COy(n) by COe whenever possible. In this approximation the energy of the oscillator is equal to E = hCOen. Further on the non-linearity of the oscillator is displayed only in the dependence of the oscillator frequency COy and the frequency detuning 6CO on the quantum number n (or on the oscillator energy E). Let us consider some properties of eq. (5) in the case when u = 0, and so eq. (5) can be rewritten in the following form: --+-~t au
3. The first averaging
dV1
d~ cos
-
~v
W
+
av \ 3u
W
=0,
where
v
H = H(u, v)
q = (2h/mCOe)l/2(u cos COet + O sin COet),
u2+v 2
p = -(2hmcoe)l/2(u n COet - v cos COet),
(4)
and insert a new notation n = u 2 + v2 ; f = F/(2hmCOe)l/2; N = T/hCOe. The steady-state distribution function does not depend on time if slow variables are used: w(t, q, p) = w(u, v). Although we consider the classical problem, it is convenient to use values of the quantum theory. We admit that slow variables do not change during the period of the external force; after averaging eqs. (2) and (3) during this period we get
=~f
dnl aCO(nl) +
~fu.
The equations of motion of the oscillator using slow variables can be represented in the hamiltonian form [9] with the hamiltonian H:
du/dt = -3Hfi)v,
dv/dt = 3H/~u.
If at t = 0 the function W(t = O, u, v) = a(u - Uo ) a ( v - Vo),
aw/at -
1
aCO(n)va W / a u FOz W
+ ( ½ f + aCO(n)u)3 w / a v
OW
v ~v
(7)
0
then at t > 0 the function
W(t;u, v) = 6(u - u(t))6(o
v(t)),
i.e. the representation point of the oscillator moves
(8)
KN. Sazonov and S. V. Zatsepin / Statistics o f the ensemble o f oscillators ~oot}
307
with a period T = oo. In the present work we assume that eq. (9) determines the only trajectory in the (u, v) plane, i.e. there are no separatrices on this plane.
%
4. The second I
Fig. 1. Dependence of the oscillator self-frequency w v on the quantum number n or on the energy E = t/wen. The external frequency is shown by a d o t t e d line. The region of the quasicontinuum is n > nqc.
along some definite trajectory (u(t), v(t)) in the (u, v) plane. This trajectory is determined by the condition
H(u, v) = const.
(9)
A possible dependence of coy(n) is shown in fig. 1. The trajectories of the oscillator in the (u, v) plane are represented in fig. 2. The time T of returning of the representation point over the trajectory to the initial position is the pulsation period of oscillator. In the general case T depends on the trajectory. For a linear oscillator T = 27r/16co1 does not depend on the trajectory. In some cases eq. (9) determines more than one closed trajectory in the (u, v) plane for the same value of constant. Then a separatrix exists - a trajectory
l
lY
averaging
To account for the influence of the heat bath we shall treat the collision frequency v as the small parameter. In this case the transition time of representation points from one trajectory to another is much larger than the pulsation period. It is convenient to rewrite eq. (9) in the following form:
at
au,, ~-v w+jc" + a u ~
w+ic° =o,
where
]c,, = -¼vN aW/au - ~vuW,
icy = - I v N a W / a o - ½voW. We again introduce new variables (H, S), where S is the length of the arc of a trajectory from an arbitrary point. By definition W(H, S) is equal to
W(H, S) = W(u(H, S), v(H, SD)IJJ, where J = D(u, v)/D(H, S) is the jacobian of transformation (u, v) -+ (H, 5"). We chose the direction of the increase of S so that J > 0. After some necessary calculations with the help of the following relations:
3H/Ou = (l/J) Oo/OS, OH/Oo= - ( 1 / 3 ) 3u/3S, aS/au = - ( l / J ) av/aH,
aSlav = (1/J) au/aH,
we get
aw/at + (a/as)(w/J) + aicH/aH + a/es/aS = O,
(10)
where ]cH = ]cu o r ~ a s - ]cv a u / a s ,
lcS =Jcv Ou/OH- Jcu Or~all. Fig. 2. Trajectories in the (u, v) plane corresponding to the dependence of ~ov on n as shown in fig. 1. In this case ~co(n) = we - wv(n) < 0 and H3 < H2 < Ha < 0. The arrows show the direction of the representation p o i n t m o t i o n and the direction of the increase of S.
