The Principle of Eurhythmy: A Key to the Unity of Physics

Abstract

The unification of the basic physical laws, from classical physics, quantum physics to gravitic physics seems now possible. The key for this unity is the principle of eurhythmy, literally from the Greek, the principle of the most adequate path. It will be shown that Heron principle of minimum path, Fermat’s principle of minimum time and de Broglie’s guiding principle are no more than mere particular cases of the principle of eurhythmy. Furthermore, it will be shown, with concrete examples, from classical physics, quantum physics to gravitic physics, that all these branches of physics can be unified and understood in a causal way as particular cases of this general principle.

Appendices

Derivation of Snell’s Refraction Law from Fermat’s Principle of Minimum Time

This derivation can be found in almost all good textbooks on optics. Nevertheless for the sake of reference we shall present here a simplified version. In order to do that, consider the following sketch where the light is emitted from point S and later is detected at point P.

The time t taken by the light in the course from point S to point P is equal to the time in the incidence region t _i plus the time in the transmission medium t _t,

$$ t = {t_i} + {t_t}. $$

(A1.1)

These times being given by the travelled path divided by the velocity, which must be expressed in terms of the variable x,

$$ \left\{ {\begin{array}{*{20}{c}} {{t_i} = \dfrac{{\bar{S}\!\bar{O}}}{{{v_i}}} = \dfrac{1}{{{v_i}}}{{\left( {{x^2} + y_i^2} \right)}^{{ \frac{1}{2} }}}} \\[9pt] {{t_t} = \dfrac{{\bar{O}\!\bar{P}}}{{{v_t}}} = \dfrac{1}{{{v_t}}}{{({{(h - x)}^2} + y_t^2)}^{{ \frac{\hbox{$\scriptstyle 1$}}{\hbox{$\scriptstyle 2$}} }}}} \end{array} } \right. $$

(A1.2)

Since the total possible times is now a function of the variable x, t = t(x), in order to find the minimum of all those possible times we use the habitual calculus technique

$$ \frac{{dt(x)}}{{dx}} = 0, $$

(A1.3)

which gives the condition

$$ \frac{1}{{{v_i}}}\frac{x}{{({x^2} + y_i^2) \frac{\hbox{$\scriptstyle 1$}}{\hbox{$\scriptstyle 2$}} }} = \frac{1}{{{v_t}}}\frac{{(h - x)}}{{({{(h - x)}^2} + y_t^2) \frac{\hbox{$\scriptstyle 1$}}{\hbox{$\scriptstyle 2$}} }} $$

or

$$ \frac{1}{{{v_i}}}\frac{x}{{\bar{S}\!\bar{O}}} = \frac{1}{{{v_t}}}\frac{{(h - x)}}{{\bar{O}\!\bar{P}}}. $$

(A1.4)

From the sketch Fig. A1.1 we see that

$$ \sin {\theta_i} = \frac{x}{{\bar{S}\!\bar{O}}}\quad {\text{and}}\quad \sin {\theta_t} = \frac{{(h - x)}}{{\bar{O}\!\bar{P}}}, $$

(A1.5)

which by substitution in (A1.4) gives

$$ \frac{1}{{{v_i}}}\sin {\theta_i} = \frac{1}{{{v_t}}}\sin {\theta_t}, $$

(A1.6′)

or, multiplying by the constant c, it gives finally Snell’s refraction law

$$ {n_i}\sin {\theta_i} = {n_t}\sin {\theta_t}, $$

(A1.6)

where n _i  = c/v _i and n _t  = c/v _t are the indexes of refraction in the incident and transmission mediums.

Fig. A1.1

Schetch for derivation of Snell’s law from Fermat principle of minimum time

Derivation of Snell’s Refraction Law from the Principle of Eurhythmy

In the derivation of Snell’s refraction law from the principle of Eurhythmy we start from the following assumptions: Basic:

• 1. The space available to the corpuscle is only and only the theta wave field. No corpuscles are found where the theta wave field is null.

• 2. In this approach there is energy conservation of the corpuscle. More complex cases where energy dissipation is important are not considered here.

• 3. The natural velocity of the corpuscle v _N is enormous, v _N  >>> c. Still the observed velocity, the average, velocity can be very small even null.

• 4. The corpuscle moves preferentially to the zones where the theta wave field has higher intensity.

Formal:

• 1. The space available to the corpuscle is divided into cubic cells of linear size ℓ ₀ .

• 2. In each transition, which takes a very short time t ₀ , the corpuscle may move only to the adjacent cells.

• 3. The explicit form of the probabilities of transition are, for all practical purposes, the mathematical expression of the principle of eurhythmy.

In this context the natural velocity has the form

$$ {v_N} = {v_0} = \frac{{{\ell_0}}}{{{t_0}}} > > > c. $$

(A2.1)

where c is the habitual constant standing for velocity of the light in a medium with index of refraction one.

