6638
J. Phys. Chem. B 2001, 105, 6638-6641
Quantum Smoluchowski Equation II: The Overdamped Harmonic Oscillator† Philip Pechukas,*,‡ Joachim Ankerhold,§ and Hermann Grabert§ Department of Chemistry, Columbia UniVersity, New York, New York 10027, and Fakulta¨ t fu¨ r Physik, Albert-Ludwigs-UniVersita¨ t Freiburg, Hermann-Herder-Strasse 3, 79104 Freiburg, Germany ReceiVed: January 11, 2001; In Final Form: March 29, 2001
The quantum Smoluchowski equation is an equation for the coordinate-diagonal elements of the density operator, and it is identical to the classical Smoluchowski equation. Here, the leading finite-friction correction to the quantum Smoluchowski equation is derived, for the particular case of the overdamped harmonic oscillator. It is in fact a quantum correction, different from the well-known classical correction, and it dominates the classical correction in the strong friction regime.
Introduction Recently we looked at the strong friction (Smoluchowski) limit of the Caldeira-Leggett model1,2 for quantum Brownian motion; this work will be referred to here as “QSE-I”.3 We found that in this limit the density matrix of the Brownian particle is essentially diagonal in coordinate and the diagonal element F(Q,t): ) 〈Q|Fˆ (t)|Q〉 satisfies the classical Smoluchowski equation. In this sense, the strong friction limit of quantum dissipation is classical dissipation: quantum effects are “hidden” in the momentum P of the Brownian particle, which must have almost infinite dispersion to make the density operator almost diagonal in Q. The leading correction to the Smoluchowski equation, for classical Brownian motion with finite friction, is well-known.4 Here, we determine the leading correction to the quantum Smoluchowski equation of QSE-I, for the particular case of the damped harmonic oscillator. It is in fact a quantum correction, different from the classical, and it dominates the classical correction in the strong friction regime. In essence, the leading correction to QSE-I is that needed to get the equilibrium dispersion of oscillator position Q correct: the quantum dispersion, unlike the dispersion of a classical oscillator, depends on the friction constant γ, varying from the dispersion of the isolated quantum oscillator when γ ) 0 to the dispersion of the classical oscillator as γ f ∞. Our strategy is the same as that in QSE-I. We start from the Caldeira-Leggett Hamiltonian
H ) P2/2M + MΩ2Q2/2 +
∑[pj2/2m + mωj2(qj - Q)2/2]
(1)
We solve the classical (Newton) and quantum (Heisenberg) equations of motion for Q(t), P(t), and {qj(t), pj(t)}; the classical and quantum equations are formally identical, and they are linear, so Q(t) can be expressed as a linear combination of initial values/operators Q, P, and {qj, pj}. We look at the classical problem and discover how the Smoluchowski equation, together
with its leading finite-friction correction, emerges from the requirement
∫eiuQ(t)F(Q,P,{qj, pj},0) dQ dP d{qj,pj} ) ∫eiuQF(Q,P,{qj, pj},t) dQ dP d{qj,pj} ≡ ∫eiuQF(Q,t) dQ
(2)
where we use F to denote both the probability density in the full phase space of Brownian particle plus oscillator bath and the probability density of the Brownian particle position Q alone. We then attempt the same calculation with the corresponding quantum solution for Q(t). Classical Smoluchowski From Hamiltonian (1), we find in standard fashion the generalized Langevin equation:
Q ¨ (t) ) -Ω2Q(t) -
∫0t dτ ∑(mωj2/M) cos[ωj(t - τ)]Q˙ (τ) +
∑[(mωj2/M) cos(ωjt)(qj - Q) + (ωj/M) sin(ωjt)pj]
(3)
by solving the equations of motion for {qj(t)} in terms of the trajectory Q(τ), 0 e τ e t, and substituting in the equation of motion for Q(t). We write Q(t) and P(t) as linear combinations of the initial values Q, P, and {qj,pj}:
Q(t) ) A(t)Q + B(t)P +
∑[aj(t)(qj - Q) + bj(t)pj]
P(t) ) MQ˙ (t) ) MA˙ (t)Q + MB˙ (t)P + ...
