Brownian motion at absolute zero

PHYSICAL REVIEW B VOLUME 45, NUMBER 14 1 APRIL 1992-II Brownian motion at absolute zero Supurna Sinha Department of Physics, Syracuse Syracuse, New York 13244 113-0 University, Rafael D. Sorkin Enrico Fermi Institute, 5640 S. Ellis Avenue, Chicago, Illinois, 60637 (Received 5 November 1991) We derive a general quantum formula giving the mean-square displacement of a diffusing particle as a function of time. Near 0 K we find a universal logarithmic behavior (valid for times longer than the relaxation time), and deviations from classical behavior can also be significant at larger values of time and temperature. Our derivation depends neither on the specific composition of the heat bath nor on the strength of the coupling between the bath and the particle. An experimental regime of microseconds and microdegrees Kelvin would elicit the pure logarithmic diffusion. — — The so-called fluctuation-dissipation theorem which relates the thermal fluctuations of a variable x to the response of that variable to a weak external force is the Smoluchowskiusually described as generalizing Einstein relation for Brownian motion, D =kTp; but it is not easy to find in the literature any explicit derivation of this relation as a direct corollary of the theorem. In this paper we will provide such a derivation under the assumption that the times involved are long compared to the relaxation time ~, as defined below. But, because the fluctuation-dissipation theorem is really a quantummechanical relationship, it will tell us something more than just the laws of classical diffusion, which will emerge only in the limit Pi~0, or equivalently in the limit of long times and high temperatures. In the opposite limit where kTht (&A, the usual linear dependence hx will turn out to give way to a universal behavior hx -lnht, which probably should be interpreted as a diffusion driven by quantal zero-point motions rather than by thermal kinetic energy. The logarithmic behavior will follow from a general formula (12) for (hx ), which will hold for all times long compared to ~, given the assumption of constant mobility p. In what follows we will derive this general formula, discuss the limiting cases just alluded to, and show that some deviations from classical behavior may be observable on the basis of current experimental technique. In recent years there have been several efforts' to understand the dynamics of a quantum particle coupled to a heat bath. Insofar as our work overlaps those efforts, our results appear to agree. The main difference is that the cited papers make far-reaching assumptions about the nature of the medium (heat bath) in which the particle moves, and require the coupling between particle and bath to be linear (meaning in eff'ect that the coupling is weak). In contrast, we only use that the response to a weak external perturbation is linear, allowing the coupling of the particle to the bath and/or environment itself to be strong, as it will in fact be in most situations. On the other hand we will predict only the mean-square displacement, whereas the more special treatments can in principle yield the full density operator as a function of ht. THE FLUCTUATION-DISSIPATION THEOREM IN THE TIME DOMAIN The fluctuation-dissipation theorem, as usually stated, refers to the Fourier transforms of the autocorrelation and response functions. Let x(t) be some dynamical variable (operator) in the Heisenberg picture, and let (t) be an infinitely weak external force applied to x at time t. (We will not need the more general form of the theorem in which the external coupling is to a different variable y. } the response function R (t) is defined by the relation f I (x(t) )f —(x )o= R (t s)f (s)ds, — -ht 45 where ( )& denotes the expectation value in the presence of the force, assuming the system of which x is a variable to have been in thermal equilibrium with temperature T at early times; and ( )o is the same expectation value for zero force. Also let C(t}= '(x(t)x(0)+x(0)x(t) ) —, be the "autocorrelation" or "two-point" function in equilibrium at temperature T. [Or, if you prefer, you can sub«a«off (x(t))(x(0)) =(x(0)) from this definition without invalidating what follows. This would be equivalent to working with x —(x ) in place of x. ] Then the fluctuation-dissipation theorem stated in the frequency domain is (with P= I lkT) ImR(v)=iri 'tanh(nPA'v)C(v) [We are using the following form 7=( . . ): . definition of fourier trans- P(v)= Jdtl "P*(t), where 1"=e "'".] Our first job is to transform 8123 1992 this relation to the time The American Physical Society BRIEF REPORTS 8124 domain. To that end, let us introduce in place of R (t) (which vanishes for t by virtue of causality) the equivalent odd function (0 It is then easy to check that 2iImV(R) =V(R ), whence (2) can be written in the equivalent ) 2l =— tanh(iraqi}iv)V(C) . (3) C(v)=( —iA/2)coth(irPA'v)[V(R )](v)+c5(v), (4) where c is a constant and where, for definiteness, the P(cothx) principal part of coth may be taken: =d/dxln sinh~x~. The ambiguity in 1/tanh (irPiilv) is just a term proportional to 5(v), which would drop out of (4) anyway, since it would be multiplying the odd function V(R ). ] The Fourier transform of (4) reads iA V(cothirgfiv) e R 2 V' + c, determining C, up to an additive constant, in terms of the