_{1}

^{*}

“ = everything flows”, Eraclitus, (Ephesus, 535-475 B.C.). If really in Nature everything changes and progresses, then at least two questions arise: 1) how can be these changes entropic but nonetheless somehow predictable without risk of oxymoronic behavior; 2) how can Science conform itself to follow this requirement of the Nature. To attempt an answer to these questions, the present paper introduces an ab initio theoretical model aimed to show that physical information is actually nothing else but straightforward quantum and relativistic implication of the concept of evolution.

Many physicists have emphasized the unreasonable effectiveness of mathematics in describing the physical world; among them the most authoritative one is Wigner [

Even Bertrand Russel was concerned about the link between mathematics and physics [

Nevertheless, the outcomes of the natural sciences are subjected to experimental tests: what is false or true is definable regardless of hungers and geometrical distresses. On the one hand, abstract numbers express reliable physical laws describing properties and predicting behavior of Nature. On the other hand, however, this epistemological shortcut in fact leaves unexplained the link between science and reality, calculation and experiment, mental ideas and actual story of the Universe. Quoting Einstein “the most incomprehensible thing of the Universe is that it is comprehensible”.

Paradoxically, it is easy to understand the correlation between mathematical algorithm and natural event assuming first deterministic evolution of systems according to the old classical physics: once having selected properly the initial conditions, the successive evolution is in principle uniquely determinable. In practice any deterministic model requires a suitable number of descriptive parameters exactly known of a whole system, whose time evolution is codified and described via appropriate functions of these parameters; the mathematical definitions valid at a given time

Actually however the problem is much more complicated.

The task of guessing the evolution of a physical system from a given initial condition must settle up with the probabilistic frame of the quantum theory: uncertainty relationships imply the impossibility of knowing simultaneously couples of conjugate dynamical variables. This constrain at the time

The predictive ability of science becomes further at stake considering also the relativistic theory, according which

Moreover the link between quantum and relativistic theory is still a hard challenge even today.

To approach gradually the epistemological problem raised by these short considerations, suppose preliminarily that a given event K is allowed to occur in a given R at the arbitrary time

being all dynamical variables known by definition, regard the coefficients

In principle it is possible that the coefficients

It is clear that further sets of six experimental data obtained measuring the same quantities at five additional times with respect to the initial condition, yield a system of six equations with six unknowns. Now the system admits a unique solution for all

Anyway take for granted that, by definition, all coefficients

On the one hand, is comprehensible the interest to describe the system at subsequent times after that of the initial condition for completeness of information. On the other hand, however, since in general the descriptive parameters are functions of time, e.g. the dynamical variables of the various particles, the evolution of the system during a given time range becomes in fact essential requirement for the mathematical approach: repeating the same numerical procedure at

the system of equations removes the indeterminacy inherent a unique observation time and contextually describes how a given observable of the system changes at various times

Every column of the matrix

Moreover

On the one hand this procedure, seemingly sterile, deserves attention as it shows that the link between numerical representation of the reality and physical events is in fact plausible: mathematics has its own rules to elaborate numbers; if these rules are implemented to reproduce the results of measurements, then the efforts of scientists are addressed to convert this empirical analysis of data, correct by definition, into rational information to be understood. So Wigner’s doubts are bypassed regarding in fact the empiricism as an intermediate step between mere observation and profound knowledge of the reality, which however remains implicitly hidden in the raw data.

On the other hand all previous considerations evidence three key requirements necessary for any theoretical attempt to bridge abstract numbers and informative interpretation of results: 1) it must be holistic, 2) it must have space time structure, 3) it must inherently have evolutionary character. These three points prospect the non-trivial heuristic worth of

In principle is difficult to discern, on the basis of a linear combination of parameters only, whether for example two arbitrary time ranges

The idea is at this point to bypass the best fit approach, valid by definition, by introducing a general function

the index r stands for the set of three space coordinates and related vector components of all dynamical variables characterizing the system, e.g. possible internal and external vector fields

having nested into X all possible descriptive parameters implicitly governing the physical state of the system.

