u,t. Note a notation change with respect to u,t will obey the following continuity equation u-space t = −k u,t ln t is the nonequilibrium entropy per unit volume, s t = Nm r, t = e

82 A. Gadomski, I. Santamaria-Holek, N. Kruszewska et al. inverse powerly in time. Bear in mind that this quantity is proportional to the interstitial- fluid pressure, cf. [124, 232]. The dynamic friction coefficient f t of the mAC during a load of duration t is naturally redefined in a non-CA form to be f t = 1 K t , 80 thus being an inverse of the tribomicellization kernel function of the mAC, K t, given above. It is then easy to recover, at least in a qualitative way, some experimental curves by Ateshian et al. [124, 238], displaying the dynamic microscopic behavior of our systems in a { η t, f t} plane, for a given time interval t. A detailed fitting of the curves displayed on Fig. 5 from [124] is proposed to be left for another study. It is noteworthy to underline that the model view presented designs both functions as algebraic functions of time power laws, excellent for really simple fitting procedure to be applied. A more realistic fitting can be done by departing from the plane, for a given time interval. A detailed fitting of the curves displayed on Fig. 5 from [124] is proposed to be left for another study. It is noteworthy to underline that the model view presented designs both functions as algebraic functions of time power laws, excellent for really simple fitting procedure to be applied. A more realistic fitting can be done by departing from the d F = 1 most effective behavior, that is, by relaxing the straight ion-channel assumption employed above, which is by the way fairly idealized towards realizing the purpose of this work; an additional option appears to be to vary surely, according to the constraints 77-78 γ and d RW exponents to reproduce well step by step the more realistic temporal behavior in { η t, f t} experimental plane, cf. [124]. It is noteworthy to explore further the already sketched above non-CA avenue of re- search on the AC, with interstitial-fluid pressurization as the appropriate mechanism for facilitated AC-biolubrication, because such channels involved, have been already detected to exist, and are directly named as the voltage-gated proton channels, being also of SAPL- involving nature [239]. Aggregation vs shear, a MNET-type picture - When a shear stress is applied laterally to the system through a boundary, as for example in the case of a Couette flow [88], it induces a velocity gradient ∇ v on the system. This is a typical situation in which the AC could be involved. To estimate the influence of the shear flow on the viscoelastic and even the non- Newtonian properties of the synovial liquid, one may propose different approaches, see for example Ref. [112]. In this case the description must be performed at smaller times since, in general, the shear stress introduces a new time scale associated with the magnitude of | ∇ v | = ˙ γ . In this physical situation, one must extend the space of variables accounting for the state of the system, by also considering the instantaneous velocity u of the particle. Hence, the PDF will be now of the form: f

r, u,t. Note a notation change with respect to

previous PDF notation, designed by ρ . In similar form as in Sec. 2.1.4., f

r, u,t will obey the following continuity equation

in the

r, u-space

∂ ∂ t f + ∂ ∂ r · u f = − ∂ ∂ u · f V u , 81 where the explicit form of the diffusion current f V u in u-space can be found by following Can Modern Statistical Mechanics Unravel Some Practical Problems . . . 83 the MNET formalism [33, 76]. In this case the irreversibility criterion is of the form [77, 33] ρ s

r, t = −k

B Z f

r, u,t ln

f f le du + ρ s le , 82 where ρ s

r, t is the nonequilibrium entropy per unit volume, s

le the entropy at local equi- librium and ρ

r, t = Nm

R f u, r, tdu the mass density field. N is the total number of Brownian particles. In this case, the reference state is characterized by the PDF f le

u, r, t = e

m kBT [µ le − 1 2 u−v 2 − φ T ] , 83 where now 1 2 u−v 2 is the specific kinetic energy of the particle with respect to the imposed flow v and φ T r has the same interpretation as Eqs. 22 or 23. Equation 43 can be used to compute the entropy production of the system during its evolution in time [77, 76]. In this form, by taking the time derivative of 43, using 81 and integrating once by parts, we obtain σ = − m T Z f V u · ∂ ∂ u k B T m ln f f le + µ le du − Z f u · ∇φ T du , 84 which is the sum of the products of the probability currents f V u and f u times their gen- eralized forces conjugated. Following the usual linear law scheme and taking into account memory effects, we have f V u = ζ −1 u − v f + D e f f · ∂ f ∂ u + ε f ∇φ T , 85 where we have used the explicit expression of f leq , defined the effective diffusion coefficient in velocity space a second order tensor D e f f = k B T m ~ ~ ξ t where ~ ~ ξ = [1 − ζε∇ v ] ζ −1 t. The Onsager coefficient ε t characterizes the coupling between the momentum current in velocity space with the forces exerted on the particle [240]. From a fundamental point of view, the expression for D e f f shows that when an external flow is applied on the system, the fluctuation-dissipation theorem is not further valid [240]. The correction factor proportional to the velocity gradient implies that the system of parti- cles take energy to perform their Brownian motion not only from the thermal energy of the bath, but also from the energy transferred to the system via the boundary. This energy may play a significant role in the nucleation processes that may take place in the system, since it facilitates that the crystal units overcome the corresponding energetic barriers [241]. When Eq. 85 is substituted into 81, one obtains the generalized FPS type equation ∂ ∂ t f + ∇ · u f + ε∇φ T · ∂ f ∂ u = ζ −1 u − v f + D e f f · ∂ f ∂ u , 86 that incorporates memory effects into the description through the time dependent coeffi- cients ζ t and ε t. 84 A. Gadomski, I. Santamaria-Holek, N. Kruszewska et al. Hydrodynamics, viscoelasticity and non-Newtonian behavior - From Eq. 86 it is possible to calculate the balance equations for the central quantities of hydrodynamics: mass, momentum and pressure tensor [76]. The first of this set of equations is ∂ρ ∂ t = − ∇ · ρ v , 87 where the diffusion current is defined by ρ v r, t ≡ R u f du. Eq. 87 can be obtained by taking the time derivative of the definition of ρ

r, t, using Eq. 86 and performing an