It is convenient to make a sketchy representation of the space in order to write the explicit form for the probabilities of transition. In two dimensions we have Fig. A2.1 where ${p_{{i \to i + 1,j}}}$ corresponds to the probability of transition from cell C_i,j to cell C_i+1,j and ${p_{{i \to i,j}}}$ is the probability of remaining in the same cell C_i,j. Since there is no danger of confusion for notation simplicity reasons the index not corresponding to the transition is dropped. So in the following we make ${p_{{i \to i + 1,j}}} \equiv {p_{{i,i + 1}}}$, ${p_{{i \to i,j}}} \equiv {p_{{i,i}}}$, and successively. For the other probabilities their meaning can be inferred from the sketch. Naturally, for the generic cell C_i,j the sum of the exit probabilities plus the probability of remaining must be one, that is

Fig. A2.1

Generic cell in a two-dimensional field with transition probabilities

$$ {p_{{i,i - 1}}} + {p_{{i,i + 1}}} + {p_{{i,i}}} + {p_{{j,j - 1}}} + {p_{{j,j + 1}}} = 1. $$

(A2.2)

Note that for notation simplification the index j was dropped in the horizontal representation and i in the vertical. As the following To explicit the analytic form of the probabilities of transition it is convenient to consider the following cases:

• 1. The corpuscle is in an initial condition such that is without any preferential direction for the motion. In this case the form of the probabilities stems directly from the principle of eurhythmy giving

  $$ {p_{{i,i - 1}}} = \frac{{{I_{{i - 1,j}}}}}{{{I_{{i - 1,j}}} + {I_{{i + 1,j}}} + {I_{{i,j}}} + {I_{{i,j - 1}}} + {I_{{i,j + 1}}}}} $$

  (A2.3-1)
  $$ {p_{{i,i}}} = \frac{{{I_{{i,j}}}}}{{{I_{{i - 1,j}}} + {I_{{i + 1,j}}} + {I_{{i,j}}} + {I_{{i,j - 1}}} + {I_{{i,j + 1}}}}} $$

  (A2.3-2)
  $$ {p_{{i,i + 1}}} = \frac{{{I_{{i + 1,j}}}}}{{{I_{{i - 1,j}}} + {I_{{i + 1,j}}} + {I_{{i,j}}} + {I_{{i,j - 1}}} + {I_{{i,j + 1}}}}} $$

  (A2.3-3)
  $$ {p_{{j,j - 1}}} = \frac{{{I_{{i,j - 1}}}}}{{{I_{{i - 1,j}}} + {I_{{i + 1,j}}} + {I_{{i,j}}} + {I_{{i,j - 1}}} + {I_{{i,j + 1}}}}} $$

  (A2.3-4)
  $$ {p_{{j,j + 1}}} = \frac{{{I_{{i,j + 1}}}}}{{{I_{{i - 1,j}}} + {I_{{i + 1,j}}} + {I_{{i,j}}} + {I_{{i,j - 1}}} + {I_{{i,j + 1}}}}} $$

  (A2.3-5)

  where I _i,j is the intensity of the theta wave in the cell C_i,j and respectively.

• 2. The corpuscle due to some initial preliminary process in the surrounding theta wave field behaves as if it has a preferred tendency for motion.

In this case the expression for the probabilities of transition need to have two terms:

• (a) One due only to the principle of eurhythmy depending only on the relative intensity of the field.

• (b) The other resulting from the direct modification of the field in the neighborhood of the corpuscle, which usually is traduced by the so called initial velocity. In this approach the absolute value of the initial velocity is assume constant.

In such conditions for the generic form of the probabilities we got

$$ p = \tilde{p} + {p_0},\;\quad {p_0} = {\text{constant}} $$

(A2.4)

where $\tilde{p}$ is the direct contribution of the principle of eurhythmy and p ₀ resulting from the initial velocity. In this situation the equation of conservation of probability (A2.2) assumes the form

$$ {\tilde{p}_{{i,i - 1}}} + {q_{{oh}}} + {\tilde{p}_{{i,i + 1}}} + {p_{{oh}}} + {\tilde{p}_{{i,i}}} + {\tilde{p}_{{j,j - 1}}} + {q_{{ov}}} + {\tilde{p}_{{j,j + 1}}} + p_{{ov}} = 1, $$

(A2.5)

with q _oh corresponding to the term for the initial motion backwards, p _oh to the motion forward, for the vertical initial component, q _ov downwards and p _ov upwards. Naturally

$$ {q_{{oh}}} + {p_{{oh}}} + {q_{{ov}}} + {p_{{ov}}} = {\text{const}} $$

(A2.6)

with each one also constant.

The master stochastic equation describing the number of the corpuscles in the generic cell assumes the form

$$ {n_{{i,j,\,t + 1}}} = {p_{{i - 1,i}}}\,{n_{{i - 1,j,\,t}}} + {p_{{i + 1,i}}}\,{n_{{i + 1,j,\,t}}} + {p_{{i,i}}}\,{n_{{i,j,\,t}}} + {p_{{j - 1,j}}}\,{n_{{i,j - 1,\,t}}} + {p_{{j + 1,j}}}\,{n_{{i,j + 1,\,t}}}, $$

(A2.7)

with n _i,j,t representing the number of singularities in the cell C_i,j at time t.

According to the concrete physical situation the transition probabilities assume a fixed form allowing consequently to precise the structure of the master stochastic equation.

3.2.1 Homogeneous Medium

In the case of Snell’s refraction law and the Fermat’s principle of minimum time we had two optical different mediums. In such a case we can study the behaviour of the master stochastic equation in a homogeneous medium.

