Linear Response in Classical Physics

Transcription

1 Linear Response in Classical Physics Wouter G. Ellenbroek Technische Universiteit Eindhoven Notes (section ) by Fred MacKintosh (Vrije Universiteit), used previously in the 11 DRSTP course 1 Introduction The theory of linear response deals with the question how the measurable properties of a physical system change when it is slightly taken away from its equilibrium state. Traditionally, the term refers to linear response of thermal systems to slight deviations from thermodynamic equilibrium. In the first part of this course, we will deal with the question how this response is related to the equilibrium thermal fluctuations in the system. The notes for this first part have been provided by Fred MacKintosh, who taught a similar course in a DRSTP school a few years ago. In the second part, we will consider an athermal system that is in mechanical equilibrium, but not in thermodynamic equilibrium, and study its (linear) response to mechanical perturbations. The course will be followed up with a course on quantum linear response theory, taught by Rembert Duine. Fluctuations and Linear Response.1 Brownian Motion: Random Walks in 1d We begin with arguably the simplest problem illustrating the effect of fluctuations, e.g., in a fluid: Brownian motion. This term derives from the botanist Robert Brown who observed random motions of pollen grains in a fluid in 188. This kind of motion is general to sufficiently small particles suspended in a fluid. By observing different kinds of particles, he concluded that the motion was not dependent on the type of particles. We now know that this motion is due to the ever-present thermal fluctuations in the fluid itself. In a series of papers beginning in 195, Einstein gave a theoretical basis for Brownian motion. About the same time, Smoluchowski derived the same results in a somewhat different way. Here, we derive the properties of what is usually referred to as the random walk problem for a single particle. Despite the seeming simplicity, this constitutes a rich example that will help to illustrate many of the principles of fluctu- 1

2 ation about equilibrium generally. Examples of such random walks include the Brownian motion of a particle in a fluid, as well as the random conformations of long polymer molecules under some conditions. In the first of these examples, the essential physical aspects of this problem are: the existence of randomly fluctuating forces pushing the particle in random directions, the viscous drag of the particle in the fluid, and the inertia of the particle. As we shall see later, the the drag of the particle (or equivalently, its mobility when subject to an external force) is fundamentally related to the strength of the random forces that it experiences in the fluid. We ll see later a statement of this deep connection between fluctuations and dissipation in the form of the so-called Fluctuation- Dissipation Theorem, and it is largely with this in mind that we shall examine the basic problem of Brownian motion in such great detail. We first look at Brownian motion in just one spatial dimension. Consider a particle that can move just left (L) and right (R) along a line by steps of equal length l. Steps to the left and right occur randomly, with equal probability p = 1/. Because the probability to jump to the left and to the right are equal, the average displacement of a particle during the i-th step is zero: x i = 1 (+l) + 1 ( l) =. (1) However, the mean squared displacement is non zero: x i = 1 (+l) + 1 ( l) = l. () Next consider average displacement after N steps: X = N x i =. (3) i=1 The mean squared displacement after N steps is: X = N i=1 j=1 N x i x j. (4) The jump directions on different steps are not correlated, therefore x i x j = for i j. (5) For i = j, we have x i x j = l. If we insert this in the expression for X we get N N X = l δ ij. (6) where δ ij is a Kronecker delta: i=1 j=1 δ ij 1 for i = j = for i j.

3 Hence: X = Nl. (7) If the time interval between successive jumps of the particle is δt, then the number of jumps in a time interval t is equal to t/δt and X (t) = l t/δt. (8) In other words, the mean squared displacement increases linearly with time. We should compare this with the ballistic motion of a particle with a velocity v, in which case X(t) = vt, and hence X (t) = v t. For the simple case of one-dimensional Brownian motion we can compute not just the mean-squared displacement of a particle, but the complete probability distribution to find a particle at a distance x from its original position, after N steps. We denote by N L and N R the number of steps that occur to the left and right. The total number of steps is then N = N L + N R, while the net length of the walk (to the right) is x = l (N R N L ). The number of such walks total steps N, of which N L steps are to the left and N R to the right is Ω (x, N) = N! N L!N R!. (9) Then, the probability of a such a walk of net length x is P (x, N) = N! N L!N R! ( ) N 1 = ( N x/l N! )! ( N+x/l By using Stirling s approximation n! = πnn n e n, we find that )! ( ) N 1. (1) P (x, N) N N e N R+N L πn = ( ) [N+x/l]/ ( ) [N x/l]/ N e N N+x/l N x/l πnr πnl πn = (1 + x/nl) [ N+x/l ] (1 x/nl) [ N x/l ] π(n + x/l) π(n x/l) = ( x/l (1 (x/nl) ) N/ (1+x/Nl) (1 x/nl)) Nπ(1 x /(Nl) ) Now we make use of the fact that and that ln (1 (x/nl) ) N/ exp( x /(Nl )) ( ) x/l (1 + x/nl) (x/l) (x/(nl) + x/(nl)) = x /(Nl ) (1 x/nl) 3

