Abstract
A nonlinear stochastic Liouville equation postulated phenomenologically and used recently for the description of the interplay of damping and nonlinearity is shown to arise in a natural manner as an approximate consequence of a set of Langevin equations introduced for the analysis of the Brownian motion of a nonlinear quantum dimer. The validity of the phenomenological equation is discussed in the light of the approximate derivation. A more physical transport equation for the system density matrix is obtained and shown, on the one hand to reduce to the phenomenological equation in an appropriate high temperature limit, and on the other to predict novel behavior. The new nonlinear stochastic Liouville equation we present is capable of unifying the description of the nonlinear transport of quasiparticles in several different parameter regimes and we suggest that it should be regarded and used in the same role in the study of the physics of nonlinear transport phenomena that the linear stochastic Liouville equation does in fields such as (linear) exciton dynamics in molecular crystals and aggregates.
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Tsironis, G.P., Kenkre, V.M., Finley, D.: Phys. Rev. A37, 4474 (1988)
Kenkre, V.M., Campbell, D.K.: Phys. Rev. B34, 4959 (1986); Kenkre, V.M., Tsironis, G.P., Campbell, D.K.: In: Nonlinearity in condensed matter. Bishop, A.R., Campbell, D.K., Kumar, P., Trullinger S.E. (eds.). Berlin, Heidelberg, New York: Springer 1987
Reineker, P.: In: Exciton dynamics in molecular crystals and aggregates. Springer Track in Modern Physics. Vol. 94, Hohler, G. (ed.). Berlin, Heidelberg, New York: Springer 1992
Haken, H., Strobl, G.: Z. Phys.262, 135 (1973); Haken, H., Reineker, P.: Z. Phys.249, 253 (1972)
Silbey, R.J.: Ann. Rev. Phys. Chem.27, 203 (1976)
Hemenger, R.P., Lakatos-Lindenberg, K., Pearlstein, R.M.: J. Chem. Phys.60, 3271 (1974). This paper contains a series of insightful applications of the linear stochastic Liouville equation which have been missed, and sometimes rediscovered, by later authors
Kenkre, V.: In: Exciton dynamics in molecular crystals and aggregates. Springer Track in Modern Physics. Vol. 94, Hohler, G. (ed.). Berlin, Heidelberg, New York: Springer 1982; see also V.M., Kenkre, D., Brown: Phys. Rev. B31, 2479 (1985)
Kenkre, V.M.: In: Singular behavior and nonlinear dynamics. Vol. II, Pnevmatikos, St., Bountis, T., Pnevmatikos, Sp. (eds.) Singapore: World Scientific 1989; see also Kenkre, V.M.: In: Disorder and nonlinearity. Vol. 39, Bishop, A.R., Campbell, D.K., Pnevmatikos, S. (eds.). Berlin, Heidelberg, New York: Springer 1989
Kenkre, V.M., Wu, H.-L.: Phys. Rev. B39, 6907 (1989); Kenkre, V.M., Wu, H.-L.: Phys. Lett. A135, 120 (1989)
Grigolini, P., Wu, H.-L., Kenkre, V.M.: Phys. Rev. B40, 7045 (1989)
Wu, H.-L., Grigolini, P., Kenkre, V.M.: J. Phys. G2, 4417 (1990)
Eilbeck, J.C., Lomdahl, P.S., Scott, A.C.: Physica D16, 318 (1985)
Cruzeiro-Hansson, L., Christiansen, P.L., Elgin, J.N.: Phys. Rev. B37, 7896 (1988)
Feynman, R.P., Vernon, F.L., Hellwarth, R.W.: J. Appl. Phys.28, 49 (1957)
Scott, A.C., Christiansen, P.L.: Los Alamos National Laboratory. (Preprint)
See, e.g., Kerson Huang: Statistical mechanics. 2nd edn., p. 171. New York: Wiley 1987
Grigolini, P., Kenkre, V.M., Scully, M.: (unpublished)
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Kenkre, V.M., Grigolini, P. A new nonlinear stochastic Liouville equation. Z. Physik B - Condensed Matter 90, 247–253 (1993). https://doi.org/10.1007/BF02198161
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DOI: https://doi.org/10.1007/BF02198161