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The paper introduces a simple theoretical model aimed to provide a possible derivation of the quantum fluctuations of the black body radiation. The model offers the chance of inferring and linking contextually quantum and relativistic results.

In 1859, Kirchhoff had the remarkable idea that a small hole in the side of a massive body of material containing a large cavity was the best experimental approximation of the concept of total absorber: the radiation penetrating through the hole was correctly assumed bouncing between the internal walls of the cavity with a little probability of escaping outside. With this viewpoint, still today acknowledged [

The problem of the quantum fluctuations of black body radiation is still today debated for its theoretical interest [

While focusing on the radiation field only seems reductive, the variety of phenomena involved when a black body system is out of the equilibrium suggests the usefulness of a comprehensive approach to the problem and introduces the three main motivations of this paper:

1) To propose a model where the photon interaction with the walls of the cavity appears as a natural consequence of the theoretical approach underlying the black body physics.

2) To highlight the thermodynamic aspects of the black body fluctuations with reference to their quantum basis, in particular the uncertainty principle.

3) To show that relativistic results are also obtainable in the frame of a unique conceptual model.

After a preliminary outline of the main dynamical variables prospectively implicated in the problem, the model is specifically addressed to introduce not only the fluctuation but also the main physical laws expectedly useful to describe it. Despite the inherent complexity of the problem, the exposition is organized in order to be as simple, gradual and self-contained as possible.

Consider one free particle of mass m moving within a space range

where n is an arbitrary real number. The second position emphasizes that the range

In principle, nothing hinders to express the numerical parameter n as the ratio

Defining the link between time and space range sizes via c, in order to ensure that any massive particle is effectively confined within

yield

The first equation simply rewrites

Simple considerations show that these positions are physically sensible. The first equation reads indeed

i.e.

so that

yields

In other words, the matter wave propagating at rate

so, through the position

To check this point, replace in the first Equation (6)

At the left hand side appears the Lagrangian L of the free particle, which correctly results as a difference of two energies. Indeed by definition

This limit holds for a photon wave, in which case Equation (6) yields

If

These conclusions are inferred regarding in particular

according to Equation (2); i.e., in agreement with the dual behavior of matter, the range size is related via

Furthermore, an interesting consequence follows regarding

Consequently the minimum energies

Implement to this purpose the case of a particle bouncing elastically back and forth against either boundary wall that delimits the confinement range, Equation (11); the momentum change of the particle reads thus

the subscript

which yields

This is not a hypothesis “ad hoc”, as the Planck units have fundamental worth, being based on dimensional relationships involving fundamental constants of nature. It is immediate to describe in this respect the particular case of photon confinement taking the limit for

which expresses the condition even for a photon to be trapped inside any

Start eventually from the identity (1)

Consider now the identity

Apart from the simplicity of reasoning, is remarkable the fact that the most typical feature of the quantum physics, the Heisenberg inequalities, has been obtained from the relativistic Equation (6).

Equation (3) and other results of this section have been inferred directly from general considerations about the properties of the space time [

This section introduces the fluctuation of all variables previously introduced, with the aim of finding possible links between these variations. Differentiate

being

These positions relate in general

where

In general

emphasizes that the variables of the problem are four and that only Y depends upon

defines

of course

by consequence k is also a conversion factor such that

Among the possible values of Y, calculate Equation (15) with the specific value

This result yields:

These equations are obtained simply averaging the ratios of Equation (19). It worth emphasizing that

So far the first Equation (3) is the only equation correlating E and

It is evident that a hypothesis has been introduced regarding

is the sought second equation linking E and v; the notation emphasizes that the integration constant

which yields at the first order of approximation

if the position (23) is correct, then even this lowest order of approximation should give a sensible result. The validity of Equation (24) is preliminarily proven recalling Equations (3), according which v yields

and then

so, neglecting preliminarily the third addend at the right hand side, this result reads

Before proceeding, it is useful to verify further the validity of the equations hitherto inferred, in particular as concerns the physical meaning of the series expansion (23) of Equation (15). A simple one-dimensional approach is still enough for the present purposes.

