Abstract
A new type of Coulomb gas is defined, consisting of n point charges executing Brownian motions under the influence of their mutual electrostatic repulsions. It is proved that this gas gives an exact mathematical description of the behavior of the eigenvalues of an (n × n) Hermitian matrix, when the elements of the matrix execute independent Brownian motions without mutual interaction. By a suitable choice of initial conditions, the Brownian motion leads to an ensemble of random matrices which is a good statistical model for the Hamiltonian of a complex system possessing approximate conservation laws. The development with time of the Coulomb gas represents the statistical behavior of the eigenvalues of a complex system as the strength of conservation‐destroying interactions is gradually increased. A ``virial theorem'' is proved for the Brownian‐motion gas, and various properties of the stationary Coulomb gas are deduced as corollaries.
REFERENCES
- 1. E. P. Wigner, Proceedings of the 4th Canadian Mathematics Congress (Toronto University Press, Toronto, Canada, 1959), p. 174; Google Scholar
C. E. Porter and N. Rosenzweig, Suomalaisen Tiedeakat. Toimituksia A VI, No. 44 (1960); Google Scholar
C. E. Porter and N. Rosenzweig, Phys. Rev. 120, 1698 (1960). Google ScholarCrossref - 2. A proof of this theorem for is given by Porter and Rosenzweig (reference 1). Their argument can easily be extended to the cases Google Scholar
- 3. J. Wishart, Biometrika 20, 32 (1928). Google ScholarCrossref
- 4. F. J. Dyson, J. Math. Phys. 3, 140 (1962). Google ScholarScitation
- 5. A convenient summary of the theory of Brownian motion is contained in G. E. Uhlenbeck and L. S. Ornstein, Phys. Rev. 36, 823 (1930), Google ScholarCrossref
and in M. C. Wang and G. E. Uhlenbeck, Revs. Modern Phys. 17, 323 (1945). , Google ScholarCrossref
These two papers are reprinted in Noise and Stochastic Processes, edited by N. Wax (Dover Publications, New York, 1954). , Google Scholar - 6. See Wang and Uhlenbeck, reference 5, Sec. 8. Google Scholar
- 7. Except for a misprint, this is Eq. (15) of Uhlenbeck and Ornstein (reference 5). Google Scholar
- 8. F. J. Dyson, J. Math. Phys. 3, 1199 (1962), following paper. Google ScholarScitation
- 9.Here use is made of the identitycot
, which holds when - 10. M. L. Mehta, Nuclear Phys. 18, 395 (1960). Google ScholarCrossref
- 11. A. Lenard, J. Math. Phys. 2, 682 (1961): Google ScholarScitation
S. F. Edwards and A. Lenard, J. Math. Phys. 3, 778 (1962). , J. Math. Phys. , Google ScholarScitation
The Edwards‐Lenard paper describes a Brownian motion model which has some similarity to ours; however, their model differs fundamentally from ours in identifying the fictitious time variable t with the space coordinate x. , Google Scholar - © 1962 The American Institute of Physics.
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