The Green Kubo Relations

In 1828 the botanist Robert Brown observed the motion of pollen grains suspended in a fluid. Although the system was allowed to come to equilibrium, he observed that the grains seemed to undergo a kind of unending irregular motion. This motion is now known as Brownian motion. The motion of large pollen grains suspended in a fluid composed of much lighter particles can be modelled by dividing the accelerating force into two components: a slowly varying drag force, and a rapidly varying random force due to the thermal fluctuations in the velocities of the solvent molecules. The Langevin equation as it is known, is conventionally written in the form,

(4.1)

Using the Navier-Stokes equations to model the flow around a sphere it is known that the friction coefficient ζ-6πηd∕m, where η is the shear viscosity of the fluid, d is the diameter of the sphere and m is its mass. The random force per unit mass F_R, is used to model the force on the sphere due to the bombardment of solvent molecules. This force is called random because it is assumed that ⟨v(0)⋅F_R(t)⟩=0,∀t. A more detailed investigation of the drag on a sphere which is forced to oscillate in a fluid shows that a non-Markovian generalisation (see §2.4), of the Langevin equation (Langevin, 1908) is required to describe the time dependent drag on a rapidly oscillating sphere,

(4.2)

In this case the viscous drag on the sphere is not simply linearly proportional to the instantaneous velocity of the sphere as in (4.1). Instead it is linearly proportional to the velocity at all previous times in the past. As we will see there are many transport processes which can be described by an equation of this form. We will refer to the equation

(4.3)

as the generalised Langevin equation for the phase variable A(Γ). K(t) is the time dependent transport coefficient that we seek to evaluate. We assume that the equilibrium canonical ensemble average of the random force and the phase variable A, vanishes for all times .

, (4.4)

The time displacement by t₀ is allowed because the equilibrium time correlation function is independent of the time origin. Multiplying both sides of (4.3) by the complex conjugate of A(0) and taking a canonical average we see that,

(4.5)

where C(t) is defined to be the equilibrium autocorrelation function,

(4.6)

Another function we will find useful is the flux autocorrelation function f(t)

(4.7)

Taking a Laplace transform of (4.5) we see that there is a intimate relationship between the transport memory kernel K(t) and the equilibrium fluctuations in A. The left-hand side of (4.5) becomes

and as the right-hand side is a Laplace transform convolution,

(4.8)

So that

(4.9)

One can convert the A autocorrelation function into a flux autocorrelation function by realising that,

Then we take the Laplace transform of a second derivative to find,

(4.10)

Here we have used the result that Ċ(0)-0. Eliminating between equations (4.9) and (4.10) gives

(4.11)

Rather than try to give a general interpretation of this equation it may prove more useful to apply it to the Brownian motion problem. C(0) is the time zero value of an equilibrium time correlation function and can be easily evaluated as k_BT∕m, and where F is the total force on the Brownian particle.

(4.12)

where

(4.13)

is the Laplace transform of the total force autocorrelation function. In writing (4.13) we have used the fact that the equilibrium ensemble average denoted ⟨...⟩, must be isotropic. The average of any second rank tensor, say ⟨F(0)F(t)⟩, must therefore be a scalar multiple of the second rank identity tensor. That scalar must of course be ⅓Tr{⟨F(0)F(t)⟩}=⅓⟨F(0)⋅F(t)⟩.

In the so-called Brownian limit where the ratio of the Brownian particle mass to the mean square of the force becomes infinite,

(4.14)

For any finite value of the Brownian ratio, equation (4.12) shows that the integral of the force autocorrelation function is zero. This is seen most easily by solving equation (4.12) for C^F and taking the limit as s→0.

Equation (4.9), which gives the relationship between the memory kernel and the force autocorrelation function, implies that the velocity autocorrelation function Z(t)≡⅓⟨v(0)⋅v(t)⟩ is related to the friction coefficient by the equation,

(4.15)

This equation is valid outside the Brownian limit. The integral of the velocity autocorrelation function, is related to the growth of the mean square displacement giving yet another expression for the friction coefficient,

(4.16)

Here the displacement vector ∆r(t) is defined by

(4.17)

Assuming that the mean square displacement is linear in time, in the long time limit, it follows from (4.15) that the friction coefficient can be calculated from

(4.18)

This is the Einstein (1905) relation for the diffusion coefficient D.

It should be pointed out that the transport properties we have just evaluated are properties of systems at equilibrium. The Langevin equation describes the irregular Brownian motion of particles in an equilibrium system. Similarly the self diffusion coefficient characterises the random walk executed by a particle in an equilibrium system. The identification of the zero frequency friction coefficient 6πηd∕m, with the viscous drag on a sphere which is forced to move with constant velocity through a fluid, implies that equilibrium fluctuations can be modelled by nonequilibrium transport coefficients, in this case the shear viscosity of the fluid. This hypothesis is known as the Onsager regression hypothesis (Onsager, 1931). The hypothesis can be inverted: one can calculate transport coefficients from a knowledge of the equilibrium fluctuations. We will now discuss these relations in more detail.