The Irving-Kirkwood procedure has given us microscopic expressions for the thermodynamic fluxes in terms of ensemble averages. At equilibrium in a uniform fluid, the Irving-Kirkwood expression for the pressure tensor is the same expression as that derived using Gibbs' ensemble theory for equilibrium statistical mechanics. If the fluid density is uniform in space, the O_ij operator appearing in the above expressions reduces to unity. This is easier to see if we calculate microscopic expressions for the fluxes in k-space rather than real space. In the process we will better understand the nature of the Irving-Kirkwood expressions.

In this section we derive instantaneous expressions for the fluxes rather than the ensemble based, Irving-Kirkwood expressions. The reason for considering instantaneous expressions is two-fold. The fluxes are based upon conservation laws and these laws are valid instantaneously for every member of the ensemble. They do not require ensemble averaging to be true. Secondly, most computer simulation involves calculating system properties from a single system trajectory. Ensemble averaging is almost never used because it is relatively expensive in computer time. The ergodic hypothesis, that the result obtained by ensemble averaging is equal to that obtained by time averaging the same property along a single phase space trajectory, implies that one should be able to develop expressions for the fluxes which do not require ensemble averaging. For this to be practically realisable it is clear that the mass, momentum and energy densities must be definable at each instant along the trajectory.

We define the Fourier transform pair by

      (3.134)

In the spirit of the Irving-Kirkwood procedure we define the instantaneous r-space mass density to be,

(3.135)

where the explicit time dependence of ρ(r,t) (that is the time dependence differentiated by the hydrodynamic derivative ∂∕∂t, with r fixed) is through the time dependence of r_i(t). The k-space instantaneous mass density is then

(3.136)

We will usually allow the context to distinguish whether we are using ensemble averages or instantaneous expressions. The time dependence of the mass density is solely through the time dependence of r_i, so that

(3.137)

Comparing this with the Fourier transform of (2.4) (noting that d∕dt|_k in (3.137) corresponds to ∂∕∂t|_r in (2.4)) we see that if we let J(r,t)=ρ(r,t)u(r,t) then,

(3.138)

This equation is clearly the instantaneous analogue of the Fourier transform of the Irving-Kirkwood expression for the momentum density. There is no ensemble average required in (3.137). To look at the instantaneous pressure tensor we only need to differentiate equation (3.138) in time.

(3.139)

We can write the second term on the right hand side of this equation in the form of the Fourier transform of a divergence by noting that,

(3.140)

Combining (3.139) and (3.140) and performing an inverse Fourier transform we obtain the instantaneous analogue of equation (3.123). We could of course continue the analysis of §3.7 to remove the streaming contribution from the pressure tensor but this is more laborious in k-space than in real space and we will not give this here. We can use our instantaneous expression for the pressure tensor to describe fluctuations in an equilibrium system. In this case the streaming velocity is of course zero, and

(3.141)

The k-space analysis given provided a better understanding of the Irving-Kirkwood operator O_ij. In k-space it is not necessary to perform the apparently difficult operation of Taylor expanding delta functions.

Before we close this section we will try to make the equation for the momentum density, J(r,t)=ρ(r,t)u(r,t), a little clearer. In k-space this equation is a convolution,

(3.142)

Does this definition of the streaming velocity u, make good physical sense? One sensible definition for the streaming velocity u, would be that velocity which minimises the sum of squares of deviations from the particle velocities v_i. For simplicity we set t=0, and let R, be that sum of squares,

(3.143)

If u(r) minimises this sum of squares then the derivative of R with respect to each of the Fourier components u(k_m), must be zero. Differentiating (3.143) we obtain,

(3.144)

This implies that

(3.145)

Both sides of this equation can be identified as k-space variables,

(3.146)

So that

(3.147)

This is the Fourier series version of equation (3.142).

We can use the same procedure to calculate an expression for the heat flux vector. As we will see this procedure is very much simpler than the Irving-Kirkwood method described in §3.7. We begin by identifying the instantaneous expression for the instantaneous wavevector dependent energy density in a fluid at equilibrium,

(3.148)

This is instantaneous, wavevector dependent analogue of (3.112). To simplify notation in the following we will suppress the time argument for all phase variables. The time argument will always be t. If we calculate the rate of change of the energy density we find,

(3.149)

Where we use the notation Φ_i=½∑_jΦ_ij. If we denote the energy of particle i as e_i and F_ij as the force exerted on particle i due to j then (3.149) can be rewritten as,

(3.150)

This equation involves the same combination of exponents as we saw for the pressure tensor in (3.140). Expanding exponentials to first order in k, and using equation (2.24) we find that the wavevector dependent heat flux vector can be written as

(3.151)

In r-space rather than k-space the expressions for the instantaneous pressure tensor and heat flux vector become,

(3.152)

(3.153)

Our procedure for calculating microscopic expressions for the hydrodynamic densities and fluxes relies upon establishing a correspondence between the microscopic and macroscopic forms of the continuity equations. These equations refer only to the divergence of the pressure tensor and heat flux. Strictly speaking therefore we can only determine the divergences of the flux tensors. We can add any divergence free quantity to our expressions for the flux tensors without affecting the identification process.