It is clear that the strategy of implementing the form (1.4) as a starting point, requires to extract successively from

Try to simplify the problem: although in principle the following considerations hold even for

the summation over i accounts for the arbitrary number I of terms of the series, that on j reproduces the same number of terms of the linear combination (1.1), the index

the additive term is assumed known, being the initial boundary condition of the problem. Each j-th term is still related to the respective parameter

The next step to overcome the legitimate Wigner doubts is just the time correlation (1.5), which does exist indeed and involves space time ranges as they appear in (1.6), not the local

The worth of this information appears just from these equations comparing the particular cases where

Hence the first order and second order terms of the series do not imply merely two different degrees of numerical approximations in calculating

Moreover the Equation (1.6) introduces contextually the concept of evolution regarding in the same way also the initial configuration of the system through products of ranges

This is the first hint to reproduce the coefficients

In effect it will be found in the following that

Anyway, apart from mathematical details, the known value of any

Thus the basic idea is that a general function,

On the one hand if the function

Clearly the second question concerns the development of science and has heuristic valence in describing anything effectively allowed to happen in a changing Universe.

The purpose of the present paper is to highlight some straightforward hints towards this aim, i.e. how in principle could a single function

For simplicity and brevity of exposition the model is deliberately one dimensional: this choice does not represent a conceptual limit, it merely aims to simplify the theoretical approach with mathematical formalism as simple as possible. Also, the model purposely considers scalar quantities: for example v is the component of the velocity vector

To add a further step forwards, consider more closely the particular space time interval introduced by (1.6)

as

To describe self-consistently size and position of

the first definition relates

The actual value of

However, the fact that space time terms

and, increasing again

the former defines

where

neglecting the higher order terms.

Here and in the following x and

The remainder of the paper concerns these points through an “ab initio” theoretical model whose exposition aims to be as self-contained as possible. Such model aims to deduce both well known results, as a validation, and new achievements, as innovative implications: in both cases, however, the assessment benchmark is its conceptual root in the Equations (1.11) and (1.12) only.

Despite for sake of brevity and clarity of exposition physical properties like energy and momentum have been taken for granted and explicitly mentioned as well acknowledged concepts in this introductory section, actually all of them will be inferred self-consistently themselves uniquely through (1.11) and (1.12); this holds also for quantities like charge and mass that apparently have nothing to do with the concept of evolution defined by these equations. Although seemingly trivial and innocuous, these two equations are unique source of information and unique input enough to infer all considerations exposed below in a consequential way, while overcoming Wigner’s doubts and renouncing to any hints from physics theories currently existing. For completeness, when necessary, are also shortly sketched some results previously published to emphasize their connection with the present conceptual frame.

To infer information of physical interest from the initial positions (1.11) and (1.12), the simplest idea is to relate appropriately

The ratio at the left hand side introduces a new concept implied by

The significance of this result, which follows the Equation (1.11) only, appears rewriting both sides according to the Equation (1.12) i.e. implementing likewise the identity

The explicit physical meaning of these identities appears when

All this makes sense, as in fact the symbols

has physical meaning while the aforesaid limits imply contextually

This first example has emphasized how to infer information about one specific physical system through the local extrapolation

It is easy to generalize this result to the case where the string is non-homogeneous simply considering another possible chance of defining the link between

yields the further identity according to (1.12)

and thus, dividing both sides by

formally the Equation (2.7) results from two steps, taking first the changes (2.6) of the quantities at both sides of (2.5), which are subsequently related to

being

Now it is possible to infer from the Equation (2.9) the pertinent differential equation once more via the position

The particular result with

The outcomes (2.4) and (2.10) highlight the strategy of the present paper: the arbitrary function

These results are not accidental outcomes inherent the explanatory examples just carried out; in effect no “ad hoc” hypotheses have been made on the concerned systems, e.g. homogeneous or non-homogeneous string, having simply introduced two different ways of describing the local change, i.e. the evolution, of

Let us exemplify further possible ways to handle

which suggests the following definitions according to (2.8)

As the unique Equation (2.9) cannot specify both

By definition

i.e.

