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# Quantum Effects Through a Fractal Theory of Motion

Written By

M. Agop, C.Gh. Buzea, S. Bacaita, A. Stroe and M. Popa

Submitted: 24 September 2012 Published: 03 April 2013

DOI: 10.5772/54172

From the Edited Volume

## Advances in Quantum Mechanics

Edited by Paul Bracken

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## 1. Introduction

Scale Relativity Theory (SRT) affirms that the laws of physics apply in all reference systems, whatever its state of motion and its scale. In consequence, SRT imply [1-3] the followings:

1. Particle movement on continuous and non-differentiable curve (or almost nowhere differentiable), that is explicitly scale dependent and its length tends to infinity, when the scale interval tends to zero.

2. Physical quantities will be expressed through fractal functions, namely through functions that are dependent both on coordinate field and resolution scale. The invariance of the physical quantities in relation with the resolution scale generates special types of transformations, called resolution scale transformations. In what follows we will explain the above statement.

Let F(x) be a fractal function in the interval x[a,b] and let the sequence of values for x be:

xa=x0,x1=x0+ε,xk=x0+kε,xn=x0+nε=xbE1

We can now say that F(x,ε) is a –scale approximation.

Let us now consider as a ε¯-scale approximation of the same function. Since F(x) is everywhere almost self-similar, if ε and ε¯ are sufficiently small, both approximations F(x,ε) and must lead to same results. By comparing the two cases, one notices that scale expansion is related to the increase dε of ε, according to an increase dε¯ of ε¯. But, in this case we have:

dεε=dε¯ε¯=dρE2

situation in which we can consider the infinitesimal scale transformation as being

ε'=ε+dε=ε+εdρE3

Such transformation in the case of function F(x,ε), leads to:

F(x,ε')=F(x,ε+εdρ)E4

respectively, if we limit ourselves to a first order approximation:

F(x,ε')=F(x,ε)+F(x,ε)ε(ε'ε)=F(x,ε)+F(x,ε)εεdρE5

Moreover, let us notice that for an arbitrary but fixed ε0, we obtain:

ln(ε/ε0)ε=(lnεlnε0)ε=1εE6

situation in which (5) can be written as:

F(x,ε')=F(x,ε)+F(x,ε)ln(ε/ε0)dρ=[1+ln(ε/ε0)dρ]F(x,ε)E7

Therefore, we can introduce the dilatation operator:

D=ln(ε/ε0)E8

At the same time, relation (8) shows that the intrinsic variable of resolution is not ε, but ln(ε/ε0).

The fractal function is explicitly dependent on the resolution (ε/ε0), therefore we have to solve the differential equation:

dFdln(ε/ε0)=P(F)E9

where P(F) is now an unknown function. The simplest explicit suggested form for P(F) is linear dependence [2]

P(F)=A+BF,A,B=const.E10

in which case the differential equation (9) takes the form:

dFdln(ε/ε0)=A+BFE11

Hence by integration and substituting:

B=τ,E12
AB=F0E13

we obtain:

F(εε0)=F0[1+(ε0ε)τ]E14

We can now generalize the previous result by considering that F is dependent on parameterization of the fractal curve. If p characterizes the position on the fractal curve then, following the same algorithm as above, the solution will be as a sum of two terms i.e. both classical and differentiable (depending only on position) and fractal, non-differentiable (depending on position and, divergently, on ε/ε0)

F(p,ε/ε0)=F0(p)[1+ξ(p)(ε0ε)τ(p)]E15

where ξ(p) is a function depending on parameterization of the fractal curve.

The following particular cases are to be considered:

1. in asymptotic small scale regime εε0, τ is constant (with no scale dependence) and power-law dependence on resolution is obtained:

F(p,ε/ε0)=T(p)(ε0ε)τaT(p)=F0(p)Q(p)bE16
1. in the asymptotic big scale regime εε0, τ is constant (with no scale dependence) and, in terms of resolution, one obtains an independent law:

F(p,ε/ε0)F0(p)E17

Particularly, if F(p,ε/ε0) are the coordinates in given space, we can write

X(p,ε/ε0)=x(p)[1+ξ(p)(ε0ε)τ]E18

In this situation, ξ(p) becomes a highly fluctuating function which can be described by stochastic process while τ represents (according to previous description) the difference between fractal and topological dimensions. The result is a sum of two terms, a classical, differentiable one (dependent only on the position) and a fractal, non-differentiable one (dependent both on the position and, divergently, on ε/ε0). This represents the importance of the above analysis.

By differentiating these two parts we obtain:

dX=dx+dξE19

where dx is the classical differential element and dξ is a differential fractal one.

1. There is infinity of fractal curves (geodesics) relating to any couple of points (or starting from any point) and applied for any scale. The phenomenon can be easily understood at the level of fractal surfaces, which, in their turn, can be described in terms of fractal distribution of conic points of positive and negative infinite curvature. As a consequence, we have replaced velocity on a particular geodesic by fractal velocity field of the whole infinite ensemble of geodesics. This representation is similar to that of fluid mechanics [4] where the motion of the fluid is described in terms of its velocity field v=(x(t),t), density ρ=(x(t),t) and, possibly, its pressure. We shall, indeed, recover the fundamental equations of fluid mechanics (Euler and continuity equations), but we shall write them in terms of a density of probability (as defined by the set of geodesics) instead of a density of matter and adding an additional term of quantum pressure (the expression of fractal geometry).

2. The local differential time invariance is broken, so the time-derivative of the fractal field Q can be written two-fold:

Both definitions are equivalent in the differentiable case dtdt. In the non-differentiable situation, these definitions are no longer valid, since limits are not defined anymore. Fractal theory defines physics in relationship with the function behavior during the “zoom” operation on the time resolution δt, here identified with the differential element dt (substitution principle), which is considered an independent variable. The standard field Q(t) is therefore replaced by fractal field Q(t,dt), explicitly dependent on time resolution interval, whose derivative is not defined at the unnoticeable limit dt0. As a consequence, this leads to the two derivatives of the fractal field Q as explicit functions of the two variables t and dt,

Notation “+” corresponds to the forward process, while “-” to the backward one.

1. We denote the average of these vectors by dx±i, i.e.

dX±i=dx±i,i=1,2E22

Since, according to (19), we can write:

dX±i=dx±i+dξ±iE23

and it results:

dξ±i=0E24
1. The differential fractal part satisfies the fractal equation:

d±ξi=λ±i(dt)1/DFE25

where λ±i are some constant coefficients and DF is a constant fractal dimension. We note that the use of any Kolmogorov or Hausdorff [1, 5, 6-8] definitions can be accepted for fractal dimension, but once a certain definition is admitted, it should be used until the end of analyzed dynamics.

1. The local differential time reflection invariance is recovered by combining the two derivatives, d+/dt and d/dt, in the complex operator:

d^dt=12(d++ddt)i2(d+ddt)E26

Applying this operator to the “position vector”, a complex velocity yields

V^=d^Xdt=12(d+X+dXdt)i2(d+XdXdt)=V++V2iV+V2=ViUE27

with:

V=V++V2aU=V+V2bE28

The real part, V, of the complex velocity V^, represents the standard classical velocity, which does not depend on resolution, while the imaginary part, U, is a new quantity coming from resolution dependant fractal.

