Norton and Thévenin's theorems are of fundamental importance in electrical circuit theory (Brophy, 1966). They prove that any network of resistors and power supplies can be analysed in terms of equivalent circuits which include either ideal current or ideal voltage sources. These two theorems are an example of the macroscopic duality that exists between what are generally recognised as thermodynamic fluxes and thermodynamic forces - in the electrical circuit case, electrical currents and the electromotive force. Indeed in our earlier introduction to linear irreversible thermodynamics (Chapter 2), there was an apparent arbitrariness with respect to our definition of forces and fluxes. At no stage did we give a convincing macroscopic distinction between the two.
Microscopically one might think that there is a clear and unambiguous distinction that can be drawn. For an arbitrary mechanical system subject to a perturbing external field the dissipation can be written as, dH0 ad∕dt ≡ -J(Γ)Fe(t). The dissipative flux is the phase variable J(Γ) and the force is the time dependent independent variable, Fe(t).This might seem to remove the arbitrariness. However, suppose that we complicate matters a little and regard the external field Fe(t), as a Gaussian multiplier in a feedback scheme designed to stop the flux J(Γ), from changing. We might wish to perform a constant current simulation. In this case the imposed external field Fe(t), is in fact a phase variable, Fe(t). Even microscopically the distinction between forces and fluxes is more complex than is often thought.
In this section we will explore the statistical mechanical consequences of this duality. Until recently the Green-Kubo relations were only known for the conventional Thévenin ensemble in which the forces are the independent state defining variables. We will derive their Norton ensemble equivalents. We will then show how these ideas have been applied to algorithms for isobaric molecular dynamics simulations. This work will provide the necessary background for the derivations, in Chapter 9, of fluctuation expressions for the derived properties of nonequilibrium steady states including the nonlinear inverse Burnett coefficients.
From the colour Hamiltonian (6.9) we see that the equations of motion for colour conductivity in the Thévenin ensemble are,
(6.65)
These equations are the adiabatic version of (6.18 & 6.19). We will now treat the colour field as a Gaussian multiplier chosen to fix the colour current and introduce a thermostat.
Our first step is to redefine the momenta (Evans and Morriss, 1985), so that they are measured with respect to the species current of the particles. Consider the following set of equations of motion
(6.66)
where α is the thermostatting multiplier and λ is the current multiplier. These equations are easily seen to be equivalent to (6.18) and (6.19). We distinguish two types of current, a canonical current J defined in terms of the canonical momenta,
(6.67)
and a kinetic current I, where
(6.68)
We choose λ so that the canonical current is always zero, and α so that the canonical (ie. peculiar) kinetic energy is fixed. Our constraint equations are therefore,
(6.69)
and
(6.70)
The Gaussian multipliers may be evaluated in the usual way by summing moments of the equations of motion and eliminating the accelerations using the differential forms of the constraints. We find that
(6.71)
and
(6.72)
If we compare the Gaussian equations of motion with the corresponding Hamiltonian equations we see that the Gaussian multiplier λ can be identified as a fluctuating external colour field which maintains a constant colour current. It is however, a phase variable. Gauss' principle has enabled us to go from a constant field nonequilibrium ensemble to the conjugate ensemble where the current is fixed. The Gaussian multiplier fluctuates in the precise manner required to fix the current. The distinction drawn between canonical and kinetic currents has allowed us to decouple the Lagrange multipliers appearing in the equations of motion. Furthermore setting the canonical current to zero is equivalent to setting the kinetic current to the required value I. This can be seen by taking the charge moment of (6.66). If the canonical current is zero then,
(6.73)
In this equation the current, which was formerly a phase variable has now become a possibly time dependent external force.
In order to be able to interpret the response of this system to the external current field, we need to compare the system's equations of motion with a macroscopic constitutive relation. Under adiabatic conditions the second order form of the equations of motion is
(6.74)
We see that to maintain a constant current I(t) we must apply a fluctuating colour field E eff,
(6.75)
The adiabatic rate of change of internal energy H0 is given by
(6.76)
As the current, J =J(Γ) is fixed at the value zero, the dissipation is -I(t)•λ(Γ). As expected the current is now an external time dependent field while the colour field is a phase variable. Using linear response theory we have
(6.77)
which gives the linear response result for the phase variable component of the effective field. Combining (6.77) with (6.75) the effective field is, therefore,
(6.78)
where the susceptibility χ is the equilibrium λ autocorrelation function,
(6.79)
By doing a Fourier-Laplace transform on (6.78) we obtain the frequency dependent colour resistance, E=RI
(6.80)
To compare with the usual Green-Kubo relations which have always been derived for conductivities rather than resistances we find,
(6.81)
This equation shows that the Fourier-Laplace transform of χ(t) is the memory function of the complex frequency dependent conductivity. In the conjugate constant force ensemble the frequency dependent conductivity is related to the current autocorrelation function
(6.82)
From equations (6.79) - (6.82) we see that at zero frequency the colour conductivity is given by the integral of the Thévenin ensemble current correlation function while the resistance, which is the reciprocal of the conductivity, is given by the integral of the colour field autocorrelation function computed in the Norton ensemble. Thus at zero frequency the integral of the Thévenin ensemble current correlation function is the reciprocal of the integral of the Norton ensemble field correlation function. Figure 6.2 gave a comparison of Norton and Thévenin algorithms for computing the colour conductivity. The results obtained for the conductivity are ensemble independent - even in the nonlinear regime far from equilibrium.
In Figure 6.13 we show the reduced colour conductivity plotted as a function of frequency (Evans and Morriss, 1985). The system is identical to the Lennard-Jones system studied in Figure 6.2. The curves were calculated by taking the Laplace transforms of the appropriate equilibrium time correlation functions computed in both the Thévenin and Norton ensembles. Within statistical uncertainties, the results are in agreement. The arrow shows the zero frequency colour conductivity computed using NEMD. The value is taken from Figure 6.2.