To examine either chance, calculate with the help of (2.12), (2.2) and (2.9)

Regarding separately the addends at the initial and final left and right hand sides, this chain of equations is consistent:

i.e.

The well known Equation (2.15) will be inferred again later; these short notes aim to justify preliminarily the positions (2.12) according which, regarding from now on

So the finite value of c follows as a corollary.

Also, it is not surprising that the energy is defined an arbitrary constant apart; it will be shown shortly, however, that the constant has in this context a peculiar physical meaning. If

With

since it is certainly possible to introduce an arbitrary function g such that

Whatever the function

This is just the general form of diffusion equation in a homogeneous and isotropic medium in the absence of internal sources or sinks. But diffusion of what? Although

It is possible to multiply

where C is an appropriate function describing the local value of mass density, concerns the matter transport function under non-equilibrium concentration gradient. It is known that other important phenomena fulfill (2.20); in fact the extension to these cases, e.g. the Fourier heat diffusion, is also possible in an analogous way. Implementing a different dimensional factor to the local function

where now with usual notation K replaces D to express the heat diffusion coefficient simply identifying

Note that the present strategy to infer information about physical systems reveals unexpected links between seemingly different laws: it is significant the fact that elementary manipulations of the equation of vibrating string lead to the diffusion equations.

The Equation (2.8) reads according to (2.9)

because of course the positions (2.12) still hold also in this particular case. This equation can be implemented in two ways.

The first way is

and thus the second equality yields

Here v still appears because the ratio

this result reads therefore

where p and

The second way is highlighted rewriting (2.25) as

which yields

the constant

A few remarks help to simplify the notations in the following:

− the subscripts of

− the velocities v and c are profoundly different, as the former is defined as ratio of two range sizes whereas the latter is a universal constant of the Nature;

− the definitions of two “new” quantities, momentum p and energy

The lack of specific assumptions on p and

Consider now the Equations (2.28) and (2.26): the former concerns ranges, the latter local values. Let us show that relevant physical information is obtainable merging these equations. Multiplying side by side

one finds

thus (2.30) is compatible with

So follow three relevant equations

Introduce now the boundary condition

so m is the rest mass. Calling c the constant velocity

Clearly the particular case

The second equation is compatible with

Note that in addition to the concepts of mass, momentum and energy, follow from (1.11) and (1.12) the constancy of light speed and Lorentz transformations of energy and momentum.

A problem however arises now about why the first (2.34) is consistent with

where obviously

In both cases, dimensional considerations confirm the validity of the three positions (2.36), regarding in particular

It is interesting the fact that the Equations (2.36), pillars of quantum mechanics, are obtained contextually to the relativistic expressions of momentum, energy and rest mass.

Write (2.28) as

rewriting left hand side via (2.27) with

and thus

Also now the general concept of energy takes physical meaning via the limit

So

as all of this is coherent with

one finds

in agreement with the well known definition of action S. Moreover, the second (2.39) yields

owing to Euler’s theorem of homogeneous functions. Hence

It is immediate to conclude that (2.42) yields the Hamilton function.

As the Equations (2.3) and (2.9) have sensible implications, (2.4) and (2.10), whereas (2.39) and (2.42) allow describing correctly the dynamics of any particle, the present approach appears significant: a relationship between space and time ranges

Instead of attempting to explain some particular physical event on the basis of the intuition about its presumed theoretical foundation, we started from arbitrary changes of an introductory function,

In (2.2) v is defined by the time range

where

It is immediate to show that also the positions (2.36), in particular the third one, allow calculating consistently the group velocity of a matter wave packet through the following simple chain of equations. Implementing

whatever

Eventually, note that the third Equation (2.36) alone is enough itself to confirm this result. Write

as

whence

Therefore

yields for

i.e. the well known group velocity of a matter packet wave.

