Noise and Brownian motion. So far, the equations we have been studying in the context of nonequilibrium statistical

Phys504 Statistical Physics Fall 2006 Lecture #20 Professor Anthony J. Leggett Department of Physics, UIUC Noise and Brownian motion So far, the equations we have been studying in the context of nonequilibrium statistical mechanics have been deterministic, as a result of averaging over the statistically described behavior of the relevant components. 1 However, there are many situations in which we are interested in the fluctuations of various quantities around their average values. Typically, we may be interested explicitly in the behavior of a particular subsystem e.g. a heavy Brownian particle in a liquid suspension, or the nuclear spins in a solid and the fluctuations then may be thought of as coming from interaction with, or buffeting by, an environment whose behavior we cannot control. Alternatively, we may be interested in a single macroscopic degree of function of a macroscopic body say the pressure of a gas on one wall of its container, or the voltage across a bar of metal which is the result of the collective behavior of the many microscopic degrees of freedom composing the body; in this case, stretching terms a little, we may regard the microscopic degrees of freedom as the environment of the macroscopic one. There is one very general statement we can make about the fluctuations in any physical quantity A in a physical system, which is a generalization of the limited form of the FD theorem we already met in l. 15. Consider the symmetrized correlation function C A (t) = 1 2 { Â(0)Â(t) + Â(t)Â(0) } ( 1 { Â(t)Â(0) + Â( t)â(0) } (1) 2 where Â(t) is taken in the Heisenberg representation and we define as usual ˆx T rˆρˆx, ˆρ Z 1 exp βĥ. According to the definition of the quantity S AA(ω), (eqn. (26) of l. 11), the Fourier transform of C AA (t) is just 1 2 (S AA(ω) + S AA ( ω)), and this in turn is by eqn. 1 For example, when solving the Boltzmann equation, linearized or not, we implicitly assumed that the number of particles 2 with which a given particle 1 can collide is given by the thermal equilibrium distribution (the Stosszahlanantz of Boltzmann). 1

(30) of l. 15 equal to the expression coth(β hω/2)imχ AA (ω). Thus C AA (ω) = coth(β hω/2)imχ AA (ω) (2) which is evidently consistent with eqn. (32) of l. 15. Thus, if we can calculate χ AA (ω) in terms of various phenomenological parameters, we can immediately obtain the fluctuations of A at arbitrary frequencies in terms of those same parameters. This is the (generalized) FD theorem. The best-known example of the theorem is the Nyquist-Callen-Welton result for the fluctuations of the current (or voltage) in a metallic wire. In this case, we set A equal to the total current through the wire, and note that the perturbation in the Hamiltonian which involves this quantity is of the form δh = I(t) A(t)dl (3) Since the time derivative of A(t) dl is the potential difference V across the ends of the wire, and ( I/ V ) ( ω) is by definition the inverse of the generalized ohmic conductance R(ω), we have χ II (ω) = iωr 1 (ω) (4) In the usual case of interest, the frequency of measurement ω is much smaller than the characteristic relaxation times, etc., of the system, and thus R(ω) is given to a good approximation by its dc limit R. Further, the quantity C II (ω) is just the spectral density of the current fluctuations, i.e. if we write I(t) = dωe iωt I(ω) (5) then C II (ω) = I(ω) 2, where I(ω) 2 dω is the power spectrum of the fluctuations in ω. Hence we can write I(ω) 2 dω = hω coth β hω/2 R 1 dω (6) By a similar argument, since V/ I is simply R, the voltage fluctuations are given by V (ω 2 dω = hω coth β hω/2 R dω (7) 2

