Can Modern Statistical Mechanics Unravel Some Practical Problems . . .
35 therein. In fact, this local equilibrium PDF can be written in a more general and suitable
form by expressing it in terms of the minimum work necessary to change the state of the system [67], see Eq. 22, exactly in the form of Eq. 23 where
∆φ
T
r ≡
∆
W
. This ex- pression also includes, among others, the presence of activated volumes
∆ V p the pressure
and surface effects through the surface tension σ
. Now, using Eqs. 42-44 and 23, it is possible to derive a generalized FPS type
equation in the position space for ρ
r, t [8, 11, 33, 14]. To this end, one may first calculate
the entropy production of the system by taking the time derivative of Eq. 43, after using 42, and integrating by parts assuming that the fluxes vanish at the boundaries. Once the
entropy production is obtained, linear laws can be assumed for the relation between forces and fluxes
V
r
= − ζ
t ∇φ
T
− Dt ∇ρ
r, t, 45
where ζ
t is the time dependent Onsager coefficient entering through the linear law as- sumption [11, 33, 14]. Since
ζ t plays the role of an effective mobility, we have in-
troduced, by a fluctuation-dissipation formula, the time dependent diffusion coefficient D
t = β
−1
ζ t, which constitutes a generalization of the ES formula, the eES formula.
The substitution of Eq. 45 into 42 yields ∂
∂ t
ρ =
ζ t
∇ · [
ρ∇φ
T
] + β
−1
ζ t
∇
2
ρ ,
46 which is the desired generalized FPS type equation. The time dependence of the transport
coefficients introduces memory effects in the description [65, 28], whereas the first term at the right-hand side of the equation accounts for external and bath interactions with the
particle, and thus is suitable to be used in both unconfined and confined motion, and in IFS and EFS conditions.
For test particles moving under IFS in an unconfined space the physically more simple case, and with linear dimensions sufficiently large when compared with the length char-
acterizing the heat bath, for instance the characteristic length of the polymer network in polymer solutions, the host complex fluid can be assumed as a continuum. In this case, at
the mesoscopic level of description, the interactions between the test particle and the other components of the bath can be assimilated into the time dependence of the diffusion or ef-
fective friction coefficients. Notice however that, from a microscopic point of view, such interactions are responsible for the anomalous subdiffusion performed by the particle.
Under these assumptions, it is not necessary to take into account in explicit way the interaction potential:
∆φ
T
r = 0. As a consequence, the evolution equation for the test
Brownian particle becomes ∂
∂ t
ρ
r, t =
β
−1
ζ t
∇
2
ρ
r, t.
47 The mesoscopic effective properties of the viscoelastic medium are often determined by
analyzing the time dependence of the mean square displacement MSD hr
2
it or, equiva- lently through the creep compliance
χ
cc
t or the complex shear modulus G
′′ msd
ω , with
ω the frequency [64, 68, 69, 11, 8]. The subindex msd stands for the fact that this quantity is
obtained by using the generalized ES relation [78, 64], i.e., by measuring the MSD of the
36 A. Gadomski, I. Santamaria-Holek, N. Kruszewska et al.
test particle by means, for instance, of diffusing-wave spectrometry techniques or video- based methods [69, 68, 74, 75]. As an example, it is convenient to mention that in a certain
range of frequencies, it has been shown in experiments that the complex shear modulus follows the scaling behavior G
′′ msd
∼ ω
α
with α
1 [64, 74, 75]. Bear in mind that this power-law behavior is the cornerstone of microrheological viscosities.
These experimental results can be explained in the context of hydrodynamics for which it has been proven that the mobility has the general form ˆ
ζ ω
= ζ
τ
D
1 + a λ
−1
ω
−1
, [79, 80, 76], Here we have introduced
τ
D
as the characteristic diffusion time in order to keep the correct dimensions. Here,
ζ is the inverse Stokes friction coefficient and
λ is the
so called viscous penetration length λ
−1
= pi
ω ν
K
, where ν
K
is the kinematic viscosity of the host fluid [80]. When the host fluid is viscoelastic, the kinematic viscosity becomes
frequency dependent ν
K
ω and then one may assume the form 48. For frequencies lying
in the range 1 τ
D
ω ≪
β , i.e., times satisfying
τ
D
t ≥ β
−1
, the mobility coefficient can be written as
ˆ ζ
ω ≃
ζ τ
−1 D
τ
D
ω
−
1 −
δ 2
. 48
The exponent δ
characterizes the subdiffusion. At this level of description, its value can be justified in terms of the characteristic dimension of diffusing particle [8, 71] or by simple
comparison with the experiment. In a more detailed description, it can be justified that its value is determined by both, the elastic forces of the viscoelastic medium and the hydrody-
namic interactions [11].
Here, it is essential to point out that the inverse Laplace transform of Eq. 48 yields a memory function ˜
ζ t, this shall not be confused with the time dependent mobility coeffi-
cient ζ
t [65, 81]. Both quantities are related through the relation ζ
t =
t
Z
t
o
˜ ζ
zdz. 49
which represents a time average of the memory function. It is interesting to notice that this interpretation also arises when considering the non-
Markovian FPS equation in the complete ordering prescription COP, that is, in which memory effects are introduced through memory functions [65, 82]. The actual descrip-
tion involving time-dependent coefficients [technically called partial ordering prescription POP], can be related with the COP by taking into account that, in the case of slowly vary-
ing fields, the integral becomes a temporal average of the memory function. The relation between both prescriptions is treated in detail in Refs. [65] and [63].
In view of these considerations, we have to accept that the inverse Laplace transform of Eq. 48 gives the memory function
˜ ζ
t ≃ ζ
τ
−1 D
1 Γ
1 −
δ 2
t
−
1 +
δ 2
τ
−
δ −1
2
D
, 50
which in turn can be integrated over time in order to obtain the effective time dependent mobility
ζ t ≃
2 ζ
τ
−1 D
1 − δ
Γ
1 −
δ 2
t τ
D −
1 +
δ 2
. 51
Can Modern Statistical Mechanics Unravel Some Practical Problems . . .
37 This expression is useful since it can be compared with experimental results in order
to obtain the value of the exponent δ
for each viscoelastic medium. This can be done by examining the time dependence of the MSD, defined by
hr
2
it =
R
r
2
ρ
r, tdr. The