When the influence of the heat bath is negligibly small, one can put in (10)]cs =]cH = 0. According to (10) in this case we have the following equations of motion of the representation point: dH/dt = 0
V.N. S a z o n o v a n d S. V. Z a t s e p i n / S t a t i s t i c s o f t h e e n s e m b l e o f oscillators
308
and dS/dt =J-~. It is necessary for us now to make a more precise definition of the variable S. It is possible to measure the length of the arc of a trajectory with the help of such a coordinate system that the velocity of the representation point over the trajectory will depend neither on time nor on S. Moreover, let the interval of changing of S along an arbitrary trajectory be (0, 1). Then
where T(/-/) is determined by the value of H. Now we can take into account the influence of the heat bath. The currentsjcH and ]cs are proportional to u, so according to (10) we can write -
T(H)(a/cHIOH + alcs/as) ~ vT(H).
Here and below w is the stationary solution of (10) i.e. a steady-state distribution function. Thus, if v T ( H ) < < 1,
(12)
the dependence w(H, S) on S is negligibly small. Let us average eq. (10) over the variable S, i.e. we apply the operator flo dS to both sides of this equation. After the averaging we get ajcH/aH = 0, where fct4 = flo JcH dS. Due to the evident relation/~tt(H = ~o) = 0 the equation a-fcH/aH = 0 is equivalent to ]-cn = 0, where t
JcH = ½P(wlT)(OdS - UOS) -- (uN/4T)[(v's) z + (u's) 2 ] (alaH)(w/T) t
t
+ (uN/4T)(vSU H
+
t
t
USUH)(O/OS)(w/T ).
(13)
The last term can be neglected due to relation (12). During the integration of (13) over the trajectory the variable S changes in the same direction as the representation point because dS/dt > 0. Then 1
f
d S f = sign(H)
0
1
--J dS(v aulaS -
u
av/as)= - 2
sign(H)
0
f du dr, H
1
(I/T)
f dS[OulaS)z +
(arias) 2 ]
0
(11)
dS/dt = 1/J = lIT(H),
aw/aS =
the following relations
2 sign(H) f d u dv (d/dn)(naoa(n)). H
The integrals on the right-hand sides of these relations are taken over the region of the (u, v) plane restricted by the trajectory for a given value of H. After the averaging of (13) with the help of the preceding relations we get Jell = 0. So the equation for the steadystate distribution function w(H) has the following form: 1 _d_ ( _w] / ' d u d v - - d [n6co(n)] + [ -w- t [ ' d u d v 2 dH\T]J H dn \ T ] aH
= O. (15)
This equation can be easily solved in quadratures.
5. The approximation of a weak force
The integrals in eq. (15) can be easily evaluated in the weak force approximation. We shall use parametric equations of trajectories in the form of u = u l + r cos O,
v = r sin O,
where 0 is a parameter. According to (7) we get with an accuracy of first order in f r 2 = n(H),
ul = -f/26aa(n(H)),
where n(H) is determined by the relation
fdu fOu/aS)-' 1
tf
dn ~ko(n) = H. 0
= sign(H) f d o f(ao/as) -1 ,
(14)
H
where f is an arbitrary function of (u, U);~fH is the integral over the trajectory with a given value of H; the integral is taken in the positive direction; the function sign(H) equals +1 ( - 1 ) for a positive (negative) direction of motion of the representation point. Using (14), (11), (8)and the Green theorem we get
The relations obtained are valid only in the case of a weak force, i.e. lull < < r . Now we have f du du
nn(If),
H
f du dv (d/dn) [n6 o3(n)] = rrn(H) 6 o3(n(tt)). tt
V.N.
