ABSTRACT
The purpose of this chapter is chiefly to summarize the formal results obtained so far, and extend them to the important case where we are interested in the correlations among several physical quantities. In the simple example of spin diffusion, we were only concerned with the “autocorrelation function” S MM ( r → t , r → ′ t ′ ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429493683/5d5d4b71-9588-46a4-97b3-71883ae9a3a0/content/eq228.tif"/> of the magnetization the probability essentially of finding the magnetization at the space-time point r → , t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429493683/5d5d4b71-9588-46a4-97b3-71883ae9a3a0/content/eq229.tif"/> , t if you know its value at the point r → ′ , t ′ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429493683/5d5d4b71-9588-46a4-97b3-71883ae9a3a0/content/eq230.tif"/> . Now in a liquid, there are several quantities of interest: the particle density n ( r → , t ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429493683/5d5d4b71-9588-46a4-97b3-71883ae9a3a0/content/eq231.tif"/> , the momentum density g → ( r → , t ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429493683/5d5d4b71-9588-46a4-97b3-71883ae9a3a0/content/eq232.tif"/> , the energy density ε ( r → , t ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429493683/5d5d4b71-9588-46a4-97b3-71883ae9a3a0/content/eq233.tif"/> , and may be others. And these are dynamically coupled. A local imbalance in the energy density (i.e., o temperature inhomogeneity) will result in a spatially varying particle density as well, for example. We are therefore led to consider such correlation functions as S n ε ( r → t , r → ′ t ′ ) = [ < n ( r → t ) ε ( r → ′ t ′ ) > − < n ( r → t ) > < ε ( r → ′ t ′ ) > ] https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429493683/5d5d4b71-9588-46a4-97b3-71883ae9a3a0/content/eq234.tif"/> . Or else, since we found that the averaged commutator was a little closer to the action, such response functions as χ ″ n ε ( r → t , r → ′ t ′ ) = < ( 1 / 2 ℏ ) [ n ( r → t ) , ε ( r → ¢ t ′ ) ] > https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429493683/5d5d4b71-9588-46a4-97b3-71883ae9a3a0/content/eq235.tif"/> . We will therefore consider the general properties of multivariate correlation functions, most of which are obtained by a perfectly straightforward extension from the case of a single variable. We shall treat the general, quantum-mechanical, case which is formally a little easier to handle in fact, and indicate classical limits where appropriate. This chapter follows in much detail Martin 1968; see also Berne and Harp 1970.
