Generalized equations of thermodynamics with nonspecific hidden variables

Abstract

For the collective gravithermodynamic Gibbs microstates the connection between all thermodynamic potentials and parameters of matter have been found. This connection is realized with the help of four hidden wave functions that can take arbitrary values with certain probability. The possibility of obtaining the known equations of thermodynamic state of real gases is shown based on the use of both their the limit velocities of individual (separate) motion and the mathematical expectations precisely of these four nonspecific hidden parameters (wave functions) and functions of them. It is substantiated that in a quasi-equilibrium state, a real gas has spatial homogeneity not only of its entropy but also of the resulting extensive parameter (an indicator the compressibility coefficient). But the radial values of resulting intensive parameter (an indicator of hierarchical complexity and of quasi-equilibrium of cooling down) of a real gas are invariant in time.

1. Introduction

Equations of state of matter are a necessary complement to the laws of thermodynamics. They allow the application of the laws of thermodynamics to specific substances and systems, since the laws of thermodynamics by themselves do not provide complete information about the state of the system. Equations of state cannot be derived from the laws of thermodynamics alone. They are obtained experimentally or theoretically, using ideas about the structure of matter, for example, methods of statistical physics.

The most famous equations of state for real gases are the generalized Clapeyron–Mendeleev equation, the van der Waals virial equation (1873) [1], the Dieterici equation (1898) [2], the Berthelot equation (1900–7) [3], the Kamerlingh-Onnes virial equation (1901), the Beattie–Bridgeman equation (1927) [4, 5], the Benedict–Webb–Rubin equation (1940–42) [6 – 9], the Redlich–Kwong equation (1949) [10], the Soave–Redlich–Kwong equation (1972) [11, 12], the Peng-Robinson EOS (1976) [13], etc. [14 – 35].

Studies of the spatially uniform compressibility coefficient of gases and liquids are important [21, 23, 27, 36 – 39]. It may also be important for the analysis of the cooling process of the hot Universe (when the Universe was uniformly filled mainly with hydrogen) to find out the value of the time-invariant intensive thermodynamic parameter . The most popular, practical, and perfect are the van der Waals virial equation [1, 15 – 17, 24] and the Benedict–Webb–Rubin equation of state [6 – 9, 18, 20, 25 – 35]. But they are also purely empirical and artificial. After all, they are based on the use of only coefficients and corrections, and not thermodynamic nonspecific hidden parameters, which are wave functions capable of taking any values with a certain probability. Therefore, these equations of the thermodynamic state of matter do not allow us to obtain a set of multitude Gibbs microstates for matter. Moreover, they do not allow us to obtain equations of spatially inhomogeneous quasi-equilibrium thermodynamic states of astronomical gas clusters that gradually cool down. It is to the solution of these important and urgent problems that the proposed in the article results of careful theoretical research of the author are devoted.

Internal energy U of real gases, liquids and solid matters depends on many pairs of their intensive  and extensive  thermodynamic specific hidden parameters. And there are a very large number of those parameters in solid substances, and therefore the internal energy in these substances is very significant. Those facts prompt us (in the general relativity (GR)) to falsely identify the inert free energy  of matter with the multiplicative component  of its thermodynamic internal energy due to the use (in the GR) of the eigenvalue of the hybrid enthalpy  that is invariant along the radial coordinate r (in the gravitational quantum time of the matter [41]). Similarly, in relativistic gravithermodynamics (RGTD), the ordinary rest energy of matter is identified with the multiplicative component G₀ of the Gibbs thermodynamic free energy G [41]. Here:  and  are the eigenvalues of the mass of matter that is not under pressure and the true mass of the matter, respectively; ; ,  and  are eigenvalues of the multiplicative component U₀ of the internal energy U, of the pressure p and of the molar volume V of the matter, respectively; v_cv and v_l are the coordinate pseudo-vacuum velocity of light of the GR and the equivalent (but not identical) limit velocity of individual (separate) motion of matter (which at the same point in space may be different in the RGTD for different matters) respectively.

However, what is considered here are not at all specific hidden parameters characterizing thermodynamic macrostates of matter, but rather non-specific hidden variables that are mutually related to thermodynamic natural parameters (pressure, molar volume, temperature, and entropy). In addition, non-specific hidden variables that form Gibbs microstates, unlike specific internal macroscopic parameters, can instantly take any values with a certain probability.

