We will show that for an arbitrary phase variable A(Γ), evolving under equations of motion which preserve the equilibrium distribution function, one can always write down a Langevin equation. Such an equation is an exact consequence of the equations of motion. We will use the symbol iL, to denote the Liouvillean associated with these equations of motion. These equilibrium equations of motion could be field-free Newtonian equations of motion or they could be field-free thermostatted equations of motion such as Gaussian isokinetic or Nosé-Hoover equations. The equilibrium distribution could be microcanonical, canonical or even isothermal-isobaric provided that if the latter is the case, suitable distribution preserving dynamics are employed. For simplicity we will compute equilibrium time correlation functions over the canonical distribution function, fc,
(4.19)
We saw in the previous section that a key element of the derivation was that the correlation of the random force, FR(t) with the Langevin variable A, vanished for all time. We will now use the notation first developed in §3.5, which treats phase variables, A(Γ), B(Γ), as vectors in 6N-dimensional phase space with a scalar product defined by ∫dΓf0(Γ)B(Γ)A*(Γ), and denoted as (B,A*). We will define a projection operator which will transform any phase variable B, into a vector which has no correlation with the Langevin variable, A. The component of B parallel to A is just,
(4.20)
This equation defines the projection operator P.
The operator Q=1-P, is the complement of P and computes the component of B orthogonal to A.
(4.21)
In more physical terms the projection operator Q computes that part of any phase variable which is random with respect to a Langevin variable, A.
Figure 4.1. The projection operator P, operating on B produces a vector which is the component of B parallel to A
Other properties of the projection operators are that,
PP=P, QQ=Q, QP=PQ=0 (4.22)
Secondly, P and Q are Hermitian operators (like the Liouville operator itself). To prove this we note that,
(4.23)
Furthermore, since Q=1-P where 1 is the identity operator, and since both the identity operator and P are Hermitian, so is Q.
We will wish to compute the random and direct components of the propagator eiLt. The random and direct parts of the Liouvillean iL are iQL and iPL respectively. These Liouvilleans define the corresponding random and direct propagators, eiQLt and eiPLt. We can use the Dyson equation to relate these two propagators. If we take eiQLt as the reference propagator in (3.100) and eiLt as the test propagator then,
(4.24)
The rate of change of A(t), the Langevin variable at time t is,
(4.25)
But,
(4.26)
This defines the frequency iΩ which is an equilibrium property of the system. It only involves equal time averages. Substituting this equation into (4.25) gives,
(4.27)
Using the Dyson decomposition of the propagator given in equation (4.24), this leads to,
(4.28)
We identify eiQLtiQLA as the random force F(t) because,
(4.29)
where we have used (4.22). It is very important to remember that the propagator which generates F(t) from F(0) is not the propagator eiLt, rather it is the random propagator eiQLt. The integral in (4.28) involves the term,
as L is Hermitian and i is anti-Hermitian, 
, (since the equations of motion are real). Since Q is Hermitian,
(4.30)
Figure 4.2. Schematic diagram of the frequency- and wavevector dependent viscosity and stress autocorrelation function. We can resolve the wavevector dependent momentum density into components which are parallel and orthogonal to the wavevector, k.
where we have defined a memory kernel K(t). It is basically the autocorrelation function of the random force. Substituting this definition into (4.28) gives
(4.31)
This shows that the Generalised Langevin Equation is an exact consequence of the equations of motion for the system (Mori, 1965a, b; Zwanzig, 1961). Since the random force is random with respect to A, multiplying both sides of (4.31) by A*(0) and taking a canonical average gives the memory function equation,
(4.32)
This is essentially the same as equation (4.5).
As we mentioned in the introduction to this section the generalised Langevin equation and the memory function equation are exact consequences of any dynamics which preserves the equilibrium distribution function. As such the equations therefore describe equilibrium fluctuations in the phase variable A, and the equilibrium autocorrelation function for A, namely C(t).
However the generalised Langevin equation bears a striking resemblance to a nonequilibrium constitutive relation. The memory kernel K(t) plays the role of a transport coefficient. Onsager's regression hypothesis (1931) states that the equilibrium fluctuations in a phase variable are governed by the same transport coefficients as is the relaxation of that same phase variable to equilibrium. This hypothesis implies that the generalised Langevin equation can be interpreted as a linear, nonequilibrium constitutive relation with the memory function K(t), given by the equilibrium autocorrelation function of the random force.
Onsager's hypothesis can be justified by the fact that in observing an equilibrium system for a time which is of the order of the relaxation time for the memory kernel, it is impossible to tell whether the system is at equilibrium or not. We could be observing the final stages of a relaxation towards equilibrium or, we could be simply observing the small time dependent fluctuations in an equilibrium system. On a short time scale there is simply no way of telling the difference between these two possibilities. When we interpret the generalised Langevin equation as a nonequilibrium constitutive relation, it is clear that it can only be expected to be valid close to equilibrium. This is because it is a linear constitutive equation.