We shall often refer to averages over equilibrium distribution functions f₀ (we use the subscript zero to denote equilibrium, which should not be confused with f(0), a distribution function at t=0). Distribution functions are called equilibrium if they pertain to steady, unperturbed equations of motion and they have no explicit time dependence. An equilibrium distribution function satisfies a Liouville equation of the form

(3.79)

This implies that the equilibrium average of any phase variable is a stationary quantity. That is, for an arbitrary phase variable B,

(3.80)

We will often need to calculate the equilibrium time correlation function of a phase variable A with another phase variable B at some other time. We define the equilibrium time correlation function of A and B by

(3.81)

where B* denotes the complex conjugate of the phase variable B. Sometimes we will refer to the autocorrelation function of a phase variable A. If this variable is real, one can form a simple graphical representation of how such functions are calculated (see Fig. 3.4).

Because the averages are to be taken over a stationary equilibrium distribution function, time correlation functions are only sensitive to time difference between which A and B are evaluated. C_AB(t) is independent of the particular choice of the time origin. If iL generates the distribution function f₀, then the propagator exp(-iLt) preserves f₀. (The converse is not necessarily true.) To be more explicit f₀(t₁)=exp(-iLt₁)f₀=f₀, so that C_AB(t) becomes

for samples in the sum to be independent, τ should be chosen so that

 

(3.82)

In deriving the last form of (3.82) we have used the important fact that since iL=dΓ∕dt•∂∕∂Γ and the equations of motion are real it follows that L is pure imaginary. Thus, (iL)*=iL and (e^iLt)*=e^iLt. Comparing (3.82) with the definition of C_AB(t), above we see that the equilibrium time correlation function is independent of the choice of time origin. It is solely a function of the difference in time of the two arguments, A and B. A further identity follows from this result if we choose t₁=-t. We find that

(3.83)

So that,

(3.84)

or using the notation of section 3.3,

(3.85)

The second equality in equation (3.85) follows by expanding the operator exp(-iLt) and repeatedly applying the identity

The term iLf₀ is zero from equation (3.79).

Over the scalar product defined by equation (3.81), L is an Hermitian operator. The Hermitian adjoint of L denoted, L^◊ can be defined by the equation,

(3.86)

Comparing (3.86) with (3.85) we see two things: we see that the Liouville operator L is self adjoint or Hermitian (L=L^†); and therefore the propagator e^iLt, is unitary. This result stands in contrast to those of §3.3, for arbitrary distribution functions.

We can use the autocorrelation function of A to define a norm in Liouville space. This length or norm of a phase variable A, is defined by the equation,

(3.87)

We can see immediately that the norm of any phase variable is time independent because

(3.88)

The propagator is said to be norm preserving (Fig. 3.5). This is a direct result of the fact that the propagator is a unitary operator. The propagator can be thought of as a rotation operator in Liouville space.

A phase variable whose norm is unity is said to be normalised. The scalar product, (A,B^*) of two phase variables A, B is simply the equilibrium average of A and B^* namely <AB^*>₀. The norm of a phase variable is simply the scalar product of the variable with itself. The autocorrelation function C_AA(t) has a zero time value which is equal to the norm of A. The propagator increases the angle between A^* and A(t), and the scalar product which is the projection of A(t) along A^*, therefore decreases. The autocorrelation function of a given phase variable therefore measures the rate at which the 6N-dimensional rotation occurs.

We will now derive some relations for the time derivatives of time correlation functions. It is easy to see that

(3.90)