4 and hence We can then write ( ) x/l (1 + x/nl) = exp(x /(Nl )). (1 x/nl) P (x, N) 1 = πn exp( x /(Nl) ) exp(x /(Nl )) 1 x /(Nl) πn exp( x /(Nl )) (11) Of course, it comes as no surprise that this probability distribution is symmetric and that the most likely walk returns to the origin (x = ). Thus, the average x =. The mean-square displacement for such random walks, however, is nonzero. This is most easily calculated by treating P (x, N) above as a continuous distribution in the variable x. In fact, for small enough x/l, the distribution above varies slowly over the discrete values of x that are possible for a given large value of N. Thus, x x e x /(Nl ) dx = e x /(Nl ) dx = Nl. (1) Here, we have assumed that the particle moves with independent random steps of length l. If we assume further that these steps occur at equally-spaced time intervals δt, then t = Nδt and x = tl /δt. (13) This behavior x t or x t is characteristic of diffusion. Before leaving this simple one-dimensional random walk problem, we can make another observation. This problem is instructive for other physical situations, such as the conformations that a long polymer chain can make. The resulting conformations are referred to as Gaussian. We ignore the fact that the polymer chains cannot overlap in space. (Although an unphysical microscopic assumption, there are real situations where polymer conformations are nevertheless Gaussian.) Then, Ω(x, N) above represents the number of such conformations, subject to a given end-to-end separation x. Thus, by the fundamental assumption of statistical mechanics (that microscopic states of the same energy are equally likely), the probability of having a given value of x is proportional to Ω(x, N), the number of such conformations or states of the system. Since, for this simple problem, there are N states in all (N independent steps, with choices at each), the normalized probability is just P (x, N) above.. Diffusion One can also study Brownian motion from the point of view of diffusion. Here, we denote the density of (identical) particles suspended in the fluid by a concentration variable n( r, t), which is a function of both position r and time t. 4

5 We also denote by j( r, t) = n( r, t) v( r, t) the current density of particles at r and time t, where v is the velocity. In thermal equilibrium, and in the absence of other forces, the average density n( r, t) will be constant. Furthermore, by the Brownian motion of particles in the fluid, an initial inhomogeneity in the density will relax over time toward this average, uniform density. Thus, spatial variations (gradients) in n result in net particle motion, i.e., a particle current. For small variations in n, we expect that the corresponding current j is small. Thus, it is natural to assume a linear relationship: j = D n, (14) which is known as Fick s law. As we shall see, the minus sign agrees with our physical intuition that Brownian motion will lead over time to a uniform particle density. A positive sign here would lead to unphysical results. Thus, we see immediately that Brownian motion or diffusion is essentially dissipative and irreversible. The constant D here is known as the diffusion constant, which has units of square length per time. There is one more physical aspect to Brownian motion that we have neglected in this treatment so far, namely the fact that particles are conserved. If no particles are created or destroyed while they fluctuate around in the fluid, then the particle current and density are related by conservation n( r, t) t + j( r, t) =, which is simply a mathematical expression of the fact that a net particle current into (out of) any volume results in an increase (decrease) in the number of particles in that region. This, together with Fick s law leads to the diffusion equation, n( r, t) = D n( r, t). t A particular solution to this equation is n( r, t) = N e r 3/ (4πDt) /(4Dt). (15) This is, in fact, the solution 1 to the initial condition that N particles are placed at the origin at time t =, i.e., n( r, ) = N δ( r). This result is for a number N of particles in the fluid. Provided that these do not interact with each other, which is valid if the density is not too high, then we could have equivalently described things in terms of the probability density P ( r, t) for a single particle, which is given by Eq. (15) with N = 1. We then find that r(t) = ; r (t) = r P ( r, t)d 3 r = 6Dt t, 1 See Appendix A. 5

6 which is in agreement with our earlier result for the one-dimensional random walk: x(t) = ; x (t) = tl /δt = Dt t, where D = l /(δt). The general result depends on the number of dimensions d: r (t) = ddt. Even without solving the diffusion equation explicitly, we can relate x (t) to D. We do this as follows. We start with the one-dimensional diffusion equation for the probability density P (x, t): P (x, t) t = D P (x, t) x. We now multiply both sides of this equation by x and integrate over x: P (x, t) dx x = t We can write the left hand side of this equation as P (x, t) dx x = t t dx x D P (x, t) x. dx x P (x, t) = x (t) t The right-hand side can be computed by partial integration (making use of the fact that P (x, t) and its derivative vanish at x = ± : dx x D P (x, t) x P (x, t) = dx xd = +D dx P (x, t). x But as P (x, t) is normalized to one, the result is simply D and therefore: This we can integrate to yield x (t) t = D (16) x (t) = Dt where we have used the fact that at t =, the probability distribution is a δ-function. We can write similar expressions for y (t) and z (t). Adding these, we obtain r (t) = x (t) + y (t) + z (t) = 6Dt. In reality, for the example of a particle suspended in a fluid, we expect that the kicks that the particle experiences in the fluid are random. Even so, since the particle has a mass, there will be some effect of its inertia. So, we expect that the motion is not truly random, but will be correlated for short times. For such short times, the particle will move ballistically, and we expect that the displacement grows linearly with time. Thus, we expect that x t, rather than x t 6

7 for short enough times. Actually, this must be true. We can already see that there is a problem with the Eq. (13), because it would imply that x t, which leads to an unphysical divergence of the velocity at short times. Thus, only for times long compared with some microscopic correlation time (which we can be more quantitative about below) will diffusion, characterized by Eq. (13), be valid..3 Velocity correlation function x(t) is the distance that a particle has traveled in the x-direction in a time interval t. If we denote the instantaneous velocity in the x-direction by v x (t), then we can write x(t) = dt v x (t ). We can then write the mean-squared displacement as x (t) = dt dt v x (t )v x (t ). The quantity v x (t )v x (t ) is an example of a time correlation function - in this case, the velocity auto-correlation function (VACF). This quantity measures the degree of correlation between the velocity of a particle at two different times t and t. When t = t, the value of the of the velocity correlation function is simply v. At sufficiently long times, when the particle has lost all memory of its original velocity, v x (t )v x (t ) = v x (t ) v x (t ) = (because the average velocity of a particle in equilibrium is zero). Time correlation functions, such as the VACF are properties of the system in equilibrium. Their value can therefore never depend on the times t and t individually, but only on the difference. This is so because the properties of a system in equilibrium are time invariant, i.e.: If we choose τ = t, we get v x (t )v x (t ) = v x (t + τ)v x (t + τ) ( τ) v x (t )v x (t ) = v x (t t )v x (), which shows that the VACF depends only on the difference t t. Another important property of (classical) time correlation functions is that we can permute v x (t ) and v x (t ): v x (t )v x (t ) = v x (t )v x (t ). Because of this symmetry, we can write dt dt v x (t )v x (t ) = dt dt v x (t )v x (t ). 7