In Nyquist s original (1928) calculation, it was assumed that we are in the classical limit hω kt, so the formulas reduce to I(ω) 2 dω = R 1 k B T dω, V (ω) 2 dω = k B T R dω (8) Let us now consider the more general noise problem. We may set the problem up schematically as follows: Assume for simplicity that there is only a single operator of the system, say A, which couples to the environment, and that the coupling is linear in A: thus H S env = Â(t)ˆΩ(t) where Ω is an operator on the states of the environment. Then we may think of Ω as a sort of conjugate field for A; for example, for the NMR problem A is the z-component of spin and Ω is the environment-originated field H z which acts on it. Rather generally, a complete description of the effects of the environment will require us to specify the complete statistics of ˆΩ(t), i.e. not just the expectation value Ω, mean-square value ˆΩ 2, etc. but arbitrary correlations of the form K(t 1 t 2 t 3...t n ) = ˆΩ(t 1 ) ˆΩ(t 2 )..ˆΩ(t n ) (9) for arbitrarily large n. (Alternatively, if we regard Ω(t) as a classical random field, we will have to specify the probability functional P (Ω(t 1 )Ω(t 2 )..Ω(t n )), again for arbitrarily large n.) This is a great deal of information. Fortunately it turns out that in real life, for most of the cases of practical interest, there are two major simplifying factors. First, it almost invariably turns out that the field Ω can be written as the sum of contributions from many independent modes of the environment, i.e. in the form of a sum of operators ξ ν which are uncorrelated: Ω = µ c µ ξ µ ξ µ ξ ν δ µν (10) 3

(I will try to make this idea more precise in l. 21 when I deal with the explicit quantummechanical description.) The nice thing about the description (10) is that in the limit that the number of different µ s tends to infinity, the central limit theorem applies which means that ˆΩ itself is Gaussian-distributed, i.e., a knowledge of the quantity ˆΩ(t 1 )ˆΩ(t 2 ) ˆΩ(0)ˆΩ(t) suffices for a calculation of arbitrary expressions of the form (9). The second useful feature of many interesting real-life noise problems is that the timescales associated with the motion of the system, including those involving dissipative effects, is very long compared with the timescales of the environment. As an obvious example, in a liquid we expect that the time interval between different collisions of the same molecule is of order at most 10 12 (secs); however, the Brownian motion of a small particle suspended in the liquid takes place over a timescale ( 1 sec) long enough for it to be observable under a microscope in real time. Similarly, in many NMR problems (though not all) the correlation time of the random field may be, again, as short as 10 12 sec. while the resulting relaxation time of the spins may be microseconds or even milliseconds. This turns out to simplify the problem very considerably, since it means that over timescales of the order of these characteristic of the system the dynamics of the environment may often be describable in terms of a few simple phenomenological parameters. In this lecture, let s try to get some orientation by analyzing a problem which has the virtue of being analytically solvable, namely that of a simple linear harmonic oscillator (or its limit, a free particle) linearly coupled to an environment of (many) other harmonic oscillators. (As we shall see in l. 21, while choice of a harmonic oscillator as the system is very specific, 2 the representation of the environment as a bath of harmonic oscillators is surprisingly generic in its applicability.) Thus, the complete Hamiltonian of the universe ( system + environment) is Ĥ = ĤS + ĤE + ĤSE (11) Ĥ S = (P 2 /2M) + 1 2 Mω2 ox 2 (12) 2 Though it does include Brownian motion as the special case ω o 0, see below.. 4