The eurhythmic probabilities of transition in a homogeneous medium, where I _i,j  = constant, are, of course, also constant, as can be seen

$$ {\tilde{p}_{{i,i - 1}}} = \frac{{{I_{{i - 1,j}}}}}{{{I_{{i - 1,j}}} + {I_{{i + 1,j}}} + {I_{{i,j}}} + {I_{{i,j - 1}}} + {I_{{i,j + 1}}}}} = p = {\text{constant}}{.} $$

(A2.8)

Assuming that the components due to the initial velocity are such that

$$ {q_{{oh}}} = {q_{{ov}}} = 0, $$

(A2.9)

In such conditions the probabilities of transition can be written

$$ {p_{{i - 1,i}}} = {p_h},\quad {p_{{i + 1,i}}} = {q_{{h,\quad }}}{p_{{i,i}}} = \delta, \quad {p_{{j - 1,j}}} = {p_v},\quad {p_{{j + 1,j}}} = {q_v}, $$

(A2.10)

with

$$ {p_h} = p + {p_{{oh}}},\quad {q_h} = p,\quad \delta = p,\quad {p_v} = p + {p_{{ov}}},\quad {q_v} = p, $$

(A2.11)

which by substitution in the stochastic master equation (4.7) gives

$$ {n_{{i,j,\,t + 1}}} = {p_h}\,{n_{{i - 1,j,\,t}}} + {q_{{h\,}}}{n_{{i + 1,j,\,t}}} + \delta \,{n_{{i,j,\,t}}} + {p_{{v\,}}}{n_{{i,j - 1,\,t}}} + {q_{{v\,}}}{n_{{i,j + 1,\,t}}}. $$

(A2.12)

This discrete equation can be approached to a continuous equation of the form

$$ \begin{gathered} n(x,y,t + {\tau_0}) = \hfill \\ {p_h}\,n(x - {\ell_0},y,t) + q{}_h\,n(x + {\ell_0},y,t) + \delta \,n(x,y,t) + {p_v}\,n(x,y - {\ell_0},t) + {q_v}\,n(x,y + {\ell_0},t) \hfill \\ \end{gathered} $$

(A2.13)

that can be expanded in Taylor series giving

$$ {\fontsize{9.3}{10}\selectfont \begin{aligned} &n(x,y,t) + {\tau_0}\frac{{\partial n(x,y,t)}}{{\partial t}} + \frac{1}{{2!}}\tau_0^2\frac{{{\partial^2}n(x,y,t)}}{{\partial {t^2}}} + \ldots \hfill \\[3pt] &\hspace{3pt} = {p_h}n(x,y,t) - {p_h}{\ell_0}\frac{{\partial n(x,y,t)}}{{\partial x}} + \frac{1}{{2!}}{p_h}\ell_0^2\frac{{{\partial^2}n(x,y,t)}}{{\partial {x^2}}} - \frac{1}{{3!}}{p_h}\ell_0^3\frac{{{\partial^3}n(x,y,t)}}{{\partial {x^3}}} + \ldots \hfill \\[3pt] &\quad + {q_h}n(x,y,t) + {q_h}{\ell_0}\frac{{\partial n(x,y,t)}}{{\partial x}} + \frac{1}{{2!}}{q_h}\ell_0^2\frac{{{\partial^2}n(x,y,t)}}{{\partial {x^2}}} + \frac{1}{{3!}}{q_h}\ell_0^3\frac{{{\partial^3}n(x,y,t)}}{{\partial {x^3}}} + \ldots \hfill \\[3pt] &\quad + \delta \,n(x,y,t) \hfill \\[3pt] &\quad + {p_v}n(x,y,t) - {p_v}{\ell_0}\frac{{\partial n(x,y,t)}}{{\partial y}} + \frac{1}{{2!}}{p_v}\ell_0^2\frac{{{\partial^2}n(x,y,t)}}{{\partial {y^2}}} - \frac{1}{{3!}}{p_v}\ell_0^3\frac{{{\partial^3}n(x,y,t)}}{{\partial {y^3}}} + \ldots \hfill \\[3pt] &\quad + {q_v}n(x,y,t) + {q_v}{\ell_0}\frac{{\partial n(x,y,t)}}{{\partial y}} + \frac{1}{{2!}}{q_v}\ell_0^2\frac{{{\partial^2}n(x,y,t)}}{{\partial {y^2}}} + \frac{1}{{3!}}{q_v}\ell_0^3\frac{{{\partial^3}n(x,y,t)}}{{\partial {y^3}}} + \ldots \raisetag{-5pt}\end{aligned}} $$

(A2.14)

since the linear size ℓ ₀ of the cell is very small and furthermore the velocity of transition is very big implying that

$$ {\ell_0} > > > {t_0} \equiv {\tau_0} $$

(A2.15)

it is reasonable to cut the expression (A2.14) from $\tau_0^2$ and $\ell_0^3$ giving

$$ \begin{gathered} n + {\tau_0}\frac{{\partial \,n}}{{\partial \,t}} = ({p_h} + {q_h} + \delta + {p_v} + {q_v})\,n \hfill \\ \quad \quad \quad - {\ell_0}({p_h} - {q_h})\frac{{\partial \,n}}{{\partial \,x}} + \frac{1}{2}\ell_0^2({p_h} + {q_h})\frac{{{\partial^2}n}}{{\partial \,{x^2}}} \hfill \\ \quad \quad \quad - {\ell_0}({p_v} - {q_v})\frac{{\partial \,n}}{{\partial \,y}} + \frac{1}{2}\ell_0^2({p_v} + {q_v})\frac{{{\partial^2}n}}{{\partial \,{y^2}}} \hfill \\ \end{gathered} $$