8 We make use of this result in evaluating x (t) t = t = = dt dt v x (t)v x (t ) dτ v x (τ)v x (), dt v x (t )v x (t ) where we have defined τ t t. If we insert the above result in Eqn. 16, we obtain D = dτ v x (τ)v x (). (17) This equation is only valid for times much longer than the decay time of correlations in the velocity, and hence we may replace the upper limit in the integral by +. The expression that results is D =.4 Langevin Theory dτ v x (τ)v x (). (18) In order to see at a deeper level the connection between the random fluctuations in the fluid and the dissipative nature of the resulting particle motion, we turn now to an approach suggested by Langevin. This will also solve a problem noted above, namely that there must be a failure of our prior analysis of Brownian motion at short times. Our approach here will be to more directly account for the fluctuating forces acting on the particles by the surrounding fluid. We shall see how thermal equilibrium (e.g., the expected uniform spatial distribution of particles) can be established by these fluctuations. We begin with the simple equation of motion satisfied by the particle of mass m m d v dt = F(t), where F is the force acting on the particle. We might be tempted at this point to assume that this force is random, and independent of the particle motion. After all, these forces are due to the fluctuating solvent, right? In fact, there are two physically distinct parts to this force: (i) the viscous drag on the particle, which depends on v and (ii) the randomly fluctuating forces ζ(t). We assume here that the first of these can be described by the Stokes formula for the drag on a spherical particle in a viscous fluid: 6πηa v, where η is the fluid viscosity and a is the radius of the sphere. This is valid for small particles and small velocities. It is important to note that this force 8

9 is random and has zero mean, only to the extent that the particle velocity is random with zero mean. Thus, we can say that, in the absence of external forces, it has a zero time average. It is important also to distinguish another kind of average that we have been a bit sloppy about so far: the ensemble average. This represents an average over a large number of similarly prepared systems (an ensemble). We can consider, for instance, the motion of a particle with some initial condition, say velocity v() at time t =. We expect that this initial velocity will change over time, and that it will become uncorrelated with the velocity at much later times t. Let τ denote this correlation time, after which memory of the initial conditions is lost. As a specific example, if this initial velocity were large compared with the expected thermal velocities, we would expect that it would decay toward zero for times large compared with τ. But, the precise way in which it will do this depends on the details of the fluctuating force ζ due to the fluid. We are not interested in the exact trajectory that an individual particle makes due to each and every collision with individual fluid molecules. Rather, we want to know what to expect for the way the particle velocity decays in a typical sample of a certain kind of solvent (e.g., with known viscosity η) at a certain temperature. Thus, we want to know how the particle velocity behaves on the average, i.e., the ensemble average. However, the ensemble average of the random force is zero ζ(t) =, while the ensemble average of the particle velocity is not zero, at least for times smaller than τ. Mathematically, we could say that the ensemble corresponds to a large set of possible functions ζ(t) consistent with a given fluid at a given temperature. The properties of these functions are yet to be determined, but that the average ζ(t) over this set is zero for all times t is clear by construction. At any time t, ensemble average quantities can be obtained by averaging over the possible functions ζ(t). We rewrite the equation of motion as follows m d v(t) = 6πηa v(t) + ζ(t). dt (19) If we take the average of this over the ensemble, we obtain which implies that m d v(t) dt = 6πηa v(t), v(t) = v()e t/τ, where we can now identify τ = m/(6πηa) as the relaxation time for effects of the initial condition to die out. Here again the essentially dissipative nature of this result is evident by the fact that this result cannot be reversed in time, although the precise connection of this irreversibility to the fluctuating forces ζ is still unclear. This is what we want to establish next. We can take the scalar product of (19) with the instantaneous position r(t). In so doing, we note that r d v dt = d dt ( r v) v. The result is m d dt r v = 6πηa r v + m v. () 9

10 We have used the fact that r ζ =. This last point is not as obvious as it may appear at first sight. Although r ζ =, it can be shown that v ζ. In the equation above, if we further assume that the Brownian particle has attained thermal equilibrium, then we can replace v by the equipartition value 3kT/m. (The kinetic energy mv / of the particle involves three independent degrees of freedom, in which the energy is quadratic. Classical Statistical Mechanics then predicts that this energy is 3 kt on average.) The resulting simple differential equation can be solved: r v = Ce t/τ + kt πηa. But, the constant C can be determined by the initial condition, which we take to be that r = at t = : r v = kt ( ) 1 e t/τ. πηa We can integrate this one more time, to find that r = kt [ ( )] t τ 1 e t/τ, πηa d since r v = 1 dt r. At long times, for which t τ, we find again that r = 6Dt. Furthermore, we can now identify the diffusion coefficient with the dissipative mechanism of the fluid viscosity: D = kt 6πηa. (1) We also find, as expected, that for short times, r t, thus resolving the problem with our earlier analysis for short times. More generally, there is a fundamental relationship between the particle diffusion constant D and the mobility µ or response of the particle to external forces. If we apply an external force f to the particle, say due to gravity, then the resulting drift velocity is v = µ f. This defines what we mean by the mobility. For our Stokes sphere in the viscous fluid, the full equation of motion becomes m d v(t) dt Thus, the mobility in this case is µ = 1 6πηa, and = 6πηa v(t) + f + ζ(t). () D = µkt. (3) This is known as the Einstein relation, and is more general than this particular problem of a sphere in a viscous fluid. It is also an example of the fundamental 1