In summary, relevant equations of physics are simply inferred and described through various chances of changing an arbitrary function

The starting point is the first Equation (2.34), which must be rearranged in order to find a sum rule between two arbitrary velocities

in this way one has introduced

so that the sought result is

Accordingly any v summed to or subtracted from c still yields c.

The results so far obtained are enough to get four relevant consequences, exposed below.

Write (2.27) as

being n an arbitrary integer. The reason of this definition is to make (2.27) independent of a specific reference system. Suppose that (3.1) holds for ranges defined in R whereas

Is evident the hint to the well known Planck units, whose choice implies

It implies that with this choice of measure units, the statistical formulation of quantum uncertainty reads simply

the stars indicate arbitrary real numbers, n is instead an arbitrary real integer number. This reasoning shows that in fact the Equations (2.27) hold regardless of any reference system; otherwise stated, the problem of specifying the reference system where are defined the four uncertainty ranges is physically meaningless, provided that the local dynamical variables are systematically replaced by respective uncertainty range totally unknown in any physical problem. This holds also for the derivatives, which are defined in the present model as mere ratios of uncertainty ranges arbitrary, unknown and conceptually unknowable: for example is meaningless to inquire whether

It is usually assumed that the quantum problems are tackled via the operator formalism of wave mechanics, introducing operators and wave equations. For comparison purposes, this section sketches very shortly results concerning one case where the wave equation can be exactly solved: the non relativistic hydrogenlike atom. The aim is to show that identical information is obtainable via a a “corpuscular approach”, which does not require solving any wave equation; it is enough to replace

The starting point is the classical component of

which introduces a range of possible values for

Just this conclusion on the physical uniqueness of ^{2} is now inferred as well. The components averaged over the possible states summing

Consider the quantum system formed by a particle in a central force field, e.g. an electron around a nuclear charge; the concept of force will be justified in the conceptual frame of (1.12) and (1.11). Assuming the origin O of R on the nucleus, let

Two numbers of states, i.e. two quantum numbers, are expected because of the radial and angular uncertainties. In effect the Equations (2.1) and the quantum M^{2} yield

Minimize

and thus

The physical meaning of

Consider now the identity

where the last equation of the chain introduces the momentum p by dimensional reasons and reads

It shows the link between De Broglie momentum, Planck energy and condition

The first chain of equalities will be explained in the next section 6, in particular as concerns the evident link of pc and

These results confirm that the operator formalism and the uncertainty equations are equivalent in describing the quantum systems. As concerns the spin, the paper [

after having added

and that in general these consideration introduce the spin component

Besides its inherent worth, the hydrogenlike model has been explicitly quoted here because it also provides useful information about the characteristic lengths in the atom, the first of which is of course the Bohr radius inferred in (3.7). The first powers of

whose values are

the Bohr radius scales

Owing to (2.28),

This result is more expressively rewritten in the form

The physical meaning of this result, the dependence of m on v via the constant

The strategy is still that followed to find (2.15) and to infer (2.31) and (2.32) from (2.29). Consider the Equation (2.25) and (2.26) rewritten in the particular case

the former equation defines the maximum energy range

However just the fact that the (3.14) appears suitable to be directly linked to (2.34) rises a quantum problem. Replace (2.36) in the Equation (2.34) via the positions

The subsection 3.2 has been explicitly enclosed in the present exposition to emphasize that the quantum eigenvalues leave out any information about the range sizes; the Equations (3.3) to (3.12) elucidate this assertion. In other words the previous results obtained implementing

The chance of demonstrating the actual effectiveness of this reasoning has heuristic worth in demonstrating the close connection between quantum and relativistic theories.