H E = 1 (p 2 i /m i + m i ωi 2 x 2 i ) (12b) 2 i I shall assume that the spectrum of the oscillators is very dense in the sense that ρ(ω) i δ(ω ω i ) N, N (13) As we will see, the quantity ρ(ω) itself plays no role. As to the choice of the systemenvironment coupling, the first shot choice is one which is bilinear in the S and E coordinate variables separately: H (fs) SE = xω, Ω i c i x i (14) where to obtain system behavior independent of N the (real) coupling constants C i must be taken proportional to N 1/2 (cf. below). With the choice (14) the quantity Ω is the effective field exerted on the system by the environment. Now, we are not interested in the environment for its own sake, but only for its effects on the system, and it will turn out that all these effects can be calculated once we know a single function, the spectral function J(ω) defined by J(ω) i (C 2 i /m i ω i )δ(ω ω i ) (15) which is by construction independent of N in the limit N. For the moment we will make only a few qualitative assumptions about J(ω), namely (a) that J(ω) is finite for all ω, and tends to zero for ω 0; (b) that the quantity o J(ω)dω/ω exists, and (c) that the characteristic scale of frequency over which J(ω) has appreciable structure is some frequency ω c which is very large compared both to the bare system frequency ω o and to the damping rate γ which will emerge below. An example of a form of J(ω) has satisfying all these requirements is J(ω) = const. ω n e ω/ωc, n > 0. Note that our model now incorporates both the qualitative features identified above as characteristic of typical noise problems (many independent environmental modes, and separation of frequency scales of S and E). 5

Before proceeding, however, we need to note a technical difficulty related to the choice of coupling given in eqn. (14) as it stands: We want the coupling to provide a mechanism of dissipation and noise for the system, but the coupling (14) actually does more than that: it shifts the effective potential seen by the system. To see this, we evaluate the total potential energy when X is fixed but the x i are allowed to adjust to their optimal values for that X, namely (C i /m i ω 2 i )X. The result is to add to the original system potential energy V (X) 1 2 Mω2 ox 2 an extra term V (X) = 1 (Ci 2 /m i ωi 2 )X 2 1 2 i 2 X2 o J(ω) dω (16) ω (in a chemical-physics context this would be called the solvation energy). In many contexts, for example, Brownian motion, this term is not physical, as it is advisable to subtract it from eqn. (14) by hand, i.e. to write (14) with a counterterm V (X)(> 0) : H SE = XΩ V (X) (17) which is equivalent to replacing the potential-energy term in (12b) by the expression 1 2 m i ωi 2 (x i (C i /m i ωi 2 )X) 2 (18) i In future, I shall assume the counterterm has been added without always commenting explicitly, and write δωo 2 M 1 2 V (x)/ x 2 +M 1 o J(ω)dω/ω. In the present lecture, I shall be discussing the classical theory of the model described by eqns. (11), (12) and (17). It should be emphasized that for a real-life model of this nature the classical description is completely applicable only when k B T is large compared to the zero-point energy of all the oscillators, i.e. when k B T hω c ; however, for present purposes it is useful to imagine that h 0, so that the classical limit holds for all T including zero. We start, then by considering the dynamics of the system at T = 0 (which means, as we shall see below, that the noise vanishes in the classical case). Using the standard responsefunction technique, we start with S and E both at rest and perturb S with an external force F (t) (so that the perturbation to the Hamiltonian is F (t)x). Thus the equation of motion of X and the x i are (with the counterterm inserted) 6

MẌ = M(ω2 o + δω 2 o)x + i C i x i + F (t) (19a) m i ẍ = m i ω 2 i x 2 i + CX (19b) and we impose the boundary conditions x i (t = 0) = X(t = 0) = 0. (19c) We take Fourier transforms, eliminate the x i and express the result in terms of the response function χ(ω) (the Fourier transform of δx(t) δf (t )): χ(ω) = 1 M 1 [(ω 2 o + δω 2 o) ω 2 ] K(ω) (20) when the quantity K(ω), which contains the complete effect of the environment, is given by K(ω) i (C 2 i /mω 2 i )(ω 2 i ω 2 ) 1 o ω J(ω ) ω 2 ω 2 dω (21) Let us examine the structure of the function K(ω). First, we recall that in the RF technique the frequency variable ω should have an infintesimal positive imaginary part (ω ω + iη); thus the imaginary part of K(ω) is Im K(ω) = π 2 J(ω) (22) As to the real part, we see that according to our definition of δω 2 o(above) we have δω 2 o = M 1 ReK(0) ( M 1 K(0)) (23) Hence, χ(ω) = M 1 can be written in the form χ(ω) = M 1 (ω 2 o ω 2 (iπ/2m)j(ω) M 1 K(ω)) 1 (24) K(ω) K(ω) K(0) = ω 2 dω J(ω ) P o ω (ω 2 ω 2 ) (25) 7