11 relationship that exists between a dissipative response and fluctuations. In this case, the response is that of a particle to an external field. This response is characterized by the mobility µ. The fluctuating quantity here is the particle position r(t). There is another way to derive the Einstein relation that makes the relation between the mobility µ and the diffusion constant D more obvious. Consider solution in a closed volume V. If there is a concentration gradient of the dissolved species, this will result in a diffusion current j. The diffusion is equal to the number density ρ of the dissolved species, multiplied by the average velocity of these particles. j = ρ v. Now suppose that the dissolve particles are subject to an external potential U(x). If this potential is not constant, there will be a net force acting on the particles: f x = U(x), x where we have assumed that the potential ia a function of x only. The average velocity of a particle due to this force is v x = µf x = µ U(x) x As the particles move under the influence of this force, the density will become inhomogeneous. But once the density is not constant, there will also be a diffusion current. When the system reaches equilibrium, the diffusion current is exactly balanced by the current due to the external force, i.e.:. = ρ v x = ρ(x)µ U x D ρ(x) x. (4) But we also know that, in equilibrium, the probability to find a particle at a position x must be proportional to the Boltzmann factor exp( U(x)/kT ): ρ(x) = ρ exp( U(x)/kT ). If we insert this expression in Eqn. 4, we get This equation is satisfied for all x if = ρ v x = ρ(x)µ U x + D kt ρ(x) U(x). (5) x D = µkt, that is, the Einstein relation. We can actually establish a more detailed connection between dissipation (specifically, the drag coefficient α = 6πηa here) and the fluctuating force ζ(t). To do this, we rewrite the equation of motion, dividing out by the particle mass d dt v(t) = 1 τ v(t) + 1 m ζ(t). (6) 11

12 This, we rearrange slightly and multiply by e t/τ : e t/τ d dt v(t) + 1 τ et/τ v(t) = d ( ) e t/τ v(t) = e t/τ 1 ζ(t). dt m (7) We now change t to t and integrate from to t, with the result that v(t) = v()e t/τ + e t/τ 1 m e t /τ ζ(t )dt. (8) Again, we see that if we take the ensemble average of this equation, we obtain v(t) = v()e t/τ, as we found before. This is because ζ(t ) = for any t. If we square both sides of (8) and then take the ensemble average, however, we find that v = v() e t/τ + e t/τ 1 m e (t +t )/τ ζ(t ) ζ(t ) dt dt. (9) Again, a term had been dropped because ζ =. The correlation function K(t, t ) = ζ(t) ζ(t ), however, is non-zero in general. This is where the ugly details of the fluctuating molecules in the fluid begin to appear directly in our analysis. We finally have a measurable quantity that would appear to depend directly on the random forces ζ. These appear through the correlation function K(t, t ) = ζ(t) ζ(t ), which has several general properties that we can identify. First of all, so long as we are looking at an equilibrium system that is not evolving in any macroscopic sense in time, then this function can only depend on the time interval s = t t : K(t, t ) = K(s). Furthermore, provided that we confine our attention to classical variables (as opposed to possibly non-commuting quantum mechanical operators), we can reverse the order of ζ(t) and ζ(t ), which tells us that K(s) is symmetric, i.e., that it depends only on s. On simple physical grounds, we also expect that K(s) decreases as the time interval s increases, since we expect the force ζ(t) to lose memory of its earlier values. More precisely, we expect that ζ(t) and ζ(t ) become decorrelated as t t grows, i.e., that K(s) = ζ(s) ζ() ζ(s) ζ() = as s grows. Furthermore, we can see that K(s) attains its maximum value at s =. In fact, since [ ζ(s) ± ζ() ] = ζ (s) + ζ () ± ζ(s) ζ() = (K() ± K(s)), the function K(s) is bounded by the limits K() and K(). In particular, K(s) K(). This all leads us to the inevitable conclusion that K(s) = ζ(s) ζ() is a function peaked at s =, from which it decays rapidly toward zero in a short 1

13 time τ, which is determined by the fast dynamics of the individual molecules of the fluid surrounding the Brownian particle. We expect a very short time scale for this, of order the time between collisions with the individual molecules. More generally, it should be the time scale that characterizes the rate with which the degrees of freedom responsible for the random forces ζ relax to internal equilibrium after a displacement of the particle. In fact, for all practical purposes, this correlation function decays so quickly away from s = that we may take it to be proportional to the delta function, that is that it can be taken to be non-zero only for s =, but with finite integral over s: C = K(s)ds. Returning to the problem that led us to consider this function K(s) in the first place, namely the evaluation of the mean-square velocity, we note that the double integral in (9) is appreciably non-zero only for t = t. Thus, v = v() e t/τ + e t/τ C m e t /τ dt = v() e t/τ + e t/τ C m τ ( e t/τ 1 where we have made the approximation suggested above: K(s) = Cδ(s). ), (3) In particular, however, Eq. 3 is true for times t large compared with τ, at which point all memory of the initial conditions (e.g., v()) should have been forgotten. Then, the ensemble average v = 3kT/m, consistent with the Equipartition Theorem. The only way this can be true is if Cτ m = 3kT, i.e., that the drag coefficient α is fundamentally related to the random forces ζ: α = 1 6kT ζ(s) ζ() ds. (31) The above result can be obtained in a few lines, using the techniques described in Appendix B (see, in particular, section B.1). The relation between the drag coefficient and the random forces acting on a particle is special case of what we will later learn generally as the Fluctuation-Dissipation theorem. For now, we note that the drag that a particle experiences in the fluid is due, at a microscopic level, to the randomly fluctuating forces that the particle feels at finite temperature. The random forces are at the origin of macroscopic dissipation. We began this analysis of the motion of a particle in a dissipative medium by considering the equation of motion m d v(t) dt = α v(t) + f + ζ(t), (3) 13

