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Entropy and the Time Evolution of Macroscopic Systems$

Walter T. Grandy, Jr.

Print publication date: 2008

Print ISBN-13: 9780199546176

Published to Oxford Scholarship Online: September 2008

DOI: 10.1093/acprof:oso/9780199546176.001.0001

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(p.189) Appendix C Analytic Continuation of Covariance Functions

(p.189) Appendix C Analytic Continuation of Covariance Functions

Source:
Entropy and the Time Evolution of Macroscopic Systems
Publisher:
Oxford University Press

To study further the properties of the linear covariance function, it is convenient to adopt a scenario similar to that of (6.16) and presume information is to be provided on <F(t)> in the infinite past, −∞ < t ≤ 0. We are interested in the causal situation in which <C(t)> is to be predicted for t > 0. In linear approximation, the predicted value of the deviation ΔC(t)> = C(t) − <C>0 at t > 0 is

(C.1)
< Δ C ( t ) > = 0 λ ( t ) K C F 0 ( t t ) d t , t > 0 = t λ ( t τ ) K C F 0 ( τ ) d τ = 0 θ ( τ t ) λ ( t τ ) θ ( τ ) K C F 0 ( τ ) d τ ,
where in the third line we have inserted step functions to indicate explicitly that θ ( τ ) K C F 0 ( τ ) is a causal function, even though necessarily τ > t. Note that the lower limit of the integral in the third line of (C.1) can be extended to −∞, resulting in a convolution integral. Fourier transformation as in (6.31a) yields
(C.2)
< Δ C ( ω ) > = 0 e i ω t λ ( t ) d t 0 e i ω t K C F 0 ( τ ) d τ = Λ ( ω ) K C F 0 ( ω ) + .
Note that K C F 0 ( ω ) + is a causal transform, and that Λ(ω) involves λ(t) over the only interval on which it is defined. The full Fourier transform of K C F 0 ( t ) has the form
(C.3a)
e i ω t K C F 0 ( t ) d t = 0 e i ω t K C F 0 ( t ) d t + 0 e i ω t K C F 0 ( t ) d t ,
or
(C.3b)
K C F 0 ( ω ) = K C F 0 ( ω ) + + K C F 0 ( ω ) .
Both K C F 0 ( t ) and K C F 0 ( ω ) are real, but that is not necessarily true of K C F 0 ( ω ) ± .

The form of the causal transform in (C.2) suggests that it may be profitable to study a more general function by continuation into the complex plane. The (p.190) Fourier-Laplace transform is defined as

(C.4)
K ˜ C F 0 ( z ) 0 e i z t K C F 0 ( t ) d t , Im z > 0 = 1 2 π K C F 0 ( ω ) ω z d ω , Im z 0.
The function K ˜ C F 0 ( z ) is regular in the upper half-plane and can be continued to the lower half-plane where it is also regular; it has a branch cut along the entire real axis. The real function K ˜ C F 0 ( ω ) is essentially the discontinuity in K ˜ C F 0 ( z ) across the cut.

Equation (C.1) represents the linear response of the system at time t to the behavior of <F(t)> in the past, and this is characterized by K ˜ C F 0 ( ω ) . The physical, or causal, response is the boundary value on the real axis as it is approached from above, or from the physical sheet. This can be extracted from (C.4) with the help of the identity,

(C.5)
lim γ 0 + 1 x ± i γ = P ( 1 x ) i π δ ( x ) ,
where P( ) denotes a principal value. Hence,
(C.6)
lim γ 0 + K ˜ C F 0 ( ω + i γ ) = P K C F 0 ( u ) u ω d u 2 π + i 2 K C F 0 ( ω ) .
The dissipative part of the response is thus the full Fourier transform:
(C.7)
K C F 0 ( ω ) = 2 Im K ˜ C F 0 ( ω + i 0 + ) ,
providing the physical interpretation of the transformed covariance function.

We can now relate the covariance function to the ordinary time correlation function, after developing two further relations. The identity (A. 13) can be applied to the time variables to provide the relation

(C.8a)
d d t K C F 0 ( t t ) = 1 i β < [ F , C ( t t ) ] > 0 ,
and its Fourier transform
(C.8b)
i ω K C F 0 ( ω ) = 1 i β < [ F , C ( ω ) ] > 0 .

The second relation stems from the identity

(C.9)
< C ( 0 ) F ( t ) > 0 e i ω t d t = e β ω < F ( t ) C ( 0 ) > 0 e i ω t d t ,
(p.191) for any two Hermitian operators C and F. One proves this by writing out both sides in a representation in which H is diagonal. This identity can now be manipulated into the relation
(C.10)
< [ C ( 0 ) , F ( t ) ] > 0 e i ω t d t = ( 1 e β ω ) < F ( t ) C ( 0 ) > 0 e i ω t d t .

With the relations (C.8b) and (C.10), along with the definition (6.35b) of the time correlation function Γ C F 0 ( r , τ ) = < Δ C ( r , τ ) Δ F ( 0 ) > 0 , we find for the full transform

(C.11)
K C F 0 ( k , ω ) = 1 e β ω β ω Γ C F 0 ( k , ω ) ,
as advertised in (6.35a). For βℏω ≪ 1 we see that K C F 0 ( k , ω ) Γ C F 0 ( k , ω ) , which can only happen if [F,H] = 0; that is, when the Kubo transform can be neglected. The possible noncommutativity, of course, is how the Kubo transform arises in the first place. (p.192)