J. Phys. Chem. B 2005, 109, 21293-21295
21293
How Good Is Langevin† N. G. van Kampen Institute for Theoretical Physics of the UniVersity Utrecht, Netherlands ReceiVed: May 12, 2005
The Langevin equation is universally applied to describe noise in physical systems of all kinds. To derive it, rather than postulate it, one has to introduce a noise source such as an environment or “reservoir”. The solution requires that at some initial time the reservoir is in equilibrium and uncorrelated with the system. This is unphysical and, moreover, in the quantum mechanical case leads to inconsistencies. The way to overcome this anomaly is to consider the equilibrium of system and noise source combined. Then the resulting timedependent equations for the correlation of various operators can be given. To find out whether they can be reproduced by a Langevin equation, the result is applied to the explicitly solvable system of a harmonic oscillator in a harmonic bath. Conclusion: Langevin cannot describe quantum noise but belongs to the realm of mesoscopic physics.
1. Introduction
2. Standard Theory
In 1908, Langevin published an equation for the treatment of a Brownian particle in a fluid. The velocity of the particle was supposed to obey
In this section we describe the customary way of deriving the equations from basic physics. The system S interacts with an outside bath B, both quantum mechanical of course. Together they constitute a closed total system described by a total density matrix FT obeying the evolution equation
dV -γV + ξ(t) dt
(1)
(d/dt) FT ) -i [HT, FT],
where the first term is friction and the second term is due to the collisions of the surrounding fluid molecules, described quite reasonably by a rapidly varying force ξ(t). The details of its behavior would be known only if the microscopic equations of all molecules could be solved, but the redeeming feature is that this is unnecessary. When the Brownian particle is heavy, all that matters are averages over small time intervals ∆t, and some simple assumptions could be made about those. Later the theory was formalized mathematically by describing the force as a stochastic function, assumed Gaussian and having the second moment1
〈ξ(t)ξ(t′)〉 ) 2Dδ(t - t′)
(2)
The success was tremendous; it provided a way of introducing both fluctuations and irreversibility in a simple way. It invited the application to all kinds of problems, not only in physics but also in biology and even economics. All one has to do is take a familiar macroscopic equation and merely add a “Langevin force,” characterized by eq 2 and accompanied by a term for the corresponding dissipation. The question whether the underlying physics of the system justifies these assumptions is regarded as nit-picking. It is usually answered by a verbal appeal to interaction with a surrounding world. A more conscientious treatment introduces the surroundings in the shape of a reservoir or bath, which is then endowed with the required properties, in particular those needed for irreversiblity.2 The question whether this reservoir is not itself a physical system and therefore also subject to the fundamental property of reversibility is answered by an appeal to the entire universe. †
Part of the special issue “Irwin Oppenheim Festschrift”.
HT ) HS + HB + λHI
(3)
where λ measures the strength of the interaction between system and bath. This equation is formally solved for a given initial FT at time t ) 0 by
FT(t) ) e-itHT FT(0) eitHT
(4)