## 2. Covariant total derivative

Let us now assume that curves describing particle movement (continuous but non-differentiable) are immersed in a 3-dimensional space, and that X of components Xi(i=1,3¯) is the position vector of a point on the curve. Let us also consider a fractal field Q( X, t ) and expand its total differential up to the third order:

d±Q=Qtdt+QdX±++122QXiXjd±Xid±Xj+163QXiXjXkd±Xid±Xjd±XkE29

where only the first three terms were used in Nottale’s theory (i.e. second order terms in the motion equation). Relations (29) are valid in any point both for the spatial manifold and for the points X on the fractal curve (selected in relations 29). Hence, the forward and backward average values of these relations take the form:

d±Q=Qtdt+Qd±X+122QXiXjd±Xid±Xj+163QXiXjXkd±Xid±Xjd±XkE30

The following aspects should be mentioned: the mean value of function f and its derivatives coincide with themselves and the differentials d±Xi and dt are independent; therefore, the average of their products coincides with the product of averages. Consequently, the equations (30) become:

d±Q=Qtdt+Qd±X+122QXiXjd±Xid±Xj+163QXiXjXkd±Xid±Xjd±XkE31

or more, using equations (23) with characteristics (24),

d±Q=Qtdt+Qd±X+122QXiXj(d±xid±xj+d±ξid±ξj)+163QXiXjXk(d±xid±xjd±xk+d±ξid±ξjd±ξk)E32

Even if the average value of the fractal coordinate d±ξi is null (see 24), for higher order of fractal coordinate average, the situation can still be different. Firstly, let us focus on the averages d+ξid+ξj and dξidξj. If ij, these averages are zero due to the independence of d±ξi and d±ξj. So, using (25), we can write:

d±ξid±ξj=λ±iλ±j(dt)(2/DF)1dtE33

Then, let us consider the averages d±ξid±ξjd±ξk. If ijk, these averages are zero due to independence of d±ξi on d±ξj and d±ξk. Now, using equations (25), we can write:

d±ξid±ξjd±ξk=λ±iλ+jλ±k(dt)(3/DF)1dtE34

Then, equations (32) may be written as follows:

d±Q=Qtdt+d±xQ+122QXiXjd±xid±xj+122QXiXjλ±iλ±j(dt)(2/DF)1dt+163QXiXjXkd±xid±xjd±xk+163QXiXjXkλ±iλ±jλ±k(dt)(3/DF)1dtE35

If we divide by dt and neglect the terms containing differential factors (for details on the method see [9, 10]), equations (38a) and (38b) are reduced to:

d±Qdt=Qt+V±Q+122QXiXjλ±iλ±j(dt)(2/DF)1+163QXiXjXkλ±iλ±jλ±k(dt)(3/DF)1E36

These relations also allow us to define the operator:

d±dt=t+V±+122XiXjλ±iλ±j(dt)(2/DF)1+163XiXjXkλ±iλ±jλ±k(dt)(3/DF)1E37

Under these circumstances, let us calculate (Q/t). Taking into account equations (26), (27) and (37), we shall obtain:

Qt=12[d+Qdt+dQdti(d+QdtdQdt)]==12Qt+12V+Q+λ+iλ+j14(dt)(2/DF)12QXiXj+λ+iλ+jλ+k112(dt)(3/DF)13QXiXjXk++12Qt+12VQ++λiλj14(dt)(2/DF)12QXiXj+λiλjλk112(dt)(3/DF)13QXiXjXki2Qti2V+Qλ+iλ+ji2(dt)(2/DF)12QXiXjλ+iλ+jλ+ki12(dt)(3/DF)13QXiXjXk++i2Qt+i2VQ+λiλji2(dt)(2/DF)12QXiXj+λiλjλki12(dt)(3/DF)13QXiXjXk==Qt+(V++V2iV+V2)Q+(dt)(2/DF)14[(λ+iλ+j+λiλj)i(λ+iλ+jλiλj)]2QXiXj++(dt)(3/DF)112[(λ+iλ+jλ+k+λiλjλk)i(λ+iλ+jλ+kλiλjλk)]3QXiXjXk==Qt+VQ+(dt)(2/DF)14[(λ+iλ+j+λiλj)i(λ+iλ+jλiλj)]2QXiXj+(dt)(3/DF)112[(λ+iλ+jλ+k+λiλjλk)i(λ+iλ+jλ+kλiλjλk)]3QXiXjXkE38

This relation also allows us to define the fractal operator:

^t=t+V^+(dt)(2/DF)14[(λ+iλ+j+λiλj)i(λ+iλ+jλiλj)]2XiXj++(dt)(3/DF)112[(λ+iλ+jλ+k+λiλjλk)i(λ+iλ+jλ+kλiλjλk)]3XiXjXkE39

Particularly, by choosing:

λ+iλ+j=λiλ_j=2DδijE40
λ+iλ+jλ+k=λiλ_jλ+k=22D3/2δijkE41

the fractal operator (39) takes the usual form:

^t=t+V^iD(dt)(2/DF)1Δ+23D3/2(dt)(3/DF)13E42

We now apply the principle of scale covariance and postulate that the passage from classical (differentiable) to “fractal” mechanics can be implemented by replacing the standard time derivative operator, d/dt, with the complex operator ^/t (this results in a generalization of Nottale’s [1, 2] principle of scale covariance). Consequently, we are now able to write the diffusion equation in its covariant form:

Qt=Qt+(V)QiD(dt)(2/DF)1ΔQ+23D3/2(dt)(3/DF)13Q=0E43

This means that at any point on a fractal path, the local temporal tQ, the non-linear (convective), (V)Q, the dissipative, ΔQ, and the dispersive, 3Q, terms keep their balance.

## 3. Fractal space-time and the motion equation of free particles in the dissipative approximation

Newton's fundamental equation of dynamics in the dissipative approximation is:

mVt=ΦE44

where m is the mass, V the instantaneous velocity of the particle, Φthe scalar potential and

t=t+(V)iD(dt)(2/DF)1ΔE45

is the fractal operator in the dissipative approximation.

In what follows, we study what happens with equation (44), in the free particle case (Φ = 0), if one considers the space-time where particles move changes from classical to nondifferentiable.

According to Nottale [11], the transition from classical (differentiable) mechanics to the scale relativistic framework is implemented by passing to a fluid-like description (the fractality of space), considering the velocity field a fractal function explicitly depending on a scale variable (the fractal geometry of each geodesic). Separating the real and imaginary parts, (44) becomes:

Vt+VV=0UV=DΔVE46

where V^=Vi U is the complex velocity defined through (27) and D defines the amplitude of the fractal fluctuations (D=D(dt)(2/DF)1).

Let us analyze in what follows, the second equation (46) which, one can see, may contain some interesting physics. If we compare it with Navier-Stokes equation, from fluid mechanics [12]

DvDt=vt+vv=ν2v E47

we can see the left side of (46) gives the rate at which V is transported through a 'fluid' by means of the motion of 'fluid' particles with the velocity U; the right hand side gives the diffusion of V, (D which is the amplitude of the fractal fluctuations, plays here the role of the 'cinematic viscosity' of the 'fluid'). One can notice, in those regions in which the right hand side of (47) is negligible, Dv/Dt = 0. This means that in inviscid flows, for instance, V is frozen into the 'particles of the fluid'. Physically this is due to the fact that in an inviscid 'fluid' shear stresses are zero, so that there is no mechanism by which V can be transferred from one 'fluid' particle to another. This may be the case for the transport of V by U in the second equation (II.3).

If we consider the flow of V induced by a uniform translational motion of a plane spaced a distance Y above a stationary parallel plane (Fig. 1), and if the 'fluid' velocity increases from zero (at the stationary plane) to U (at the moving plane) like in the case of simple Couette flow, or simple shear flow, then

rate of shear deformation =dVdy=UYE48

For many fluids it is found that the magnitude of the shearing stress is related to the rate of shear proportionally:

T=ηdVdy=ηUYE49

Fluids which obey (49) in the above situation are known as Newtonian fluids, which have a very small coefficient of viscosity. When such 'fluids' flow at reasonable velocities it is found that viscous effects appear only in thin layers on the surface of objects or surfaces over which the 'fluid' flows. That is, if one continues the analogy, and questions how is V transported by the motion of 'fluid' particles with the velocity U, in second equation (46), one can assume that the mechanism of transfer of V from one particle of 'fluid' to another is achieved over small distances (in thin layers, as stated above).

We study an important case, of the one-dimensional flow along the Ox axis :

V=ζ(x) kE50

To resume, the model considered here consists in analyzing the transport of V, along a small elementary distance Λ, by the 'particles' of a Newtonian fluid moving with velocity U, where the stress tensor obeys (49), i.e.

dζ(x)dx=ζ(x)ΛE51

like in the case of simple Couette flow, or simple shear flow.

Consequently, the second eq. (46) reduces to the scalar equation

ζ(x)+K2(x)ζ(x)=0E52

which is the time independent Schrödinger equation, and

K2(x)=1ΛDU(x)E53

with Λ and D having the significance of a small elementary distance and of the 'cinematic viscosity' (or amplitude of the fractal fluctuations), respectively, and U(x) is the velocity of the 'Newtonian fluid', which is nothing but the imaginary part of the complex velocity [13]. In what follows, we solve this equation accurately by means of the WKBJ approximation method with connection formulas.