Recalling Equations (10) and (1), trivial manipulations show that Equation (19) reads

The first equation links

From Equation (25) follow interesting consequences. Rewrite

where

Note that

The dimensional analysis suggests that

First the well known law

Moreover

where C is the concentration of m in

Eventually, the third equation defines the chemical potential; this clarifies the physical meaning of

In effect, the diffusion equations are contextually obtainable. Dividing both sides of Equation (25) by

i.e. this equation defines a force F acting on m. It is easy to convert force into mass flux J, having physical dimensions of mass per unit time and surface, dividing both sides by

This is the well known Fick diffusion law, from which also follows the second Fick law with the help of an appropriate continuity equation that excludes mass sinks or sources within

subjected to the condition

i.e. the definition of mass flux and the one dimensional second Fick law.

Eventually, Equation (26) reads with the help of Equations (3) and (27) as follows

Suppose now that m is the j-th mass in a system constituted of a number

Since by definition

This equation defines the entropy S a function

note that

All this is linked to the further information provided by Equation (25). Noting that

this equation relates

Combine now Equations (24) and (6) to eliminate v; as

the result is

So

Put preliminarily

Equations (29) and (30) concern both arbitrary square energies, a scale factor apart for the three quantities characterizing the initial

Consider now that in Equation (29)

this term having the form

The fact that

in effect

In effect

and thus

Since

i.e. is admissible Equation (21) with the right hand side having the form

Calculate via the second Equation (6)

Split this equation putting by definition

The second position, allowed in principle by dimensional reasons, allows to handle the first equation as follows with the help of the first Equation (6)

Note that there is no reference to

This result is confirmed by the second Equation (32), which yields with the help of the third Equation (6)

this equation reads

being

It appears in conclusion that the term

Implement then Equation (21) in the simplest form

Comparing with Equation (19),

the physical meaning of this equation is to consider the averages of all possible

In conclusion, to the four variables appearing in (17) correspond three Equations (3), (21) and (36); the free parameter k introduced in (18) is a freedom degree of the problem as a function of which are in principle determinable various E, m and v, i.e. n.

These results have been hitherto obtained without specific reference to the black body cavity and even regardless of the Planck formula. The next section concerns just this topic.

To specify the previous results in the case of radiation in a black body cavity of arbitrary volume V, it is useful to consider first the Planck law. Noting that this law reads

let us examine the three factors that define

The degeneracy factor 2 of the Bose statistical distribution of photons with the same energy corresponds to the orthogonal polarizations of light [

The factor

The notation

With these hints, is really easy to infer the Planck result even in the present physical frame only.

First of all,

Note that

This point does not need further comments. Here, with the minus sign and putting

The number density

Let the cavity contain

where clearly

is fulfilled because V has not yet been specified. Whatever V might be, the sum over the various

In this case one would find

It is known in effect that the steady wavelengths

with n integer; in other words, the electric field of an electromagnetic wave must vanish at the boundaries of its physical volume of confinement, correspondingly to wave nodes at the boundaries.

The seemingly innocuous position (39) implies thus the energy quantization in the cavity. Equations (2) and (3) yield indeed

i.e.

thus

To highlight the physical meaning of the differentials

All frequencies allowed in the cavity contribute to

defines the radiation energy density per unit frequency

If Equation (44) leads to the correct formulation of the Planck law, then it also proofs indirectly that the photon thermalization mechanism occurs at the surface of the cavity.

The integration of

Noting that

Equation (45) reads then

This expression can be considerably simplified because

has a maximum as a function of

Therefore, the plain Planck law corresponds to the particular set of frequencies that, among the ones allowed in the cavity, maximize the number density of photons with a given energy at a fixed T.

Actually, however, no physical reason requires

In the present model it appears therefore that:

・ The interaction between degenerate photon clusters and internal walls of the cavity is responsible for the thermalization mechanism.

・ The fluctuations are inferred contextually to the Planck law itself.

