t; cf. Eq. 14. Since the PDF is normalized, it will t ln [28]. These interactions can be considered separately from those due

34 A. Gadomski, I. Santamaria-Holek, N. Kruszewska et al. instance, the growth process of crystals or biomolecules in whose conformation memory, elastic and finite-size effects play an important role [42, 39]. At certain time scales, the main feature of their properties is the power-law behav- ior of the complex shear modulus [68], the creep compliance and the diffusion coefficient [69]. The viscoelasticity of these systems can be studied by means of microrheological techniques, such as the diffusing wave spectroscopy or video based methods which may characterize them in terms of the mean square displacement MSD of test particles that undergo subdiffusion [74, 75]. The MSD of the particle manifests a power-law dependence on time in which, in the case of small particles, the exponent can depend on the aspect ratio between the particle radius and the characteristic length of the polymer network [11] for certain values of these quantities. In the case when the linear dimension of the particle i.e., its radius a and its mass are much larger than the polymers surrounding it, an apparently universal 3 4 exponent is found [74]. Mesoscopic nonequilibrium approach to viscoelasticity - Consider the motion of a testing spherical Brownian particle macroion, spherule of radius a through a complex fluid composed by other Brownian particles macroions, spherules or by polymer molecules. The presence of these particles introduce spatial non-homogeneities and act on the test particles through electrostatic and elastic forces [11, 8]. As we have mentioned previously, at diffusion times the relevant microscopic variable determining the state of the test particle is the position vector r. Hence, the dynamics can be described by means of the PDF ρ

r, t; cf. Eq. 14. Since the PDF is normalized, it will

obey the continuity equation ∂ ∂ t ρ = − ∂ ∂ r · ρ V r , 42 where the explicit form of the probability diffusion current ρ V r can be found by assuming local equilibrium and using the rules of MNET [33, 76]. To obtain ρ V r , this thermo-kinetic formalism uses an irreversibility criterion based upon the generalized Gibbs entropy postu- late [77, 33] s t = −k B Z ρ

r, t ln

ρ ρ le dr + s le , 43 cf. Eq. 21, where s t is the nonequilibrium entropy, s le the entropy at local equilibrium. In Eq. 43, the reference state is characterized through the local equilibrium PDF of Gibbs- Boltzmann form ρ le r = e β [µ le − ∆φ T ] , 44 where µ le is the chemical potential at local equilibrium, ∆φ T r = φ e + φ B the total interac- tion potential related with external forces φ e r and with interactions between the particle and the bath φ B r, [28]. These interactions can be considered separately from those due to external agents, because of the two well known “opposite” roles the bath plays in the dynamics of the particle: Supplying thermal energy and introducing dissipation. Note that, since energy dissipation is due to surface forces, in general it depends on the size of the particle. Specifically, the potential ∆φ T r may in general involve energetic as well as entropic barriers [67] and, as a consequence, it could be responsible for Krames’ type dynamics of the system and thus useful to describe aggregation process, see Ref. [14] and references 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