### 3.1. Solving the Schrödinger type equation by means of the WKBJ approximation method

Let us re-write (53) in the form

K2(x)=1ΛDU(x)=2m2c2(χγ(x))E54

where we take D=/2m and consider the small elementary distance the Compton length Λ=/mc [14]. Therefore, the Schrödinger equation (52) splits into:

d2dx2ζ(x)+k2(x)ζ(x)=0,   χ>γord2dx2ζ(x)ρ2(x)ζ(x)=0,   χ<γE55

where

k(x)=2μ(χγ(x))2,ρ(x)=2μ(γ(x)χ)2withμ=m2cE56

χ is a limit velocity and γ(x) a 'velocity potential'.

Let us try a solution of the form ζ(x)=A exp ((i/ħ)S(x)). Substituting this solution into the time-independent Schrödinger equation (52) we get:

id2Sdx2(dSdx)2+2k2=0orid2Sdx2(dSdx)22ρ2=0E57

Assume that ħcan, in some sense, be regarded as a small quantity and that S(x) can be expanded in powers of ħ, S(x) = S0(x)+ ħ S1(x) +....

Then,

iddx[dS0dx+dS1dx+...](dS0dx+dS1dx+...)2+2k2=0 ,     (χ>γ(x))E58

We assume that |dS0dx|>>|dS1dx| and collect terms with equal powers of ħ.

[dS0dx]2+2k2=0        S0=±xk(x)dxE59
id2S0dx22dS0dxdS1dx=0        S1=12ilnk(x)E60

We have used:

iddx(dS0dx)=2dS0dxdS1dx ,    idkdx=2kdS1dx,   dS1=i2dkkE61

Therefore, for χ>γ(x)

ζ(x)=Ak12e±ixk(x)dxE62

In the classically allowed region S0=±xk(x)dx counts the oscillations of the velocity wave function. An increase of 2π ħ corresponds to an additional phase of .

Similarly, in regions where χ<γ(x) we have:

ζ(x)=Aρ12e±xρ(x)dxE63

For our first order expansion to be accurate we need that the magnitude of higher order terms decreases rapidly. We need |dS0dx|>>|dS1dx| or |k|>>|12kdkdx|. The local deBroglie wavelength is λ = 2π/k. Therefore, |λ4πdλdx|<<λ , i.e. the change in λ over a distance λ/4π is small compared to λ. This holds when the velocity potential γ(x) varies slowly and the momentum is nearly constant over several wavelengths.

Near the classical turning points the WKBJ solutions become invalid, because k goes to zero here. We have to find a way to connect an oscillating solution to an exponential solution across a turning point if we want to solve barrier penetration problems or find bound states.

### 3.2. Velocity potential γ (x) and the bound states

We want to find the velocity wave function in a given velocity potential well γ(x). Assuming that the limit velocity of the particle isχ and that the classical turning points are x1 and x2, x1<x2, i.e. we have a velocity potential well with two sloping sides (Fig. 2).

For x < x1 the velocity wave function is of the form:

ζ1(x)=A1ρ12exρ(x)dxE64

For x > x2 the velocity wave function is of the form:

ζ3(x)=A3ρ12exρ(x)dxE65

In the region between x1 and x2 it is of the form:

ζ2(x)=A2k12e+ixk(x)dx+A2k12eixk(x)dxE66

At x = x1 and x = x2 the velocity wave function ζ and its derivatives have to be continuous. Near x1 and x2 we expand the velocity potential well γ(x) in a Taylor series expansion in x and neglect all terms of order higher than 1. Near x1 we have γ(x)=χK1(xx1), and near x2 we have γ(x)=χ+K2(xx2).

In the neighborhood of x1 the time-independent Schrödinger equation then becomes:

d2ζdx2+2μK12(xx1)ζ=0E67

and in the neighborhood of x2 the time-independent Schrödinger equation becomes:

d2ζdx22μK22(xx2)ζ=0E68

Let us define z=(2μK12)13(xx1). Then we obtain d2ζdz2zζ=0 near x1. The solutions of this equation which vanish asymptotically as z → ∞ or x → -∞ are the Airy functions. They are defined through:

Ai(z)=1π0cos(s33+sz)dsE69

which for large |z| has the asymptotic form

Ai(z)~12πz14exp(23z32) ,     (z>0)E70

and

Ai(z)~1π(z)14sin(23(z)32+π4) ,     (z<0)E71

If the limit velocity χ is high enough, the linear approximation to the velocity potential well remains valid over many wavelengths. The Airy functions can therefore be the connecting velocity wave functions through the turning point at x1.

If we define z=(2μK22)13(xx2) then we find d2ζdz2zζ=0 near x = x2 and the Airy functions can also be the connecting velocity wave functions through the turning point at x2. Here z → ∞ or x → ∞.

In the neighborhood of x1 we have

k2=ρ2=(2μK12)13(xx1)=(2μK12)13zE72

Therefore

x1xρdx=(2μK12)13x1xzdx=0xzdz=23z32E73

Similarly

x1xkdx=(2μK12)13x1xzdx=0xzdz=23(z)32E74

By comparing this with the asymptotic forms of the Airy functions we note that

ζ1(x)=A1ρ12e+x1xρ(x)dx(x<x1)E75

must continue on the right side as

ζ2(x)=2A1k12sin(x1xkdx+π4)(x>x1)E76

In the neighborhood of x2we similarly find that

ζ3(x)=A3ρ12ex2xρ(x)dx(x>x2)E77

must continue in region 2 as

ζ2(x)=2A3k12sin(xx2kdx+π4)(x<x2)E78

Both expressions for ζ2(x) are approximations to the same eigenfunction. We therefore need

2A1k12sin(x1xkdx+π4)=2A3k12sin(xx2kdx+π4)E79

For (79) to be satisfied, the amplitudes of each side must have the same magnitude, and the phases must be the same modulo π :

|A1|=|A3|x1xkdx+π4=xx2kdxπ4+nπE80

Knowing that x1x2=x1x+xx2, we have

x1x2kdx=(n12)π,       n=1,2,3,...E81

This can be re-written as

x1x2Πdx=(n12)h2orΠdx=(n12)hE82

with

Π=[2μ(χγ(x))]1/2=[2μU(x)]1/2=mc[2U(x)c]1/2E83

Here denote an integral over one complete cycle of the classical motion. The WKBJ method for γ(x) velocity potential well with soft walls, therefore, leads to a Wilson-Sommerfeld type quantization rule except that n is replaced by n-1/2. It leads to a quantization of the complex velocity U(x).

The factor of π/2 arises here due to the two phase changes of π/4 at x1 and x2. In case where only one of the walls is soft and the other is infinitely steep the factor of 1/2 is replaced by 1/4 in (81). If both walls are infinitely steep, the factor of 1/2 in (81) is replaced by 0.

WKBJ approximation is a semi classical approximation, since it is expected to be most useful in the nearly classical limit of large quantum numbers. The method will not be good for, say, lowest limit velocity states χ, so in order to overcome this shortcomings there is a need for a modified semi classical quantization condition. For oscillations between the two classical turning points x1 and x2, we obtain the semi classical quantization condition by requiring that the total phase during one period of oscillation to be an integral multiple of 2π; [15] such that

2x1x2kdx+ϕ1+ϕ2=2πnE84

where ϕ1 is the phase loss due to reflection at the classical turning point x1 and ϕ2 is the phase loss due to reflection at x2. Taking ϕ1 and ϕ2 to be equal to π/2leads to the modified semiclassical quantization rule, i.e.

x1x2kdx=(nm4)πE85

where mis the Maslov index [15], which denotes the total phase loss during one period in units of π/2. It contains contributions from the phase losses ϕ1and ϕ2 due to reflections at points x1 and x2, respectively. It is pertinent to note that taking ϕ1 = ϕ2 = π/2 and an integer Maslov index m = 2 in (85), we have the familiar semi classical quantization rule, i.e. (81).

Let us apply the constraint equation (81) to an harmonic oscillator. The condition then is (passing without loss of generality to the limits -a to +a)

a+a[2μ(χω x)]1/2dx=(n12)πE86

where the energy Wc of the oscillator U(x) with the pulsation ωwrites

Wc=12mω2x2=12mγ2(x)E87

and we get the expression for the x dependence of the velocity term, γ(x)=ωx.

Theleft side term of (86) is an elementary integral and we find:

(χ+υ)3/2(χυ)3/2=3π2a(2μ)1/2(n12)υE88

where υ=ω a is the liniar velocity (see the graphic in Fig. 3).