In practice, to generalize the standard relativistic result (2.34), implement again the first (3.14) with the same steps from (2.29) to (2.31) and then to (2.32), but rewriting the third and fourth positions as

where in fact

which yields

and then

Hence, reasoning as before, this result implies:

As hold for (3.18) the same considerations carried out for (2.34), because also the new terms

The notation

(3,19) reads

As expected, thanks to the higher order terms

then

First of all, replace

hence (3.20) is compatible with the quantum condition (2.36) even for

Moreover rewrite the second (3.15) with the help of (2.36) as

To recognize the physical meaning of this equation under the condition that

The first one is an identity, whose left hand side is simply rewritten introducing the Compton length

For sake of generality the notation emphasizes that

Now it is necessary to express the fact that

It appears that if

which yields

Next, inserting the positions (3.23) in (3.24) trivial manipulations yield

Clearly

It is known that (3.20) is a valuable equation of quantum gravity able to solve three cosmological paradoxes [

The subsection 3.2 has shown that the corpuscular approach to quantum mechanics provides sensible results in agreement with the wave formalism. This subsection shows that also the wave formalism enters in the conceptual frame hitherto exposed. Implement the quantum relativistic Equation (3.20), noting that

Admitting that even the single factors at the right hand side have physical meaning, it is possible to introduce imaginary momentum

being simply required

The correct correspondence of signs in (3.28) is indeed such that

in this way the Equation (3.27) turns again into a real form. Introduce now the positions

being

which are justified soon below. In principle the positions (3.30) are compatible each other because

The addend

the first equation is still the precursor (2.3) of the D’Alembert Equation (2.4) and is clearly an identity

so that one finds

then the position

yields by consequence

Reasonably therefore the positions (3.30) imply (3.32), which yield (3.35) in agreement with (2.35). Hence

It is useful to introduce now the local limit

obtained equating the left hand sides of (3.35) and (3.36), are both fulfilled by

being

An interesting corollary of (3.38) follows from

Although

Hence

Eventually note that

is a relativistic invariant in different inertial reference systems. Moreover, dividing both sides by

With

This equation is nothing else but the Doppler shift of frequencies reciprocally moving at rate v along their sight line.

As (3.39) shows that both signs of (3.30) are admissible, consider now separately either sign of the Equations (3.36).

1) The negative sign yields

which of course confirm (3.28); so, for

In these equations the physical meaning of

Note that holds for (3.44) and (3.45) the same remark carried out for (2.3) and (2.4): also now the left hand side of (3.44) are in fact not calculable explicitly because are indeterminate not only

In this case

The relativistic equations (3.44) are implied by the invariant xt of (3.39), as shown in (3.42); obviously, replacing xt with another function

for simplicity of notation, the symbols of the constants have been kept unchanged. Hence the first two Equations (3.44) turn into

Put eventually

Clearly these expressions, suggested by the outcomes (2.23), agree with (3.28) and specify via the limit

these results are the well known equations of the old quantum theory; the subscript “ef” stands for “eigenfunction”. The modern quantum physics was born postulating these crucial equations, whence the importance of having found them as corollaries: the present theoretical approach brings back just to early formulation of quantum mechanics and its basic assumptions.

2) Consider now also the plus sign of (3.36), which yields

The Equations (3.64) correspond to the Equations (3.28), whereas the Equations (3.48) read

Note eventually that the Equations (3.37) are well known in the operator formalism

In conclusion this simple approach has found the operator formalism and contextually the uncertainty equation, both compatible with relativistic concepts. These outcomes have several further corollaries, the most relevant of which are shortly summarized in the following. Final remark to close this section. The range products

the “new” quantity F, so far not explicitly concerned but only anticipated in Section 3.2 for exposition purpose only, takes in this way justification and physical meaning, it is usually known as force. The concept of pressure and energy density also follow from this result dividing both sides by the arbitrary surface

Are concerned in this section several interesting outcomes still hidden in the approach hitherto outlined.