Sazonov
and S. V. Zatsepin
/Statistics
of the
ensemble
of oscillators
309
This gives according to (15), 44 u>= w?yT(H) Wu,u)
1 =WN) exp I-n(H(u, u))/Nl, = A exp -(2/N) j [
~/wMl)
whereA is a constant. By definition is n(H(u,u)) 1s
dn &w(n) = 1 j
0
0
lC*+V* dn SW(H) t ifu,
With the help of this relation we get n(H(u, u)) with an accuracy of first order in f
n(H(u,u))=u* + II* +fu/sw(u* + u*>. Thus w&u)=
(A/N) exp[-n/N-
fu/NGo(n)],
Fig. 3. The dotted line shows the dependence of the population w(n) on the quantum number n in the equilibrium case, when w(n) = we(n), where WG(n) is Gibbs distribution. The pointed line shows w(n), when excitation by an external force is taken into account and 6 w(n) = 6 w(O) = const corresponding to the linear oscillator case. The solid line shows w(n), when both excitation and non-linearity are taken into account. The second maximum of the solid line is due to the sharp decrease of 18 w(n)l in the region n > ngc (see fig. 1).
(16) 2n. This gives
wheren=u*+v*.
6. Discussion
0
In the caseof a linear oscillator the exact solution of (5) is known [lo]. In this casethe steady-state distribution function has the form of (16) irrespective of the value off, if v << 1&w/. The restriction v<< /Sol coincides with (12) because the period T in the case of linear oscillator,equals T = 277/164. Thus the non-linearity of oscillator both in the initial equation (5) and in the steady-state distribution function can be taken into account by replacement of 60 by 6w(n). It is interesting to note that formula (16) in the linear oscillator case can be rewritten in the form of wh 4 = V/N) exp
L-Q@,u)P”l ,
where T’ is the temperature of the heat bath, Q(u, u) is the oscillator quasi-energy averaged over the period of the external force (see ref. [ Ill; in this work the concept of “quasi-energy” was extended from the quantum theory [ 121 to the classical one). Let us obtain the distribution as a function of 12 with the help of (16). We put u = nl’* cos cpand u = +“2 sin cpand integrate w over dq from 0 up to
where
Z(n) =AI,Cfi”*/Nl6w(n)l).
(17)
The function w&r) is’the equilibrium normalized Gibbs distribution. The population at energy E = Aw,n distinguishes from the equilibrium population by a factor I(n). The modified Bessel function 1, increases rapidly, if the argument increases [ 131. Thus, the better condition (1) is fulfilled, the more the excessive population is. Fig. 3 shows schematically the dependence w(n) corresponding to fig. 1. The present calculations confirm our conclusion about the appearance of excessive population in the resonant regions of phase space, which was drawn earlier [ 111 with the help of qualitative reasoning. The statistics of the ensemble of quantum oscillators (under some special assumptions) was investigated in ref. [ 141. It was shown that excessive population appears at levels n > n, when
@‘nr+l - EnrP = 0, .
310
V.N. Sazonov and S. V. Zatsepin / Statistics o f the ensemble o f oscillators
This formula corresponds to the classical condition (1). The dependence coy(n), corresponding to the decrease of 15w(n)l as n increases, is shown in fig. 1. This dependence describes the quasicontinuum in high levels of the energy spectrum of polyatomic molecules. There is a resonant pair of levels for an arbitrary laser frequency in the region of the quasicontinuum (i.e. when n ~> nqc ). According to (17) the quantity of the additional population [factor l(n)] increases and can raech a significant value at a large quantum number n. At the same time l(n) ~ 1 for small n. This shows that tile external force can exert a significant stimulating influence on the chemical reactions with a high activation energy, whereas the reactions with a low activation energy will not be stimulated. Specific features of polyatomic molecules are described in our model by decreasing tSw(n)l, when n becomes larger than nqc. Another way is to take into account the stochasticity of intramolecular motion (see e.g. refs. [ 1 5 - 1 9 ] ) . Eqs. (2), (3) and (5) in our work can describe stochasticity too. It arises if a separatrix exists. Indeed, condition (12) does not surely hold if the trajectory approaches the separatrix, because in this case T ~ oo. Thus, even for an arbitrary small collision frequency v the stochastic layer in the (u, v) plane exists. In this layer the motion has an irregular character. Such a motion is described by a rapid diffusion of the distribution function in the neighbourhood of the separatrix. The validity of our model is restricted, mainly, by two assumptions: (1) Assumption (12) can be rewritten as r > > 15col-1, where 6 co is a characteristic frequency detuning value, r is a characteristic relaxation time of resonant molecules. Even if 16601 = 10 -3 cm -1 , 7"> > 3 × 10 -a s. This condition is carried out for V - T relaxation time in many gas mixtures. (2) The external force F ( t ) in (2) was assumed to be a harmonic one: F ( t ) = F cos Wet. So we may formally consider only the time interval 0 ~< t < < (Acoe) -1 , where Awe is the spectral width of the laser radiation. This restriction is formally very strong: if Awe ~ 3 × 10 -2 cm -1 , then t < < 10 -9 s. At the same time the characteristic length of the laser radiation action in refs. [ 1 - 4 ] was 10 - 2 - 1 0 3 s. But actually formula (17), which expresses the main result of our
work, remains correct, if substitution 16~)(n)f-* max{lfico(n)i, r -1 , Awe} is made. Thus, we can conclude: (1) Laser radiation exerts a considerable excessive population at high levels of the energy spectrum of resonant polyatomic molecules. (2) The excessive population factor I [see (17)] increases when the quantum number n increases and (or) the temperature of the heat bath T = ]Vhcoe decreases. We hope these qualitative conclusions are model independent.