We try to estimate a value for the limit velocity χ. Let us expand the left side term of (88) in series and keep the first term. If we replace μ from (56) and take a = Λ (the Compton length), we get:

χπ228μa2(n12)2=π232(2n1)2c={π232c,  9π232c,  25π232c, }E89

It is interesting to note that only the first velocity in (89) is less than the velocity of light, c.

Let us analyze now, one more bound state, the velocity wave function in a given velocity double well potential γ(x).

We begin by deriving a quantization condition for region 2 analogous to (81). Again, applying the boundary condition for region 1 leaves only the exponentially growing solution. Applying the connection formula at x1 then gives an expression for the velocity wave function in region 2 :

ζ1(x)=2Ak1/2sin[x1xkdx'+π4]E90

However, the solution in region 3 must have both growing and decaying solutions present. Considering the region 3 solutions in terms of x2 and letting BL and CL be the amplitudes of the decaying and growing solutions respectively, the connection formulas give another expression for the velocity wave function in region 2:

ζ2(x)=2BLk1/2cosθ+CLk1/2sinθE91

with

θ=xx2kdx'π4E92

We equate the two expressions (90), (91) for the velocity function in region 2 and cancel common factors giving

2Asin[x1xkdx'+π4]=2BLcosθ+CLsinθE93

Using trigonometric identities to simplify the right hand side, gives

2Asin[x1xkdx'+π4]=(4BL2+CL2)1/2sin(θ+π2ϕL)E94

where

ϕL=cos1[2BL(4BL2+CL2)1/2]E95

The magnitude of the sin function must be equal, and the magnitude of the phases must be equal modulo π :

4A2=4BL2+CL2E96
x1xkdx'+π4=xx2kdx'π4+ϕL+nπE97

Simplifying and combining the integrals gives the quantization condition for region 2:

θ12x1x2kdx=π(n12)+ϕLE98

with n = 1, 2,....

A similar treatment for the turning point x3 yields the condition for region 4:

θ34x3x4kdx=π(m12)+ϕRE99

with m = 1, 2,... and ϕR given by:

ϕR=cos1[2CR(4CR2+BR2)1/2]E100

where BR and CR are the amplitudes of the decaying and growing region 3 solutions in terms of x3.

We now have the quantization conditions (98, 99) for regions 2 and 4, but they contain the free parameters ϕL and ϕR. To eliminate these free parameters, we consider the WKBJ solution in region 3. The coefficients BL, CL, BR, CR define two expressions for solution, which must be equal:

ζ3=BLρ1/2exp[x2xρdx']+CLρ1/2exp[x2xρdx']E101
ζ3=BRρ1/2exp[xx3ρdx']+CRρ1/2exp[xx3ρdx']E102

Equations (101) and (102) each contain a term that grows exponentially with x and a term that decays exponentially with x. Equating the growing terms from each equation and the decaying term from each equation gives two constraints:

BLexp[x2xρdx']=BRexp[xx3ρdx']E103
CLexp[x2xρdx']=CRexp[xx3ρdx']E104

Combining the integrals in these constraints gives

BLBR=CRCL=exp(θ23)E105

with

θ23x2x3ρdx'E106

The constraints (98, 99, 105) may be combined to give a single quantization condition for the allowed WKBJ velocity limits χ for a double-well velocity potential γ(x). Applying trigonometric identities to (95) and (100), and plugging into (105) gives

tanϕLtanϕR=(CL2BL)(BR2CR)=14exp(2θ23)E107

Equation (107) may be combined with (98) and (99) to give the WKBJ quantization condition for a double-well potential in terms of the phase integrals θ12 and θ34 :

ctgθ12ctgθ34=14exp(2θ23)E108

confirming the results given in [16].

Equation (108) is a nonlinear constraint approximately determining the allowed velocity levels χ of a double-well velocity potential γ(x) (see Fig. 4) and can be written (taking ϕR = ϕL = π/4 in (98) and (99), i.e. the velocity quarter-wave shift in the connection formulas, which is known to optimize the tunneling effect between two oscillating waves [17] ) as :

1x2x3[2μ(γ(x)χ)]1/2dx=ln{4ctg[π(n12)+ϕL]ctg[π(m12)+ϕR]}1/2=ln2     m,n=1, 2, 3, ...E109

In terms of the momentum Π we have :

x2x3Πdx=ln2orΠdx=2ln2E110

where denotes an integral over one complete cycle of the classical motion, this time

Π=[2μ(γ(x)χ)]1/2=[2μU(x)]1/2=mc[2U(x)c]1/2E111

since γ(x)>χfor the integration limits, i.e. region 3 (see Fig. 5). We get again a quantization of the complex velocityU(x), where the levels are equally spaced at a value of ħ ln2.

In 1961, Landauer [18] discussed the limitation of the efficiency of a computer imposed by physical laws. In particular he argued that, according to the second law of thermodynamics, the erasure of one bit of information requires a minimal heat generation kBT ln2, where kBis Boltzmann’s constant and T is the temperature at which one erases. Its argument runs as follows. Since erasure is a logical function that does not have a single-valued inverse it must be associated with physical irreversibility and therefore requires heat dissipation. A bit has one degree of freedom and so the heat dissipation should be of order kBT. Now, since before erasure a bit can be in any of the two possible states and after erasure it can only be in one state, this implies a change in information entropy of an amount −kBln2.

The one-to-one dynamics of Hamiltonian systems [19] implies that when a bit is erased the information which it contains has to go somewhere. If the information goes into observable degrees of freedom of the computer, such as another bit, then it has not been erased but merely moved; but if it goes into unobservable degrees of freedom such as the microscopic motion of molecules it results in an increase of entropy of at least kBln2.

Inspired by such studies, a considerable amount of work has been made on the thermodynamics of information processing, which include Maxwell’s demon problem [20], reversible computation [21], the proposal of the algorithmic entropy [22] and so on.

Here, considering a double-well velocity potential γ(x) and the velocity quarter-wave shift in the connection formulas, a quanta of ħ ln2 for the complex velocityU(x) of the moving Newtonian 'fluid' occurs. It can be argued that it can be put into a one-to-one correspondence to the quanta of information Landauer and other authors discussed about [23, 24].

Furthermore, one gets an interesting result when taking ϕR = ϕL = π/2, i.e. the velocity half-wave shift in the connection formulas, when singularities occur in (II.66). We try to solve this case by making use of the vortices theory. Benard in 1908 was the first to investigate the appearance of vortices behind a body moving in a fluid [12]. The body he used was a cylinder. He observed that at a high enough fluid velocity (or Reynolds number based on the cylinder diameter), which depends on the viscosity and width of the body, vortices start to shed behind the cylinder, alternatively from the top and the bottom of the cylinder.

Consequently, we write (109) in the form

x[2μ(γ(x)χ)]1/2dx=ln2ln{tg[πl(xx0)]}E112

where we use ctg(α + π/2) = - tg(α), take m = n, make the notations x=nl,  x0=l/2 and consider again the one-dimensional case, motion along the Ox axis.

Solving (112) one gets

U(x)=c(1+ctg2[2πΛ(xx0)])E113

where we assume l = Λ (the Compton length), U(x) = [γ(x) - χ]/2π2 and replace μ=m2c, where c is the velocity of light. When plotting (113) (see Fig. 6) we see that indeed, singularities are obtained for x - x0 = Λ/2 and for x - x0 = Λ/4we get for U(x) minima of value the velocity of light, c.

Usually, at some distance behind a body placed in a fluid, vortices are arranged at a definite distance l apart and with a definite separation h between the two rows. The senses of the rotation in the two rows are opposite (see Fig. 7).

In 1912 von Karman expounded a theory of such vortex streets and the drag which a cylinder would experience due to their formation [12]. Since we considered here the one-dimensional case, we get the solution of a single row of rectilinear vortices, which has already been referred to as characterizing a surface of discontinuity (see Fig. 8).

A typical bound state in a double-well velocity potential has two classically allowed regions, where the velocity potential γ(x)is less than the limit velocity χ. These regions are separated by a classically forbidden region, or barrier, where the velocity potential is larger than the limit velocity. As we can see, quantum mechanics predicts that a velocity wave ζ(x) travelling in such a potential is most likely to be found in the allowed regions. However, unlike classical mechanics, quantum mechanics predicts that this velocity wave can also be found in the forbidden region. This uniquely quantum mechanical behavior allows a velocity wave, initially localized in one potential well, to penetrate through the barrier, into the other well (as we will see in what follows).