The key equations are (3.1) and (2.25). Consider an arbitrary time range

the second position expresses that the time interval is arbitrary but fixed by definite time boundaries within which hold the following considerations. Since

Hence, integrating along an element

so the Fermat principle, also expressible identically as

In an analogous way one finds the Maupertuis principle. Calculate

The right hand sides involves

i.e.

The reasoning already carried out for a beam of particles, see (2.46), is extended here considering a light beam propagating in a dispersive medium at rate

of course

It is instructive to examine closer the Equation (4.2) in order to evidence that a further aspect of the motion of a corpuscle of mass m is describable by a wave packet moving as a whole with at rate

and replacing

Require now purposely

being

so that

then, for

The key step of the reasoning is the well defined amount energy

Think now one Planck frequency (2.37) as that included in a packet of waves of different wavelengths propagating in a dispersive medium with different λ-dependent velocities: in effect, the Equations (3.1) regard

The equations now obtained directly from the Equation (2.34) emphasize a new implication: neither

According to (3.64) and (2.35), if

Regard m of (2.33) as a particular case of a general dynamical variable related to p through v and examine how the new concept of mass could tend to zero correspondingly to

which regards m as a constant mass while introducing a new mass

In conclusion, according to the quantum uncertainty the behavior of a corpuscle of mass m should inherently have a wave-like propagation too, whereas the fact that

that introduces the refractive index of the medium where propagates an electromagnetic wave at velocity

takes into account that

so that the right hand side represents kinetic energy. On the one hand

The second equation is direct consequence of the first one; it emphasizes that the concerned velocity v is actually

These considerations rise however three questions.

The first one can be formulated as follows: as (4.7) is made by mass and massless terms, what determines either property of matter? Obviously the immediate answer points to the kind of experiment made on the particles constituting the body of matter. Also this is the non-real essence of quantum mechanics, which actually regards the matter neither as a packet of waves nor as a cluster of corpuscles, but as an undefined state of probabilistic mixing of both states until some experiment “creates” either state. The electron diffraction in the two slit experiment and the Thomson experiment inspired by the Millikan result elucidate the physical meaning of the addends of (4.7). To this equation is also related the physical meaning of the EPR thought paradox, showing that the quantum properties are not pre-definable outcomes according to some principles of classical mechanics, rather they are created by the experiment itself. In effect (3.1) exclude not only the concept of trajectory, but also that of distance and velocity; as shown in 3.2 the local space time coordinates must be replaced by the respective ranges, so concepts like “superluminal” distance are actually unphysical. In this sense the EPR paradox shouldn’t even be formulated: replacing systematically

The second one concerns the addition of velocities. Consider an electromagnetic wave that appears in the point where

The third question concerns the energy fluctuation necessary to account for the mass change when

The corpuscle/wave dualism has been accepted as compelling experimental evidence since the early experiments of electron diffraction, simply acknowledging that either behavior depends on the kind of experiment. Yet this shortcut leaves in fact unexplained why mass appears explicitly in (3.64) whereas it is hidden in the proportionality constants (2.36), despite both concern momentum and energy of a free particle. The fact that both equations have been inferred in the frame of a unique model based on the definitions (1.11) and (1.12) stimulates one to think that even this duality could find rational explanation, i.e. explainable by a logical physical reasoning in the conceptual frame of the present model, without need of supplementary “ad hoc” hypotheses. This hope is supported by the probabilistic character of (4.7), direct consequence of the concept of velocity dependent mass elucidated in the form (4.4): in effect the chances

The results of the point 4.3 have been obtained considering initially a particle of mass m that displaces at rate v; next has been considered also its probability of mass, i.e. energy, fluctuation, which eventually turns it into massless electromagnetic wave or matter wave traveling at rates

