In this section we calculate formally exact fluctuation expressions for other derived properties including the specific heat at constant pressure and external field, C_p,Fe, and the compressibility, χ_T,Fe ≡ -∂lnV∕∂p)_T,Fe. The expressions are derived using the isothermal Kawasaki representation for the distribution function of an isothermal isobaric steady state.

The results indicate that the compressibility is related to nonequilibrium volume fluctuations in exactly the same way that it is at equilibrium. The isobaric specific heat, C_p,Fe, on the other hand, is not simply related to the mean square of the enthalpy fluctuations as it is at equilibrium. In a nonequilibrium steady state, these enthalpy fluctuations must be supplemented by the integral of the steady state time cross correlation function of the dissipative flux and the enthalpy.

We begin by considering the isothermal-isobaric equations of motion considered in §6.7. The obvious nonequilibrium generalisation of these equations is,

(9.14)

In the equations dε∕dt is the dilation rate required to precisely fix the value of the hydrostatic pressure, p =Σ (p²∕m + q.F)∕3V. α is the usual Gaussian thermostat multiplier used to fix the peculiar kinetic energy, K. Simultaneous equations must be solved to yield explicit expressions for both multipliers. We do not give these expressions here since they are straightforward generalisations of the field-free (F_e=0), equations given in §6.7.

The external field terms are assumed to be such as to satisfy the usual Adiabatic Incompressibility of Phase Space (AIΓ) condition. We define the dissipative flux, J, as the obvious generalisation of the usual isochoric case.

(9.15)

This definition is consistent with the fact that in the field-free adiabatic case the enthalpy I₀ ≡ H₀ +pV, is a constant of the equations of motion given in (9.14). It is easy to see that the isothermal isobaric distribution, f₀, is preserved by the field-free thermostatted equations of motion.

(9.16)

It is a straightforward matter to derive the Kawasaki form of the N-particle distribution for the isothermal-isobaric steady state. The normalised version of the distribution function is,

(9.17)

The calculation of derived quantities is a simple matter of differentiation with respect to the variables of interest. As was the case for the isochoric specific heat, the crucial point is that the field-dependent isothermal-isobaric propagator implicit in the notation f(t), is independent of the pressure and the temperature of the entire ensemble. This means that the differential operators ∂∕∂T and ∂∕∂p₀ commute with the propagator.

The pressure derivative is easily calculated as,

(9.18)

If we choose B to be the phase variable corresponding to the volume then the expression for the isothermal, fixed field compressibility takes on a form which is formally identical to its equilibrium counterpart.

(9.19)

The limit appearing in (9.19) implies that a steady state average should be taken. This follows from the fact that the external field was 'turned on' at t=0.

The isobaric temperature derivative of the average of a phase variable can again be calculated from (9.17).

(9.20)

In deriving (9.20) we have used the fact that ∫dV∫dΓ f(t) B(0) = < B(t) >. Equation (9.20) can clearly be used to derive expressions for the expansion coefficient. However setting the test variable B to be the enthalpy and remembering that

(9.21)

leads to the isobaric specific heat,

(9.22)

This expression is of course very similar to the expression derived for the isochoric specific heat in §9.2.

In contrast to the situation for the compressibility, the expressions for the specific heats are not simple generalisations of the corresponding equilibrium fluctuation formulae. Both specific heats also involve integrals of steady state time correlation functions involving cross correlations of the appropriate energy with the dissipative flux. Although the time integrals in (9.13) & (9.22) extend back to t=0 when the system was at equilibrium, for systems which exhibit mixing, only the steady state portion of the integral contributes. This is because in such systems, lim(t→∞) <ΔB(t) ΔJ(0)> = <ΔB(t)><ΔJ(0)> = 0. These correlation functions are therefore comparatively easy to calculate in computer simulations.