(4)
with initial conditions
A(0) ) 1, A˙ (0) ) 0; B(0) ) 0, B˙ (0) ) 1/M aj(0) ) a˘ j(0) ) bj(0) ) b˙ j(0) ) 0
(5)
†
Part of the special issue “Bruce Berne Festschrift”. * To whom correspondence should be addressed. E-mail: pechukas@ chem.columbia.edu. Fax: (212) 932-1289. ‡ Columbia University. § Albert-Ludwigs-Universita ¨ t Freiburg.
We assume strict Ohmic damping
∑(mωj2/M) cos(ωjt) ) 2γδ(t)
10.1021/jp010101z CCC: $20.00 © 2001 American Chemical Society Published on Web 05/03/2001
(6)
Quantum Smoluchowski Equation II
J. Phys. Chem. B, Vol. 105, No. 28, 2001 6639
The equations of motion for A, B, and {aj,bj} are
A¨ (t) ) -Ω2A(t) - γA˙ (t), B¨ (t) ) -Ω2B(t) - γB˙ (t) a¨ j(t) ) -Ω2aj(t) - γa˘ j(t) + (mωj2/M) cos(ωjt) b¨ j(t) ) -Ω2bj(t) - γb˙ j(t) + (ωj/M) sin(ωjt)
(7)
We need solutions to the homogeneous overdamped oscillator equation g¨ + γg˘ + Ω2g ) 0. Single-exponential solutions are exp(-λ1t) and exp(-λ2t) where λi2 - γλi + Ω2 ) 0 implies λ1 + λ2 ) γ and λ1λ2 ) Ω2. We label the exponents so that λ1 , λ2 in the strongly overdamped regime γ . Ω; then, λ1 = Ω2/γ and λ2 = γ - Ω2/γ. We also need the particular solution with g(0) ) 1 and g˘ (0) ) -γ; it is
g(t) ) [λ2 exp(-λ2t) - λ1 exp(-λ1t)]/(λ2 - λ1)
A(t) ) [λ2 exp(-λ1t) - λ1 exp(-λ2t)]/(λ2 - λ1)
∫0 dτ g(t - τ)(mωj/M) sin(ωjτ) t
here 〈...〉 denotes thermal oscillator averages. Then
∫
) [-u S˙ (t)/2]{...} + 2
∫
exp[-u2S(t)/2] iuA˙ (t)QeiuA(t)QF(Q,0) dQ
(9)
∫
) [-u2S˙ (t)/2] eiuQ(t)F(...,0) d... +
∫0t dτ g(t - τ)(ωj/M) sin(ωjτ)
(10)
The second term we evaluate as follows:
∫
∫
iuA˙ (t) QeiuQ(t)F(...,0) d... ) [iuA˙ (t)/A(t)] {A(t)Q +
which follow directly from eq 9 and
∑(mωj/M) sin(ωjt) ) γ
∫
iuA˙ (t) QeiuQ(t)F(...,0) d... (17)
∫0t dτ g(t - τ)(mωj2/M) cos(ωjτ)
b˙ j(t) )
(16)
d/dt {exp[-u2S(t)/2] eiuA(t)QF(Q,0) dQ}
We note the useful relations
for t > 0
(11)
which is just the integral version of eq 6. Now for the Smoluchowski regime, γ . Ω. After a short time of order 1/γ, we have to good approximation
A(t) = λ2 exp(-λ1t)/(λ2 - λ1)
(12)
During this short time, Q suffers a small displacement, roughly P/Mγ, associated with the particular initial velocity of the Brownian particle; as in QSE-I, we ignore this displacement, which amounts to setting B(t) ) 0 (but see the discussion in Appendix B). The approximation for Q(t) that we use in eq 2 is then
