Nonlinear Response Theory

In Chapter 6 we saw that nonequilibrium molecular dynamics leads inevitably to questions regarding the nonlinear response of systems. In this chapter we will begin a discussion of this subject.

It is not widely known that in Kubo's original 1957 paper (Kubo, 1957), he not only presented results for adiabatic linear response theory, but that he also included a formal treatment of the adiabatic nonlinear response. The reason why this fact is not widely known is that, like many treatments of nonlinear response theory that followed, his formal results were exceedingly difficult to translate into a useful, experimentally verifiable forms. This difficulty can be traced to three sources. Firstly, his results are not easily transformable into explicit representations that involve the evaluation of time correlation functions of explicit phase variables. Secondly, if one wants to study nonequilibrium steady states, the treatment of thermostats is mandatory. His theory did not include such effects. Thirdly, his treatment gave a power series representation of the nonlinear response. We now believe that for most transport processes, such expansions do not exist.

We will now give a presentation of Kubo's perturbation expansion for the nonequilibrium distribution function, f(t). Consider an N-particle system evolving under the following dynamics,

(7.1)

The terms C_i(Γ) and D_i(Γ) describe the coupling of the external field F_e to the system. In this discussion we will limit ourselves to the case where the field is switched on at time zero, and thereafter remains at the same steady value. The f-Liouvillean is given by

(7.2)

where iL₀ is the equilibrium Liouvillean and i∆L* is the field dependent perturbation which is a linear function of F_e. The Liouville equation is,

(7.3)

To go beyond the linear response treated in §5.1, Kubo assumed that f(t) could be expanded as a power series in the external field, F_e,

(7.4)

where, f_i(t) is i^th order in the external field F_e The assumption that f(t) can be expanded in a power series about F_e=0 may seem innocent, but it is not. This assumption rules out any functional form containing a term of the form, F_e^α, where α is not an integer. Substituting (7.4) for f(t), and the expression for iL , into the Liouville equation (7.3), and equating terms of the same order in F_e, we find an infinite sequence of partial differential equations to solve,

(7.5)

where i≥1. The solution to this series of equations can be written as,

(7.6)

To prove that this is correct, one differentiates both sides of the equation to obtain (7.5). Recursively substituting (7.6), into equation (7.4), we obtain a power series representation of the distribution function

(7.7)

Although this result is formally exact, there are a number of difficulties with this approach. The expression for f(t) is a sum of convolutions of operators. In general the operator i∆L* does not commute with the propagator, exp[iL₀t], and no further simplifications of the general result are possible. Further, as we have seen in Chapter 6, there is a strong likelihood that fluxes associated with conserved quantities are non-analytic functions of the thermodynamic force, F_e. This would mean that the average response of the shear stress, for example, cannot be expanded as a Taylor series about F_e(=γ)=0. In Chapter 6 we saw evidence that the shear stress is of the form, ⟨P_xy⟩=-γ(η₀+n₁γ^½) (see §6.3). If this is true then f₂(t)≡½γ²(∂²f(γ)∕∂γ²)γ=0 must be infinite.