24 A. Gadomski, I. Santamaria-Holek, N. Kruszewska et al.
counterpart corresponds to Eq. 11 correlation strength with γ
→ 0. The real challenge that can happen here is, however, that Eq. 12 is fully equivalent to a Smoluchowski-type equa-
tion [36] in R–configurational space that can be solved with suitable viz reflecting boundary conditions characteristic of a two-state dynamic process with a surmountable Kramers-type
energetic barrier [47]. This barrier, given by Φ
, can be determined exactly, and both its height and shape can be controlled by the fluctuations [45] of the radius of the SPC-A given
by the diffusion function involved in the Smoluchowski-type dynamics, D R, t = D[Rt],
defined still under t t
o
as
D [Rt] =
∞
R
R
2
ρ R, tdR
2t ,
13 whereby the above relation is the Einstein-type relation in the one-dimensional phase space
R.
2.1.4. Smoluchowski SPC-A Scenario in
R–space
In a picturesque way one can see the Smoluchowski dynamics as a cluster-cluster forma- tion. Its first stage will certainly rely on creation in a random way a single nuclei, see
Fig. 4A. Next, many nucleus can also be formed, see Fig. 4B. The process as a whole can be described by the FPS formalism.
Thus, the global mass conservation law for the protein-spheroid object grown in the entropic milieu e.g., lysozyme non-Kossel crystals or spherulites in a water solution [39]
eventually yields the Smoluchowski-type equation for the probability density ρ
R, t, also named the probability distribution function PDF
∂ ∂
t ρ
R, t = ∂
∂ R
D [Rt]
∂ρ R, t
∂ R
+ β
D [Rt]
∂Φ R
∂ R
ρ R, t
, 14
where Φ
R ≡ Φ
[Rt] becomes the free energy of the thermodynamic process, ultimately contributing to the determination of the Kramers’ barrier, and
β = 1k
B
T with T the temper- ature. It is completely equivalent to Eq. 12 with Eq. 9 and Eq. 10. Now, we may speak
of an ensemble-average Rt =
∞
R
R ρ
R, tdR which is a well-defined quantity. This is also the case of
R
2
t =
∞
R
R
2
ρ R, tdR which stands for the mean-squared dis-
placement of the Φ
–drifted superdiffusive process in the R–space [36]. Thus, the reduced variance,
κ t, given by
κ t =
R
2
t Rt
2
− 1, 15
as a measure of the fluctuations in the R–space, can be determined too. Moreover, one is able to derive
Φ R in a Boltzmann-like form [39] as
Φ R = −
1 β
ln [
σ
R ne
−1
], 16
Can Modern Statistical Mechanics Unravel Some Practical Problems . . .
25
Figure 4. Two consecutive snapshots of the FPS type pictures in two different stages A - early stage, B - late stage, showing how small randomly walking clusters, or large
monomers, are merged into some bigger ones. In the stage, B, under inhomogeneous solu- tion’s regime, there is a possibility of creating a ripe viz thermodynamically stable nucleus
from which the crystal formation may likely start to develop. Note, that the situation de- picted by the stage A could possibly be seen as a formation of a disordered cluster under
diluted solution’s condition - for it the possibility of creating such nucleus markedly dimin- ishes in the course of time. By the envelopes seen around each charged object the double
layers, expected to emerge as depletion zones in electrorheological solutions, are marked [39, 42]. A liquid-liquid phase separation, enabling to form precrystalline aggregates, is
also more likely to occur in the stage B, cf. [48].
with σ
R ne
−1
determinable from Eq. 8, this quantity is suit to be the system’s nonequi- librium supersaturation. D
[Rt] is also determinable from the MNET-type proposal just offered, and reads ultimately
D [Rt] = Dt[
σ
R ne
]
−2
, 17
where D t is generally given by Eq. 10 a general time-correlational proposal, or by
Eq. 11 with the limit of γ
→ 0, i.e. when the specific ballistic-type proposal pointing to the fully hydrodynamic stable mode of the overall SPC-A process taking place within the
Stern-type double layer is envisaged. Note that by comparing Eq. 17 with the Einstein- type definition of D
[Rt] := R
2
t 2t Eq. 13, one has to have for Φ
R 6= 0, and
∆Φ R 0 thereby indicating a natural thermodynamic course of the process, that
is a superdiffusion in our R–space. The case of Φ
= 0 will particularly correspond to the standard Einstein diffusion. When
Φ R 6= 0 again but
∆Φ R 0 appears to hold because
of some unusual, e.g., auxetic-type boundary elastic effects [39, 49], one may detect some thermodynamic-kinetic anomalies either at the crystal surface, or within the interface [41],
apparently in a certain conjunction with the protein-velocity correlational field, coupled to the curvatures-involving crystal’s border.
26 A. Gadomski, I. Santamaria-Holek, N. Kruszewska et al.
2.1.5. Towards a Morphological Phase Diagram