Q(t) = λ2 exp(-λ1t)Q/(λ2 - λ1) +
∑[aj(t)(qj - Q) + bj(t)pj]
(13)
In eq 2, we need the initial distribution of the CaldeiraLeggett oscillators {qj,pj} attached to the Brownian particle; we assume that at t ) 0 the oscillators are in thermal equilibrium with respect to the instantaneous position of the Brownian particle, Q:
F(Q,P,{qj,pj},0) ) F(Q,P,0)
∑[aj(t)2〈(qj - Q)2〉 + bj(t)2〈pj2〉] ) ∑[aj(t)2/βmωj2 + mbj(t)2/β]
S(t): )
t
∫0 dτ g(t - τ)(1/M)[1 - cos(ωjτ)]
a˘ j(t) )
where
∫eiuQ ∂F(Q,t)/∂tdQ )
B(t) ) [exp(-λ1t) - exp(-λ2t)]/M(λ2 - λ1)
bj(t) )
∫eiuQF(Q,t) dQ ) ∫eiuQ(t)F(Q,P,{qj,pj},0) dQ dP d{qj,pj} ) ∫exp{iuA(t)Q + iu∑[aj(t)(qj - Q) + bj(t)pj]}F(...,0) d... (15) ) exp[-u2S(t)/2]∫eiuA(t)QF(Q, 0) dQ
(8)
In the formulas below, g(t) always denotes this particular solution. The solutions to eq 7 with initial conditions (5) are
aj(t) )
The initial distribution of P does not matter, for within the approximation of eq 13 P has disappeared from Q(t): the marginal distribution of Q at time t is entirely determined, via eq 2, by the marginal distribution of Q at time 0. By direct integration over the initial oscillator distribution, we calculate
∏Zj-1 exp[-βpj2/2m - βmωj2(qj - Q)2/2]
(14)
∑[aj(t)(qj - Q) + bj(t)pj]}e F(...,0) d... -[iuA˙ (t)/A(t)]∫∑[aj(t)(qj - Q) + bj(t)pj]eiuQ(t)F(...,0) d... ) -iuλ1∫Q(t) eiuQ(t)F(...,0) d... + iuλ1[iuS(t)]∫eiuQ(t)F(...,0) d... ) λ1∫[-iuQ - u2S(t)]eiuQF(Q,t) dQ (18) iuQ(t)
We have used eq 12, so this calculation is valid only after a short time of order 1/γ. From eqs 17 and 18, then, we have
∂F(Q,t)/∂t ) ∂/∂Q {λ1QF(Q,t) + [S˙ (t)/2 + λ1S(t)] ∂F(Q,t)/∂Q} (19) Now the crucial point (see Appendix A): after a short time of order 1/γ, S(t) decays to its ultimate value, S(∞), as exp(-2λ1t)
S(t) = S(∞) + (...)exp(-2λ1t)
(20)
S˙ (t)/2 +λ1S(t) is therefore time-independent and equal to λ1S(∞). Further, from eq 15, we expect that S(∞) should be simply the thermal dispersion of Q, S(∞) ) 〈Q2〉 ) 1/βMΩ2; after all, F(Q,t) should go to thermal equilibrium as t f ∞, while A(t) f 0. That S(∞) ) 1/βMΩ2 can also be verified by direct
6640 J. Phys. Chem. B, Vol. 105, No. 28, 2001
Pechukas et al.
