We shall now address the important question of how the various linear susceptibilities described in Table 5.1, relate to one another. For simplicity let us assume that the initial unperturbed ensemble is canonical. In this case the only difference between the adiabatic, the isothermal, the isoenergetic and the Nosé susceptibilities is in the respective field free propagators used to generate the equilibrium time correlation functions. We will now discuss the differences between the adiabatic and isothermal responses, however the analysis of the other cases involve similar arguments. Without loss of generality we shall assume that the dissipative flux J and the response phase variable B are both extensive and have mean values which vanish at equilibrium. The susceptibility is of order N.

The only difference between (T.5.1) and (T.5.2) is in the time propagation of the phase variable B,

(5.80)

and

(5.81)

In equations (5.80) and (5.81) the Liouvillean iL_N is the Newtonian Liouvillean, and iL_T is the Gaussian isokinetic Liouvillean obtained from the equations of motion (5.23), with α given by the F_e→0 limit of equation (5.20). In both cases there is no explicit time dependence in the Liouvillean. We note that the multiplier α, is intensive.

We can now use the Dyson equation (3.102), to calculate the difference between the isothermal and adiabatic susceptibilities for the canonical ensemble. If ⇒ denotes the isothermal propagator and → the Newtonian, the difference between the two relevant equilibrium time correlation functions is

(5.82)

where we have used the Dyson equation (3.102). Now the difference between the isothermal and Newtonian Liouvillean is

(5.83)

Thus

(5.84)

where α is the field-free Gaussian multiplier appearing in the isothermal equation of motion. We assume that it is possible to define a new phase variable B' by

(5.85)

This is a rather unusual definition of a phase variable, but if B is an analytic function of the momenta, then an extensive phase variable B' always exists. First we calculate the average value of B'(t).

(5.86)

Unless B is trivially related to the kinetic energy K, ⟨B'(t_N)⟩=0. Typically B will be a thermodynamic flux such as the heat flux vector or the symmetric traceless part of the pressure tensor. In these cases ⟨B'(t_N)⟩ vanishes because of Curie's Principle (§2.3).

Assuming, without loss of generality, that ⟨B(t_N)⟩=0, then we can show ,

(5.87)

This is because ⟨J⟩=⟨α⟩=0. Because J, B and B' are extensive and α is intensive, equation (5.87) can be expressed as the product of three zero mean extensive quantities divided by N. The average of three local, zero mean quantities is extensive, and thus the quotient is intensive. Therefore, except in the case where B is a scalar function of the kinetic energy, the difference between the susceptibilities computed under Newton's equations and under Gaussian isokinetic equations, is of order 1∕N compared to the magnitude of the susceptibilities themselves. This means that in the large system limit the adiabatic and isokinetic susceptibilities are equivalent. Similar arguments can be used to show the thermodynamic equivalence of the adiabatic and Nosé susceptibilities. It is pleasing to be able to prove that the mechanical response is independent of the thermostatting mechanism and so only depends upon the thermodynamic state of the system.

Two further comments can be made at this stage: firstly, there is a simple reason why the differences in the respective susceptibilities is significant in the case where B is a scalar function of the kinetic energy. This is simply a reflection of the fact that in this case B, is intimately related to a constant of the motion for Gaussian isokinetic dynamics. One would expect to see a difference in the susceptibilities in this case. Secondly, in particular cases one can use Dyson decomposition techniques, (in particular equation (3.107)), to systematically examine the differences between the adiabatic and isokinetic susceptibilities. Evans and Morriss (1984) used this approach to calculate the differences, evaluated using Newtonian and isokinetic dynamics, between the correlation functions for each of the Navier-Stokes transport coefficients. The results showed that the equilibrium time correlation functions for the shear viscosity, for the self diffusion coefficient and for the thermal conductivity and independent of thermostatting in the large system limit.