(A2.16)

recalling that

$$ {p_h} + {q_h} + \delta + {p_v} + {q_v} = 1, $$

and making

$$ {\mu_h} = {p_h} - {q_h} = {p_{{oh}}},\quad {\mu_v} = {p_v} - {q_v} = {p_{{ov}}}, $$

(A2.17)

$$ {D_h} = {p_h} + {q_h} = 2p + {p_{{oh}}},\quad {D_v} = {p_v} + {q_v} = 2p + {p_{{ov}}} $$

which are the drift $\mu$ components and the diffusion coefficients D, along the horizontal and vertical axes. In this situation Eq. A2.16 can be written in the condensed form

$$ {D_h}\,\frac{{{\partial^2}n}}{{\partial {x^2}}} + {D_v}\,\frac{{{\partial^2}n}}{{\partial {y^2}}} - \frac{{2{\mu_h}}}{{{\ell_0}}}\frac{{\partial n}}{{\partial x}} - \frac{{2{\mu_v}}}{{{\ell_0}}}\frac{{\partial n}}{{\partial y}} = \frac{2}{{{\ell_0}{v_0}}}\frac{{\partial n}}{{\partial t}}. $$

(A2.18)

3.2.1.1 The Drift Describes a Preferential Motion in a Stochastic Process

In order to see that the drift indeed describes a preferential sense for the motion in a stochastic process we are going to write the two-dimensional stationary stochastic equation for the case when the drift has only components along the horizontal axis.

In this situation looking at Fig. A2.1, and recalling that the number of corpuscles in cell (i,j) is given by the number of those that remain plus the ones that enter the cell, we are led to write the balance equation

$$ {n_{{i,j}}} = {p_{{h\,}}}{n_{{i - 1,j}}} + {q_{{h\,}}}{n_{{i + 1,j}}} + \delta \,{n_{{i,j}}} + {p_{{v\,}}}{n_{{i,j - 1}}} + {q_{{v\,}}}{n_{{i,j + 1}}}, $$

(A2.20)

which can also be written

$$ n(x,y) = {p_{{h\,}}}n(x - {\ell_0},y) + {q_{{h\,}}}n(x + {\ell_0},y) + \delta \,n(x,y) + {p_{{v\,}}}n(x,y - {\ell_0}) + {q_{{v\,}}}n(x,y + {\ell_0}). $$

(A2.21)

This equation can be developed in Taylor power series giving after the habitual cutoff processes

$$ \begin{gathered} n = {p_{{h\,}}}n - {\ell_0}{p_{{h\,}}}\frac{{\partial \,n}}{{\partial \,x}} + \frac{1}{2}\ell_0^2{p_{{h\,}}}\frac{{{\partial^2}n}}{{\partial \,{x^2}}} \hfill \\ + {q_{{h\,}}}n + {\ell_0}{q_{{h\,}}}\frac{{\partial \,n}}{{\partial \,x}} + \frac{1}{2}\ell_0^2{q_{{h\,}}}\frac{{{\partial^2}n}}{{\partial \,{x^2}}} \hfill \\ + \delta \,n \hfill \\ + {p_{{v\,}}}n - {\ell_0}{p_{{v\,}}}\frac{{\partial \,n}}{{\partial \,y}} + \frac{1}{2}\ell_0^2{p_{{v\,}}}\frac{{{\partial^2}n}}{{\partial \,{y^2}}} \hfill \\ + {q_{{v\,}}}n + {\ell_0}{q_{{v\,}}}\frac{{\partial \,n}}{{\partial \,y}} + \frac{1}{2}\ell_0^2{q_{{v\,}}}\frac{{{\partial^2}n}}{{\partial \,{y^2}}} \hfill \\ \end{gathered} $$

(A2.22)

or

$$ \frac{1}{2}\ell_0^2({p_h} + {q_h})\frac{{{\partial^2}n}}{{\partial {x^2}}} + \frac{1}{2}\ell_0^2({p_v} + {q_v})\frac{{{\partial^2}n}}{{\partial {y^2}}} - {\ell_0}({p_h} - {q_h})\frac{{\partial n}}{{\partial x}} - {\ell_0}({p_v} - {q_v})\frac{{\partial n}}{{\partial y}} = 0 $$

giving

$$ {D_h}\frac{{{\partial^2}n}}{{\partial {x^2}}} + {D_v}\frac{{{\partial^2}n}}{{\partial {y^2}}} - \frac{{2{\mu_h}}}{{{\ell_0}}}\frac{{\partial n}}{{\partial x}} - \frac{{2{\mu_v}}}{{{\ell_0}}}\frac{{\partial n}}{{\partial y}} = 0, $$

(A2.23)

expression that, as expected, is equal to (A2.18) when the number of corpuscles does not depend explicitly on time.