calculation (see Appendix A). Using λ1λ2 ) Ω2, we find that eq 19 becomes
∂F/∂t ) λ2-1 ∂/∂Q [Ω2QF + (1/βM) ∂F/∂Q]
(21)
This differs from the Smoluchowski equation only in time scale: it’s λ2-1 on the right of eq 21 rather than γ-1, so relaxation according to eq 21 is slightly faster than Smoluchowski would predict. Expanding λ2-1 in powers of γ-1, we confirm that the leading correction to Smoluchowski, the γ-3 term, agrees with the known general result.4 Quantum Smoluchowski
∫ d{qj} 〈Q,{qj}|Fˆ (t)|Q,{qj}〉
(22)
For initial conditions, we assume the quantum analogue of eq 14
〈Q|Fˆ (0)|Q〉 ) F(Q,0)Fˆ eq osc(Q)
(23)
Fˆ eq osc(Q)
Here is the thermal density operator for the collection of oscillators, each oscillator being centered at Q. We shall calculate
∑ (Ω2 + νn2 + |νn|γ)-1 n)-∞
∫ dQ eiuQF(Q,t)
(24)
Within the approximation of eq 13, Q ˆ (t) commutes with Q ˆ , so to calculate Tr{exp[iuQ ˆ (t)]Fˆ (0)}, we need only Q-diagonal elements of Fˆ (0). For each Q, the calculation requires a trace over the oscillators, 〈•〉osc: ) Tr[•Fˆ eq osc(Q)]; we have
∑[aj(qˆ j - QIˆ) + bjpˆ j]}〉osc ) exp{-(u2/2)∑[aj2〈(qˆ j - QIˆ)2〉osc + bj2〈pˆ j2〉osc]} ) exp{-(u2/2)∑(βpωj/2) coth(βpωj/2) × 〈exp{iu
where νn ) 2nπ/pβ. The leading finite-friction correction to the “pure” Smoluchowski equation comes from this term; it is a ln(γ)/γ correction
valid for all γ sufficiently large that βpγ . 1, γ/Ω . βpΩ. Concluding Remarks What we have done is a nice exercise in linear analysis, but the damped harmonic oscillator is of limited interest. Much more interesting are finite-friction corrections to the “pure” Smoluchowski equation for anharmonic potentials; here pathintegral techniques are necessary, the guiding principle being still that the Brownian particle density matrix is essentially diagonal in coordinate, and one finds quantum corrections to both the spatial diffusion constant and the potential in which the Brownian particle diffuses.5 Appendix A We first calculate the classical sum S(t), eq 16, using eqs 9-11:
∑[aj(t)a˘ j(t)/βmωj2 + mbj(t)b˙ j(t)β] t t ) (2m/βM2)∑ωj ∫0 dτ ∫0 dτ′ g(t - τ)g(t - τ′) ×
{sin(ωjτ) cos(ωjτ′) + [1 - cos(ωjτ)] sin(ωjτ′)}
) (2/βM)
∑ ∫0 dτ ∫0 dτ′ g(t - τ)g(t - τ′)(mωj/M) sin(ωjτ′) t (A1) ) (2γ/βM)[∫0 dτ g(t - τ)]2 t
t
∫0t dτ g(t - τ) ) [exp(-λ1t) - exp(-λ2t)]/(λ2 - λ1)
The only difference between this calculation and that of the previous section is that classical thermal oscillator averages are replaced by quantum. The quantum versions of eqs 15-19 can be written down immediately, eq 20 remains true, and the quantum version of eq 2l is
ˆ 〉β,γ ∂F/∂Q] ∂F/∂t ) (Ω /λ2) ∂/∂Q [QF + 〈Q
(28)
Because (eq 8)
[aj2/βmωj2 + mbj2/β]} (25)
2
(27)
S˙ (t) ) 2
Tr{exp[iuQ ˆ (t)]Fˆ (0)} ) Tr[exp(iuQ ˆ )Fˆ (t)] )
2
+∞
〈Q ˆ 2〉β,γ ) (1/Mβ)
〈Q ˆ 2〉β,γ = 1/MβΩ2 + (p/πMγ) ln(βpγ/2π)