14 where, for instance, the drag coefficient α = 6πηa for a sphere of radius a moving in a fluid with viscosity η. Here, f represented a possible external force applied to the particle (e.g., by an electric or gravitational field). We established a fundamental connection between dissipation (specifically, the drag coefficient α here) and the fluctuating force ζ(t). For this, it was convenient to rewrite the equation of motion (6) in the absence of external forces. This allowed us to solve for the velocity in terms of an initial value v() and the fluctuating force ζ(t). We thus found a measurable quantity (specifically, v ) that depends directly on the random forces ζ. This dependence, however, is only through the correlation function K(t, t ) = ζ(t) ζ(t ), which has several general properties that identified above. Finally, we found that the drag coefficient α is fundamentally related to the random forces ζ, via Equation 31, which expresses the fact that the drag that a particle experiences in the fluid is due, at a microscopic level, to the randomly fluctuating forces that the particle feels at finite temperature. In other words, if we were to pull the particle through the fluid, the drag force experienced can be entirely expressed in terms of the forces ζ, which are presumably due to the random equilibrium fluctuations of the fluid! These random forces are at the origin of macroscopic dissipation..5 Spectral (Fourier) analysis of fluctuations When dealing with fluctuating quantities such as ζ and correlation functions such as K(t, t ) it is often most convenient to characterize them in terms of frequency. We have already noted that each specific realization of the random force ζ(t) (i.e., for each specific member of an ensemble of similar systems) is a very rapidly varying function, with a characteristic time scale of τ, which is very short compared with other time scales of interest, such as that of the motion of the particle (which also fluctuates). ζ(t) varies much more rapidly than does, for instance, v(t). Thus, ζ is expected to contain very high-frequency components (details) that are not apparent in the motion of our hypothetical particle moving around in the fluid. As we shall see, for all practical purposes, we can say that the frequencies characteristic of ζ are unbounded, or that it has components that vary arbitrarily rapidly in time. In order to make these notions more precise, let s look at the Fourier transforms of fluctuating quantities such as ζ(t). This can be done by using the orthogonality property of complex exponential functions (see Appendix C), 1 e iω(t t ) dω = δ (t t ), (33) π where δ(t t ) is the Dirac δ function, which is zero for t t, yet which integrates to unity: δ(t)dt = 1. 14

15 Consider some fluctuating quantity f(t) for a particular member of the ensemble. This could be, for instance, ζ itself, but we ll suppress any vector aspects. We can write In other words, where f(t) = = 1 π δ(t t )f(t )dt dt e iω(t t ) f(t )dω. f(t) = 1 f(ω)e iωt dω, π f(ω) = f(t )e iωt dt, (34) which define the Fourier transform and inverse Fourier transform. Here, f(ω) describes the frequency content of f(t). Assuming that f is real, then because the complex conjugate satisfies f (t) = f(t), (35) we also have that f (ω) = f( ω). (36) It is also worth noting that we have made some serious assumptions above concerning convergence of the various integrals. This will not pose a practical problem, but one might worry about this, since the function ζ is presumably nonzero and is fluctuating over the same range for arbitrarily large times. Thus, for instance, ζ(ω) might not be convergent. One could instead define the transforms for a modified function ζ Θ (t) that agrees with ζ(t) for t < Θ, and vanishes for t > Θ. One could then consider a limiting procedure in which Θ. We can do the same transform for the correlation function K(s) = ζ(t + s)ζ(t). We call the Fourier transform of this C(ω), and the two are related by and K(s) = 1 π C(ω) = C(ω)e iωs dω K(s )e iωs ds. (37) But, based on simple and general physical considerations, we showed before that K(t, t ) = ζ(t)ζ(t ) not only depends just on the time interval s = t t, but K(s) is also symmetric. It is also, of course, real. It is not hard to show that this also means that C(ω) is a real function and is symmetric in ω. We also argued before that K(s) is a function peaked at s = and rapidly decaying to zero within a time interval s of order τ, a very short molecular time scale. Thus, if ω is much smaller than 1/τ, then the argument of the exponential 15

16 in (37) is small wherever the K(s) is appreciable. This suggests that C(ω) is nearly independent of frequency ω for ω less than 1/τ, a very high frequency. In fact, such (molecular) frequencies are higher than any relevant frequency for the motion of our sphere in the fluid. So, we say that the noise spectrum is white. Just as we argued before that we can say that a constant. Note in particular that and C() = K(s) = Cδ(s), C(ω) = C, K(s )ds = 6kT α, K() = ζ() ζ() = 1 C(ω)dω π is of order C/τ. We can also relate the correlation functions K(s) and C(ω) to the Fourier transform of the noise ζ(ω). First of all, we note that (for most systems) we can replace the ensemble average for a single realization of the fluid or member of the ensemble in the definition of K by a time average over a wide enough time interval Θ < t < Θ: K(s) = ζ(t + s)ζ(t) In other words, = 1 Θ = 1 Θ = 1 Θ Θ Θ Θ Θ ζ(t + s)ζ(t) dω dω dt π e iω(t+s) ζ(ω) π e iω t ζ(ω ) dω π e iωs ζ(ω) ζ( ω). C(ω) = 1 Θ ζ(ω). Here, we have used the orthogonality of the complex exponential functions and the fact that ζ is a real function. It may seem strange that a measurable quantity C(ω), which we have seen is directly related to the drag coefficient, and thus the diffusion constant, seems to depend inversely on the time over which we observe the system Θ. However, as we have also noted, the Fourier transform ζ(ω) may not be well defined (e.g., finite). In fact, one should use the modified functions ζ Θ (t) described above. These have finite transforms that grow with Θ. But, the expression relating C(ω) to ζ Θ (ω) can be expected to become independent of Θ for large enough Θ (see Appendix D.1). 16

