I. SAMKHAN 43

The problems coming about thereby cannot be cleared

away completely in the Gibbs statistics too. In this statis-

tics one considers the distribution of the molecules (to be

more specific, the distribution of their representation

points) in the imaginary multidimensional phase space

generated by the product of the coordinates of the mole-

cules and their momenta.

In the source prerequisites of these statistics as well as

in the Maxwell-Boltzmann statistical model the principle

of independence of the molecule distributions by coordi-

nates and momenta for closed systems is adopted.

However, despite a formal denial of such dependence,

it nevertheless is partially provided for, though in an im-

plicit form, in the ultimate results obtained thereby. Thus,

for example, the entropy in the Gibbs statistics, as may

be seen from Equation (1), is a function of both the mo-

menta and the concentration of particles, what is achi-

eved owing to a correction of the primary results obtain-

ed thereby by inserting into them additional assump-

tions, specifically the assumption of a necessity of im-

proving the initial distributions by application of correc-

tion multiplier æ.

Moreover, the thesis on the independence of the mo-

lecular distributions in the coordinate space (of the parti-

cle concentrations) and in the momenta space does not

agree with the great canonical Gibbs distribution for the

systems with a variable number of particles. In that event,

the probability distribution density of particles

,

NE ,

even in the absence of external force fields, is represen-

ted as a function

,e

E

kT

EС

fN, (30)

depending both on the chemical potential

and the

number of particles in the system and on the energy

of the subsystem ii

N

NE

incorporating i number

of particles with N

i

energy (C is here the normaliza-

tion multiplier).

However, in this case also the known statistical meth-

ods cannot explain sufficiently enough the interrelation

of the entropy with the changes of the molecular distri-

butions in the adiabatic and polytropic thermodynamic

processes.

4. Conclusion

The performed analysis shows that some concepts of the

statistical and phenomenological methods of describing

the classical systems do not quite correlate with each

other. Particularly, in these methods, various caloric ideal

gas equations of state are employed, while the possibility

of obtaining the same distributions both due to a change

of concentrations and owing to a change of temperature

in the thermodynamic processes is difficult to explain

from the standpoint of a statistical method.

The above-mentioned difference of the equations of

state may be cleared away using in the statistical func-

tions corresponding to the canonical Gibbs equation a

variable scaling factor instead of the Planck’s constant.

The proposed factor depends on the parameters of a sys-

tem and coincides with the Planck’s constant in a par-

ticular case in going of a classical system to the degener-

ate state. Under such an approach, the statistical entropy

is transformed into one of the forms of heat capacity,

what correlates with the determination of the entropy in

representing normal molecular distributions with the use

of the normalized coordinates.

In its turn, the agreement of the methods under consid-

eration in the question as to the dependence of the mo-

lecular distributions on the concentration of particles,

apparently, will call for further refinement of the physical

model of ideal gas and the techniques for its statistical

description.

In this regard, it is interesting to note that as one of the

causes of difficulties arising in the harmonization of ther-

modynamics can be considered the deterministic approach

based on the possibility of determining the coordinates

and momenta of individual particles at every instant by

application of the equations of the classical mechanics.

Presumably, these assumptions cannot be met enough

strictly in all possible thermodynamic systems owing to a

randomness of the heat motion occurring with a possibil-

ity of the initiation thereby also of the forces of collective

interaction in case of a departure of the particle’s para-

meters from their average statistical values.

To further refine the statistical conception of classical

systems, the development of the statistical models based

on the idea of the molecular distributions as a result of

thermal fluctuations of the particle matter, i.e. on the id ea

of a random departure of the particles’ parameters from

their equilibrium values, can turn out to be more prefer-

ential.

In this instance, the functional relation between the ge-

neralized coordinates

,,xyz and momenta

,,

yz

ppp of the particles, which is characteristic for

the known classical models, is not required, and the iden-

tical molecules of the classical systems, just as in the

quantum statistics, will be considered as indistinguish-

able.

REFERENCES

[1] M. Kardar, “Statistical Physics of Particles,” Cambridge

University Press, Cambridge, 2007.

doi:10.1017/CBO9780511815898

[2] I. Samkhan, Phisiks-Doklady, Vol. 41, 1996, pp. 16-18.

[3] P. W. Atki ns, “P hys i cal Ch emis try,” Oxfo rd Un iv ers it y P ress,

Oxford, 1978.

[4] I. Samkhan, The Open Fuels & Energy Science Journal,

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