From this the density matrix of the system itself is obtained by taking the trace over the bath: FS(t) ) TrBFT(t). The time dependence of FS(t) does not obey a differential equation by itself and therefore one cannot find it without solving the total evolution eq 4. Also, it is not sufficient to choose an initial state FS(0) of S since one needs to know an initial state FT(0) of the total system. In the standard treatment, these problems are met by assuming that the bath B is near equilibrium at all times. If it were exactly in equilibrium its effect on the motion of system S would average out. The presence of S, however, modifies the state of B to order λ. This modification acts back on S, with the result that S is affected to order λ2. The initial-value problem is dismissed by assuming that initially the bath is in equilibrium and uncorrelated with the system,
FT(0) ) FS(0) × Feq B
(5)
To find an approximate solution of eq 4 with this initial condition, one assumes that the precise nature of the bath is not essential and that it may be taken to consist of a set of harmonic oscillators, which makes it posible to determine explicitly the operator e-itHB. Also, the eigenfrequencies of the harmonic oscillators must form a sufficiently dense set on the positive real axis. One may then expand e-itHT to second order
10.1021/jp0581633 CCC: $30.25 © 2005 American Chemical Society Published on Web 09/28/2005
21294 J. Phys. Chem. B, Vol. 109, No. 45, 2005
van Kampen
in λ and take the trace over FB so as to obtain FS for small time ∆t. Subsequently, one interprets [FS(∆t) - FS(0)]/∆t as a derivative (d/dt)FS(t). The result is
d F (t) ) -i[HS,FS] dt S λ2
S
B
S
B
3. Objections Unfortunately eq 6 is unacceptable as an evolution equation for the density matrix FS(t) because it does not obey the criterion for the density matrix to remain positive definite in the course of time.4 This flaw is caused by the artificial condition eq 5.5 There is no valid reason for this condition: nobody believes that it is necessary for understanding Brownian motion to assume that at some time in the past the Brownian particle was uncorrelated to the surrounding fluid. One is free to postulate this condition at one time t ) 0 and compute the evolution of FS during the first interval ∆t, but in the course of time a correlation between S and B builds up and the same formula, eq 6, no longer holds. The conclusion is that FS(t) does not really obey a differential equation in time, it does not constitute a semigroup with eq 6 as infinitesimal generator. Hence there does not exist a Langevin eq 1 with delta-correlated force (eq 2). Approach6
This is our reason for eliminating assumption of eq 5, the culprit of these difficulties. Brownian motion, rather than starting at one initial moment, is a fluctuation phenomenon taking place while the total system is in equilibrium. We therefore start from the total equilibrium distribution -1 -βHT e , Z ) TrT e-βHT Feq T )Z
(7)
One now has to compute fluctuations and their correlations occurring in this equilibrium state. When G,F are two operators acting on the system, the correlation between their values at two times t1,t2 is
〈G(t1)F(t2)〉eq ) Z-1TrTe-βHT eit1HTG e-i(t1-t2)HTF e-it2HT (8) This can be computed to second order in λ. First, expansion of Z yields the identity
∫0
β
dβ′
∫0
β′
∫0β dβ′ ∫0β′ dβ′′ 〈HI(-iβ′)HI (-iβ′′)〉eq0 }
∫0τ dt′ ∫0t′ dt′′ 〈G[HI(t′′), [HI(t′), F(τ)]]〉eq0
(6)
Although the upper limit of integration is ∆t, one usually argues that the integrand is rapidly varying so that the integral converges in an even shorter time, and therefore this upper limit may be taken +∞, when ∆t is not too small. For instance, in the case of a Brownian particle ∆t should cover a number of successive individual collisions, but on the other hand it must still be small enough that the velocity of the heavy particle is barely changed.3