17 An important historical application of these ideas was to simple electrical circuits, which satisfy an equation similar to that of a particle in a dissipative medium: L d dt I(t) = V ext RI(t) + V (t), (38) where L is the inductance, I is the current, V ext represents a (constant or slowly varying) external applied field (emf), R is the resistance, and V is the fluctuating field related to dissipation. Taking the ensemble average of this equation yields the familiar law relating the voltage, current, and resistance: I = 1 R V ext. The resistance here is analogous to the drag coefficient α above, and a similar analysis to what was done above leads to R = 1 kt V (t + s)v (t) = C V V () kt. Here, the factor of rather than 6 is due to the fact that we are dealing with an inherently one-dimensional problem rather than a three-dimensional one. Furthermore, if the fluctuations of V are very fast (i.e., characterized by a short correlation time τ, here corresponding to the time between collisions of electrons in the wires, for instance), as for the fluid, then the spectrum of these fluctuations is very broad range C V V (ω) = V (t + s)v (t) e iωs ds = kt R for ω in a range of order 1/τ to 1/τ. This general relationship between the voltage fluctuations and the resistance in a circuit is known as Nyquist s theorem, and is yet another special case of the fluctuation-dissipation theorem. For a step-by-step derivation, see Appendix D.3. In Appendix D., we discuss another important noise type, namely shot noise. We have seen so far that the macroscopic drag coefficient, and thus the mobility and diffusion coefficient, of a particle in a simple liquid is related directly to the spectrum of microscopic fluctuations within the liquid. This also means that the physics governing the response of a system that is taken out of equilibrium (e.g., by dragging a sphere through a liquid) can be described entirely in terms if the fluctuations of the system about the equilibrium state. This is the fundamental observation behind the Fluctuation-Dissipation Theorem and the somewhat earlier Regression Hypothesis made by Lars Onsager. Before deriving these general relationships between fluctuations and response, however, let s examine one last aspect of our simple Langevin treatment of particle motion (fluctuations) within liquids. Namely, let us see how the mobility of the particle (e.g., when subjected to an external driving force) relates to the random fluctuations of the particle motion rather than the random fluid motion. We ll do this in just one dimension, and will thus neglect vector notation. 17

18 Consider the position of the particle, which we assume to start at t = from the origin. This satisfies Thus, r (t) = r(t) = Here, what we see is a correlation function v(t )dt. K vv (t, t ) v(t )v(t ) dt dt v(t )v(t ). (39) that is reminiscent of K we saw before. In fact, K vv satisfies all the properties that we previously identified for K = K ζζ, except that we can expect to find a longer correlation time of order τ. In particular, K vv should be a (symmetric) function only of the time interval between t and t. We can anticipate a result based on (39). Since we expect K vv (s) to decay toward zero for times long compared with the correlation time τ, if we are interested in times long compared with this time, then we expect that we can also approximate K vv (s) by a δ function, resulting in where r (t) = tcvv, C vv = K vv (s)ds is some constant. In other words, we expect to find diffusion for long times. We can, however, do better than approximating K vv (s) by a δ function. We know from before that Thus, v(t) = v()e t/τ + e t/τ 1 m v(t + s)v(t) = v() e (t+s)/τ + e (t+s)/τ m = v() e (t+s)/τ + e (t+s)/τ m where we have made the approximation made before. Continuing, we find that e t /τ ζ(t )dt. (4) (t+s) C ζζ ζ(t )ζ(t ) = C ζζ δ(t t ) v(t + s)v(t) = v() e (t+s)/τ + e (t+s)/τ m = v() e (t+s)/τ + e (t+s)/τ m 18 C ζζ C ζζ τ (t+s) e (t +t )/τ ζ(t )ζ(t ) dt dt min(t+s,t) e (t +t )/τ δ(t t )dt dt, e t /τ dt, { e t/τ 1 if s > e (t+s)/τ 1 otherwise

19 For times t long enough ( τ) so that all memory of the initial value of the velocity is lost, v(t + s)v(t) kt m e s /τ. (This is a one-dimensional result.) Indeed, for t τ, we find that where C vv = r (t) tc vv, v(t + s)v(t) ds = kt τ m, Which agrees with our prior result for the three-dimensional diffusion, once we account for the appropriate factors of 3: r (t) = 6Dt, where D = kt/α. Continuing in one dimension, however, we note the complementary expressions for the drag coefficient α and the mobility µ: and α = 1 kt µ = 1 α = 1 kt ζ(t + s)ζ(t) ds (41) v(t + s)v(t) ds. (4) For the electric circuit problem we introduced before, the analogous results for the resistance R and conductivity Σ are: and R = 1 kt Σ = 1 kt V (t + s)v (t) ds (43) I(t + s)i(t) ds, (44) where V is the (fluctuating) voltage and I is the corresponding current. The second of these relationships is usually written in terms of the current density j and the (local) conductivity σ. This is known as the Kubo formula σ = 1 kt j(t + s)j(t) ds. (45) In each of these cases, we have a pair of corresponding quantities: force and velocity, voltage and current,.... In each case, the pair of quantities is related in the sense that some generalized force (force or voltage) results in a response (drift velocity or current). In each case, the corresponding correlation functions, e.g. K ζζ and K vv, are related by an equation of the form ( ) ( ) ζ(t + s)ζ(t) ds v(t + s)v(t) ds = (kt ). (46) 19