Owing to (2.34), consider thus the energy change

The energy range

The Equation (4.9) yields

these positions are easily understood; the respective energies proportional to

Are the Equation (4.8) along with its premises and implications true indeed? To support the validity of (4.7) and thus (4.8) itself, is now tested their direct consequence, the Equation (4.9), in three particular cases of major physical interest. Write first with the help of (3.1) and (2.25)

noting that

The coefficient

1) Putting first

that introduces with the minus sign the concept of mass flux J, i.e. mass transferred per unit surface and time through the volume V; so

The double sign of v is obvious, being it a velocity component. For simplicity and brevity v and D have been regarded not dependent on x, to make quickly recognizable the link of these results with well known concepts of elementary diffusion theory; also, the diffusion process has been assumed at the constant temperature T. With the minus sign in (4.13), positive D, one acknowledges once more the definition of chemical potential

Eventually the plus sign in (4.13), which instead corresponds to negative D, describes phenomena like the spinodal decomposition of alloys of appropriate composition [

2) Putting next

i.e. the famous Einstein equation of one dimensional Brownian motion.

3) The validity of the Equation (4.8) is further checked implementing the property

simply requiring

as in principle n can take any value from 1 to ∞, the number of terms of the sum is arbitrary. The Equation (4.17) is well known and reported in all standard textbooks concerning the fluctuations of thermodynamic systems: it yields

An interesting question concerning (3.1) is the following: is

The first equality reads

the second equality reads

Now fulfill the idea that

in this way the sign of

the double sign on the one hand emphasizes that both p and

in lack of any information about the ranges, both inequalities are actually possible. Regarded in this way, i.e. implementing range boundaries arbitrary and independent each other, the notation (4.23) effectively defines

It is immediate to link (4.24) and (4.22), noting that the former defines at both sides ratios with physical dimensions of reciprocal time range. Multiplying both sides by

define energies that, in agreement with (2.42) and (2.38), correspond respectively to

Hence simple considerations on the range boundaries imply the concepts of Hamiltonian and Lagrangian according to the previous Equations (2.38) and (2.42):

for the following reason. According to the quantum uncertainty, the left hand side of (3.1) reads

yields

the range

yields then

according to the first (4.25)

The last results have somehow linked the relativistic Equation (2.25) to important results of classical statistical thermodynamics. The importance of this topic is shortly highlighted in the following three subsections.

Let us implement once more the Equation (2.26), and calculate the change of

Since by definition

being

Consider now preliminarily the case of an ideal gas of non-interacting free particles/atoms/ions/molecules and let

moreover it is also possible to sum terms like this of each particle over all particles of the system, so that it is possible to write

whence

being N the number of particles of the system. Note that this result is actually more general than prospected here. Suppose first two interacting particles only; in this case we expect

where

where

Actually nothing compels these positions, which in effect are purposely introduced to plug the present considerations into the realm of statistical mechanics. In practice

This is nothing else but the Boltzmann definition of dimensionless entropy: note that

This simple procedure has introduced the function S as sum of consistent functions

It is possible to ask at this point whether this kind of equation is uniquely referable to the property

The starting point is now (3.38) with the minus sign. The way to implement this equation is similar to that just described for the Equation (5.1): any space time factor

The algebraic steps are listed one by one after rewriting (3.38) as

1) On the basis of the section 5.1, define

being

2) regard

3) sum over all allowed states j accessible during an assigned

whereas the factor

4) the last Equation (5.9) defines a “new” quantity T called temperature

uniquely defined for a body of matter at the thermal equilibrium. Note that the first equation has been written introducing at the left hand side the summation over

i.e.

the j-th addend contributing to

and then, summing over j, owing to (5.8) one finds

Normalizing via

If

This probabilistic interpretation is possible if

having expressed

and thus

In this way

Clearly this is just the second law of thermodynamics because, as written, it concerns an isolated system; the conclusion is in effect true if no external action perturbs the system. If not so, then any action altering substantially the configuration of the system modifies by consequence the j-th range size too; in general different

