On Brownian Movement and irreversibility

MacDonald,

Physica

D.K.C.

28

409-416

1962

ON BROWNIAN

MOVEMENT

AND IRREVERSIBILITY

by D. K. C. MACDONALD Division

of Pure

Physics

National

Research

Council,

Ottawa,

Canada

synopsis The

distinction

examining reversal.

between

whether We consider

spontaneous

here instead

fluctuations

are rather

when

hf > kT.

viscosity, time

intervals,

in terms

In particular

the physical

Movement

fundamental

friction,

irreversible systems is usually made by are invariant or otherwise under time

a criterion

to consider

of the Brownian

there

and

of motion”

of the time-dependence

of the

of the system.

This leads us subsequently modifications

reversible

the “equations

difficulties we suggest

etc. in a thermal

formulae, in dealing that

significance with

simple

an irreversible

environment

of the usual quantum

and we are led to suggest

can only

irreversible

description

be valid

that

systems

in terms

of

over long enough

t, such that at least t > h/kT.

1. Introduction. The problem of irreversibility appears to be of considerable interest at the present time. It has been remarked that there is more than one problem of irreversibility. I wish to consider in this note some aspects of irreversibility in relation to the spontaneous fluctuations of a system, and to enquire about the significance of quantum modifications to the theoretical expressions for Brownian movement of electricity. A familiar problem in the analysis of irreversibility is to ask specifically how we can arrive at the Boltzmann equation, irreversible in time, for the distribution function of a system from the starting point of mechanical (and reversible) equations. From this point of view a model irreversible system will be ideally one where, in a steady state, the rate of displacement (let us say the “flow” or “current”) of some coordinate (e.g. displacement, x, or electric charge, 9) is at any instant proportional to the applied “fol .P_” (e.g. Newtonian force or electric potential difference*). A small particle of negligible inertia moving at constant speed under an applied force in a homogeneous fluid medium, or an electrical resistance where at all times the current flow is proportional to the potential difference, would thus be examples of idealized irreversible systems. On the other hand a Newtonian particle subject only to an inertial or purely elastic force, or purely “reactive” electrical elements such as pure inductance or capacitance, would be ex__~ *) The system will be linear and irreversible if the “flow” is directly proportional to the “force”. We shall consider

only sucl. systems

here.

-

409

-

410

D.

K.

C. MACDONALD

amples of perfectly reversible systems. It is then usually suggested that the necessary distinction between reversible and irreversible behaviour may readily be made by examining whether the equations of motion are invariant or otherwise under “time reversal”. Here I wish rather to consider the Brownian

movement

of certain

simple

systems in relation to reversibility and also to link this tentatively with the concept of predictability, and to ask one or two questions about the significance of quantum modifications to the theoretical expressions for Brownian movement. 2. Analysis Einsteini) 2) showed that if we observe the motion small rigid Brownian particle of negligible mass, under no elastic straint in a medium, then <{x(t) -

X(O)}2) = 2KTB&

of a con-

(1)

where x(t) is the displacement of the particle at time t along some chosen coordinate axis, B is the mobility of the particle in the medium, i.e. the mean velocity acquired under a unit applied force; and the outer brackets signify an ensemble average. If now, however, the particle under observation has appreciable inertia equation (1) is no longer adequate. If the mass of the Brownian particle be m, then as various workers (cf. Langevina), Ornsteina) Fiirth5) have shown, we should now write more correctly <(z(t) -

~(0))~) = 2kTB{t

-

mB(1

-

eC’lmB)}.

(2)

Thus if t > mB <{x(t) as in equation

.(0)}2>

M 2kTBL,

(I), but on the other hand if t < mB then

<{x(t) - x(O))2)

5%

;

t2.

(3)

If a particle with displacement, x(O), and velocity, va, at t = 0 moves mechanically and predictably for a short *) interval, t, then x(t) = x(0) + vat. Thus for an ensemble moving reversibly

of such particles <{x(t) -

which agrees with equation *) i.e. short

enough

ttat

@))a>

with different = t2,

(4) values of va, but each (5)

(3) if we choose (u$> = kT/m, i.e. the so-called

any acceleration

effects

may be ignored.

ON BROWNIAN

MOVEMENT

AND

IRREVERSIBILITY

.