References [ 1] G.N. Basov, E.P. Markin, A.N. Oraevsky and A.V. Pankratov, Dokl. Akad. Nauk SSSR 198 (1971) 1043; G.N. Basov, A.N. Oraevsky and A.V. Pankratov, Kvantovaya Elektron. 3 (1976) 814 [Sov. J. Quantum Electron. 6 (1976) 4431. [2] H.R. Bachmann, R. Rink, H. NiSth and K.L. Kompa, Chem. Phys. Letters 33 (1975) 261; 45 (1977) 169. [3] R.N. Zitter and D. Koster, J. Am. Chem. Soc. 99 (1977) 5491. [4] V. Slezak, J. Caballero, A. Burgos and E. Quel, Chem. Phys. Letters 54 (1978) 105. [5] R.V. Ambartzumian and V.S. Letokhov, in: Chemical and biochemical applications of lasers, Vol. 3, ed. C.B. Moore (Academic Press, New York, 1977). [6] R.V. Khokhlov, Dokl. Akad. Nauk SSSR 97 (1954) 411; Usp. Fiz. Nauk 87 (1965) 17. [7] T.K. Melik-Barkhudarov, Zh. Eksp. Teor. Fiz. 75 (1978) 97. [8] S. Mukamel, Chem. Phys. 37 (1979) 33. [9] V.I. Gorchakov and V.N. Sazonov, Zh. Eksp. Teor. Fiz. 70 (1976)467 (Sov. Phys. JETP 43 (1976) 241). [10] B.Ya. Zel'dovich, A.M. Perelomov and V.S. Popov, Zh. Eksp. Teor. Fiz. 55 (1968) 588; 57 (1969) 196. [ 11 ] V.N. Sazonov and S.V. Zatsepin, Teor. Mat. Fiz. 40 (1979) 311. [12] Ya.B. Zel'dovich, Zh. Eksp. Teor. Fiz. 51 (1966) 1492 [Soy. Phys. JETP 26 (1967) 861]; Usp. Fiz. Nauk 110 (1973) 139 [Usp. 16 (1973) 427]; V.I. Ritus, Zh. Eksp. Teor. Fiz. 51 (1966) 1544 [Sov. Phys. JETP 26 (1967) 1041]. [13] I.S. Gradshtein and I.M. Ryzhik, Tablitsy integralov (Moscow, 1971). [14] V.N. Sazonov, Zh. Eksp. Teor. Fiz. 77 (1979) 1767. [15] E.V. Suryak, Zh. Eksp. Teor. Fiz. 71 (1976) 2039 [Sov. Phys. JETP 44 (1976) 1070]. [16] D.W. Oxtoby and S.A. Rice, J. Chem. Phys. 65 (1976) 1676. [17] D.W. Noid, M.L. Koszykowski, R.A. Marcus and J.D.
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McDonald, Chem. Phys. Letters 51 (1977) 540; D.W. Noid and J.R. Stine, Chem. Phys. Letters 65 (1979) 153. [18] K.D. H~insel, Chem. Phys. Letters 57 (1978) 619; Chem. Phys. 33 (1978) 35; J. Chem. Phys. 70 (1979) 1830. [19] F.M. Israilev, J. Gazati, J. Ford and B.V. Chirikov, Insti-
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