### 3.3. Velocity potential γ (x) and the quantum barrier

We already know at the points where χγ(x)=0, special treatment is required because k is singular. The way of handling the solution near the turning point is a little bit more technical, but the basic idea is that we have a solution to the left and to the right of the turning point, and one needs a formula that interpolates between them. In other words, in the vicinity of the turning point one approximates 2μ(χγ(x))/2 by a straight line over a small interval and solves TISE (time independent Schrodinger equation) exactly. This leads to the following connection formulas:

Barrier to the right ( x = b turning point )

2kcos[xbk(x)dxπ4]1ρebxρ(x)dxE114
1ksin[xbk(x)dxπ4]1ρebxρ(x)dxE115

Barrier to the left ( x = a turning point )

2kcos[axk(x)dxπ4]1ρexaρ(x)dxE116
1ksin[axk(x)dxπ4]1ρexaρ(x)dxE117

The connection formulas enable us to obtain relationships between the solutions in a region at some distance to the right of the turning point with those in a region at some distance to the left [25-27].

One of the most important problems to which connection formulas apply is that of the penetration of a potential barrier. The barrier is shown in Fig. 4 and the limit velocity χ is such that the turning points are at x = a and x = b.

Suppose that the motion is incident from the left. Some waves will be reflected and some transmitted, so that in region III we will have:

ζ3(x)=1keibxkdxiπ4=1kcos[bxkdxπ4]+iksin[bxkdxπ4]E118

The phase factor is included for convenience of applying the connection formulas.

In region II (using (114) and (115) on (118)) we have:

ζ2(x)=121ρexbρdxi1ρexbρdxE119

Now using

xbρdx=xaρdx+abρdx=axρdx+αE120

we can write

ζ2(x)=121ρeaxρdxeαi1ρeaxρdxeαE121

Again, using the connection formulas for the case barrier to the right (using (116) and (117) on (121)), we get for region I:

ζ1(x)=12eα1ksin[xakdxπ4]i2keαcos[xakdxπ4]=         =121keαsin(u)i2keαcos(u)=                  =ik[eiu(eα+14eα)+eiu(eα14eα)]E122

Hence

{ζ1inc(x)=ik(eα+14eα)eixakdxiπ4ζ1ref(x)=ik(eα14eα)eixakdx+iπ4E123

Having obtained the expression for ζ1inc(x) and ζ1ref(x) we are now in position to calculate the transmission coefficient using:

T=|ζ3(x)ζ1inc(x)|2=e2α(1+14e2α)2E124

To summarize, for a barrier with large attenuation e-2α→0, the tunneling probability equals

T=e2α(1+14e2α)2e2α=exp(2abρdx)=exp(2ab[2μ(γ(x)χ)]1/2dx)E125

The reflection coefficient is:

R=|ζ1ref(x)ζ1inc(x)|2(eα14eα)2(eα+14eα)2,       T+R=1E126

and also in the same large attenuation limit, we have:

R1e2α=1exp(2abρdx)=1exp(2ab[2μ(γ(x)χ)]1/2dx)E127

One can see from (125) and (127) thatthe velocity wave ζ(x) on small distances, with the same order of magnitude as Λ, may be influenced by U(x),i.e. it can be transmitted, attenuated or reflected at this scale length. In other words, we get from the calculus, that the velocity field V is indeed transported by the motion of the 'Newtonian fluid' particles with the velocity U(x) (the imaginary part of the complex velocity [13]).

## 4. Casimir type effect in scale relativity theory

In recent years, new and exciting advances in experimental techniques [28] prompted a great revival of interest in the Casimir effect, over fifty years after its theoretical discovery (for a recent review on both theoretical and experimental aspects of the Casimir effect, see Refs. [29-31]). As is well known, this phenomenon is a manifestation of the zero-point fluctuations of the electromagnetic field: it is a purely quantum effect and it constitutes one of the rare instances of quantum phenomena on a macroscopic scale.

In his famous paper, Casimir evaluated the force between two parallel, electrically neutral, perfectly reflecting plane mirrors, placed a distance L apart, and found it to be attractive and of a magnitude equal to:

FC=cπ2A240L4E128

Here, A is the area of the mirrors, which is supposed to be much larger than L2, so that edge effects become negligible. The associated energy EC

EC=cπ2A720L3E129

can be interpreted as representing the shift in the zero-point energy of the electromagnetic field, between the mirrors, when they are adiabatically moved towards each other starting from an infinite distance. The Casimir force is indeed the dominant interaction between neutral bodies at the micrometer or submicrometer scales, and by modern experimental techniques it has now been measured with an accuracy of a few percent (see [28] and references therein).

Since this effect arises from long-range correlations between the dipole moments of the atoms forming the walls of the cavity, that are induced by coupling with the fluctuating electromagnetic field, the Casimir energy depends in general on the geometric features of the cavity. For example, we see from (129) that, in the simple case of two parallel slabs, the Casimir energy ECis negative and is not proportional to the volume of the cavity, as would be the case for an extensive quantity, but actually depends separately on the area and distance of the slabs. Indeed, the dependence of ECon the geometry of the cavity can reach the point where it turns from negative to positive, leading to repulsive forces on the walls. For example [29], in the case of a cavity with the shape of a parallelepiped, the sign of ECdepends on the ratios among the sides, while in the case of a sphere it has long been thought to be positive. It is difficult to give a simple intuitive explanation of these shape effects, as they hinge on a delicate process of renormalization, in which the finite final value of the Casimir energy is typically expressed as a difference among infinite positive quantities. In fact, there exists a debate, in the current literature, whether some of these results are true or false, being artifacts resulting from an oversimplification in the treatment of the walls [33].

There are three well-known technical types of derivation of the Casimir force for different geometries including the simplest geometry of two parallel, uncharged, perfectly conducting plates firstly explored by Casimir. One modern method is the quantum field theoretical approach based on the appropriate Green's function of the geometry of problem [34]. The other technical type is the dimensional regularization method that involves the mathematical complications of the Riemann zeta function and the analytical continuation [34]. The last (the most elementary/the simplest) method is based on modes summation by using the Euler-Maclurian integral formula [35-37].

The problem of finding the Casimir force, not only for the simplest geometry of two plates or rectangular prism, that we want to study here, but also for other more complicated geometries, indispensably/automatically involves some infinities/irregularities; thus, one should regularize the calculation for arriving at the desired finite physical result(s). In the Green' function method, one uses the subtraction of two terms (two Green's functions) to do the required regularization. In the dimensional regularization method, although there isn't an explicit subtraction for the regularization of the problem, as is clear from its name, the calculation is regularized dimensionally by going to a complex plane with a mathematically complicated/ambiguous approach. In the simplest method in which the Euler-Maclurian formula is used, the regularization is performed by the subtraction of the zero-point energy of the free space (no plates) from the energy expression under consideration/calculation (e.g. summation of the interior and exterior zero-point energies of the two parallel plates).

Navier-Stokes equations in scale relativity theory predict that the (vector) velocity field V and/or the (scalar) density field ρ, on small distances (the same magnitude as the Compton length) behave like a wave function and are transported by the motion of the Newtonian fluid with velocity U.

Furthermore, when considering vacuum from the Casimir cavity, a non-differentiable, Newtonian, 2D non-coherent quantum fluid whose entities (cvasi-particles) assimilated to vortex-type objects, initially non-coherent, become coherent (the coherence of the quantum fluid reduces to its ordering in vortex streets) due to the constraints induced by the presence of slabs. Casimir type forces are derived which are in good agreement with other theoretical results and experimental data, for both cases: two metallic slabs, parallel to each other, placed at a distance d apart, that constitute the plates of the cavity and a rectangle of sides d1, d.

In other words, non-differentiability and coherence of the quantum fluid due to constraints generate pressure along the Ox and Oy axis.

For viscous compressible fluids, Navier-Stokes equations

ρDvDt=ρXp+μ2v+μ3(v)E130

together with the equation of continuity

DρDt+ρv=0,E131

where ρ is the density, ν the velocity of the fluid, X the body force, p the pressure, μ the shear viscosity and D/Dtd/dt+ν the Eulerian derivative, apply to Newtonian (or near) fluids, that is, to fluids in which the stress is linearly related to the rate of strain (as will be assumed further in this section) [12].

Let us see first, what happens with the set of equations (130) and (131), if one considers that the space-time, where particles move, changes from classical to non-differentiable.