Internal energy can also be shown as a sum of internal energy of hypothetic ideal gas (liquid)  and output of multiplication of resulting intensive  and extensive  thermodynamic parameters:

Conclusion

Now we know for sure about the four hidden thermodynamic parameters (k, , m, and n) and functions on them (, , , , , ), and functions of the explicit thermodynamic parameters (, , Z, , , , ,, , and etc), as well as critical (, ,, ) and spatially homogeneous (, , , ) thermodynamic parameters. That is why it is now possible to experimentally determine at the points of phase transitions the critical values, and at the control points the standard values of the mathematical expectations of the hidden parameters, as well as of the corresponding explicit thermodynamic parameters. And on the basis of all this it is possible to obtain exact equations of thermodynamic state of real gases and liquids without using corrections. And this is facilitated by the spatial homogeneity of the hidden thermodynamic parameters, which correspond to the spatially inhomogeneous states of real matter. Of course, this will not deny the expediency of using also approximate equations of thermodynamic state of matter based on the use of corrections.

Enthalpy, which consists of the Lagrangian of its own multiplicative component and additive compensation of its multiplicative representation, is de facto the total energy of matter since it includes even the released thermal energy and the released kinetic energy of its motion. Enthalpy of matter (as well as Gibbs free energy, which multiplicative component is identical to the ordinary rest energy of matter and is equivalent to its gravitational mass) is equal in all FRs of bodies that move inertially relatively to it. And exactly this is the guarantee of Lorentz-invariance of all thermodynamic potentials and parameters of matter. Since matter motion is accompanied by the all-sided conformally-gaugely self-contraction of its size in background Euclidean space of the Universe the rate of the intrinsic time of inertially moving body is not dilated but, quite the contrary, remains invariant, despite the presence of gravitational decreasing of the rate of intrinsic time for nearby static objects. De facto the motion of the matter as well as its gravitational self-contraction in background Euclidean space of the Universe leads to its advance over unobservable in people’s world evolutionary self-contraction of the conventionally motionless matter in the Universe. That is why the release of kinetic energy is always accompanied by the decreasing of limit velocity of matter individual (separate) motion (that is equivalent to coordinate velocity of light in GR) and the decreasing of its inert free energy.

The ordinary rest energy of matter is bonded in a different ways in different physical processes. That is why we have various free energies in different processes. Both the change of the inert free energy of matter (caused by its inertial motion) and its evolutionary decrease in CFREU do not directly influence the thermodynamic parameters of matter that are changed only in thermodynamic processes. That is why it is fundamentally unobservable in intrinsic FRs of matter in the similar way as evolutionary and caused by motion reduction of molar volume of matter is unobservable in comoving with expanding Universe FR. The gravitational reduction of molar volume of matter when approaching the gravitational attraction center is also unobservable directly in intrinsic FRs of matter. However, we still can say about its presence in Euclidean space of CFREU due to the presence of gravitational curvature of intrinsic space of matter.

The hidden thermodynamic parameters discovered here (which, similarly to wave functions, can take on any values with a certain probability) confirmed Gibbs's idea of the presence of a multitude of instantaneous thermodynamic microstates in matter that is in an invariable quasi-equilibrial state. The another important thing is the substantiation of the fact that the limit velocity of individual (separate) motion of matter and the indicators of the relativistic-gravitational decrease in the molar volume of matter in the background Euclidean space do belong to the hidden internal thermodynamic parameters [50]. Therefore, an external relativistic interpretation of thermodynamics is not needed. And therefore, all thermodynamic parameters and potentials are invariant both gravitationally and relativistically. After all, all gravitational and relativistic influences on them are already contained in their formation as hidden parameters.

The acceptance of the fact of existence of an extensive parameter (that is spatially homogeneous in the gravitational field), characterizing the compressibility of gases, and a spatially inhomogeneous intensive thermodynamic parameter (that is invariant during the gas leisurely cooling process) allows us to consider the gravitational field as a consequence of a spatially inhomogeneous gravithermodynamic state of both any continuous matter and arbitrarily rarefied gas-dust matter of the cosmic vacuum.

The proved here equivalence of the Hamiltonian of the inert free energy to the inertial mass of matter, and the equivalence of Lagrangian of the ordinary rest energy (and the multiplicative component of the Gibbs free energy identical to it) of matter to a much larger gravitational mass [41, 42] have solved the problem of the shortage in baryonic mass in distant galaxies of the Universe. And it is this, together with the logarithmic gravitational potential [52, 53], that allows us to abandon the need for non-baryonic dark matter in the Universe.

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