Fourier transform V( cothirPhv ) = ( i IPA') coth( n. t IPA') . (6) In Eq. (6), the coth on the right-hand side is also to be understood as a principal part, but unlike before, this choice is forced on us, because the addition of any 5(t) piece to coth hatt/Pfi would spoil its oddness, in disagreement with the oddness of the left-hand side of (6). Understanding all coth's to be principal parts, then, we have finally (in view also of the definition of R ) the following explicit formula for C (t) in terms of R (t): C(t)= J dt'sgn(t' t)R '(Ax ) =C(0) C— (ht) . (8) —, form: [In fact it is actually this form, rather than (2), that comes out initially in the most straightforward derivation of the fiuctuation-dissipation theorem; it is thus more appropriate to view (2) as a consequence of (3) rather than vice versa. ] By taking the Fourier transform of (3) we could now express R (t) as a convolution of C(t), but our main interest here is to do the opposite. Let us therefore solve (3) for C, obtaining C= J or R(t)=sgn(t)R(~t~) . V(R (bx') = ( [x (b t) —x(0)]'& = (x(bt)'&+ (x(0)') —( [x(At ), x (0) ) =2C(0) —2C(ht), ( ~t' t~ 00 X c toh(m't'IPA')+c Combining this result with (7) gives us a general equation for ( b, x ) in terms of the response function R: '(bx ) = 2P . [The appearance of the undetermined constant c is due to the possibility of redefining the zero of x without afFecting (1). By working with the alternative definition of C(t) mentioned just before Eq. (2), we would remove this ambiguity, and correspondingly could set c =0, given some assumptions on the asymptotic behavior of x and R.] THE MEAN-SQUARE DISPLACEMENT ( bx 2 ) Now the mean-square displacement of x due to equilib6 t is ( b,x ), where rium fluctuations in time hx =x (t+bt) —x (t). Taking t =0 for convenience, we have (since the equilibrium state is time independent) o —cothQ(t'+t) —cothII(t' —t)], set f1 = it/Piii. Here, (9) where for brevity we have as before, the principal part of the coth is to be understood. Notice that the undetermined constant c in (7) has dropped out of this result. QUANTUM BROWNIAN MOTION At this stage, let us specialize x to be a Cartesian coordinate of an otherwise free particle immersed in a homogeneous medium with temperature T. For an idealized inertialess Brownian particle, the response to a weak external force would be immediate motion at velocity R would U =pf, p being the "mobility;" in other words, be the step function R (t)=@8(t) Howe. ver this idealization is plainly too unrealistic, because it leads to a divergent result in (9). [In this sense we might say that the theorem knows that particles have Auctuation-dissipation inertia. ] A more reasonable Ansatz for R must incorporate a "relaxation time" or "rise time" ~ representing the time it takes the particle to accommodate itself to any sudden change in (t). Such an Ansatz is, for example, f R(t)=p(1 —e ' ')8(t), (10) which describes the classical motion of a particle subject to viscous friction. %"ithout making so specific a choice, however, we will employ a cruder cutoff which should be adequate for times much greater than v". )— — dt'R (t')[2 cothQt' —, R (t)=@8(t r) . — With this R, (9) can be integrated exactly [using the disto protributional identity, P(cothx)=d/dxlnsinh~x~] equation of quantum duce the following fundamental Brownian motion: piif &sinhn~t 7T —r~sinhn~t+r~ sinhQ~ (d~ t »~), ) (12) = m. /PR . where again Now strictly speaking, there is the inconsistency in our derivation of (12) that C(t) is ill defined for a particle moving in an unbounded space, because ( x ) in equilibrium would be infinite, and (8) would therefore assume the indeterminate form (b, x ) = ~ —co. To overcome this problem, one could confine the particle in a very 1ong "box" (confining potential), it being intuitively clear that this could alter neither (bx 2) nor R (t) in the limit of an infinitely large such box. 0— BRIEF REPORTS 45 THREE LIMITING CASES OF THE GENERAL FORMULA (12) The possible limiting cases of (12) are determined by the relative magnitudes of the three times r, pi}i, and ht, which we may call, respectively, the relaxation time, the A priori, "quantum time, and the "diffusion time. there would be essentially 3t=6 distinct cases, but since in order to apply (12), we will limit we must have b t ourselves to only three of them. [It is nonetheless instructive to notice that (12) becomes self-contradictory for ht near ~ since it then equates an intrinsically positive expression to a negative right-hand side. This implies that (11) could not be the exact response function for any system, even in principle. More generally, one can derive from (7) and the definition of C(t), a positivity criterion which any putative response function must fulfill in order to be physically viable. We do not know how restrictive this criterion is in practice, but we have checked that the R of (10) yields a mean-square displacement which is non-negative for all times, as one might have expected. ] Case 1: pfi«r«b, t This .is the classical limit, and (12) reduces to the classical relation " " spreading which follows the quantum law (14) up to the time t& =p— fi/2m, and thereafter continues according to the classical law (13), with the second term in (15) remaining forever as a kind of residue of the quantum era. In order for this residue to