where P denotes the principal part. In evaluating the RhS of (25) we should be careful to do the calculation for small but nonzero ω and only then take the limit ω 0. In general, K(ω) is nonzero and may for example renormalize the system effective mass. However, life simplifies considerably if we remember that the reason we are studying this model is in the hope of understanding Brownian motion and related phenomena, and that in these cases the phenomenological (deterministic) equations of motion are of the form Mẍ + ηẋ + Mω 2 ox = F (t) (26) (where ω o may or may not be zero), where η is a friction coefficient. We immediately infer from (26) the response function χ(ω) = M 1 (ω 2 o ω 2 iωη/m) 1 (27) To make the imaginary parts of (24) and (27) consistent we must set 3 J(ω) = (2/π)η ω (28) Miraculously, with this choice we find dω K(ω)α P o ω 2 ω 2 = 1 2 dω ω ω = 1 2 dω ω 0 (29) Of course, this exact cancellation only occurs if we assume (28) holds for all ω, not just for the low values relevant to Brownian motion. However, it is easy to see that if J(ω) departs from the form (28) over a scale ω c, then the integral is of order (ω/ω c ) 2, and thus the mass renormalization (etc.) induced by K(ω) at Brownian frequencies is negligible and we may safely set this quantity equal to zero. We then see that with the choice (28) for ω ω c the output of our microscopic model, eqn. (24), agrees exactly with the phenomenological equation (24). In particular, in the limit ω o 0 both techniques yeild the simple form of χ(ω) appropriate to Brownian motion, namely χ(ω) = M 1 (ω 2 + iωγ) 1 γ η/m (30) 3 This general form of J(ω)(J(ω)α ω) is sometimes denoted in the literature by the term ohmic dissipation. 8

Note that the correlation function of the system momentum P = Mẋ is just M 2 ω 2 χ(ω) and hence is given by χ pp (ω) = Mω 2 /(ω 2 + iωγ) (31) Now let us consider the situation at nonzero temperature (with our existing assumption that h 0 so that we can use classical mechanics). At zero temperature we started at t = 0, according to eqn. (19c), not only with X = 0 but with x i = 0, so that the only reason for a nonzero value of the x i was that it was induced, according to eqn. (19b), by the motion of the system. At nonzero T, even if we constrain the system to be at rest at the origin at t = 0, the environmental oscillators have a nonzero probability of being displaced from their origins because of thermal fluctuations (which we cannot control). In fact, the probability of finding the value x i is given by the standard Gibbs-Maxwell formula P (x i ) = const. exp 1 2 m ix 2 i /k B T (32) The average value of x i is zero, the mean-equal value is given by the equipartition result m i ωi 2 x 2 i = k B T (33) and, more relevantly for our purposes, the autocorrelation function x i (0)x i (t) is given (in the absence of coupling to S) by x i (0)x i (t) = x 2 i cos ω i t = (kt/m i ωi 2 ) cos ω i t (and x i (0)x j (t) 0, i j) (34) Thus the average value of the noise operator Ω(t) i C i x i (t) would be zero (in the absence of coupling to the system denoted by un ) and its correlation function would be S(t) Ω(0)Ω(t) un = kt i (C i /m i ω 2 i ) cos ω i t kt o J(ω) ω cos ωt dω (35) In the ohmic case described by eqn.(28) we have (since o cos ωt dω = πδ(t)) S(t) Ω(0)Ω(t) un = 2ηkT δ(t) (36) so that the Fourier transform S(ω) of the noise correlation function is flat (independent of ω); this case is known as the case of white noise. Thus, in the context of classical physics 9