Since we have assumed that the drift was defined only along the horizontal axis,

$$ {\mu_h} = \mu, \quad {\mu_v} = 0 $$

(A2.24)

we have finally

$$ {D_h}\frac{{{\partial^2}n}}{{\partial {x^2}}} + {D_v}\frac{{{\partial^2}n}}{{\partial {y^2}}} - \frac{{2\mu }}{{{\ell_0}}}\frac{{\partial n}}{{\partial x}} = 0. $$

(A2.25)

A solution to this differential equation has the form

$$ n(x,y) = A\,{e^{{({k_1}x + ik_1^{'}x) - {k_2}|y|}}} + B = \phi + B, $$

(A2.26)

as can easily be seen. In fact:

$$ \frac{{\partial n}}{{\partial x}} = ({k_1} + ik_1^{'})\phi, \quad \frac{{{\partial^2}n}}{{\partial {x^2}}} = (k_1^2 - k_1^{{'2}} + 2i{k_1}k_1^{'})\phi, \quad \frac{{{\partial^2}n}}{{\partial {y^2}}} = k_2^2\phi, $$

(A2.27)

which by substitution in (A2.25) gives after separating the real and imaginary parts

$$ \left\{ {\begin{array}{*{20}{l}} {D_x}(k_1^2 - k_1^{{'2}}) + D{}_yk_2^2 - \dfrac{{2\mu }}{{{\ell_0}}}k_1 = 0 \\[3pt] {D_x}{k_1} - \dfrac{\mu }{{{\ell_0}}} = 0 \end{array}} \right. $$

(A2.28)

giving

$$ {k_1} = \frac{\mu }{{{\ell_0}{D_x}}}, $$

(A2.29)

where we have renamed

$$ {D_x} = {D_h},\quad {D_y} = {D_v}. $$

(A2.30)

Now a possible relation that much simplifies the result comes from making

$$ k_1^{'} = {k_1} $$

(A2.31)

in such conditions the first equation of (A2.28) transforms into

$$ {D_y}k_2^2 - \frac{{2\mu }}{{{\ell_0}}}\frac{\mu }{{{\ell_0}{D_x}}} = 0 $$

giving

$$ {k_2} = \sqrt {{\frac{2}{{{D_x}{D_y}}}}} \frac{\mu }{{{\ell_0}}}, $$

(A2.32)

consequently a solution to the balance equation (A2.25) can then be written explicitly

$$ n(x,y) = A\,{e^{{\frac{\mu }{{{\ell_0}{D_x}}}x + i\frac{\mu }{{{\ell_0}{D_x}}}x - \frac{{\sqrt {2} \,\mu }}{{{\ell_0}\sqrt {{{D_x}{D_y}}} }}\,|y|}}} + B, $$

(A2.33)

which overall graphic representation of the real part is seen in Fig. A2.2

Fig. A2.2

Plot of the real part of the solution stationary stochastic equation

Needless to say that the drift $\mu \geqslant 0$ must be very small otherwise there would be a practically complete deterministic motion. On the other hand the $\vec{k}$ wave vector is only along the xx direction, which means, as we have said before, that the drift for all average purposes characterizes indeed the preferential direction of the corpuscles, meaning that

$$ \vec{\mu} \propto \vec{k}. $$

(A2.34)

In such circumstances we can draw the graphic representing average motion of the corpuscles in two homogeneous mediumswhere the drift has the form

$$ \vec{\mu } = {\mu_x}{\vec{e}_x} + {\mu_y}{\vec{e}_y}. $$

(A2.34)

Fig. A2.3

Average motion of the corpuscles in two optical mediums

From Fig. A2.3 it is seen that

$$ \left\{ {\begin{array}{*{20}{c}} {{\mu_{{1y}}} = {\mu_1}\sin {\theta_1}} \\[6pt] {{\mu_{{2y}}} = {\mu_2}\sin {\theta_2}} \end{array} } \right. $$

(A2.35)

because along the yy direction there is no change in the medium we are allowed to make

$$ {\mu_1}\sin {\theta_1} = {\mu_2}\sin {\theta_2}. \vspace{-2pt}$$

(A2.36)

Since by (A2.34)

$$ {\mu_x} = \alpha \,{k_x};\quad {\mu_y} = \alpha \,{k_y}, $$

(A2.37)

where α is a proportionality constant we also have

$$ {k_1}\sin {\theta_1} = {k_2}\sin {\theta_2}, $$

(A2.38-1)

recalling that $k = 2\pi /\lambda$, we have by substitution

$$ \frac{{2\pi }}{{{\lambda_1}}}\sin {\theta_1} = \frac{{2\pi }}{{{\lambda_2}}}\sin {\theta_2}, $$

(A2.38-2)

or

$$ \frac{T}{{{\lambda_1}}}\sin {\theta_1} = \frac{T}{{{\lambda_2}}}\sin {\theta_2}, $$

(A2.38-3)

because in a linear optical mediums T₁ = T₂ , and $v = \lambda /T$ we can write

$$ \frac{1}{{{v_1}}}\sin {\theta_1} = \frac{1}{{{v_2}}}\sin {\theta_2}, $$

(A2.38-4)

which multiplied by c gives finally Snell’s formula

$$ {n_1}\sin {\theta_1} = {n_2}\sin {\theta_2}. $$

(A2.38-5)

This formula means, according to Fermat, that light, that is the corpuscles of light, follow a path such that the time taken in the whole course from S to P is minimum.

Derivation of Galilee’s Law from the Principle of Eurhythmy

For this derivation we assume that the theta wave intensity gravitic field has a linear variation increasing when approaching the Earth. Furthermore, it is also understood that the stochastic model can be simplified from two dimensions to one single dimension which much simplifies the calculations.