We already know, according to Nottale [11], that a transition from classical (differentiable) mechanics to the scale relativistic framework is implemented by passing to a fluid-like description (the fractality of space), considering the velocity field a fractal function explicitly depending on a scale variable (the fractal geometry of each geodesic) and defining two fractal velocity fields which are fractal functions of the scale variable dt (the non-differentiability of space).

Consequently, replacing d/dt with the fractal operator (42) and solving for both real and imaginary parts, (130) and (131) become, in a stationary isotropic case, taking the body force X = 0 (constant gravitational field) and ∇U = 0 (assuming a constant density of states for the “fluid particles” moving with the velocity U – see further in this section):

VV=pρ+υ 2VaUV+D 2V=0bE132

and

Vρ+ρV=0aUρ+D 2ρ=0bE133

where V represents the standard classical velocity, which does not depend on resolution, while the imaginary part, U, is a new quantity coming from resolution dependant fractal, υ = μ/ρ the kinematic viscosity and D =ħ/2m defines the amplitude of the fractal fluctuations.

The causes of the Casimir effect are described by quantum field theory, which states that all of the various fundamental fields, such as the electromagnetic field, must be quantized at each and every point in space. In a simplified view, a "field" in physics may be envisioned as if space were filled with interconnected vibrating balls and springs, and the strength of the field can be visualized as the displacement of a ball from its rest position. Vibrations in this field propagate and are governed by the appropriate wave equation for the particular field in question. The second quantization of quantum field theory requires that each such ball-spring combination to be quantized, that is, that the strength of the field to be quantized at each point in space. Canonically, the field at each point in space is a simple harmonic oscillator, and its quantization places a quantum harmonic oscillator at each point. Excitations of the field correspond to the elementary particles of particle physics. However, even the vacuum has a vastly complex structure, so all calculations of quantum field theory must be made in relation to this model of vacuum. The vacuum has, implicitly, all of the properties that a particle may have: spin, or polarization in the case of light, energy, and so on. On average, all of these properties cancel out: the vacuum is, after all, "empty" in this sense. One important exception is the vacuum energy or the vacuum expectation value of the energy.

Let us consider here, vacuum, as a non-differentiable, Newtonian, 2D non-coherent quantum fluid whose entities (cvasi-particles) assimilate to vortex-type objects [38] (see Fig.9) and are described by the wave function Ψ [39, 40]

Ψ=cn(u_;k)E134

with

u_=Kaz_,  az_=x+iy, bKa=KbcK=0π/2(1k2sin2ϕ)1/2dϕ,dK=0π/2(1k2sin2ϕ)1/2dϕ,   ek2+k2=1fE135

and K, K’ complete elliptic integrals of the first kind of modulus k [41], form a vortex lattice of constants a, b.

Applying in the complex plane [42], the formalism developed in [13] by means of the relation Ψ=eF(z)/Γ=cn(u_;k) one introduces the complex potential

F(z_)=G(x,y)+iH(x,y)=Γln[cn(u_;k)]E136

with Γ the vortex constant. In the general case Γ = cΛ = ħ/m [38-40], the interaction scale being specified through Γ ’s value (Λ being considered as the Compton length).

Based on the complex potential (136), one defines the complex velocity field of the non-coherent quantum fluid, through the relation:

vxivy=dF(z_)dz_=ΓKasn(u_;k)dn(u_;k)cn(u_;k)E137

or explicitly, using the notations [41, 42]:

s=sn(α,k),ac=cn(α,k),bd=dn(α,k),cα=Kax,ds1=sn(β,k), c1=cn(β,k), f d1=dn(β,k),gβ=KayhE138
vxivy=ΓKa{scd[c12(d12+k2c2s12)s12d12(d2c12k2s2)](1d2s12)(c2c12+s2d2s12d12)+is1c1d1[c2(d2c12k2s2)+s2d2(d12+k2c2s12)](1d2s12)(c2c12+s2d2s12d12)}E139

Having in view that cn(u_+Ω_)=cn(u_), where Ω_=2(2m+1)K+2niKand m,n = ±1, ±2..., for k→ 0 and k’→ 1 limits, respectively, the quantum fluid, initially non-coherent (the amplitudes and phases of quantum fluid entities are independent) becomes coherent (the amplitudes and phases of quantum fluid entities are correlated [43]). In this context, from Fig. 10a,b of the equipotential curves G(xr,yr) = const., for k2 = 0,1, it results that the coherence of the quantum fluid reduces to its ordering in vortex streets - see Fig. III.2a for vortex streets aligned with the Ox axis and Fig. 10b for vortex streets aligned with the Oy axis. This process of ordering is achieved by generation of quasi-particles. Indeed, in the usual quantum mechanics the imaginary term ( ) from the energy, i.e. E= E0 + , induces elementary excitations named resonances (for details see the collision theory [44]). Similarly, by extending the collision theory to the fractal space-time [1, 45], will imply that the presence of the imaginary term H(xr,yr) in the potentialF(z_)will generate quasi-particles, as well.

Now, writing the Navier-Stokes equation (132a) and the equation of continuity (133a) in scale relativity theory for constant density (incompressible fluids) in two dimensions, one gets

px=ρD(2vxx2+2vxy2)ρ(vxvxx+vyvxy)apy=ρD(2vyx2+2vyy2)ρ(vxvyx+vyvyy)bE140
vxx+vyy=0E141

where the shear viscosity υ is replaced by D since we are dealing here with a non-differentiable quantum fluid.

Then, after some rather long yet elementary calculus one gets from (140a,b) through the degenerations :

1. k=0,  k=1,  K=π2,  K=

py(α)=p0sinh2(πd2a)1tan2αcos(2α)+cosh(πda)apx(β)=p0sin2(πd12a)1+tanh2βcos(πd1a)+cosh(2β)bE142

with

p0=2π2ρ4M2a2;aα=πx2a;bβ=πy2acE143

and

1. k=1,  k=0,  K=,  K=π2

py(α)=p0sin2(πd2b)1+tanh2αcos(πdb)+cosh(2α)apx(β)=p0sinh2(πd12b)1tan2βcos(2β)+cosh(πd1b)bE144

with

p0=2π2ρ4M2b2;aα=πx2b;bβ=πy2bcE145

Here, ρis the quantum fluid’s density, M the mass of the quantum fluid entities, d and d1 are the elementary space intervals considered along the Oy and Ox axis, respectively.

In other words, non-differentiability and coherence of the quantum fluid due to constraints, generate pressure along the Ox and Oy axis.

Moreover, one can show that the equation of continuity (141) is identically satisfied for both cases of degeneration.

Let us consider a Casimir cavity consisting of the vacuum with the vortex lattice depicted above and two metallic slabs, that constitute the plates of the cavity, placed at a distance d apart, parallel to each other and to the xOz plane (see Fig. III.1). According to the analysis from the previous section, one can see that if the quantum fluid is placed in a potential well with infinite walls (the case of the Casimir cavity analyzed here, where the two plates are the constraints of the quantum fluid), along a direction perpendicular to the walls (the Oy axis here) a coherent structure, a vortex street forms (see Fig. III.2b). Consequently, by integrating (144a,b) with (145a-c) over αr and βr, and using the result in the quantization rule:

d1d2kdx=nπ,       n=1,2,3,...E146

where d1 ~ m π a,d ~ n π b, with m, n = 1,2,...., one gets

π2pyp0=2rarctan[tan(nπ22)tanh(mπ24r)]tan1(nπ2)rtanh(mπ24r)π2pxp0=2arctan[tan(nπ24)tanh(mπ22r)]tanh1(mπ2r)+tan(nπ24)E147

where

Graphically this is presented in Fig. III.3a,b for different values of the parameters m, n = 1, 2,.... and r.

If the plates were in the yOz plane the constraints being along the Ox axis, vortex streets would form along this axis and the result in (142a,b) with (143a-c) would have been applied, i.e. the cases i) or ii) are identical, yet they depend on the geometry chosen.

Firstly, one can notice that the pressure py on the plates, given by (147a), stabilizes for great r values, is always negative and an attractive force results (see Fig. 11 a), as is the case of the Casimir force (128).

Secondly, the theory predicts, that besides the pressure py acting on the plates, there must be yet another pressure, px (see Fig. 11 b), acting along the Ox axis and given by (147b). One can see that this pressure annuls for great r values, and has a minimum for some values of the parameters m, n. This result is new and should be checked by experiments.