20 .6 The Fluctuation-Dissipation Theorem and Linear Response We have seen in the examples above that one can, at least sometimes, relate quantities that have to do with systems out of equilibrium to the spectrum of thermal fluctuations about equilibrium. These principles are based on the Fluctuation-Dissipation Theorem, which was proven in general by Callen and Welton in 195, although special cases (as we have seen) of this were understood much earlier. The basic idea behind the fluctuation dissipation theorem was also captured in the 193s by the regression hypothesis of Onsager, who argued that the physics of macroscopic relaxation of a system back to equilibrium is governed by the same physics as the relaxation of spontaneous fluctuations about equilibrium. We have seen this, for instance in the relaxation of an initial value of v() of the velocity of a particle in a viscous liquid is described by the same time constant τ that appears in the correlation function K vv above. In order to derive the general relationship between, say the relaxation of some macroscopic observable and its equilibrium fluctuations, we need to look at the microscopic evolution of the system. This can either be done in the context of quantum mechanics, as is done in the book by Chaikin and Lubensky, or in the context of classical, Hamiltonian mechanics, as we shall do below. We describe the microscopic state of the system by a point in phase space described by a set of generalized momenta {p 1, p,...} and corresponding coordinates {q 1, q,...}. Although this is a very high-dimensional space (with of order 1 3 coordinates!), we ll denote is systematically by (p, q). The only essential result of Hamiltonian mechanics that we need to recall is the fact that, given some initial point in phase space (p(), q()) at time t =, the state of the system (p(t), q(t)) at time t in the future is completely determined by the initial condition and the Hamiltonian H (p(), q()). It is convenient for us to denote this evolution of the system by some time-evolution operator T t, defined by (p(t), q(t)) = T t (p(), q()). In classical statistical mechanics, the microstates of a system in equilibrium are distributed according to a probability distribution P (p, q) = 1 Q e βh(p,q), where Q = dp dq e βh(p,q) and β = 1/(kT ). The integral is over all of phase space. Again, we use an abbreviated notation, in which dp dq indicates an infinitesimal volume element in phase space. Consider what happens if we disturb the system, taking it out of equilibrium (or, more precisely, taking it to a new equilibrium state characterized by a new Hamiltonian H ). In fact, we shall look at the reverse problem, in which a perturbing field was applied to the system in the distant past, resulting in a new equilibrium at time t = characterized by the modified Hamiltonian H. At time t =, we turn off this field, and allow the system to relax back to equilibrium

21 (characterized by H). It is this relaxation of the system back to equilibrium that we want to examine. Let A(p, q) be some macroscopic observable whose relaxation we are interested in. We shall consider only linear response theory, in which we assume the perturbation is sufficiently weak that the Hamiltonian H can be taken to be H = H+ H, where H = fa and f is the perturbing field. (It is not hard to see that with this definition, f and A are conjugate thermodynamic variables, like pressure and volume, since f = F A, where F is the free energy.) The initial state of the system corresponds to distribution P (p, q) = 1 Q e βh (p,q), where Q = dp dq e βh (p,q). The initial (macro)state of the system is one in which A = 1 Q dp dq e βh (p,q) A(p, q). After the field is turned off, however, the system (i.e., each microstate (p, q)) evolves according to the Hamiltonian H(p, q) and the time-evolution operator T, rather than H (p, q) and T. In other words, at times t >, A(t) = 1 Q dp dq e βh (p,q) A (T t (p, q)) = dp dq e βh(p,q) (1 β H) A (T t (p, q)). dp dq e βh(p,q) (1 β H) Expanding, and keeping only terms through linear order gives A(t) dp dq e βh(p,q) A (T t (p, q)) = dp dq e βh(p,q) dp dq e βh(p,q) A(p, q)a (T t (p, q)) +βf dp dq e βh(p,q) dp dq e βh(p,q) A(p, q) dp dq e βf βh(p,q) A (T t (p, q)). dp dq e βh(p,q) dp dq e βh(p,q) This means that, to linear order, A(t) A(t) = βf ( A()A(t) A ), = βf δa()δa(t), (47) where δa(t) = A(t) A and refers to an ensemble average in the unperturbed system. We have been somewhat sloppy in the notation here. The first term in this equation refers to a evolution to time t of an ensemble of not only similarly prepared systems, but systems subject to the same perturbation. 1

22 It does not strictly refer to any ensemble average in the usual sense. Thus, it might be less confusing to write A(t) = βf δa()δa(t), (48) where this expresses the average deviation A from its equilibrium value. The average is to be understood to mean over the ensemble of systems with the same perturbation. This is one form of the Fluctuation-Dissipation theorem, expressing the macroscopic evolution of a system out of equilibrium to the spontaneous fluctuations of the system about equilibrium. This is usually expressed, however, in terms of the (linear) response of the system. This is implicit in the expression above. We have seen how the response A(t) above is linear in the force (or, more generally, the perturbation) f. This perturbation may, however, be time dependent. The most general form of a linear response to a time-dependent f is A(t) = dt χ(t, t )f(t ), where χ expresses the fact that the response at time t depends on the way the system was perturbed at other times. In principle, we should allow for arbitrarily influences from distant times. Thus, the integral above should be over all times t. It would be unphysical, however, if this influence extended to future times. How, for instance, can the system anticipate the perturbation that it will experience in the future? This simple, and physically reasonable assumption of causality can be expressed mathematically as χ(t, t ) = for all t > t. Note that χ in this sort of description is assumed to be a property of the (equilibrium) system, and not itself dependent on f. Thus, we also expect that χ should depend only on the time interval between t and t, so that { χ(t, t χ(t t ) = ) if t > t otherwise. As noted already, we expect that χ is a property of the (equilibrium) system, and not itself dependent on f. We should, therefore, be able to express it in terms if the dynamics of the equilibrium system. We have examined a particular case above, in which we perturbed the system for times t <, then removed the perturbation. This corresponds to { f if t < f(t) = otherwise. Here, we found that A(t) = βf δa()δa(t), (49) But, A(t) = dt χ(t, t )f

