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.