ohmic dissipation white noise In the case of a more general form of J(ω) (which is not of direct relevance for the Brownian-motion problem but may be in other contexts, see l. 21) S(t) has a finite memory in time (i.e. is not a simple δ-function) and correspondingly its Fourier transform S(ω) is not flat: evidently, the structure in S(ω) mirrors that in J(ω), in fact we can see directly from eqn. (35) that S(ω)αJ(ω)/ω. This case is sometimes denoted colored noise. This is yet another illustration of the relation between fluctuations (S) and dissipation (J). Of course, in real life S(t) cannot be exactly a δ-function as in eqn. (36), but its extension in time is of order ωc 1, which is very much shorter than the characteristic times ( ωo 1, γ 1 ) associated with the macroscopic system and thus looks much like a δ-function on these time scales. We now come to a crucial point: Let us return to eqn. (19a) and rewrite it in the form M(Ẍ + (ω2 o + δω 2 o)x = Ω(t) + F (t) (Ω(t) i C i x i (t)) (37) Now, we can split Ω(t) into a part induced by the interaction with S and a part resulting from the intrinsic thermal fluctuations: Ω = Ω ind + Ω noise (38) Moreover, it is a characteristic of linearly coupled harmonic oscillators that the response to an external driving force is strictly linear in the force, independently of the strength of the latter or the displacement it induces. So let us use (38) to rewrite (37), with F (t) set equal to zero in the form M(Ẍ + (ω2 o + δω 2 o)x) Ω ind (t) = Ω noise (t) (39) But Ω ind (t) i C ix ind i (t), and the quantity x (ind) i (t) by definition satisfies eqn. (19b). Thus the system dynamics described by (39) and (19b) is identical to that described by eqns. (19a c), with the only differences being that the external force F (t) occurring in the latter is replaced by the fluctuating noise term Ω noise (t). Consequently, we can immediately draw the important conclusion that in the absence of an external force and 10

for any given realization of the noise term Ω noise (t) we have for the actual position X(t ) of the system at time t X(t ) = χ(t t)ω noise (t)dt (40) where χ(t t) is the Fourier transform of the χ(ω) we have calculated. In words, we can regard the thermal fluctuations of the system as resulting from its responding to the thermal fluctuations of the water just as if the latter were an externally applied force. It should be emphasized that there is no averaging implied in eqn. (40), and that the equation shall apply whether the noise is white or colored. However, average values of X, P and their correlations can be obtained by using the fact that the statistics of Ω noise (t) is Gaussian with a two-point correlation S(t) given by eqn. (36). In the case of white noise the problem simplifies, since we know that χ(ω) is identical so that which follows from Newton s second law with a damping term ηẋ. Thus we can write 4 for the momentum P = MẊ the equation (where we set as above γ η/m, consider for definiteness the case of Brownian motion, ω o = 0, and for clarity change the notation Ω noise (t) to F n (t)) P + γp = F n (t) (41) which has the solution P (t) = t o dt F n (t ) exp γ(t t ) (42) Evidently when the time elapsed since the process was allowed to start is γ, the lower limit of the integral can be extended to. The physical interpretation of eqn. (42) is clear: the Brownian particle is buffered by the random force F n (t), but any impulse acquired is lost over the scale of the relaxation time γ 1. From (42) we can calculate the correlation function P (t)p (t ) : P (t)p (t ) = t t dudu F n (u)f n (u ) exp γ{(t u) + t u } (43) t t dudu S(u u ) exp γ{(t u) + t u } 4 Eqn. (41) is actually a very simple example of a Langerin equation, see next l. 11