Equations 3-9 are equally valid as quantum equations; Q, P, and {qj,pj} are the ordinary position and momentum operators for Brownian particle and environmental oscillators. Where necessary for clarity, in this section, we shall put hats on operators, to distinguish them from the corresponding scalars, viz., operator Q ˆ and scalar Q. We shall look at the coordinate-diagonal elements of the Brownian particle density operator
F(Q,t): )
of time scale, relative to the “pure” Smoluchowski equation. More important, in the quantum case, 〈Q ˆ 2〉β,γ depends on both the temperature and the friction constant γ. Here is the formula:2
(26)
where 〈Q ˆ 2〉β,γ is the Brownian oscillator coordinate dispersion at full thermal equilibrium with the Caldeira-Leggett oscillator bath. Equation 26 is the central result of this paper. As in the classical case, eq 26 is valid only on time scales long compared to 1/γ and only in the strongly overdamped regime γ . Ω; it is also necessary (see the remark following eq A11) that γ/Ω . βpΩ. As in the classical case, there is a slight revision
(A2)
we find, integrating (A1) and then dropping transients that decay as exp(-λ2t)
S(t) = [2γ/βM(λ2 - λ1)2][1/2λ1 - 2/(λ1 + λ2) + 1/2λ2 - exp(-2λ1t)/2λ1] (A3) which is of the form promised by eq 20. Then, because λ1λ2 ) Ω2, λ1 + λ2 ) γ, and (λ2 - λ1)2 ) γ2 - 4Ω2, we have
S(∞) ) [2γ/βM(λ2 - λ1)2][1/2λ1 - 2/γ + 1/2λ2] ) 1/βMΩ2 (A4) We now recalculate this result in a different way, a way that can be used as well for the quantum calculation. Define
g˜ t(ω): )
∫0t dτ g(t - τ)eiωτ
(A5)
Quantum Smoluchowski Equation II
J. Phys. Chem. B, Vol. 105, No. 28, 2001 6641
Then
Appendix B
S(t) ) (m/βM2)
∑ ∫0 dτ ∫0 dτ′ g(t - τ)g(t - τ′) × t
t
{sin(ωjτ) sin(wjτ′) + [1 - cos(ωjτ)][1 - cos(ωjτ′)]} ) (m/βM2)
∑ ∫0 dτ ∫0 dτ′ g(t - τ)g(t - τ′){1 - cos(ωjτ) t
t
cos(ωjτ′) + cos[ωj(t - τ′)]}
) (m/βM2)
∑[g˜ t(0) - g˜ t(ωj)][g˜ t(0) - g˜ τ*(ωj)]
(A6)
Dropping transients that decay as exp(-λ2t), we have
g˜ t(ω) = iω exp(iωt)/(λ2 + iω)(λ1 + iω) + λ1 exp(-λ1t)/(λ2 - λ1)(λ1 + iω) (A7) The sum in eq A6 appears to contain a term that decays as exp(-λ1t):
[(m/βM2) exp(-λ1t)/(λ2 - λ1)] ×
∑{[1 - λ1/(λ1 + iωj)]iωj exp(-iωjt)/ (λ2 - iωj)(λ1 - iωj) + c.c.} ) -[(m/βM2) exp(-λ1t)/(λ2 - λ1)] ×
∑[ωj /(λj 2
2
+ ωj )][exp(-iωjt)/(λ2 - iωj) + c.c.] (A8) 2
∑ωj2f(ωj)
In the continuum limit f sum is proportional to the integral
(2γM/mπ)∫∞0
(A9)
∑
|g˜ t(ωj)|2
∑ +∞ f (γ/βMπ) ∫-∞ dω/(λ2 + iω)(λ2 - iω)(λ1 + iω)(λ1 - iω) ) (m/βM 2)
ωj2/(λ2 + iωj)(λ2 - iωj)(λ1 + iωj)(λ1 - iωj)
) γ/βMλ1λ2(λ1 + λ2) ) 1/βMΩ2
(A10)
To make these calculations quantum, we simply insert the factor (βpω/2) coth(βpω/2); equation A9 becomes
∫-∞+∞ dω (βpω/2) coth(βpω/2) exp(-iωt)/ (λ12 + ω2)(λ2 - iω) (A11) which still decays as exp(-λ1t) provided the pole at ω ) -iλ1 is closest to the real axis, i.e., provided λ1 = Ω2/γ e 2π/βp. Equation A10 becomes
S(∞) ) (γ/Mπ)
∫-∞+∞ dω (pω/2) coth(βpω/2)/ |λ2λ1 - ω2 + iω(λ2 + λ1)|2
) (γ/Mπ)
∫-∞+∞ dω (pω/2) coth(βpω/2)/ [(Ω2 - ω2)2 + γ2ω2] (A12)
which is an alternate expression2 for 〈Q ˆ 2〉β,γ.