It is worth remarking once more that the evolution of the physical system has implemented two subsequent time lapses

Let the change

yields

put in this form, once more the space time Equation (5.5) of W is implemented via

also here appears the space time function

This equation follows from the arbitrariness of

of (5.15): being

Entropy and Liouville theorem, both previously inferred, are the key concepts to introduce the phase space. As this topic is well known, are reported here just a few remarks aimed only to emphasize the link between space time and phase space; i.e. the concept of space time is actually the third essential ingredient to introduce “ab initio” the statistical mechanics. To this purpose consider in particular the Equations (3.39) and (3.1).

Being x and t arbitrary and independent variables, which represent for example the space coordinate of a given particle at various times in the space time, any value of xt can be obtained keeping constant either factor and allowing appropriate values of the other one; both ways of defining an arbitrary space time coordinate

whence

and thus, comparing the initial and final ranges of coordinates,

the initial equation regards the j-th space coordinate

i.e. one particle initially at any random

The diffusion coefficient D introduced in (2.19) is usually concerned in problems of matter displacement under non-equilibrium conditions, essentially due to concentration gradients; the same holds for the heat diffusion coefficient (2.22) in non-thermal equilibrium problems, typically in the presence of temperature gradients. However, the four equations from (4.13) to (4.16), as well as the next (5.20) and (5.21), suggest a more profound physical meaning of D. In this respect deserve attention the following three remarks.

1) The dimensional definition of D is

this is the usual form to express the dependence of diffusion coefficient on temperature via the activation energy

2) Assume now a body of matter of mass m in equilibrium at temperature T and implement the reasonable idea that both D and

Dimensional considerations are useful to guess an order of magnitude estimate of

Consider indeed one particle of mass m ideally delocalized between two infinite potential walls

having expressed

Anyway it is sensible that

which thus also defines

The 3D generalization of this result is obtained imagining an arbitrary amount of mass delocalized in an appropriate range

These results will be calculated later; regardless of the numerical values, however, it is possible to remark since now some interesting implications:

− The Nernst theorem is automatically fulfilled, i.e. the absolute zero actually does not exist being clearly impossible to remove the zero point energy, which is an intrinsic feature itself of any amount of confined matter.

− As expected, the related zero point temperature

− Is in principle possible the quantization of temperature, which accordingly should start from

3) The reasoning to infer (5.21) and (5.25) introduces

To show this last point, calculate the change

since the expression at right hand side reads

with notation of (6.4), then

It is possible to identify here a minimum value of

Whatever the specific value of

Accordingly, the Lorentz transformations, in particular, should actually be nothing else but the straightforward consequence of the granular nature of space time.

This section generalizes the idea of regarding the zero point energy and volume (5.23) as intrinsic properties of matter, rather than as operative thermodynamic parameters related to specific experimental conditions. According to (5.23), think the zero point volume

To justify the legitimacy of this conclusion, consider first an ideal gas inside which energy exchanges occur via direct collisions between its molecules only. Without hypothesizing specific interactions between molecules, e.g. long range Coulomb or dipole interactions between electron shells, holds between p and

the second equality introduces the time lapse

the notation is justified by the free volume

Let

where

If

Note that in general neither

This result has been obtained considering the delocalization volume of one molecule; it holds in general for any number N of molecules regarding

The volume

This equation reminds closely the characteristic terms of the Van der Waals equation, where f and

These considerations are now extended to the concept of temperature once having introduced the quantum meaning of

If effectively exists a minimum temperature

These considerations should be also extended in particular to the statistical distributions of bosons and fermions, usually written as a function of kT only: taking into account the considerations elucidated in the case of the Van der Waals equation, one should conclude that strictly speaking in the case of a solid body the simple term kT should be replaced by

Even though the value of