3.3.1 One-dimensional Approach

The probabilities of transition constant in time A3.1 are

Fig. A3.1

Transition probabilities for the generic cell in one-dimensional field

with

$$ {q_i} = {p_{{i,i - 1}}};\quad {\delta_i} = {p_{{i,i}}};\quad \quad {p_i} = {p_{{i,i + 1}}} $$

(A3.1)

and

$$ {q_i} + {\delta_i} + {p_i} = 1. $$

(A3.2)

These transition probabilities can be expressed in terms of the intensity according to the principle of eurhythmy

$$ \left\{ {\begin{array}{*{20}{c}} {{q_i} = {p_{{i.i - 1}}} = \dfrac{{{I_{{i - 1}}}}}{{{I_{{i - 1}}} + {I_i} + {I_{{i + 1}}}}}} \\[12pt] {{\delta_i} = {p_{{i.i - 1}}} = \dfrac{{{I_i}}}{{{I_{{i - 1}}} + {I_i} + {I_{{i + 1}}}}}} \\[12pt] {{p_i} = {p_{{i.i - 1}}} = \dfrac{{{I_{{i + 1}}}}}{{{I_{{i - 1}}} + {I_i} + {I_{{i + 1}}}}}} \end{array} } \right. $$

(A3.3)

The stochastic evolution equation can be written recalling that the number of corpuscles in the generic cell C_i in the instant t +1 is equal to the number of corpuscles that remain plus the ones that enter from the right and the left

$$ {n_{{i,\,t + 1}}} = {p_{{i - 1}}}{n_{{i - 1,\,t}}} + {\delta_i}{n_{{i,\,t}}} + {q_{{i + 1}}}{n_{{i + 1,\,t}}}, $$

(A3.4)

or

$$ {n_{{i,\,t + 1}}} = (1 - {q_i} - {p_i}){n_{{i,\,t}}} + {p_{{i - 1}}}{n_{{i - 1,\,t}}} + {q_{{i + 1}}}{n_{{i + 1,\,t}}} $$

(A3.4′)

$$ {n_{{i,\,t + 1}}} = {n_{{i,\,t}}} - {q_i}{n_{{i,\,t}}} - {p_i}{n_{{i,\,t}}} + {p_{{i - 1}}}{n_{{i - 1,\,t}}} + {q_{{i + 1}}}{n_{{i + 1,\,t}}}. $$

(A3.4′′)

Naming

$$ \left\{ {\begin{array}{*{20}{l}} {t_i} = {p_i}{n_i} \\ {r_i} = {q_i}{n_i} \end{array}} \right. $$

(A3.5)

the stochastic equation assumes the form

$$ {n_{{i,\,t + 1}}} = {n_{{i,\,t}}} - {r_{{i,\,t}}} - {t_{{i,\,t}}} + {t_{{i - 1,\,t}}} + {r_{{i + 1,\,t}}}, $$

(A3.6)

which can be approached to the continuous form

$$ n{(x,t + {\tau_0})} = n(x,t) - r(x,t) - t(x,t) + t(x - {\ell_0},t) + r(x + {\ell_0},t). $$

(A3.7)

This formula can be Taylor expanded, with the customary cutoff criteria giving

$$ n + {\tau_0}\,{n_t} = n - r - t + t - {\ell_0}\,{t_x} + \frac{1}{2}\ell_0^2\,{t_{{xx}}} + r + {\ell_0}\,{r_x} + \frac{1}{2}\ell_0^2\,{r_{{xx}}} $$

$$ \hspace{-54pt}{\tau_0}\,{n_t} = - {\ell_0}\,{t_x} + \frac{1}{2}\ell_0^2\,{t_{{xx}}} + {\ell_0}\,{r_x} + \frac{1}{2}\ell_0^2\,{r_{{xx}}}, $$

(A3.8)

where

$$ {n_t} = \frac{{\partial n}}{{\partial t}},\quad {t_x} = \frac{{\partial t}}{{\partial x}},\quad {t_{{xx}}} = \frac{{{\partial^2}t}}{{\partial {x^2}}},\quad \cdots $$

(A3.9)

but

$$ \left\{ {\begin{array}{*{20}{l}} t = p\,n \\ {t_x} = {p_x}n + p\,{n_x} \\ {t_{{xx}}} = {p_{{xx}}}n + 2{p_x}{n_x} + p\,{n_{{xx}}} \end{array}} \right.\quad \left\{ {\begin{array}{*{20}{l}} r = q\,n \\ {r_x} = {q_x}n + q\,{n_x} \\ {r_{{xx}}} = {q_{{xx}}}n + 2{q_x}{n_x} + q\,{n_{{xx}}} \end{array}} \right. $$

(A3.10)

then by substitution in (A3.8) we got

$$ \begin{gathered} {\tau_0}\,{n_t} = - {\ell_0}\,{p_x}\,n - {\ell_0}\,p\,{n_x} + \frac{1}{2}\ell_0^2\,{p_{{xx}}}\,n + \ell_0^2\,{p_x}\,{n_x} + \frac{1}{2}\ell_0^2\,p\,{n_{{xx}}} \hfill \\ \quad \quad + {\ell_0}\,{q_x}\,n + {\ell_0}\,q\,{n_x} + \frac{1}{2}\ell_0^2\,{q_{{xx}}}\,n + \ell_0^2\,{q_x}\,{n_x} + \frac{1}{2}\ell_0^2\,q{n_{{xx}}} \hfill \\ \end{gathered} $$

(A3.11′)

or

$$ {\tau_0}\,{n_t} = - {\ell_0}(\,{p_x} - {q_x})\,n - {\ell_0}\,(p - q)\,{n_x} + \frac{1}{2}\ell_0^2\,({p_{{xx}}} + {q_{{xx}}})\,n + \ell_0^2\,({p_x} + {q_x})\,{n_x} + \frac{1}{2}\ell_0^2\,(p + q)\,{n_{{xx}}} $$