(B1)
To carry through the (classical) calculation with this additional term, we must specify the initial distribution of P. Suppose that distribution is thermal, centered at P ) 0; the only change in the calculation is that S(t) is augmented by the term MB(t)2/β. This additional term contributes nothing to the combination S˙ (t)/2 + λ1S(t) What if we use a more appropriate initial distribution of P, Maxwellian but centered at P ) -MΩ2Q/γ? The last entry of eq 15 becomes
exp{-(u2/2)[S(t) + MB(t)2/β]}
∫ exp{iu[A(t) -
B(t)MΩ2/γ]Q}F(Q,0) dQ (B2) We differentiate with respect to t and write
which, thanks to the pole at ω ) -iλ1, itself decays as exp(-λ1t). We conclude that the term which appears to decay as exp(-λ1t) in fact decays as exp(-2λ1t). Finally tf∞
B(t) = exp(-λ1t)/M(λ2 - λ1)
dω f(ω) the
∫-∞+∞ dω exp(-iωt)/(λ12 + ω2)(λ2 - iω)
S(∞) ) (m/βM2) lim
In our calculations, we have dropped the term B(t)P from the expression (4) for Q(t). Certainly this is in the spirit of Smoluchowski: the initial momentum of the Brownian particle should not matter, for quickly, in a time of order γ-1, momentum is “slaved” to position, taking on average the value F(Q)/γ ) -MΩ2Q/γ determined by the local force F(Q) on the Brownian particle. However, one may reasonably question discarding this term when one is attempting to calculate corrections to Smoluchowski. Suppose we retain B(t)P to the same accuracy as A(t)Q
∫
iu[A˙ (t) - B˙ (t)MΩ2/γ] QeiuQ(t)F(...,0) d... ) iu[A˙ (t) -
∫
B˙ (t)MΩ2/γ]/[A(t) - B(t)MΩ2/γ] Q(t)eiuQ(t)F(...,0) d... -
∫
iu[A˙ - ...]/[A - ...] {B(t)MΩ2Q/γ + B(t)P +
∑[aj(t)(qj - Q) + bj(t)pj]}eiuQ(t)F(...,0) d...
(B3)
A(t) and B(t) both decay as exp(-λ1t), so the first term is -iuλ1∫Q(t)eiuQ(t)F(...,0) d..., as in eq 18. Doing the integrals over P and {qj,pj} in the second term, we recover -λ1u2[S(t) + MB(t)2/β]∫eiuQ(t)F(...,0) d..., as in eq 18. Again, the only change is S(t) f S(t) + MB(t)2/β, which does not affect the result, eq 21, at all. Acknowledgment. This work was supported by the National Science Foundation (USA), under Grants INT-9726203, CHE9633796, and CHE-0078632, by the Deutscher Akademischer Austauschdienst (Bonn), and by the Deutsche Forschungsgemeinschaft (Bonn) through SFB276. P.P. thanks the Rockefeller Foundation for the opportunity to work on this project at the Foundation’s Bellagio Study Center. We dedicate this work to Bruce Berne, our friend, colleague, and mentor, on the occasion of his 60th birthday. References and Notes (1) Caldeira, A. O.; Leggett, A. J. Ann. Phys. (N.Y.) 1983, 149, 374. (2) Grabert, H.; Schramm, P.; Ingold, G.-L. Phys. Rep. 1988, 168, 115. (3) Pechukas, P.; Ankerhold, J.; Grabert, H. Ann. Phys. (Leipzig) 2000, 9, 794. (4) See, for instance, the review by Ha¨nggi, P.; Talkner, P.; Borkovec, M. ReV. Mod. Phys. 1990, 62, 251. (5) Ankerhold, J.; Pechukas, P.; Grabert, H. Manuscript in preparation.
