A Description of Laser Stability near Threshold by Liapounov’s Second Method
The theory of the generalized entropy developed by Prigogine, George and Henin  in Brussels and Austin suggests a form for an N-body Liapounov function which has already been discussed for closed systems [2–4]. Starting from the Liouville-von Neumann equation for the density operator ρ, Prigogine introduces a causal or physical particle representation wherein he defines a “physical” density matrix ρ(P) which is generated from ρ by a non-unitary transformation in strongly coupled system., this representation, the retarded and advanced components of ρ(P) obey separate evolution equations. For weakly coupled systems, ρ(P) reduces to the untransformed ρ which is a solution to the Pauli equation. This theory has been applied by Hubert  to a dense hard sphere gas obeying the Enskog equation. Further, Bulsara and Schieve have demonstrated  that infinitesimally close to thermal equilibrium, the generalized form of the entropy proposed by Prigogine et al. has the same form as the Liapounov function referred to above, for weakly coupled systems, a fact which we shall exploit in section 5 of this paper.
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- 1.Prigogine, I., George, C., Henin, F. and Rosenfeld, L., Chemica Scripta 4, 5 (1973).Google Scholar
- Bulsara, A.R., and Schieve, W.C., in Int. Jl. of Quantum Chemistry, S9, 475 (1975).Google Scholar
- 6.Sargent, M., Scully, M. and Lamb, W., Laser Physics ( Addison-Wesley, Reading, Mass., 1974 ).Google Scholar
- 14.Haken, H., Handbuch der Physik, Vol. 25/2C (Springer-Verlag, Berlin, 1970 ) and in Quantum Optics, ed. Kay and Maitland ( Academic Press, New York, 1970 ).Google Scholar
- 18.deGroot, S. and Mazur, P., Non-Equilibrium Thermodynamics ( North-Holland, Amsterdam, 1969 ).Google Scholar
- Meijer, P., in Transport Processes in Statistical Mechanics (IUPAP Symposium, Brussels, 1956, I. Prigogine, editor).Google Scholar
- 20.Callen, H., in Transport Processes in Statistical Mechanics (IUPAP Symposium, Brussels,1956, I. Prigogine, editor).Google Scholar
- 21.Prigogine, I.P. and Nicolis, G., Self-Organization in Non-Equilibrium Systems (J. Wiley, New York, 1977 ).Google Scholar
- 23.Gantmacher, F.R., Theory of Matrices ( Chelsea Publ., New York, 1960 ).Google Scholar