(A3.11)

since

$$ \left\{ {\begin{array}{*{20}{l}} p - q = \mu \\ {p_x} - {q_x} = {\mu_x} \end{array}} \right.\quad \left\{ {\begin{array}{*{20}{l}} p + q = D \\ {p_x} + {q_x} = {D_x} \\ {p_{{xx}}} + {q_{{xx}}} = {D_{{xx}}} \end{array}} \right. $$

(A3.12)

we have

$$ \frac{{{\tau_0}}}{{{\ell_0}}}\,{n_t} = \left(\frac{1}{2}{\ell_0}\,{D_{{xx}}} - {\mu_x}\right)\,n + ({\ell_0}\,{D_x} - \mu )\,{n_x} + \frac{1}{2}{\ell_0}\,D{n_{{xx}}}, \vspace{3pt}$$

(A3.13)

making

$$ A(x) = \frac{1}{2}{\ell_0}D,\quad B(x) = {\ell_0}{D_x} - \mu, \quad C(x) = \frac{1}{2}{\ell_0}{D_{{xx}}} - {\mu_x}, $$

(A3.14)

and by substitution in (A3.13) it gives the generic form for the fundamental stochastic evolution equation

$$ A(x)\frac{{{\partial^2}n}}{{\partial {x^2}}} + B(x)\frac{{\partial n}}{{\partial x}} + C(x)\,n = \frac{1}{{{v_0}}}\frac{{\partial n}}{{\partial t}}. $$

(A3.15)

The explicit form for the transition probabilities can be obtained from Eqs. A3.3 which may be written

$$ \left\{ {\begin{array}{*{20}{l}} q(x) = \dfrac{{I(x - {\ell_0})}}{{I(x - {\ell_0}) + I(x) + I(x + {\ell_0})}} \\[12pt] \delta (x) = \dfrac{{I(x)}}{{I(x - {\ell_0}) + I(x) + I(x + {\ell_0})}} \\[12pt] p(x) = \dfrac{{I(x + {\ell_0})}}{{I(x - {\ell_0}) + I(x) + I(x + {\ell_0})}} \end{array}} \right. \vspace{3pt}$$

(A3.16)

when using the integration rectangle formula.

Expanding in Taylor power series and making the cutoff at the habitual positions we have

$$ \begin{aligned} I(x\pm {\ell_0}) \cong I\pm {\ell_0}\,{I_x} + \frac{1}{2}\ell_0^2\,&{I_{{xx}}} \hfill \\ I(x - {\ell_0}) + I(x) + I(x + {\ell_0}) &= I - {\ell_0}\,{I_x} + \frac{1}{2}\ell_0^2\,{I_{{xx}}} + I + I + {\ell_0}\,{I_x} + \frac{1}{2}\ell_0^2\,{I_{{xx}}} \hfill \\ &= 3I + \ell_0^2\,{I_{{xx}}} \raisetag{14pt}\end{aligned} $$

(A3.17)

that by substitution in (A3.16) gives

$$ \left\{ {\begin{array}{*{20}{l}} q(x) = \dfrac{{I - {\ell_0}\,{I_x} + \frac{1}{2}\ell_0^2\,{I_{{xx}}}}}{{3I + \ell_0^2{I_{{xx}}}}} \\[12pt] \delta (x) = \dfrac{I}{{3I + \ell_0^2{I_{{xx}}}}} \\[12pt] p(x) = \dfrac{{I + {\ell_0}\,{I_x} + \frac{1}{2}\ell_0^2\,{I_{{xx}}}}}{{3I + \ell_0^2{I_{{xx}}}}} \end{array}} \right. $$

(A3.18)

3.3.2 Linear Approximation

Now following the initial approximation that the intensity of the theta wave field as a linear variation in the domain of validity of Galilee’s law

$$ I(x) = \alpha \,x $$

(A3.19)

we have by substitution in (A3.18)

$$ \left\{ {\begin{array}{*{20}{l}} q(x) = \dfrac{{x - {\ell_0}}}{{3x}} \\[9pt] \delta (x) = \dfrac{1}{3} \\[9pt] p(x) = \dfrac{{x + {\ell_0}}}{{3x}} \end{array}} \right. $$

(A3.20)

From the concrete forms for the transition probabilities it is possible to obtain directly the drift and diffusion coefficients and its derivatives

$$ \left\{ {\begin{array}{*{20}{l}} \mu (x) = p(x) - q(x) = \dfrac{2}{3}\dfrac{{{\ell_0}}}{x} \\[9pt] D(x) = p(x) + q(x) = \dfrac{2}{3} \end{array}} \right.\quad \left\{ {\begin{array}{*{20}{l}} {\mu_x} = - \dfrac{2}{3}\dfrac{{{\ell_0}}}{{{x^2}}} \\[9pt] {D_x} = {D_{{xx}}} = 0 \end{array}} \right. $$

(A3.21)

which by substitution in Eq. A3.14 gives for the coefficients

$$ A(x) = \frac{1}{3}{\ell_0},\quad B(x) = - \frac{2}{3}\frac{{{\ell_0}}}{x},\quad C(x) = \frac{2}{3}\frac{{{\ell_0}}}{{{x^2}}}, $$