To emphasize these points, it is necessary now to link these fluctuations with Equations (19) and (20). As expected, the fluctuation is given by temperature and frequency deviations of

Now it is possible to tackle the problem of describing the cavity for

The result (25) and Equation (44) imply the involvement of the material constituting the wall of the cavity to reach the condition of thermodynamic equilibrium of photons therein confined. In particular

For sake of clarity, collect together Equations (3), (27) and (25); one finds

These equations evidence in particular

whereas Equation (19) reads

Since k has been defined as a mean value in Equation (18), let then be

being q an arbitrary constant. Then, Equation (36) yields

Since Equation (49) reads

so that merging these equations one finds

the result obtained via Equation (50) is

Therefore

yields

hence

As

Replacing Equations (51) into (53), one finds

According to the previous considerations,

The fluctuations are likely the most typical manifestation of the probabilistic character of the quantum world, while also being the most striking evidence of the quantum uncertainty. Nevertheless, elementary and straightforward considerations have shown that the equations describing the fluctuations are also compliant with relativistic corollaries: both have been concurrently inferred from Equation (1) in a unique theoretical frame. Despite the deterministic character of the relativity, the results so far outlined emphasize this seemingly surprising connection. Actually a similar conclusion was already found also in [

First of all, the present model plugs the problem of the black body radiation and its fluctuations in a wide context of physical laws having prospective interest for the non-equilibrium physics. The quantum basis of the Fick law is important because various physical properties, e.g. the heat and electrical conductivities, have analogous form; here, in particular, the diffusion equations are in principle necessary to account for the unstable concentration gradients reasonably expected in gas phase due to random concentration fluctuations of the matter evaporated from the internal surface of the cavity. In effect the dynamics of matter particles that diffuse from the walls of the cavity contributes to the thermalization process; in this respect, the model introduces concurrently even the free energy and entropy concepts useful to infer the Clausius-Clapeyron equation governing the vapor pressure and thus the amount of matter in gas phase filling the cavity together with the radiation. In view of that, the Planck law has been inferred in order to involve since the beginning the solid matter confining the photons and even their energy quantization and statistical distribution law. The interaction of photons with matter appears in fact essential to justify the thermalization mechanism. Strictly speaking, the radiation with wavelength larger than the finite size cavity should not be consistent with the standard approach to the Planck law; here however this problem is bypassed since the cavity volume V is not predetermined, rather it is determined by the radiation wavelengths themselves via the terms (39). Thus it is by definition compliant with the arbitrary size

The black body radiation field and its fluctuations have been contextually inferred merging two separate paths: the one from Equations (14) to (20) is apparently independent on that leading from Equation (45) to Equation (53). The former series of equations does not refer specifically to the black body radiation, it introduces relationships between changes of dynamical variables that hold in general. The latter series of equations describes specifically the black body radiation under the boundary condition of Equation (20), which also implies Equations (21) to (24); this second path links the frequency and mass fluctuations, in agreement with Equations (4) to (9). Then, Equation (36) introduces the thermal equilibrium of Equation (50) leading to Equation (53).

Yet other significant results are also easily inferable from the previous considerations of the Section 4.

For example, combining Equations (26) and (27) with Equation (28) one finds at constant T

The equation

yields

Owing to the first Equation (27) put then

being

This Arrhenius-like equation is a well known property of the diffusion coefficient, whose quantum origin introduces the activation energy as a consequence.

Other important equations of processes activated by the temperature follow this kind of dependence upon

A further significant result is obtained from Equation (6), assuming that the momentum p is time dependent variable. This compels regarding the wavelength

It is possible to expand in series

being

Equation (56) reads at the zero order of approximation of the series expansion according to Equation (40)

Since F is actually the component of a force along

having put

As n and

Whatever the value of

in the mere Newtonian approximation of Equation (59). If so, however, the ratios

It is possible to show the validity of these conclusions, which should hold for the Coulomb law as well, by demonstrating how to find well known results of the general relativity as a consequence of Equation (19).

To this purpose it is necessary to generalize what

Consider first just Equation (19) used to calculate the quantum fluctuations and note that the ratio at right hand side can be rewritten defining k such that

Introduce now at the right hand side the further mass

This result becomes next more familiar via a formal and elementary manipulation. Eliminate

Eventually, recalling that

At the left hand side, the energies appear through a numerical coefficient times a ratio of the respective fluctuations. Consider now the case where this factor is

one recognizes the well known formula of the perihelion precession. This identification needs however a detailed justification and explanation: helps to this purpose a further result related to the energy loss via gravitational waves, still implied by Equation (63).

