We consider the notion of number of degrees of freedom in number theory and thermodynamics. This notion is applied to notions of terminology such as terms, slogans, themes, rules, and regulations. Prohibitions are interpreted as restrictions on the number of degrees of freedom. We present a theorem on the small number of degrees of freedom as a consequence of the generalized partitio numerorum problem. We analyze the relationship between thermodynamically ideal liquids with the lexical background that a term acquires in the process of communication. Examples showing how this background may be enhanced are considered. We discuss the question of the coagulation of drops in connection with the forecast of analogs of the gas-ideal liquid phase transition in social-political processes.

Boyle temperature is interpreted as the temperature at which the formation of dimers becomes impossible. To Irving Fisher's correspondence principle we assign two more quantities: the number of degrees of freedom, and credit. We determine the danger level of the mass of money M when the mutual trust between economic agents begins to fall.

A phase transition of the first kind is a jump of a function, a phase transition of the second kind is a jump of its first derivative, a phase transition of the third kind, a jump of the second derivative. A phase transition from one statistic to another is very gradual, but finally it is as considerable as the phase transition of the first kind. However, we cannot introduce a clearly defined parameter to which this transition corresponds. This is due to the fact that the fluctuations near the critical point are huge, and this violates, in the vicinity of that point, the main law of equilibrium thermodynamics, which asserts that fluctuations are relatively small.The paper describes the transition in the supercritical fluid region of equilibrium thermodynamics from parastatistics to mixed statistics, in which the Boltzmann statistics is realized for long-living clusters. In economics this corresponds to a negative nominal credit rate. Examples of this non-standard situation are presented.

The order statistics and empirical mathematical expectation (also called the estimate of mathematical expectation in the literature) are considered in the case of infinitely increasing random variables. The Kolmogorov concept, which he used in the theory of complexity, and the relationship with thermodynamics, which was pointed out already by Poincaré, are considered. We compare the mathematical expectation (which is a generalization of the notion of arithmetical mean, and is generally equal to infinity for any increasing sequence of random variables) with the notion of temperature in thermodynamics while deploying a certain analogue of nonstandard analysis. It is shown that there is a relationship with the Van der Waals law of corresponding states. A number of applications of this concept in economics, in internet information networks, and self-teaching systems are also considered.

We introduce several new notions in mathematical statistics that bridge the gap between this discipline and statistical physics. The analogy between them is useful both for mathematics and for physics. What is more, this new mathematical statistics is adequate for the study of computer networks and self-teaching systems. The role of the web in sociological and economic research is ascertained.

We develop the unbounded probability theory on the basis of Kolmogorov complexity and show its connections to thermodynamics, economics, and its role in the study of the Web.

We show that Gödel’s negative results concerning arithmetic, which date back to the 1930s, and the ancient “sand pile” paradox pose the questions of the use of fuzzy sets and of the effect of a measuring device on the experiment. The consideration of these facts led, in thermodynamics, to a new one-parameter family of ideal gases and, in economics, to the correction, based on Friedman’s rule, to Irving Fisher’s “Main Law of Economics.” We introduce the notion of viscosity (braking) in economics. By analogy toWiener quantization, we study the Wiener (tunnel) quantization of economics as well as the tunnel geometric quantization of economics. We also consider debt crises, the stratification of society, and Islamic revolutions from the point of view of Human Thermodynamics.

We single out the main features of the mathematical theory of equilibrium thermodynamics. The theory of Bose condensate is expressed as a problem in number theory and its relation to various evolutionary processes is studied. It is proved that the points of degeneracy of the Bose gas fractal dimension in momentum space coincide with the critical points of imperfect 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 third quantization.

Our new approach to thermodynamics agrees with statistical laws of linguistics and economics.

We consider the analogy between molecules and objects of linguistics (and other semiotic systems). We establish the correspondence principle between the thermodynamics of supercritical states and linguistics. Certain laws of thermolinguistics are formulated. The problem of pairwise interaction in different systems is discussed.

We investigate the analogy which exists between the evolution of natural and artificial languages, of human society, of animal communities, of microorganisms, and of gas molecules from the point of view of the statistical ideology of the thermodynamics of gases (fluids). We define linguistic clusters, the “energy” of a text, the entropy of languages, the “temperature” of a text, its κ -potential, and we study the isotherms of texts. We write out an equation that may be called the state equation of the thermodynamics of language. We prove that under certain conditions the isotherm of a collection of books necessarily has a critical point and present its linguistic interpretation as a kind of phase transfer in which clusters are destroyed. Further, we point out some relationships and analogies between Darwin’s theory of natural selection and the evolution of animal and human communities, the evolution of language, the evolution of rules and laws in different communities and in society.