Moreover, if one tries to compute the order of magnitude of this force, and replaces in (144a) : ħ = 1.054 10-34 J.s, m = 9.1 10-31 kg, ρ~ 1021 cm-3, b = 1Ǻ (values specific to a bosonic gas, i.e. found in high-Tc superconductors [46]) and d ~ 5 b (the distance between the plates), gets a value for py 6.18 1010 N m-2 the same order of magnitude as the value calculated using (128), FC 2.08 1010 N m-2.

As a final test, let us study the case of a Casimir cavity, as a rectangle of sides d1, d. Now, the plates induce constraints along both Ox and Oy axis, thus correlations (vortex streets) form along these directions and one should use the degenerations i) and ii), simultaneously. Consequently, from (142a,b) with (143a-c) and (144a,b) with (145a-c) one gets

py rect(α,α)=2π2ρ4m2(1a2sinh2Acos2α(1+cosh2Acos2α)1+1b2sin2Bcosh2α(1+cos2Bcosh2α)1)E149

with

A=πd2a;   B=πd2bE150

and

px rect(β,β)=2π2ρ4m2(1a2sin2Acosh2β(1+cos2Acosh2β)1+1b2sinh2Bcos2β(1+cosh2Bcos2β)1)E151

with

A=πd12a;   B=πd12bE152

At every point (x, y) there is a pressure formed of the two constraints. Consequently, adding the pressures in (149) and (151) and using again the result in (146) (i.e. d1 ~ m π a,d ~ n π b, where m, n = 1,2,....) one gets:

prect(αr,βr)p0=r2sinh2(nπ22r)cos2αr(1+cosh(nπ2r)cos(2αr))1+sin2(nπ22)cosh2(αrr)(1+cos(nπ2)cosh(2αrr))1++r2sin2(mπ22)cosh2(rβr)(1+cos(mπ2)cosh(2rβr))1+sinh2(1rmπ22)cos2βr(1+cosh(mπ2r)cos(2βr))1E153

where

p0=2π2ρ4M2b2;   αr=π2xa=π2xr;   βr=π2yb=π2yr;   r=baE154

Furthermore, we integrate (153) over xr and yr, respectively, in order to find a value of the pressure acting on the sides of the rectangular enclosure. After some long, yet elementary calculus, one finds:

prectp0=4nr2arctg[tg(mπ24)th(nπ22r)]th(nπ2r)+4nrarctg[tg(nπ22)th(mπ241r)]tg(nπ2)+                            +2nr2tg(mπ24)2nr th(mπ241r)4marctg[tg(nπ24)th(mπ221r)]th(mπ21r)+4mrarctg[tg(mπ22)th(nπ24r)]tg(mπ2)+                            +2m tg(nπ24)2mr th(nπ24r)E155

Plots of (155) for various values of parameters m, n = 1, 2,.... and r are depicted in Fig. III.4a,b.

One can notice that if the two parameters m and n have close values, the force acting on the Casimir rectangle is always negative and decreases exponentially for increasing r. For parameters m and n (1,5 and 5,1, i.e. very asymmetric) the force has negative and positive domains (see Fig. 12 b) and increases exponentially for increasing r. Moreover, if one tries to find the positive and negative domains, and solve (155) for m = 5, n = 5 finds prect< 0 for 0.45753 ≤r≤ 2.18565 and prect> 0 for r> 2.18565 and r< 0.45753. This result is in agreement with the calculus of regularization using the Abel-Plana formula where E< 0 for 0.36537 ≤L/l≤ 2.73686 and E> 0 for L/l> 2.73686 and L/l< 0.36537 [47].

## 5. Fractal approximation of motion in mass transfer: release of drug from polimeric matrices

Polymer matrices can be produced in one of the following forms: micro/nano-particles, micro/nano capsules, hydro gels, films, patches.Our new approach considers the entire system (drug loaded polymer matrix in the release environment) as a type of “fluid” totally lacking interaction or neglecting physical interactions among particles. At the same time, the induced complexity is replaced by fractality. This will lead to particles moving on certain trajectories called geodesics within fractal space. This assumption represents the basis of the fractal approximation of motion in Scale Relativity Theory (SRT) [1, 2], leading to a generalized fractal “diffusion” equation that can be analyzed in terms of two approximations (dissipative and dispersive).

### 5.1. The dissipative approximation

In the dissipative approximation the fractal operator (42) takes the form [48, 49]:

t=t+ViD(dt)(2/DF)1ΔE156

As a consequence, we are now able to write the fractal “diffusion” type equation in its covariant form:

Qdt=Qt+(V^)QiD(dt)(2/DF)1ΔQ=0E157

Separating the real and imaginary parts in (157), i.e.

Qt+VQ=0

UQ=D(dt)(2/DF)1ΔQE158

we can add these two equations and obtain a generalized “diffusion” type law in the form:

Qt+(VU)Q=D(dt)(2/DF)1ΔQE159

#### 5.1.1.Standard “diffusion” type equation. Fick type law

The standard “diffusion” law, i.e.:

Qt=D ΔQE160

results from (159) on the following assertions:

1. the diffusion path are the fractal curves of Peano’s type. This means that the fractal dimension of the fractal curves is DF = 2.

2. the movements at differentiable and non-differentiable scales are synchronous, i.e.V=U;

3. the structure coefficient D, proper to the fractal-nonfractal transition, is identified with the diffusion coefficient, i.e.

DD.

#### 5.1.2. Anomalous “diffusion” type equation. Weibull relation

The anomalous diffusion law results from (IV.4) on the following assumptions:

1. the diffusion path are fractal curves with fractal dimension DF2;

2. the time resolution, δt, is identified with the differential element dt, i.e. the substitution principle can be applied also, in this case;

3. the movements at differentiable and non-differentiable scales are synchronous, i.e.V=U.

Then, the equation (IV.4) can be written:

Qt=D(dt)(2/DF)1ΔQE161

In one-dimensional case, applying the variable separation method [50]

Q(t,x)=T(t)X(x)E162

with the standard initial and boundary conditions:

Q(t,0)=0,Q(t,L)=0,Q(0,x)=F(x),0xLE163

implies:

1D(dt)(2/DF)11T(t)dT(t)dt=1X(x)d2X(x)dx2=m2=(nπL)2,n=1, 2E164

where L is a system characteristic length, m a separation constant, dependent on diffusion order n.

Accepting the viability of the substitution principle, from (164), through integration, results:

lnT=m2D(dt)2DFE165

Taking into consideration some results of the fractional integro-differential calculus [51, 52], (165) becomes:

lnT=m2DΓ(2DF+1)t2DF,aΓ(2DF)=0x(2DF)1exdxbE166

Moreover, (166a,b) can be written under the form:

T(t)=exp[m2DΓ(2DF+1)t2DF]E167

The relative variation of concentrations, time dependent, is defined as:

T(t)=QQtQE168

where Qt and Q are cumulative amounts of drug released at time t and infinite time.

From (167) and (168) results:

QtQ=1exp[m2DΓ(2DF+1)t2DF]E169

equation similar to Weibull relation QtQ=1exp(atb), a and b representing constants specific for each system that are defined by:

a=m2DΓ(2DF+1)=(nπL)2DΓ(2DF+1)ab=2DFbE170

We observe that both constants, a and b, are functions of the fractal dimension of the curves on which drug release mechanism take place, dimension that is a measure of the complexity and nonlinear dynamics of the system. Moreover, constant a depends, also, on the “diffusion” order n.

#### 5.1.3. The correspondence between theoretical model and experimental results

The experimental and Weibull curves for HS (starch based hydrogels loaded with levofloxacin) and GA (GEL-PVA microparticles loaded with chloramphenicol) samples are plotted in Fig. 13.

The experimental data allowed to determine the values of Weibull parameters (a and b), and implicitly, the value of the fractal dimension from the curve on which release takes place [55].

These values confirmed that the complexity of the phenomena determines, also, naturally, a complex trajectory for the drug particles. Most values are between 1 and 3, in agreement with the values usually accepted for fractal process; higher values denotes the fact that, either fractal dimension must be redefined as function of structure “classes”, or the drug release process is complex, involving many freedom degrees in the phase space [56]. Another observation that can be made based on this results is that the samples with DF2 manifests a “sub-diffusion” and, in the other, with DF2, the release process is of super-diffusion, classification in concordance with the experimental observation that this samples exhibit a ”faster” diffusion, with a higher diffusion rate, in respect with the other samples [55].

