Abstract
Variational principles yielding upper and lower bounds on transport coefficients can readily be applied to the Boltzmann equation, provided it has the form of a linear, inhomogeneous integrodifferential equation with a Hermitian operator acting on the deviation from equilibrium of the distribution function. In transport problems involving a magnetic field or an alternating electric field, this operator is non-Hermitian. By suitably transforming the transport equation, we show how Variational principles may still give upper and lower bounds. The bounds are used for considering the frequency-dependent conductivity associated with a general scattering operator, and the longitudinal magnetoresistivity in the relaxation time approximation for the scattering operator. Explicit results are presented for (1) the frequency-dependent conductivity of a charged Fermi liquid and (2) the longitudinal magnetoresistivity for a weakly anisotropic Fermi surface.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. M. Ziman,Electrons and Phonons, Oxford University Press, London (1960).
H. HØjgaard Jensen, H. Smith, and J. W. Wilkins,Phys. Rev. 185:323 (1969).
F. Garcia-Moliner and S. Simons,Proc. Cambridge Phil. Soc. 53:848 (1957).
N. K. Hindley and F. Garcia-Moliner,Phys. Rev. A140:343 (1965).
E. I. Blount,Phys. Rev. 131:2354 (1963).
M. Bailyn,Phys. Rev. 126:2040 (1962).
R. Gurzhi,Soviet Phys.-JETP 8:673 (1959).
D. Pines and P. Nozières,Quantum Liquids, Vol. I, W. A. Benjamin, New York (1966).
A. A. Abrikosov and I. M. Khalatnikov,Rept. Progr. Phys. 22:329 (1959).
H. HØjgaard Jensen, H. Smith, and J. W. Wilkins,Phys. Letters 27B:532 (1968).
G. Brooker and J. Sykes,Phys. Rev. Letters 21:279 (1968).
L. Davis, Jr.,Phys. Rev. 56:93 (1939).
F. Garcia-Moliner,Proc. Roy. Soc. A249:73 (1958).
M. J. G. Lee and L. M. Falicov,Proc. Roy. Soc. A304:319 (1968).
P. A. Penz and R. Bowers,Phys. Rev. 172:991 (1968).
H. Smith and J. W. Wilkins,Phys. Rev. 183:624 (1969).
W. E. Lawrence, Thesis, Cornell University, Ithaca, New York (1970).
A. J. Bennett and M. J. Rice,Phys. Rev. 185:968 (1969).
J. A. Gaunt,Trans. Roy. Soc. 228:151 (1929).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Ah-Sam, L.E.G., Jensen, H.H. & Smith, H. On the use of variational methods for solving Boltzmann equations involving non-Hermitian operators. J Stat Phys 3, 17–34 (1971). https://doi.org/10.1007/BF01012184
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01012184