be significant, we need . p, ht IP (pfi/m)ln. (PR/2nr), or ( ( pfi »r '(M ) =(p/P)ht=pkTht, (13) —, or p, kT(b, t —r) if the leading correction is retained. Case 2: r«b, t «pR. This is the extreme quantum limit, in which (time) (energy) fi for the time scale set « by the diffusion time ht and the energy scale set by the thermal energy kT. In this limit (12) reduces to (14) or 1/2 2 if somewhat more precision is desired. It is noteworthy that the temperature has disappeared entirely from this expression (except insofar as it influences p and r), suggesting a quantum Brownian motion due entirely to "zero-point" fluctuations, which are present even at absolute zero. Indeed, the striking logarithmic dependence in (14) could also have been derived by first taking the zerotemperature limit of the fluctuation-dissipation theorem itself, and only then applying it to the R of a diffusing particle. Intermediate between cases 1 Case 3: r«pfi«At. and 2, this situation might be described as one in which the relaxation occurs on quantum time-scales, although the diffusion time itself is already classically long. [A suggestive way to rewrite the inequality r «pfi is as the relation between diffusion constants, D,&,», „& &&DqzzztU~& from the "viscous damping" relation (10).] In this case (12) reduces to (b, x ) Pts. t Pfi 2 p ir which one can interpret 1 ~= pm Pfi 2m'r as the result envisaged in (15) of a two-stage 8125 pfi/m. 2 which can occur nontrivially (i.e., without reducing to big, so that case 2) only if pirt/r is exponentially ln(pfi/r) &b, t/pfi»1. Taking m =r/p as earlier, this amounts to a requirement that the particle be extremely light: m (16) ((pA/p)e REMARKS AND NUMERICAL ESTIMATES Equations (13), (14), and (15) are all special cases of the more general relation (12), which should be valid whenever z((ht. In other situations, or for more general response functions R (t), one must refer back to (9) itself, from which the spreading can always be computed as interesting A particularly long as R (t) is known. response function to treat would be (10), and another interesting case might be a particle moving in a superfluid. In the zero-temperature limit, i.e., in case 2 above, our formula (14) may be compared with a result of Ambegaokar, who used a path-integral formalism, and assumed a linear coupling between the particle and an environment comprising an infinite collection of harmonic oscillators. He obtained an expression for the mean-square displacement of a Brownian particle in the quantum regime which corresponds to our result given in (14), if we make certain identifications. According to Ambegaokar (with a presumed misprint corrected), ((bx )) =[(h/m. )/ym]ln~(toto, y)~+const. , (17) for (1/y) &t &(piri). Here, (hx ) is given by a density operator p which reduces to a 5 function at t =0, and co, defines an upper frequency cutoff beyond which the linear relationship between particle velocity and environmental friction breaks down. Also, judging from Eq. (5.3} of Ref. 4, it appears natural to identify 1/y with our ~, and therefore 1/my with our p. If we do so, and also equate co, to r ', then we recover (14} from (17} with the constant set to zero. In connection with (14) one can ask the following question: Classically, what kind of response function would lead to a logarithmic law of diffusion? If we take the A~O limit of (9), we find that the relevant response function should be proportional to 1/t, which is physically impossible. This implies that the effect described by (14) is of purely quantum-mechanical origin. Finally, let us estimate the thresholds of time and temperature at which significant deviations from classical be- 8126 BRIEF REPORTS havior should appear. In order to be in the "pure quantum regime, we need At ((ptrt, which can also be written in the time-energy form, kTht «A. Taking sec yields kTAtlR-0. 1, deg (cf Ref 6) and At —10 which ought to be well within the "pure quantum regime, meaning that (14) should apply if the relaxation time is short enough (and the "reservoir" in thermal equilibrium). For higher temperatures or longer times, deviations of the sort described by (15) might be observable if r is small enough and a condition like (16) is satisfied. ACKNOWLEDGMENTS " T-10 " 'Permanent address: Department of Physics, Syracuse University, Syracuse N. Y. 13244-1130. A. O. Caldeira and A. J. Leggett, Physica A 121, 587 (1983). H. Grabert and P. Talkner, Phys. Rev. Lett. 50, 1335 (1983). G. W. Ford, J. T. Lewis, and R. F. O'Connel, Phys. Rev. A 37, 4419 (1988). ~V. Ambegaokar, in Frontiers of Nonequilibrium Statistical 45 We would like to thank M. Cristina Marchetti for useful discussions and for making us aware of Ref. 3 and 4. Also R.D.S. would like to thank Lochlainn O'Raifeartaigh and Siddhartha Sen for their hospitality at the Dublin Institute for Advanced Studies, where parts of this paper were written. This research was partly supported by NSF grants No. PHY 9005790, No. INT 8814944, No. PHY 8918388, and No. DMR-87-17337. Physics, edited by Gerald T. Moore and Marian O. Scully (Plenum, New York, 1986), pp. 231-239. R. Balescu, Equilibrium and Nonequilibri um Statistical Mechanics (Wiley, New York, 1975), pp. 663 —669. P. D. Lett, W. D. Phillips, S. L. Rolston, C. E. Tanner, R. N. Watts, and C. I. Westbrook. J. Opt. Soc. Am. B 6, 2084 (1989).