411

value. It is thus natural to say that for short enough time “equipartition” intervals such that t < mB, the “random” influence of the environment is insufficient to disturb the “mechanical” and predictable behaviour of a particle. On the other hand for sufficiently long time intervals such that t > mB, the influence of the fluid environment at temperature T becomes dominant and the motion becomes irreversible as typified by equation (1). Moreover, if equation (1) were valid without restriction for all time intervals, t, however short (i.e. the idealized limit of Brownian movement), then indeed the individual movement of a particle becomes quite unpredictable, as is evident if we rewrite equation (1) in the form p=p, t2

2kTB t

(14

where Ax sz x(t) - x(O). The rate of change of displacement of a system (given essentially by the square root of the right hand side of equation (la)) gets greater and greater as the time interval of observation, t, is made smaller (always assuming equation (1) to hold good). This evidently frustrates any attempt at detailed prediction of the motion of a particle. This consequence of the limiting Brownian movement equation also follows naturally if we interpret the movement as a “random walk” process where the length of each “step” becomes extremely small and at the same time the frequency of “steps” becomes very high, the direction of any particular “step” being quite uncorrelated with that of any previous “step”, or with the present position of the particle, being governed solely by statistical, or probability, considerations. Consequently I would say that from the standpoint of spontaneous fluctuations the mean square “displacement” of a system over a “sufficiently short” *) interval of time would lie between two extremes, namely (AS)

(4

N t2,

in which case the motion of the system over the appropriate t, is reversible and predictable; and N t

(b)

(6) time interval, (7)

when the motion over the time interval, t, is essentially irreversible, “nonmechanical”, and unpredictable for a single system. 2.1. Electrical fluctuations. Einstein (Zoc. cd) himself generalized equation (1) and applied it to the case of electricity in a conductor, writing <(q(t) -

q(0)}2> = 2kKt,

(8)

*)

We are assuming for convenience, as mentioned at the start of section 2, that we can ignore the influence of any elastic forces on the particle which would otherwise introduce a further “mechanical” time-constant be discussed pacitance.

which we would have to take into account. Correspondingly in the electrical case to below we are, again for convenience, assuming that we can ignore the effects of ca-

412

D. K. C. MACDONALD

where q(t) is the charge which has passed any cross-section of the conductor and G is the conductance (which for an ideal resistance is equal to the inverse of the resistance, R, of the conductor concerned). In passing let me remark that, in my opinion, the direct transition from equation (1) to equation (8) is not quite so straightforward as might appear *) In particular it is not, I think, immediately obvious that we may treat the electric charge in a conductor adequately as a single variable corresponding to the displacement of the rigid Brownian particle. It implies that we have satisfied ourselves that we are entitled to ignore the “grainy” structure of electricity itself (i.e. the electronic charge) in discussing these Brownian movement fluctuations. Nontheless I shall accept here equation (8) as a valid generalization of equation (1).

I Fig. 1. Electrical

I

circuit for discussion

of spontaneous

fluctuations.

If we now have to deal with the electrical fluctuations in the circuit of fig. 1 then, corresponding to the inclusion of mechanical inertia for the Brownian particle, we may write <{q(t) -

q(0)}2)= 2kTG{t

-

LG(1 -

e-‘jLo)},

in place of equation (8); L is the self-inductance (“magnetic inertia”) of the circuit. Once again when the time interval, t, is long enough so that t > LG we regain equation (8), and we would say that the charge fluctuations in any particular circuit are essentially irreversible and unpredictable over such a time interval. On the other hand, if the time interval is sufficiently short, so that t < LG, we have

which I would interpret as essentially predictable, and reversible, circuit, corresponding to q(t) M q(0) + iot for an individual equilibrium ensemble average for the current i is given by

behaviour since the

= kT/L. Thus, from this point of view, I would say that the idealized electrical resistance R(= l/G) a 1one, gives rise to entirely irreversible behaviour (cf. equation (8)), while the inclusion of the self-inductance, L, accounts for the element of reversible and predictable behaviour, analogously to the effect of mechanical inertia on the motion of a Brownian particle. f) with

I would like here to express Professor

Einstein

my very sincere

on this topic some years

appreciation ago.

for correspondence

and discussion

ON BROWNIAN

2.2 Quantum approached the

MOVEMENT

AND

modification of problem of Brownian

413

IRREVERSIBILITY

fluctuations. Now Nyquistc) movement of electricity from an

alternative point of view. By considering the thermodynamic equilibrium of two electrical resistances connected to one another through a suitable frequency-filter (“lossless transmission line”) he derived the result = 4kTG df,

(11)

where = 4G{hf/(ehfikT - 1)) df, (12) *) which reduces to equation (11) when hf < kT **). As I have remarked, equation (11) for the frequency spectrum of the current fluctuations is *)

term but other authors [e.g. Callen himself did not include any “zero-point” 6); cf. also Weber 9) 19) 1’)) would add a term hf/2 inside the curly brackets of equation

(1) Nyquist

and Welton

(12). I am doubtful about the physical significance of such a zeropoint energy term in the expression for measurable spontaneous fluctuations; taken at face value it would seem to me to suggest that one might perhaps be able, in principle, zero, which is surely inadmissible.