It is known that an isolated orbiting system irradiates energy all around in the space; the energy loss causes the orbit shrinking closer and closer towards the central mass. The starting input to demonstrate this effect in the present context is still Equation (19), rewritten identically via Equation (35) as follows

i.e. k, whatever its specific value might be, has been split into

moreover if

Hence the first Equation (65) reads

where the right hand side is constant. Integrating now both sides over the solid angle

The square energy at the right hand side is constant; since it consists of fundamental constants only, thanks to the position assumed for

At this point the quantum uncertainty is of valuable help; it requires that

Replacing in Equation (66) h from the second equation, one infers

With the minus sign and

where F is force. Simply considering the elementary positions

one finds

The second equation is well known in the elementary Kepler problem identifying

Here

Consider now any point of the ellipse at a given time

The idea of introducing Planck units is fruitful and general, as it is confirmed also in the following reasoning.

Rewrite

thanks to Equation (3). So, owing to Equation (21), Equations (19) and (15) yield

while being owing to Equation (21)

whence

Since

where

Note that

Is really significant the fact that also this result of the general relativity is obtained implementing Equations (19) and (18), from which have been obtained Equation (50) and then the black body fluctuation Equation (53). A wider landscape of results of the general relativity is inferred via an “ab initio” theoretical model in [

On the one hand, the result (60) highlights the quantum origin of the gravity force, simply inferable admitting time dependence of De Broglie momentum wavelength. In this respect Equation (59) prospects an interesting consequence as it yields

So, owing to Equation (58), Equation (71) reduces to the trivial identity of two reciprocal surfaces A admitting the equality

The fact that both ratios are almost exactly equal to 1 is not trivial: in general one would expect simply

On the other hand, despite the simplicity of approach, the compliance of the present model with the relativity, already emphasized by the corollaries of the Section 4, does not appear accidental. This point is elucidated next by four relevant examples.

1) According to Equation (2)

By dimensional reasons, it is also possible to put

one expects both

as in effect it is true. Indeed it is possible to express

2) Let

the notations emphasize the probabilities Π_{1} and Π_{2} that m, delocalized within_{1} and Π_{2} are possible for the particle in

where again q is an arbitrary constant. Let the primed and unprimed velocity components be defined in R' and R thinking that in general

this position ensures that both

so that

In this way

This is the well known composition rule of the velocity components along the direction of motion of two reference system reciprocally displacing. The probabilistic meaning of a relevant relativistic property also appears here, already emphasized in demonstrating the perihelion precession [

3) It is easy at this point to highlight further the physical meaning of the length

then, dividing both sides of

Hence,

and write then identically

Defining an angle

This is the well known formula of the light beam bending in a gravitational field, since the angle defining the arc of circumference is equal to that between the tangents to the circumference at the boundaries of the arc, which yield the sought path deviation.

4) Note eventually that

as in this case both addends at the left hand side remain themselves identically unchanged in two different inertial reference systems in reciprocal constant motion. This expression does not consider the mass of a particle possibly present in the space time. Consider now Equation (75) and introduce an arbitrary distance

being

Replacing

Simple considerations show that the right hand side reduces to the form

Starting from elementary considerations, the present model is allowed to describe the fluctuations in a wider theoretical context that includes even relativistic implications. No “ad hoc” hypothesis has been necessary to infer relativistic results, which deserve a few final remarks. The first one emphasizes that in the present context they have been obtained regardless of any preliminary consideration about the covariancy of the physical laws and even about the metrics describing the space time deformation in the presence of matter; actually, instead, the hidden probabilistic meaning of the most famous results of the general relativity is easily acknowledgeable. The second one stresses an open point left by Equation (56) and omitted for brevity taking the absolute value of F in Equation (60), i.e. that the space time deformation inherent the time dependence of

A final remark deserves attention. With little effort and elementary mathematical formalism, Einstein could anticipate himself as done here the most significant discoveries of his general relativity: i.e., as side corollaries of Equation (54) describing the black body fluctuation. Unfortunately his paper [

Tosto, S. (2016) Black Body Quantum Fluctuations and Rela- tivity. Journal of Modern Physics, 7, 1668- 1701. http://dx.doi.org/10.4236/jmp.2016.713152