1 Introduction to Nonequilibrium Statistical Physics and Its Foundations
59
Eventually, one obtains:
O t − −O 0
t
0
dt
R(t − t
)F(t
)
(1.212)
where R(t) is the response function:
R(t) = β
˙
A
O ◦ S
t
0
= β
d f 0 (()
dA
dt
(()e
itF 0 O(()
(1.213)
Once again, the macroscopic nonequilibrium behaviour of a given system has been
related solely to the correlations of microscopic fluctuating quantities, computed
with respect to the relevant equilibrium ensemble.
Equation (1.212) suggests that even the linear response is in general affected by
memory effects. From this point of view, the Markovian behaviour seems to be either
very special or a crude approximation, implying, for instance, that all nonequilibrium
fluids have a viscoelastic behaviour. In practice, however, in normal fluids and normal
conditions the memory terms decay rapidly, so that the Markovian approximation
is by and large justified. The viscoelastic behaviour is indeed noticeable only in
complex fluids or under extreme conditions, i.e. exceedingly far from equilibrium.
For example, perturb H 0 with a spatially uniform and constant force h along the x
direction. For small h, the average velocity moderately varies in time, and the overall
current J (P, Q) =
1
m
j p
x
j , can be considered to linear order:
J t − −J 0 = β
h
m 2
t
0
dt
j,k
p
x
j
p
x
k ◦ S
t−t
0
Assuming that momenta of different particles are uncorrelated at equilibrium, and
recalling that the mobility μ is defined by
lim
t→∞
J t = lim
t→∞
J t − −J 0 = μh
one eventually obtains
μ =
β
m 2
∞
0
dt
j
p
x
j (0) p
x
j (t
)
0
(1.214)
which is the Green-Kubo relation for diffusion, related to the diffusion coefficient D
by the Einstein relation D = μ/β. Then, writing Eq. (1.214) as
59
Eventually, one obtains:
O t − −O 0
t
0
dt
R(t − t
)F(t
)
(1.212)
where R(t) is the response function:
R(t) = β
˙
A
O ◦ S
t
0
= β
d f 0 (()
dA
dt
(()e
itF 0 O(()
(1.213)
Once again, the macroscopic nonequilibrium behaviour of a given system has been
related solely to the correlations of microscopic fluctuating quantities, computed
with respect to the relevant equilibrium ensemble.
Equation (1.212) suggests that even the linear response is in general affected by
memory effects. From this point of view, the Markovian behaviour seems to be either
very special or a crude approximation, implying, for instance, that all nonequilibrium
fluids have a viscoelastic behaviour. In practice, however, in normal fluids and normal
conditions the memory terms decay rapidly, so that the Markovian approximation
is by and large justified. The viscoelastic behaviour is indeed noticeable only in
complex fluids or under extreme conditions, i.e. exceedingly far from equilibrium.
For example, perturb H 0 with a spatially uniform and constant force h along the x
direction. For small h, the average velocity moderately varies in time, and the overall
current J (P, Q) =
1
m
j p
x
j , can be considered to linear order:
J t − −J 0 = β
h
m 2
t
0
dt
j,k
p
x
j
p
x
k ◦ S
t−t
0
Assuming that momenta of different particles are uncorrelated at equilibrium, and
recalling that the mobility μ is defined by
lim
t→∞
J t = lim
t→∞
J t − −J 0 = μh
one eventually obtains
μ =
β
m 2
∞
0
dt
j
p
x
j (0) p
x
j (t
)
0
(1.214)
which is the Green-Kubo relation for diffusion, related to the diffusion coefficient D
by the Einstein relation D = μ/β. Then, writing Eq. (1.214) as