In §5.2 we considered two forms of thermostatted dynamics - the Gaussian isokinetic dynamics and the Nosé-Hoover canonical ensemble dynamics. Both of these thermostatted equations of motion can add or remove energy from the system to control its temperature. It is particularly important to incorporate thermostatted dynamics when the system is perturbed by an external field. This allows the irreversibly produced heat to be removed continuously, and the system maintained in a steady, nonequilibrium state. We now generalise the adiabatic linear response theory of §5.1, to treat perturbed thermostatted systems we have developed in §5.2. We consider (Morriss and Evans, 1985) an N-particle system evolving under the Gaussian isokinetic dynamics for t<0, but subject for to an external field F_e, for all times t>0. The equations of motion are given by

(5.66)

The term αp_i couples the system to a thermostat and we shall take

(5.67)

so that the peculiar kinetic energy, K(Γ)=∑_ip²_i∕2m=K₀, is a constant of the motion. In the absence of the field these equations of motion ergodically generate the isokinetic distribution function, f_T, equation (5.28), with β=3N∕2K₀. As we have seen, the isokinetic distribution function f_T, is preserved by the field free isokinetic equations of motion and that,

(5.68)

we use iL_T for the zero field, isokinetic Liouvillean.

To calculate the linear thermostatted response we need to solve the linearised Liouville equation for thermostatted systems. Following the same arguments used in the adiabatic case (equations (5.8-12)), the linearised Liouville equation is,

(5.69)

where iL(t) is the external field dependent, isokinetic Liouvillean and ∆iL(t)=iL(t)-iL_T. Its solution is the analogue of (5.13), namely

(5.70)

Using equations (5.8), (5.28) and (5.66), and the fact that β=3N∕2K₀, it is easy to show that

(5.71)

There is one subtle point in deriving the last line of (5.71),

(5.72)

The last line follows because K(p) is a constant of the motion for the Gaussian isokinetic equations of motion. We have also assumed that the only contribution to the phase space compression factor comes from the thermostatting term αp_i. This means that in the absence of a thermostat, that is the adiabatic case, the phase space is incompressible and

(5.73)

This assumption or condition, is known as the adiabatic incompressibility of phase space (AIΓ). A sufficient, but not necessary condition for it to hold is that the adiabatic equations of motion should be derivable from a Hamiltonian. It is important to note that AIΓ does not imply that the phase space for the thermostatted system should be incompressible. Rather it states that if the thermostat is removed from the field dependent equations of motion, the phase space is incompressible. It is essentially a condition on the external field coupling terms C_i(q,p) and D_i(q,p). It is not necessary that C_i be independent of q, and D_i be independent of p. Indeed in §6.3 we find that this is not the case for planar Couette flow, but the combination of partial derivatives in equation (5.73) is zero. It is possible to generalise the theory to treat systems where AIΓ does not hold but this generalisation has proved to be unnecessary.

Using equation (5.67) for the multiplier α, to first order in N we have

(5.74)

This equation shows that ∆iL(t)f(Γ) is independent of thermostatting. Equations (5.74) and (5.15) are essentially identical. This is why the dissipative flux J is defined in terms of the adiabatic derivative of the internal energy. Interestingly, the kinetic part of the dissipative flux, J(Γ), comes from the multiplier α, while the potential part comes from the time derivative of Φ.

Substituting (5.74) into (5.70), the change in the isokinetic distribution function is given by

(5.75)

Using this result to calculate the mean value of B(t), the isothermal linear response formula corresponding to equation (5.16), is,

(5.76)

Equation (5.76) is very similar in form to the adiabatic linear response formula derived in §5.1. The notation ⟨⟩_T,0 signifies that a field-free (0), isokinetic (T) ensemble average should be taken. Differences from the adiabatic formula are that;

The last point means that time averaging rather than ensemble averaging can be used to generate the time zero starting states for the equilibrium time correlation function on the right hand side of equation (5.76).

It can be useful, especially for theoretical treatments, to use ergodically inconsistent formulations of linear response theory. It may be convenient to employ canonical rather than isokinetic averaging, for example. For the canonical ensemble, assuming AIΓ, we have in place of equation (5.71),

(5.77)

where ∆dΦ∕dt is the difference between the rate of change of Φ with the external field turned on and with the field turned off (dΦ(F_e)∕dt-dΦ(F_e-0)∕dt). Similarly ∆α=α(F_e)-α(F_e=0)=α₁F_e (see equation 5.67). The response of a phase variable B, is therefore,

(5.78)

Using the same methods as those used in deriving equation (5.35), we can show that if B is extensive, the second integral in equation (5.78) is of order 1 and can therefore be ignored.

Thus for a canonical ensemble of starting states and thermostatted Gaussian isokinetic dynamics, the response of an extensive variable B, is given by

(5.79)

Like the isokinetic ensemble formula, the response, ⟨B(t_T)⟩_c, possesses well defined steady state limit.

It is straightforward to apply the linear response formalism to a wide variety of combinations of statistical mechanical ensembles, and equilibrium dynamics. The resultant susceptibilities are shown in the Table 5.1 below. It is important to appreciate that the dissipative flux J(Γ) is determined by both the choice of equilibrium ensemble of starting states and the choice of the equilibrium dynamics.

‡ Equilibrium dynamics: t_N, Newtonian; t_T, Gaussian Isokinetic; t_I Gaussian isoenthalpic; t_c Nosé-Hoover. Ensemble averaging:- ⟨⟩_c canonical; ⟨⟩_T isokinetic; ⟨⟩_E microcanonical; ⟨⟩_I isoenthalpic.

§ Proof of (T.5.5) can be found in a paper by Holian and Evans (1983).