Z ) Z0 {1 - λ2
eq 〈G(0)F(τ)〉eq λ ) 〈G(0)F(τ)〉0 {1 -
λ2
∫0∆t dt′ TrB{[HI, [e-it′[H +H ]HIeit′(H +H ), FeqΒ ]]} FS(t)
4. The New
ref 6, it depends of course only on the time difference t2 - t1 ) τ,
- λ2 + λ2
∫0β dβ′∫0β′ dβ′′ 〈HI (-iβ′)HI (-iβ′′)GF(τ)〉eq0
- iλ2
∫0β dβ′ ∫0τ dt′ 〈HI(-iβ′)G[HI(t′), F(τ)]〉eq0
(11)
Comparison with the standard result shows the following differences. All our integrals over β are new. They disappear for low β, high kT, see section 6. This relates to the fact that rapid fluctuations are cut off by the Planck distribution and therefore eq 2 cannot be correct since a delta function requires all frequencies. The only remaining integral is the one on the second line; it represents the standard result, except for the upper limit in the interior integral, which is t′ rather than ∆t ) ∞. The question about the initial condition and the lack of positivity no longer comes up. 5. A Soluble Model This general result is correct (to order λ2) but not very informative. Let us therefore consider a model that can be treated explicitly, namely a harmonic oscillator in a bath of harmonic oscillators, with a linearized interaction7
HT ) 1/2(P02 + Ω02Q02) + 1/2
∑k {Pk2 + k2(Qk + kQ0)2}
(12)
This may be regarded as a simplified description of a harmonically bound charged particle in a electromagnetic field. We write the same Hamiltonian in the slightly simpler form
HT ) 1/2(P02 + Ω2Q02) + 1/2
∑k (Pk2 + k2Qk2) + Q0 ∑ λkQk k
(13)
where
λk ) k2k
and
Ω2 ) Ω02 + Σk2k2 ) Ω02+ Σλk2/k2
This implies the condition Ω2 > Σλk2/k2 in order that HT be positive. The quadratic form in the Q can be diagonalized with an orthogonal transformation X,
Qk )
∑V XkVqV Q0 ) ∑V X0VqV
(14)
The eigenvalues ωV, are the zeros of the function
dβ′′〈HI (-iβ′)HI (-iβ′′)〉eq 0 + O(λ3)} (9)
G(ω) ) ω - Ω 2
where use is made of the notation
HI(-iβ′) ) e-β′(HS+HB)HIeβ′(HS+HB)
(10)
The subscript 0 indicates the case λ ) 0. In the same way, one may expand the entire expression of eq 8; the result is given in
2
∑k
λk2 ω2-k2
(15)
When the same transformation is applied to the P as well, one gets
H T ) 1 /2
∑V (pV2 + ωV2qV2)
(16)
How Good is Langevin?
J. Phys. Chem. B, Vol. 109, No. 45, 2005 21295
Accordingly the equilibrium of the combined system is characterized by
βωV ωV 〈pVpµ〉eq ) δVµ coth 2 2 〈qVqµ〉 ) δVµ
βωV 1 coth 2ωV 2
where ξ(t) obeys eq 2. The standard calculation1 yields for τ > 0
(17)
In the classical limit, coth(βωV/2) ) 2/βωV and 〈pVqµ〉 ) 0. It is now possible to evaluate correlations such as eq 11. For instance
)
〈(
(
βωV
1
)〉
sinωµτ
X0VX0µ qV qµcosωµτ + pµ ∑ Vµ
∑V X0V2 2ω
coth V
2
The classical Langevin equation for our harmonic oscillator is, with so far undetermined damping coefficient γ,
Q4 0(t) ) P0(t), P4 0(t) ) -γP0(t) - Ω2Q0(t) + ξ(t) (24)
〈pVqµ〉 ) -1/2iδVu
〈Q0(0)Q0(τ)〉 )
7. Comparison with Langevin
ωµ
cos ωVτ +
)
i sinωµτ 2
ωµ (18)
6. Inevitable Approximations The sum may be written as a path integral
〈Q0(t)Q0(t + τ)〉 )
{
}
D -γτ/2 γ e cosΩ1τ + sin Ω1τ 2 2Ω γΩ 1
(25)
where Ω21 ) Ω2 - (γ/2)2. Comparison with eq 22 shows γ ) 2Γ, and Ω1 is the same as above. Furthermore, agreement with eq 25 is obtained if, in eq 22, one sets coth 1/2βΩ1 ≈ 2/βΩ1 and D/γΩ2 ) 1/βΩ12, which is close enough to the classical fluctuation-dissipation theorem D ) γ/β ) γkT. Numerous efforts have been made to modify the Langevin equation so as to take quantum mechanics into account9 with various results. The second term in eq 22, however, is imaginary and hence of purely quantum mechanical origin; cf. eq 17. This is often overcome by the ad hoc excuse that for comparing with the classical case one should look at the symmetrized expression:
