3.3.2 Elimination of the Lagrange multipliers

The main problem is to eliminate the Lagrange multipliers, the generalized response parameters Fnð Þt . As in the case of kinetic theory, this is also possible explicitly in the case of linear response theory (LRT). With the operator relation eAþ<sup>B</sup> <sup>¼</sup> eA <sup>þ</sup> <sup>Ð</sup> 1 0 dλe<sup>λ</sup>ð Þ <sup>A</sup>þ<sup>B</sup> Beð Þ <sup>1</sup>�<sup>λ</sup> A, we get for the relevant statistical operator (150) up to first order of the nonequilibrium fluctuations Bf g<sup>n</sup>

$$\rho\_{\rm rel}(t) = \rho\_{\rm eq} + \beta \int\_0^1 \mathbf{d}\lambda \sum\_n F\_n(t) \, \mathbf{B}\_n(\mathbf{i}\hbar \beta \lambda) \, \rho\_{\rm eq}. \tag{155}$$

Here, we made use of the modified-Heisenberg picture Oð Þ¼ τ exp ið Þ Hτ=ℏ O exp ð Þ �iHτ=ℏ with τ ! iℏβλ replacing in the exponents HS by H ¼ HS � ∑cμcNc. We want to calculate expectation values of macroscopic relevant variables that commute with the particle number operator N<sup>c</sup> so that we can use both H and HS synonymously. (Mention that also the Massieu-Planck functional Φð Þt has to be expanded so that the fluctuations around the equilibrium averages <sup>B</sup><sup>n</sup> � h i <sup>B</sup><sup>n</sup> eq n o appear).