### 5.2. The dispersive approximation

Let us now consider that, in comparison with dissipative processes, convective and dispersive processes are dominant ones. In these conditions, the fractal operator (42) takes the form:

dt=t+(V^)+23D3/2(dt)(3/DFD)13E171

Consequently, we are now able to write the diffusion equation in its covariant form, as a Korteweg de Vries type equation:

Qdt=Qt+(V^)Q+23D3/2(dt)(3/DFD)13Q=0E172

If we separate the real and imaginary parts from Eq. (172), we shall obtain:

Qt+VQ+23D3/2(dt)(3/DF)13Q=0aUQ=0bE173

By adding them, the fractal diffusion equation is:

Qt+(VU)Q+23D3/2(dt)(3/DF)13Q=0E174

From Eq. (173b) we see that, at fractal scale, there will be no Q field gradient.

Assuming that |VU|=σQ with σ=constant (in systems with self structuring processes, the speed fluctuations induced by fractal - non fractal are proportional with the concentration field [55]), in the particular one-dimensional case, equation (174) with normalized parameters:

τ¯=ωt,aξ¯=kx,bΦ=QQ0cE175

and normalizing conditions:

σQ0k6ω=23D3/2(dt)(3/DF)1k3ω=1E176

take the form:

τ¯ϕ+6ϕξ¯ϕ+ξ¯ξ¯ξ¯ϕ=0E177

In relations (175a,b,c) and (176) ω corresponds to a characteristic pulsation, k to the inverse of a characteristic length and Q0 to balanced concentration.

Through substitutions:

w(θ)=ϕ(τ¯,ξ¯),aθ=ξ¯uτ¯bE178

eq.(177), by double integration, becomes:

12w2=F(w)=(w3u2w2gwh)E179

with g, h two integration constants and u the normalized phase velocity. If F(w) has real roots, equation (177) has the stationary solution:

ϕ(ξ¯,τ¯,s)=2a(E(s)K(s)1)+2acn2[as(ξ¯u¯2τ¯+ξ0¯);s]E180

where cn is Jacobi’s elliptic function of s modulus [41], a is the amplitude, ξ0¯ is a constant of integration and

K(s)=0π/2(1s2sin2ϕ)1/2dϕaE(s)=0π/2(1s2sin2ϕ)1/2dϕbE181

are the complete elliptic integrals [41].

Parameter s represents measure characterizing the degree of nonlinearity in the system. Therefore, the solution (180) contains (as subsequences for s=0) one-dimensional harmonic waves, while for s0 one-dimensional wave packet. These two subsequences define the non-quasi-autonomous regime of the drug release process [48, 49, 55], i.e. the system should receive external energy in order to develop. For s=1, the solution (180) becomes one-dimensional soliton, while for s1, one-dimensional soliton packet will be generated. The last two imply a quasi-autonomous regime (self evolving and independent [48]) for drug particle release process [48, 49, 55].

The three dimensional plot of solution (180) shows one-dimensional cnoidal oscillation modes of the concentration field, generated by similar trajectories of the drug particles (see Fig. 14). We mention that cnoidal oscillations are nonlinear ones, being described by the elliptic function cn, hence the name (cnoidal).

It is known that in nonlinear dynamics, cnoidal oscillation modes are associated with nonlinear lattice of oscillators (the Toda lattice [56]). Consequently, large time scale drug particle ensembles can be compared to a lattice of nonlinear oscillators which facilitates drug release process.

#### 5.2.1. The correspondence between theoretical model and experimental results

In what follows we identify the field Φ from relation (180) with normalized concentration field of the released drug from micro particles.

For best correlation between experimental data and the theoretical model (for each sample) we used a planar intersection of the graph in Fig. 14 [57], in order to obtain two-dimensional plots.

The highest value of the correlation coefficient (for two data sets: one obtained from the planar intersection, the other from experimental data) will represent the best approximation of experimental data with the theoretical model.

Our goal was to find the right correlation coefficient which should be higher than 0.60.7, in order to demonstrate the relevance of the model we had in view. Figs. 15 show experimental and theoretical curves that were obtained through this method, where R2 represents the correlation coefficient and η a normalized variable which is simultaneously dependent on normalized time and on nonlinear degree of the system (s parameter). Geometrically, η represents the congruent angle formed by the time axis and the vertical intersection plane.

## 6. Conclusions

1. Scale relativistic framework is implemented by passing to a fluid-like description (the fractality of space), considering the velocity field a fractal function explicitly depending on a scale variable (the fractal geometry of each geodesic) and defining two fractal velocity fields which are fractal functions of the scale variable dt (the non-differentiability of space).

An application of these principles to the motion equation of free particles leads to the occurence of a supplementary TISE (time independent, Schrödinger-type equation) and the following interesting results :

• ζ(x) behaves like a wave function on small distances (the same magnitude as the Compton length);

• for γ(x) a velocity potential well, U(x) is quantified;

• for the harmonic oscillator case, the limit velocity χ has discrete values, and only the first value is less than the velocity of light, c;

• in the double-well velocity potential, the complex velocity U(x) is again quantized, this time the levels are equally spaced at a value of ħ ln2;

• if one takes ϕR = ϕL = π/2, singularities are obtained for x - x0 = Λ/2 and for x - x0 = Λ/4 one gets minima for U(x)= c in a double-well velocity potential;

• since we considered here the one-dimensional case we get the solution of a single row of rectilinear vortices, which has already been referred to as characterizing a surface of discontinuity;

• a typical bound state in a double-well has two classically allowed regions, where the velocity potential is less than the limit velocity; these regions are separated by a classically forbidden region, or barrier, where the velocity potential is larger than the limit velocity;

• for tunneling case, there is a nonzero transmission, reflection coefficient, which leads to the proof of the transport of the V field by the motion of the Newtonian fluid with velocity U(x), on small distances (of the order of magnitude of Compton length).

1. We analyzed vacuum from the Casimir cavity, considered a non-differentiable, Newtonian, 2D non-coherent quantum fluid, by writing the Navier-Stokes equations in scale relativity theory’s framework. As a result the following results may be extracted:

• the (vector) velocity field V and/or the (scalar) density field ρ behave like a wave function on small distances (the same magnitude as the Compton length);

• the (vector) velocity field V and/or the (scalar) density field ρ are transported by the motion of the Newtonian fluid with velocity U, on small distances (the same magnitude as the Compton length);

Also, the entities assimilated to vortex-type objects from the Casimir cavity, initially non-coherent, become coherent due to constraints induced by the presence of walls and generate pressure along the Ox and Oy axis, thus one can stress out :

• the pressure py on the plates, is negative and an attractive force results, as is the case of the Casimir force;

• besides the pressure py acting on the plates, there must be yet another pressure, px, acting along the Ox axis;

• the order of magnitude of this force, py ≅ 6.18 1010 N m-2 is the same with the value of the classical Casimir force calculation, FC ≅ 2.08 1010 N m-2;

• in the case of the Casimir cavity from inside a rectangular enclosure of sides d1, d, the plates induce constraints along both Ox and Oy axis, and one can notice that if the two parameters m and n have close values, the force acting on the Casimir rectangle is always negative and for parameters m and n very asymmetric the force has negative and positive domains, in agreement with the calculus of regularization using the Abel-Plana formula.

1. Using fractional calculus, the fractal “diffusion” equation give rise to Weibull relation, a statistical distribution function of wide applicability, inclusively in drug release studies. In this approach, we consider all the simultaneous phenomena involved, equivalent with complexity and fractality, offering, in this way, a physical base to this equation and for its parameters. They are functions of fractal dimension of the curves on which drug release mechanism takes place, dimension that is a measure of the complexity and nonlinear dynamics of the system, dependent on the diffusion order.

This theory offers new alternatives for the theoretical study of drug release process (on large time scale) in the presence of all phenomena and considering a highly complex and implicitly, non linear system. Consequently, the concentration field has cnoidal oscillation modes, generated by similar trajectories of drug particles. This means that the drug particle ensemble (at time large scale) works in a network of non linear oscillators, with oscillations around release boundary. Moreover, the normalized concentration field simultaneously depends on normalized time non linear system (through s parameter).

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Written By

M. Agop, C.Gh. Buzea, S. Bacaita, A. Stroe and M. Popa

Submitted: 24 September 2012 Published: 03 April 2013