The Transient Time Correlation Function formalism (TTCF), provides perhaps the simplest nonlinear generalisation of the Green-Kubo relations. A number of authors independently derived the TTCF expression for adiabatic phase averages, (W. M. Visscher, 1974, Dufty and Lindenfeld, 1979 and Cohen, 1983). We will illustrate the derivation for isokinetic planar Couette flow. However the formalism is quite general and can easily be applied to other systems. The theory gives an exact relation between the nonlinear steady state response and the so-called transient time correlation functions. We will also describe the links between the TTCF approach and the Kawasaki methods outlined in §7.2. Finally, we will present some numerical results which were obtained as tests of the validity of the TTCF formalism.

Following Morriss and Evans, (1987), we will give our derivation using the Heisenberg, rather than the customary Schrödinger picture. The average of a phase variable, B(Γ), at time t, is,

(7.26)

where the second equality is a consequence of the Schrödinger-Heisenberg equivalence. For time independent external fields, differentiating the Heisenberg form with respect to time yields,

(7.27)

In deriving (7.27) we have used the fact that, Ḃ(t)=iLexp[iLt]B=exp[iLt]iLB. This relies upon the time independence of the Liouvillean, iL. The corresponding equation for the time dependent case, is not true. Integrating (7.27) by parts we see that,

(7.28)

The boundary term vanishes because: the distribution function f(0), appoaches zero when the magnitude of any component of any particle’s momentum becomes infinite, and because the distribution function can be taken to be a periodic function of the particle coordinates. We are explicitly using the periodic boundary conditions used in computer simulations.

Integrating (7.28) with respect to time we see that the nonlinear nonequilibrium response can be written as,

(7.29)

The dynamics implicit in B(s), is of course driven by the full field-dependent, thermostatted equations of motion ((7.1) and (7.2)). For a system subject to the thermostatted shearing deformation, is given by the thermostatted SLLOD equations of motion, (6.44).

If the initial distribution is Gaussian isokinetic it is straightforward to show that, . If the initial ensemble is canonical then, to first order in the number of particles, is βVP_xyf(0). To show this one writes, (following §5.3),

(7.30)

where P^K_xy(0) is the kinetic part of the pressure tensor evaluated at time zero (compare this with the linear theory given in §5.3). Now we note that ⟨P^K_xy(0)∆K(0)∕⟨K⟩⟩_c=0. This means that equation (7.30) can be written as,

(7.31)

As in the linear response case (§5.3), we assume, without loss of generality, that B(Γ) is extensive. The kinetic fluctuation term involves the average of three zero mean, extensive quantities and because of the factor 1∕⟨K(0)⟩, gives only an order one contribution to the average. Thus for both the isokinetic and canonical ensembles, we can write,

(7.32)

This expression relates the non-equilibrium value of a phase variable B at time t, to the integral of a transient time correlation function (the correlation between P_xy in the equilibrium starting state, P_xy(0), and B at time s after the field is turned on). The time zero value of the transient correlation function is an equilibrium property of the system. For example, if B=P_xy,then the time zero value is ⟨P²_xy(0)⟩. Under some, but by no means all circumstances, the values of B(s) and P_xy(0) will become uncorrelated at long times. If this is the case the system is said to exhibit mixing. The transient correlation function will then approach ⟨B(t)⟩⟨P_xy(0)⟩, which is zero because ⟨P_xy(0)⟩=0.

The adiabatic systems treated by Visscher, Dufty, Lindenfeld and Cohen do not exhibit mixing because in the absence of a thermostat, does not, in general, go to zero at large times. Thus the integral of the associated transient correlation function does not converge. This presumably means that the initial fluctuations in adiabatic systems are remembered forever. Other systems which are not expected to exhibit mixing are turbulent systems or systems which execute quasi-periodic oscillations.

If AIΓ (§5.3) is satisfied, the result for the general case is,

(7.33)

We can use recursive substitution to derive the Kawasaki form for the nonlinear response from the transient time correlation formula, equation (7.33). The first step in the derivation of the Kawasaki representation is to rewrite the TTCF relation using iL to denote the phase variable Liouvillean, and -iL to denote its nonhermitian adjoint, the f-Liouvillean. Thus and ∂f∕∂t=-iLf. Using this notation equation (7.33) can be written as,

(7.34)

(7.35)

where we have unrolled the first p-propagator onto the distribution function. Equation (7.3.10) can be written more simply as,

(7.36)

Since this equation is true for all phase variables B, the TTCF representation for the N-particle distribution function must be,

(7.37)

We can now successively substitute the transient correlation function expression for the nonequilibrium distribution function into the right hand side of (7.37). This gives,

(7.38)

This is precisely the Kawasaki form of the thermostatted nonlinear response. This expression is valid for both the canonical and isokinetic ensembles. It is also valid for the canonical ensemble when the thermostatting is carried out using the Nosé-Hoover thermostat.

One can of course also derive the TTCF expression for phase averages from the Kawasaki expression. Following Morriss and Evans, (1985) we simply differentiate the (7.38) with respect to time, and then reintegrate.

(7.39)

A simple integration of (7.39) with respect to time yields the TTCF relation (7.33). We have thus proved the formal equivalence of the TTCF and Kawasaki representations for the nonlinear thermostatted response.

Comparing the transient time correlation expression for the nonlinear response with the Kawasaki representation, we see that the difference simply amounts to a time shift. In the transient time correlation form, it is the dissipative flux J, which is evaluated at time zero whereas in the Kawasaki form, the response variable B, is evaluated at time zero. For equilibrium or steady state time correlation functions the stationarity of averages means that such time shifts are essentially trivial. For transient response correlation functions there is of course no such invariance principle, consequently the time translation transformation is accordingly more complex.

The computation of the time dependent response using the Kawasaki form directly, equation (7.32), is very difficult. The inevitable errors associated with the inaccuracy of the trajectory, as well as those associated with the finite grid size in the calculation of the extensive Kawasaki integrand, combine and are magnified by the exponential. This exponential is then multiplied by the phase variable B(0), before the ensemble average preformed. In contrast the calculation of the response using the transient correlation expression, equation (7.3.8), is as we shall see, far easier.

It is trivial to see that in the linear regime both the TTCF and Kawasaki expressions reduce to the usual Green-Kubo expressions. The equilibrium time correlation functions that appear in Green-Kubo relations are generated by the field free thermostatted equations. In the TTCF formulae the field is ‘turned on’ at t=0.

The coincidence at small fields, of the Green-Kubo and transient correlation formulae means that unlike direct NEMD, the TTCF method can be used at small fields. This is impossible for direct NEMD because in the small field limit the signal to noise ratio goes to zero. The signal to noise ratio for the transient correlation function method becomes equal to that of the equilibrium Green-Kubo method. The transient correlation function method forms a bridge between the Green-Kubo method which can only be used at equilibrium, and direct NEMD which is the most efficient strong field method. Because a field is required to generate TTCF correlation functions, their calculation using a molecular dynamics, still requires a nonequilibrium computer simulation to be performed.

It is also easy to see that at short times there is no difference between the linear and nonlinear stress response. It takes time for the nonlinearities to develop. The way to see this is to expand the transient time correlation function in a power series in γt. The coefficient of the first term in this series is just v⟨P²_xy⟩∕k_BT, the infinite frequency shear modulus, G_∞. Since this is an equilibrium property its value is unaffected by the strain rate and is thus the same in both the linear and nonlinear cases. If we look at the response of a quantity like the pressure whose linear response is zero, the leading term in the short time expansion is quadratic in the strain rate and in time. The linear response of course is the first to appear.