Il Nuovo Cimento (1955-1965)

, Volume 37, Issue 4, pp 1407–1421

# The high-energy limit of the statistical model

Article

## Summary

The sum over allN-particle phase space integrals forN ⩾ 3$$\Omega _R (P) = \sum\limits_{N = 3}^\infty {\frac{{(2mR^3 )^N }}{{N!}}\int {...\int {\prod\limits_{t = 1}^N {\frac{{d^3 p_i }}{{2p_{i0} }}} } } } \delta (4)(\sum\limits_{i = 1}^N {p_i - P} )$$ with interaction factorR is evaluated for high energiesW= √P2≫m, using the limit-theorem formalism of statistics. Applied to the calculation of the multiplicity our result is shown to yield asymptotically Fermi’s thermodynamic limit.

## Riassunto

Si valuta la somma sopra tutti gli integrali dello spazio delle fasi perN particelle, oonN⩾3,$$\Omega _R (P) = \sum\limits_{N = 3}^\infty {\frac{{(2mR^3 )^N }}{{N!}}\int {...\int {\prod\limits_{t = 1}^N {\frac{{d^3 p_i }}{{2p_{i0} }}} } } } \delta (4)(\sum\limits_{i = 1}^N {p_i - P} )$$con il fattoreR di interazione, per alte energieW=√P2≫m, usando il formalismo statistico del teorema limite. Si mostra che il presente risultato, applicato al calcolo della molteplicità, porta asintoticamente al limite termodinamioo di Fermi.

## References

1. (1).
E. Fermi:Progr. Theor. Phys. (Japan),1, 570 (1950).
2. (2).
P. P. Srivastava and G. Sudaeshan:Phys. Rev.,110, 765 (1958). For further literature see M. Kretzschmar :Ann. Rev. Nucl. Sci.,11, 1 (1961).
3. (3).
F. Lueçat and P. Mazur:Nuovo Cimento,31, 140 (1964).
4. (4).
A. I. Khinchik:Mathem. Foundations of Statistical Mechanics (New York, 1949).Google Scholar
5. (5).
P. Mazub and J. Van dee Linden:Journ. Math. Phys.,4, 271 (1963).
6. (6).
M. Neumann:An. Acad. Brasil. Cienc.,31, 361, 487 (1959); H. Satz:Forts,d. Phys.,11, 445 (1963); L. Van Hove:Nuovo Cimento,28, 798 (1963).Google Scholar
7. (7).
H. Joos and H. Satz:Nuovo Cimento,34, 619 (1964).
8. (8).
A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi:Higher Trascendental Functions, vol. 2 (New York, 1953).Google Scholar
9. (9).
Seee.g. L. Landau and E. Lifschitz:Statistical Physios (London, 1958).Google Scholar
10. (10).
Seee.g. M. G. Kendall :The Advanced Theory of Statistics I (London, 1952).Google Scholar
11. (11).
M. Frechet and J. Shohat:Trans. Am. Math. Soc., 33 (1931). The N-dimensional generalization is given by E. K. Haviland:Am. Journ. Math.,56, 625 (1934).Google Scholar
12. (12).
Seee.g. H. Cramée:Mathematical Methods of Statistics (Princeton, 1961).Google Scholar