In Chapter 2 we gave a brief outline of the structure of macroscopic hydrodynamics. We saw that given appropriate boundary conditions, it is possible to use the Navier-Stokes equations to describe the resulting macroscopic flow patterns. In this chapter we began the microscopic description of nonequilibrium systems using the Liouville equation. We will now follow a procedure first outlined by Irving and Kirkwood (1950), to derive microscopic expressions for the thermodynamic forces and fluxes appearing in the phenomenological equations of hydrodynamics.

In our treatment of the macroscopic equations we stressed the role played by the densities of conserved quantities. Our first task here will be to define microscopic expressions for the local densities of mass, momentum and energy. If the mass of the individual atoms in our system is m then the mass per unit volume at a position r and time t can be obtained by taking an appropriate average over the normalised N-particle distribution function f(Γ,t). To specify that the particles should be at the macroscopic position r, we will use a macroscopic delta function, δ(r-r_i). This macroscopic delta function is zero if atom i is outside some microscopic volume δV; it is a constant if atom i is inside this volume (δ is a smeared out version of the usual point delta function). We will assume that particle dynamics are given by field-free Newtonian equations of motion. The value of the constant is determined from the normalisation condition,

(3.109)

The volume V is envisioned to be infinitesimal on a macroscopic scale.

The mass density ρ(r,t) can be calculated from the following average,

(3.110)

The first line of this equation is a Schrödinger representation of the density while the second and third lines are written in the Heisenberg representation. The equivalence of these two representations is easily seen by 'unrolling' the propagator from the distribution function onto the phase variable. Since r, is a constant, a nominated position it is unchanged by this 'unrolling' procedure.

The momentum density, ρ(r,t)u(r,t), and total energy density, ρ(r,t)e(r,t), are defined in an analogous manner.

(3.111)

(3.112)

In these equations v_i is the velocity of particle i, p_i is its momentum, r_ij ≡r_j- r_i, and we assume that the total potential energy of the system, Φ is pair-wise additive and can be written as,

(3.113)

We arbitrarily assign one half of the potential energy to each of the two particles which contribute Φ_ij to the total potential energy of the system.

The conservation equations involve time derivatives of the averages of the densities of conserved quantities. To begin, we will calculate the time derivative of the mass density.

(3.114)

The fifth equality follows using the delta function identity,

We have shown that the time derivative of the mass density yields the mass continuity equation (2.4) as expected. Strictly speaking therefore, we did not really need to define the momentum density in equation (3.111), as given the mass density definition, the mass continuity equation yields the momentum density expression. We will now use exactly the same procedure to differentiate the momentum density.

(3.115)

We have used Newtonian equations of motion for the Liouvillean iL.

If we consider the second term on the right-hand side then

(3.116)

In the final term in equation (3.116), u(r,t) is independent of the particle index and can be factored outside the summation. The remaining summation is, using equation (3.110), simply equal to the mass density ρ(r,t). Combining these results it follows that

(3.117)

We will now consider the first term on the right hand side of this equation in some detail.

(3.118)

Treating themacroscopicdelta function as an analytic function, we may expand δ(r-r_j) as a Taylor series about δ(r-r_i). This gives

(3.119)

Thus the difference between the two delta functions is

(3.120)

where the operator O_ij is given by,

(3.121)

Using this equation for the difference of the two delta functions δ(r-r_i) and δ(r-r_j) leads to

(3.122)

Comparing this equation with the momentum conservation equation (2.12) we see that the pressure tensor is,

(3.123)

where F_ij=-∂Φ_ij∕∂r_i is the force on particle i due to particle j.

We will now use the same technique to calculate the microscopic expression for the heat flux vector. The partial time derivative of the energy density is (from equation (3.112))

(3.124)

In the second term, the gradient operator ∂∕∂r is contracted into v_i. Using our previous result for the difference of two delta functions, equation (3.120), gives

(3.125)

From equation (2.24) we conclude that,

(3.126)

Now the definition of the energy density, equation (3.112) gives

(3.127)

so that,

(3.128)

Similarly, from the definition of the pressure tensor P(r,t) (see equation (3.123)), we know that

(3.129)

thus we identify the heat flux vector as,

(3.130)

From the definitions of the mass density and momentum density (equations (3.110) and (3.111)) we find that

(3.131)

so there is no contribution from the u² term. Further, if we define the peculiar energy of particle i to be

(3.132)

then the heat flux vector can be written as

or,

(3.133)