In view of the fact (eqn. (36)) that S(u u ) = 2ηkT δ(u u ), this becomes (for t > t) P (t)p (t ) = exp γ(t t) t du exp 2γ(t u) 2ηk B T = exp γ(t t)(η/γ)k B T (44) while for t < t the roles of t and t are interchanged. Since η/γ = M, we finally find P (t)p (t ) = Mk B T exp γ t t. (45) The t t limit agrees with the result expected from the equipartition theorem. The process described by eqn. (41), with the force correlations given by (36), is a typical example of a Gaussian random process, in this case with white noise, i.e. a random walk. To make this more obvious, it is helpful for a moment to discretize the time into small elements t( γ ( 1) and write I n t n dtf noise (t), so that the noise created by the environment on the system appears as a set of discrete impulses I n at times t n. In view of the linearity of eqn. (41), the effect (on e.g. P (t) or X(t)) of a series of blows is simply the sum of the effects of each separately, i.e., we can write (e.g.) X = n X n when X n is the displacement due to the n th blow. Since the momentum induced, at a time t after the blow, by an impulse I n is, according to eqn. (41), exp γt I n, the total displacement X n is X n = o dt P (t)/m = I n /Mγ η 1 I n (46) Since in view of (36) I 2 n t t o o dt dt F n (t) F n (t ) = (2ηkT ) t (47) we have X 2 = (2kT/η) t (48) If the magnitude of all the X n were identical, the only difference being in the sign, then the process would be a simple random walk (in this case, one-dimensional), and we can apply 12

all the standard results. What we actually have is a random walk problem with variable step length, but it is straightforward to show that the resulting behavior is exactly identical to that of the simple case provided that the uniform step length a is replaced by the rms value X 2 1/2 It is known that in the limit that the number of steps N tends to, the simple random walk problem is equivalent to a diffusion problem with diffusion constant D = a 2 /2 t; thus in our case it is also a diffusion problem, with D = kt/η (49) Note that the step length t has dropped out of the problem, so we can at this point safely take the limit t 0, i.e. go back to our original (continuous-time) problem. Eqn. (49) recovers the correct Einstein relation between the frictional constant η and the diffusion constant D for a Brownian particle 5. A well-known consequence of the motion being described by a simple diffusion equation is that if we know that at time D the particle was at the origin, then the probability of finding it a distance X away at time t is given by the standard 1D solution. The mean-square displacement after time t is P (X, t) = (4πDt) 1/2 exp X 2 /4Dt (50) X 2 P (X, t)dx = 2Dt. (51) It is worth noting that the result (50) (and, actually, with a little labor also (50)) could have been obtained from our expression (30) for the position response function χ(ω). To evaluate the X 2 of eqn. 51, we actually have to calculate the unconstrained value of [x(t) X(0)] 2 2{ X(0)) 2 X(0)X(t) }. From the classical FD theorem we have, using (30) for the Fourier transform of X(0)X(t) X(0)X(t) ω = k B T χ(ω) ω = k BT M ωγ ω 2 (γ 2 + ω 2 ) (52) 5 To make contact with l. 18, we note that for a particle of change c η is related to the mobility η = Mµ/e. 13

so that the expression for X 2 is X 2 (t) = 2k BT M o dω (1 cos ωt) ω 2 For times γ 1 the term ω 2 in the (γ 2 + ω 2 ) can be ignored, so we get γ (53) γ 2 + ω 2 X 2 (t) = 2k BT η o (1 cos ωt) dω ω 2 = k BT η t (54) which by (49) is equivalent to (51). This derivation has the advantage of calling our attention explicitly to the face that the diffusion equation describes the behavior only over times γ 2 : such that explicit memory of most of the impulses received has vanished 6. The results derived in this lecture are no doubt self-consistent and pleasing, but there is one rather large snag: They have been derived using a model, for the environment, namely that of a set of localized 1D harmonic oscillators linearly coupled to the system, which at first sight, at least, is highly unrealistic physically. Can we justify the assumption that a more realistic model of the environment, and of the coupling of the system to it, will reproduce these results? I will address this question in the next lecture. 6 The behavior for K γ 1 is explored in problem 3, PS5. 14