to extract power via the fluctuations from a system at absolute Let me also remark that I am unaware of any enfierimental

investigation

of equation

that

to date of the validity

with or without

(2) Professor

Van

(12) for Bromnian

the “zero-point”

term equation

Hove

the

criticises

movemertt proper;

(12) reduces

assumption

to equation

of a purely

constant

we should

note

(11) when h,f < G in equation

kT. (12),

citing the case of impurity resistance with relaxation time 7 in a metal at very low temperatures (kT < h/7). In that case he argues that the left hand side of equation (12) depends little on f so long as 277f < 7-1, and thus in particular for f N kT/h. Hence on the right hand side of equation (12) G must depend strongly on f in this region. I am most grateful to Professor Van Hove tar kind and patient point,

but I have not yet been able to convince

in terms

of an idealized

resistance,

R, supposed

**) Presumably the corresponding velocity fluctuations of a Brownian

myself

independent

Brownian

movement,

although

correspondence

this invalidates of frequency

on this particular

the general in the first

discussion place.

relation to equation (12) for the frequency spectrum particle of entirely negligible mass would read: (vfs> = 4B{lf/(ehflkT

As far as I know, no such relation

that

-

I)} df.

has been made use of in connection the corresponding (vf2)

relation

to equation

of the (12al

with the analysis

of mechanical

(1 l), namely:

= 4BkT df

(lln)

has been used elsewhere by the writer (MacDonald 12)) to analyse the classical Brownian movement problem under visual observation, taking account of the limited frequency-response of the eye. Equation (12~) taken at its face value would then imply, as we shall see below, that fort < h/kT the motion of such a particle would always become quasi-mechanical ((dxz) N ts). Formally, at any rate, this would satisfy a requirement pointed out by Vineyard 13) in discussing atomic motion in liquids. However, in any real liquid the influence of inertial and elastic forces (cf. e.g. equation (3) above) would doubtless supersede equation I am very grateful to Dr R. G. Chambers

(12~) in causing the transition from (dxz) for drawing my attention to Dr Vineyard’s

t

to - ts. article.

414

D.

K.

C. MACDONALD

precisely equivalent to equation (8) for the time-like behaviour of the charge fluctuations, and correspondingly we can now derive the time-like equation for the charge fluctuations directly corresponding to Nyquist’s quantum “noise” relation (equation (12)). The appropriate transform (cf. MacD onald7)

states

that in general

(13) Hence in this case

(+I@)- dW) =

2hG os

-$r

W( 1 - cos 2nft) df - f(eh’!“’ _ 1) -.

= _!F . log,{ 77;2 This function

is plotted

si;(f;;Th) ,z

(14a) *)

).

(146) **)

in figure 2.

t/h / kT Fig.

2:

plotted

(1 /ZGkTt)

times the expression

as a function

convenience (cf. equation

arising on the right hand side of equation

of t/(h/kT). We have normalized

in plotting

so that

as

the expression

t --f 00 the function

as plotted

(14b),

in this way for tends

to unity

(15)).

For t > A/kT equation

(14) yields <{q(t) -

in direct agreement

with equation <{q(t) -

*) Assuming as before independent; see footnote **) I am very indebted

q(0)}2> w 2GkTt,

(15)

(8). On the other hand for t 5 h/kT

q(O))2>

M

2Gn;;2T2 t2.

(16)

that, at least in the first place, we can idealize R and G as freyuency5 2 to equation (12) and footnote on next page. to Dr T. H. Ii. Rarron for evaluating the integral in this closed form.

