# zbMATH — the first resource for mathematics

Dequantization, statistical mechanics and econophysics. (English) Zbl 1180.82064
Litvinov, G. L. (ed.) et al., Tropical and idempotent mathematics. International workshop TROPICAL-07, Moscow, Russia, August 25–30, 2007. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4782-4/pbk). Contemporary Mathematics 495, 239-279 (2009).
Summary: Using a rigorous statement and rigorous proof of the Maxwell distribution as an example, we establish estimates of the distribution depending on the parameter $$N$$, the number of particles. Further, we consider the problem of the occurrence of dimers in a classical gas as an analog of the Bose condensation and establish estimates of the lower level of the analog of the Bose condensation. Using dequantization principles, we find the relationship of this level to “capture” theory in the scattering problem corresponding to an interaction of the form of the Lennard-Jones potential. This also solves the problem of the Gibbs paradox.
We derive the equation of state for a non-ideal gas as a result of pair interactions of particles in Lennard Jones models and, for classical gases, discuss the $$\lambda$$ transition to the condensed state (the state in which $$V_{\text{sp}}$$ does not vary with increasing pressure; for heat capacity, this is the $$\lambda$$ point).
We use econophysics to explain the nature of a financial crisis.
For the entire collection see [Zbl 1172.00019].

##### MSC:
 82B30 Statistical thermodynamics 91B02 Fundamental topics (basic mathematics, methodology; applicable to economics in general) 81S99 General quantum mechanics and problems of quantization 82C22 Interacting particle systems in time-dependent statistical mechanics 60C05 Combinatorial probability 46N10 Applications of functional analysis in optimization, convex analysis, mathematical programming, economics 46N55 Applications of functional analysis in statistical physics 91B80 Applications of statistical and quantum mechanics to economics (econophysics)