/2{〈Q0(0)Q0(τ)〉 + 〈Q0(τ)Q0(0)〉} ) 1/(βΩ12)e-Γτ cos Ω1τ (26)
1
1 〈Q0(0)Q0(τ)〉 ) 2πi
I
dω {coth(1/2βω) cos ωτ + i sin ωτ} G(ω) (19)
with the path extending over a loop around the positive real axis in the complex ω-plane. This expression is exact, but for Brownian motion it is necessary that the bath frequencies k lie dense. Let g(k)∆k be the number of them in a small interval ∆k, each multiplied with its strength λk2. Then the values of G along the north and south shores of the loop are
G(ω ( i) ) ω2 - Ω2
∫
∞
g(k) dk
0
Ω -k 2
2
( iπ
g(Ω) 2Ω
(20)
In the last two terms ω is replaced by Ω; this is justified when λ is so small that these terms are of no importance unless ω2 ≈ Ω2. The integral is a principal value; it amounts to a shift of the resonance frequency Ω2 but will here be neglected. As a consequence one may write
G(ω ( i) ) ω2 - (Ω ( iΓ)2 ) ω2 - Ω12 - 2iΩΓ (21) where Γ ) πg(Ω)/4Ω2 is of the order λ2 and Ω12 ) Ω2 - Γ2. The two integrations in eq 19 may now be carried out by shifting the paths in such a way that only the contributions of the two poles in eq 21 contribute:
〈Q0(0)Q0(τ)〉 ) 1/(2Ω1)e-Γτ{coth 1/2βΩ1 cos Ω1τ + i sin Ω1τ} (22) Thus only the bath oscillators in near resonance with the system contribute. The coth in eq 19 has poles (i/β, which give contributions e-kT/pΩ; they must be omitted if Langevin is to be obtained, and therefore β must be small, kT .pΩ, as we saw already in eq 4. Similar expressions can be obtained8 for
〈Q0(0)P0(τ)〉, 〈P0(0)Q0(τ)〉, 〈P0(0)P0(τ)〉
(23)
This is in fact the same as the first term of eq 22. The second term in eq 25, however, is not reproduced by our exact result (eq 22) for the equilibrium correlation. The same discrepancy occurs in the other correlations mentioned in eq 23. My conclusion is that the Langevin approach is essentially classical since it deals with probabilities. More precisely, it is mesoscopic, in the original sense of that word: it describes quantum physics on the coarse-grained level, on which interference of complex probability amplitudes is negligible and only probabilities are relevant. Therefore, it is unable to reproduce the quantum mechanical expressions for the correlations in equilibrium. References and Notes (1) Uhlenbeck, G. E.; Ornstein, L. S. Phys. ReV. 1930, 36, 823. Chandrasekhar, S. ReV. Mod. Phys. 1943, 15, 1. Wang, M. C.; Uhlenbeck, G. E. ReV. Mod. Phys. 1945, 17, 323. These articles are reprinted in Selected Papers on Noise and Stochastic Processes; Wax, N., Ed.; Dover: New York, 1954. (2) Senitzky, I. R. Phys ReV. 1960, 119, 670. Łuczka, J. Physica A 1990, 167, 919. (3) van Kampen, N. G.; Oppenheim, I. Physica A 1986, 138, 231. van Kampen, N. G.; Oppenheim, I. J. Stat. Phys. 1997, 87, 1325. Oppenheim, I.; Romero-Rochin, V. Physica A 1987, 147, 184. (4) Kossakowski, A. Bull. Acad. Pol. Sci., Se´ r. Math. Astr. Phys. 1971, 20, 1021. Kossakowski, A. Rep. Math. Phys. 1972, 3, 247. Lindblad, G. Comm. Math. Phys. 1975, 40, 147. (5) Sua`rez, A.; Silbey, R.; Oppenheim, I. J. Chem. Phys. 1992, 97, 5101. (6) van Kampen, N. G. Fluct. Noise Lett. 2001, 1, C7-C13. Zero temperature has been considered by Jordan, A. N.; Bu¨ttiker, M. Phys. ReV. Lett. 2005, 92, 247901. (7) Ullersma, P. Physica 1966, 32, 27, 56, 7, 4, 90. Grabert, H.; Weiss, U.; Talkner, P. Z. Phys. B 1984, 55, 87. Haake, F.; Reibold, R. Phys. ReV A 1985, 32, 2462. (8) van Kampen, N. G. J. Stat. Phys. 2004, 115, 1057. (9) Benguria, R.; Kacˇ, M. Phys. ReV. Lett. 1981, 46. Gardiner, C. W. IBM J. Res. DeV. 1988, 32, 127. Kleinert, H.; Shabanov, S. V. Phys. ReV. Lett. A 1995, 200, 224. Shikritov, P. et al.; Semicond. Sci. Technol. 2004, 19, S232.
