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## Modern thermodynamics as a branch of mathematics (mathematical physics)

A parallel between physical derivation and mathematical proof in classical thermodynamics is drawn. A relationship between thermodynamics and analytic number theory is demonstrated.

The relationship between thermodynamics and economics has been known for a long time. The term ``thermoeconomics'' has even appeared. However, several aspects of the old thermodynamics are unacceptable in economics. For example, experts in thermodynamics believe that the diamond crystal is in the metastable state, and in due time will be transformed into graphite. However, these experts can hardly convince businessmen to part with their ancient diamonds.

The laws of economics require that the old conceptions of thermodynamicsa be mathematically scrutinized and reviewed.

The correspondence principle for quantum statistics, classical statistics and economics which associates the number of particles with the amount of money, the chemical potential with the nominal percentage, the negative pressure with debts, and the law of economic preference allowed to obtain agreement of the general theory of thermoeconomics with the latest experimental data.

Negative pressure also means negative energy and, therefore, “holes”, antiparticles. Continuation across infinity to negative energies is accomplished by using a parastatistical correction to the Bose-Einstein distribution.

It is well known that the supercritical state of a gas has great dissolving capacity. In this paper, the mathematical reason for this phenomenon is studied in great detail.

We single out the main features of the mathematical theory of noble gases. It is proved that the points of degeneracy of the Bose gas fractal dimension in momentum space coincide with the critical points of noble gases, while the jumps of the critical indices and the Maxwell rule are related to tunnel quantization in thermodynamics. We consider semiclassical methods for tunnel quantization in thermodynamics as well as those for second and ultrasecond quantization (the creation and annihilation operators for pairs of particles). Each noble gas is associated with a new critical point of the limit negative pressure. The negative pressure is equivalent to covering the (P,Z) diagram by the second sheet.

We single out the main features of the mathematical theory of noble gases. It is proved that the points of degeneracy of the Bose gas fractal dimension in momentum space coincide with the critical points of noble gases, while the jumps of the critical indices and the Maxwell rule are related to tunnel quantization in thermodynamics. We consider semiclassical methods for tunnel quantization in thermodynamics as well as those for second and ultrasecond quantization (the creation and annihilation operators for pairs of particles). Each noble gas is associated with a new critical point of the limit negative pressure. The negative pressure is equivalent to covering the (P,Z)- diagram by the second sheet.