ON BROWNIAN

The questions

MOVEMENT

AND

415

IRREVERSIBILITY

I would now like to raise concern

the significance

of e-

quation (16) ; the transition from equation (15) to equation (16) ; and the related frequency-spectrum equations (11) and (12). Let me first point out that the characteristic time interval, W/kT, involved in the transition from equation (15) to equation (16), is obviously a “universal” one dependent only on the absolute temperature, T, but not on any specified mechanism of the electrical resistance, R. I then wish to ask how we should interpret physically the indicated transition from a P-dependence (when t 5 h/kT) of the charge fluctuations, which I have previously argued is characteristic of reversible behaviour of a system, to the t-dependence of the charge fluctuations (for t > W/kT) which is characteristic of irreversible and (individually) unpredictable behaviour? I would like to make some very tentative suggestions, but I am anxious that the question I have just asked should be regarded in no way as rhetorical*). Broadly speaking I would propose that we might interpret the transition from equation (15) to equation (16) as suggesting that: (i) For events significant over a time interval 5 &t/kT a description of the electric charge behaviour in terms of a thermal environment at temperature, T, and an (irreversible) electrical resistance, R, is inadequate. More generally, I would naturally suggest that this would be an ultimate limitation for an?’ displacement variable in a thermal environment at temperature, T. (ii) More or less as a corollary, I might suggest that a pure resistance, “viscous” medium, or a like “irreversible” environment can only be regarded adequately as irreversible at best for time intervals involved which are large compared with k/kT. (iii). From (ii) I would be inclined to have some doubts about the validity of Nyquist’s relation, equation (12), for the quantum modification of the fluctuations from an electrical resistance, precisely because when hJ 5 kT I have these doubts about the adequacy of a purely thermal and irreversible description of the situation. Since this means that in general I have doubts about the validity of a quantum correction to the ordinary Nyquist law (equation (1 l)), it means that my doubts apply also to Brownian movement formulae modified for the quantum region with the inclusion of a “zero-point energy” term. In addition we might note that the results of equations (14), (15) and (16) *) The writer put forward this problem some time ago at a theoretical physics seminar at the University of Utrecht under the chairmanship of Professor Van Hove. The writer appreciated at that time the comments

of Professors

Van

Hove

and Van

Kampen

particularly,

but it appeared

that no complete answer to the problem was then forthcoming. As remarked earlier, Professor Van Hove criticises the assumption here of a constant (idealized) G, concluding that we are not faced with any problem of a (limiting) universal time interval, Si/kT, but rather that: “. . the transition from the ts region to the t region will always occur for time values depending on the dynamics of the system”. My own very tentatwe suggestion (see text) is that, quite generally, for f 2 kT/h a description of resistance or viscosity as purely irreversible in terms of a single parameter (e.g. R) fails, but naturally in any particular case it rnal’ already be inadequate for much lower frequencies.

416

OK

BROWNIAN

MOVEMENT

AND

IRREVERSIBILITY

appear to bear a resemblance to quantum mechanical time-dependent scattering calculations, but in the foregoing discussion no specific or detailed assumptions have been called for, or made, about perturbation energies, nor about the existence of “a range of neighbouring quantum levels”, etc. In the present case I would suggest that it is the existence of a thermal environment which ultimately, over “sufficiently long” obresvation times (i.e. > W/KY), leads to random behaviour or, what is presumably equivalent, to the existence of pure “transition-probabilities”, rather than anything particularly appropriate to quantum mechanics. 3. Conclztsion. Tentatively then I might suggest that a fundamental and limiting time scale (h/kT) is indicated within which a reversible description is appropriate *) (mandatory?), and beyond which the thermal environment progressively “smears out” the memory of past events. Alternatively I would say that a specification of the environment solely by a temperature, T, is adequate (only) for time intervals long compared with A/kT; it would also appear then that this time interval would determine the minimum period before we could assume that a time average for a single system involving this environment may be replaced by the usual ensemble average. I am most grateful to Professor Van Hove, Dr Acknowledgements. R. G. Chambers and Dr R. 0. Davies particularly, for their stimulating comments and criticisms of this paper. Naturally this does not imply their agreement with all, or any, of the conclusions. Received

15-l l-61. REFERENCES

1)

Einstein,

A., Ann. Phys. Lpz. 17 (1905) 549.

a

Einstein,

A., Ann. Phys. Lpz. 19 (1906’

3) 4)

Langevin, Ornstein,

P., Comptes Rendus (Paris) 146 (1908) 530. L. S., Proc. roy. Acad. Amsterdam 21 (1919) 96.

5)

Fiirth,

371.

R., Zs. f. Phys. 2 (1920) 244.

Nyquist, H., Phys. Rev. 32 (1928) 110. MacDonald, D. K. C., Phil. Mag. 40 (1949) 561. H. B. and Welton, T. A., Phys. Rev. 83 (1951) 34; (cf. also Bernard 8) Callen, Rev. mod. Phys. 31 (1959) 1017. 6) 7)

J., Phys. Rev. 9) Webcr, J., Phys. Rev. 10) Weber, J., Phys. Rev. 11) Weber, 11. K. C., 12) MacDonald, 13) Vineyard,

and Callen,

90 (1953) 977. 94 (1954) 211. 96 (1954) 556. Phil. Mag. 41 (1950) 814.

G. H., Phys.

Rev.

110 (1958) 999.

over time intervals of this order *) If it is suggested that events are perhaps not “observable” (5 /t/kT), it would seem to me to imply a corresponding doubt about whether we could observe the spectrum of the “current” fluctuations where ht 2 kT, i.e. doubt once more about the validity of a quantum

modification

to the Brownian

movement

formula?.