(A3.22)

and by substitution in Eq. A3.15 we finally obtain for the master stochastic equation for the process

$$ \frac{{{\partial^2}n}}{{\partial {x^2}}} - \frac{2}{x}\frac{{\partial n}}{{\partial x}} + \frac{2}{{{x^2}}}\,n = \frac{3}{{{\ell_0}{v_0}}}\frac{{\partial n}}{{\partial t}}. $$

(A3.23)

In order to integrate this equation it is convenient to make the transformation

$$ n = x\,\phi. $$

(A3.24)

In this case the derivatives give

$$ \left\{ {\begin{array}{*{20}{l}} \dfrac{{\partial n}}{{\partial x}} = \dfrac{{\partial }}{{\partial x}}(x\phi ) = \phi + x\dfrac{{\partial \phi \,}}{{\partial x}} \\[9pt] \dfrac{{{\partial^2}n}}{{\partial {x^2}}} = 2\dfrac{{\partial \phi \,}}{{\partial x}} + x\dfrac{{{\partial^2}\phi }}{{\partial {x^2}}} \\[9pt] \dfrac{{\partial \phi \,}}{{\partial x}} = \dfrac{{\partial }}{{\partial x}}\left(\dfrac{1}{x}n\right) = - \dfrac{1}{{{x^2}}}n + \dfrac{1}{x}\dfrac{{\partial n}}{{\partial x}} = - \dfrac{1}{2}\left( - \dfrac{2}{x}\dfrac{{\partial n}}{{\partial x}} + \dfrac{2}{{{x^2}}}n\right) \end{array}} \right. $$

(A3.25)

and inserting in (A3.23)

$$ \frac{{{\partial^2}\phi }}{{\partial {x^2}}} = \frac{3}{{{\ell_0}{v_0}}}\frac{{\partial \phi }}{{\partial t}}. $$

(A3.26)

A solution to this equation is

$$ \phi = A{e^{{ - k\,x + \omega \,t}}} + B{e^{{ + k\,x + \omega \,t}}},\quad \omega = \frac{{{k^2}{\ell_0}{v_0}}}{3},\vspace{-6pt} $$

(A3.27)

then the solution for the master stochastic equation assumes the form

$$ n = Ax{e^{{ - k\,x + \omega \,t}}} + Bx{e^{{ + k\,x + \omega \,t}}}. $$

(A3.28)

In order to fix the value of the constants we assume that at the initial time the number of corpuscles in the starting cell is

$$ n({\ell_0},0) = {n_0} $$

(A3.29)

so

$$ n({\ell_0},0) = {n_0} = A\,{\ell_0}{e^{{ - k{\ell_0}\,}}} + B\,{\ell_0}{e^{{ + k{\ell_0}\,}}}. $$

(A3.30)

On the other side it is necessary to determinate the average time that a corpuscle takes to reach the generic cell S, such that $S\,{\ell_0} = \ell$, after traveling the path of length ℓ. This situation occurs when

$$ n(\ell, t) = {n_0},\quad {\text{and}}\quad \;{n_0} > > > 1. $$

(A3.31)

This condition implies that

$$ A\,\ell {e^{{ - k\ell + \omega \,t\,}}} + B\,\ell {e^{{ + k\ell + \omega \,t\,}}} = A\,{\ell_0}{e^{{ - k{\ell_0}\,}}} + B\,{\ell_0}{e^{{ + k{\ell_0}\,}}}. $$

(A3.32)

Now the value of the constants need to be such that they contain Galilee’s law stating that the traveled spaces are proportional to the square of the times taken

$$ \ell \propto {t^2},\quad t = \beta \,{\ell^{{\frac{1}{2}}}}. $$

(A3.33)

In such conditions by substitution in (A3.32) we have

$$ A\,\ell {e^{{ - k\ell + \omega \,\beta \,{\ell^{{\frac{1}{2}}}}\,}}} + B\,\ell {e^{{ + k\ell + \omega \,\beta \,{\ell^{{\frac{1}{2}}}}\,\,}}} = A\,{\ell_0}{e^{{ - k{\ell_0}\,}}} + B\,{\ell_0}{e^{{ + k{\ell_0}\,}}} $$

then

$$ B = \frac{{{\ell_0}{e^{{ - k{\ell_0}\,}}} - \ell {e^{{ - k\ell + \omega \,\beta \,{\ell^{{\frac{1}{2}}}}\,}}}}}{{ - {\ell_0}{e^{{ + k{\ell_0}\,}}} + \ell {e^{{ + k\ell + \omega \,\beta \,{\ell^{{\frac{1}{2}}}}\,}}}}}A\,. $$

(A3.34)

Now assuming the relationship (A3.34) between the two constants for the solution (A3.28)

$$ n = Ax{e^{{ - k\,x + \omega \,t}}} + A\frac{{{\ell_0}{e^{{ - k{\ell_0}}}} - \ell \,{e^{{ - k\,\ell + \omega \,\beta \,{\ell^{{\frac{1}{2}}}}}}}}}{{ - {\ell_0}{e^{{ + k{\ell_0}}}} + \ell \,{e^{{ + k\,\ell + \omega \,\beta \,{\ell^{{\frac{1}{2}}}}}}}}}\,\,x\,{e^{{ + k\,x + \omega \,t}}} $$

(A3.35)

and by an inverse reasoning we can obtain, directly Galilee’s law

$$ \ell \propto {t^2}. $$