23 in this case, since f = for t >. By a change of variable of integration, this becomes A(t) = f t dt χ(t ). But, this only applies for t >. Thus, { β d χ(t) = dt δa()δa(t) if t > otherwise. This expresses the way in which a system responds to a perturbation (taking it out of equilibrium) in terms of the spontaneous fluctuations of the system about equilibrium. Last time, we saw how the response of the system, specifically in terms of some observable quantity A, could be described by a linear response function χ: A(t) = dt χ(t, t )f(t ), (5) where χ expresses the fact that the response at time t depends on the way the system was perturbed by some generalized force f at other times. Here, A(t) A(t) A, where A(t) represents the expected value of A at time t (this is actually an ensemble average quantity, but for a particular ensemble of similarly perturbed systems.) Based on the physical principle of causality, however, we expect the response only to depend on the history of the system and its perturbation. Furthermore, since the response function should be only an equilibrium property of the system itself (and not, for instance, dependent on f, at least if the perturbation is weak enough that the response is, indeed, linear). Thus, we expect that { χ(t, t χ(t t ) = ) if t > t otherwise. In deriving the above, we assumed a very particular form of the perturbed Hamiltonian: H = H fa. In general, however, we expect if the (generalized) force f is weak, then one can simply expand this Hamiltonian to linear order, with the result that H = H fb. That is, the linear coefficient of f need not be A, but may be some other variable B. When the system is subjected to such a perturbation, we expect to find a linear response of the variable A, as given by (5), where we shall denote the response function by χ AB (t t ), since it represents the response of A to a perturbation f that is conjugate to B. Here, f and B are conjugate in the 3

24 thermodynamic sense, in that F B = kt B log Q = kt 1 dp dq e β(h fb) Q B = f, where Q = dp dq e β(h fb) is the partition function and F = kt log Q is the free energy. This is just like the relationship between the conjugate variables volume V and pressure P : ( ) F = P. V T,N As before, if we consider a perturbing field f that is turned on in the distant past, and then switched off at time t =, we can write the evolution of (the ensemble average of) A as A(t) = 1 Q dp dq e βh (p,q) A (T t (p, q)) = dp dq e β(h(p,q) fb(p,q)) A (T t (p, q)). dp dq e β(h(p,q) fb(p,q)) An alternative way of deriving the linear response that we calculated before is to look at the derivative of this with respect to the perturbation f: = β Q dp dq e βh (p,q) B (p, q) A (T t (p, q)) A(t) f dp dq e βh (p,q) A (T t (p, q)) ( ) dp dq e βh (p,q) β dp dq e βh (p,q) B (p, q). Evaluating this derivative for f =, and noting that the point (p, q) in phase space represents the state of the system at time t =, we find that A(t) βf ( B()A(t) A(t) B() ) = βf B()δA(t). (51) Following the same analysis as before, we also find the response function χ AB (t) in terms of the fluctuating quantities δa and B: χ AB (t) = β d dt B()δA(t). (5) 4

25 Again, here, we should note that these thermodynamic averages refer to the unperturbed (i.e., equilibrium) system. Thus, they represent equilibrium fluctuations of the system, which exhibit some temporal correlations. The main example we have been looking at is the motion of a particle in a viscous fluid. Here, the perturbing field can be taken to be an external force f applied to the particle (say, by some magnetic or gravitational field). The response of the system is to develop a drift velocity v. But, the thermodynamically conjugate variable to f is the displacement x of the particle (in the direction of the force, which we take to be the x direction). Thus, H = H fx. So, the ensemble average drift velocity is v(t) = χ vx (t t )f(t )dt = f χ vx (t )dt, when, as before, a force f is applied in the distant past and switched off at time t =. But, in this case, we expect that v() = µf, where µ is the particle mobility. But, we know that µ = 1 kt v(t)v() dt = 1 kt t v(t)v() dt, where we have used the symmetry of K vv (t) = v(t)v(). This suggests that χ vx (t) = βk vv (t). This says that the dynamics associated with the response χ of a system to a non-equilibrium perturbation are governed by the same underlying principles as the fluctuations K vv (t) = v(t)v() about equilibrium. This is, indeed, the case, as Onsager suggested in his regression hypothesis. But, let s see if we can derive the relationship χ vx (t) = βk vv (t). We see from (5) that But, which cannot depend on t. Hence, χ vx (t) = β d dt x()v(t). (53) x()v(t) = x(t )v(t + t ), = d dt x(t )v(t + t ) = d dt x(t )v(t + t ) + x(t ) d dt v(t + t ) = v(t )v(t + t ) + x(t ) v(t + t ), and d dt x(t )v(t + t ) = x(t ) v(t + t ) = v(t )v(t + t ) = K vv (t). 5

26 This shows that χ vx (t) = βk vv (t) = 1 m e t/τ, for our particle in the fluid. We see that this represents what is sometimes referred to as a memory function. It shows how the response at one time depends on the history of disturbances, but with vanishing influence from distant past. That is, the memory fades over time. (As we all know too well!).7 The Damped Harmonic Oscillator The harmonic oscillator illustrates nicely many of the ideas we have been developing. It also turns out to be a very practical application of the Langevin analysis. There are many experimental situations of current interest where one has a harmonic potential for particles that would otherwise undergo brownian motion in a fluid. So, to our brownian particle above we add a harmonic, trapping potential Kx, which tends to localize the particle near x =. Our equation of motion becomes mẍ(t) + αẋ(t) + Kx(t) = f(t), where f is the force acting on the particle. We ll consider the case where this is some external driving force. Of course, there will also be our familiar random brownian forces. But, we shall treat ensemble quantities below. Fourier transforming the equation of motion leads to ( K mω iαω ) x ω = f ω, where we shall tend to employ a shorthand notation for transforms x ω of x(t), and similarly for the force. This can also be written in a way suggestive of linear response: 1 x ω = (K mω iαω) f ω, where 1 χ(ω) = K mω iαω is the Fourier transform of the response function defined by x(t) = dt χ(t t )f(t ). It is tempting at this point to anticipate the frequency-dependent version of our expression which would read χ xx (t) = β d dt x(t)x() = β d dt K xx(t), (54) χ(ω) = iωβ x ω x ω. (incorrect) 6