Abstract
Relations of the memory function (GME) theory of exciton transport to other theories and entities in statistical mechanics and condensed matter, as well as solutions of the GME theory for a specific system of special interest to experiments, the linear chain, were the content of this chapter. Our theory is built as a generalization of the theories of Förster as well as Dexter but looks quite different from those developed by Haken and Reineker, and by Grover and Silbey. It was shown that the approaches can be unified by establishing relations among them by inspecting them all in terms of memory functions. Their conceptual differences from the GME theory were also analyzed. It was argued that each method has its own particular purpose and use, separate from the others. The connection between memory functions in probability equations and correlation functions encountered more usually in statistical mechanics within the framework of linear response was analyzed. The two examples selected for this comparison were velocity correlation functions used in mobility or conductivity theory, (Kubo 1957), and scattering functions used in neutron diffraction (Van Hove 1954b). Respectively, they find application in charge mobility physics and in the study of the movement of hydrogen atoms and other interstitials in metals. The next content of the chapter was explicit solutions of the GME extended to have the simple Wigner-Weisskopf incorporation of incoherence in them. These solutions will be useful in subsequent chapters where the effect of coherence is studied on transient grating, Ronchi ruling, and sensitized luminescence observations. Solutions for the full SLE case (with hopping terms added) were also provided. Finally in this chapter, partially nonlocal memories and transfer rates were shown to arise from strong intersite coupling. This chapter thus connects memory functions, primary to the book, to more familiar entities in statistical mechanics and condensed matter physics.
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Notes
- 1.
Two experimentalists in this field who were my contemporaries and engaged in intense debates on exciton coherence made path-breaking innovations in experimental science: Mike Fayer at Stanford and (the late) Ahmed Zewail at Caltech. The latter won the 1999 Nobel prize in Chemistry.
- 2.
Along with a somewhat cumbersome terminology involving the GSHRS equation distinguished from the GSHRS formalism. However, in a sense, the reason for constructing the SLE in its two-part form, thus its focus, are then certainly lost.
- 3.
The latter authors have also commented on the important question of whether the donor and the acceptor are coupled to the same or different baths, and the issue of inhomogenous broadening of spectra.
- 4.
Lest too much be read into this, note that what the GME method was compared to in that particular investigation, was not the MIT procedure discussed here. Instead, it was a semiclassical approximation that was shown to be relatively crude. All that demonstration does in the present context is to strengthen our faith in the accuracy of the GME method. It is to be emphasized that the latter method does not undertake the second diagonalization made in the MIT procedure.
- 5.
The GME as a tool is made for problems that require the computation of probabilities, or observables that depend only on probabilities. If off-diagonal elements of the density matrix are required, one must step back and use the SLE. It is tempting to fall into the erroneous trap of regarding everything one meets with as a nail if one happens to have acquired a hammer. I know of investigators for whom memories and projections are a panacea. Fortunately, physics is richer than that and demands that we remain flexible and move from tool to tool to perform our tasks.
- 6.
Surely this reminds the reader of the well-known fairy tale in which a man went to sleep on a mountain slope for many years while the rest of the world moved on in time. Accordingly, I called the walker’s behavior Rip Van Winkle in Kenkre (1977a) after the name of the man in the fairytale.
- 7.
Related results in the area of transient nucleation and dynamics with internal states were published about the same time or a little later, independently, by Shugard and Reiss (1976), and by Landman et al. (1977), respectively. How they emerge from the expressions in Kenkre and Knox (1974a) has been shown in footnote 36 of Kenkre (1977a).
- 8.
- 9.
Lest the phrase “narrow-band” cause confusion, note that by the expression we mean here that the bandwidth is much smaller than the thermal energy k B T. The usage “narrow-band” here has nothing to do with the coherence issue which happens to be associated with the ratio of the bandwidth to a scattering rate α.
- 10.
A notational alert! So far in the book, k has represented the dimensionless quantity reciprocal to the site index m. In Eq. (4.20) it is, however, dimensioned and reciprocal to the (dimensioned) position x. Given that in most contexts x = ma where a is the lattice constant on a chain, k in Eq. (4.20) is obtained by dividing the dimensionless k by a. Having given careful (and agonizing) consideration to defining two different symbols for the two different k’s, I have decided to avoid the consequent clutter. I rely on the reader’s alertness, and indulgence, to determine by inspection (it is quite easy) which use is meant in a given context.
- 11.
Chapter 7 will force an exception on us in that we will have to use M, N as site indices again and use m, n to represent internal states of vibration.
- 12.
Of course we have put ħ = 1 as we have already been doing. Simplicity rather than wisdom is the motto in this regard. I realize this will endear me to some readers and make me a subject of scorn for other readers. Surely, that is what happens in real life as well.
- 13.
For instance, one of those earlier expressions actually used for numerical computations was
$$\displaystyle \begin{aligned} P_m(t)=\frac{\alpha}{2\pi}\int_{-\bar{\kappa}}^{\bar{\kappa}} dk\,e^{ikm}\frac{\exp{\left[t\left(-\alpha+\sqrt{\alpha^2-16V^2\sin^2(k/2)}\right)\right]}}{\sqrt{\alpha^2-16V^2\sin^2(k/2)}} \\ +\frac{1}{8\pi ^2}\int_{-\pi}^{\pi}\int_{0}^{\pi} dl\, e^{ikm}\left[8V\sin l \sin{}(k/2)\right]^2 \left[ \frac{e^{t\left(-\alpha +i4V\sin{}(k/2)\cos l\right)}}{16V^2\sin^2(k/2)\sin^2 l -\alpha^2}\right] \end{aligned} $$where the first term is present conditionally and determines the value of \(\bar {\kappa }\). Given that this expression contains two integrations and its connection with the extreme limits is by no means transparent, it is clear that our result Eq. (4.35), from Kenkre and Phatak (1984), is much preferable to use.
- 14.
Surprisingly, I have not come across elsewhere this interesting expression for transfer rates on a chain for strong coupling interactions, a very common entity, for the last four decades since it was derived. In a modest manner, the expression hints at what happens when one does not operate under the famous λ 2 t limit of van Hove and yet makes the Markoffian approximation to pass from the GME to the Master equation.
- 15.
Some of the details of the solution for the exercise may be found in Kenkre (1978d) where the time-dependence of the opposite corner site is also shown graphically. Attempt to draw conclusions about the system beyond those set out in that publication.
References
Aslangul, C., & Kottis, P. (1974). Density operator description of excitons in molecular aggregates: Optical absorption and motion. I. The dimer problem. Physical Review B, 10(10), 4364.
Avakian, P., Ern, V., Merrifield, R. E., & Suna, A. (1968). Spectroscopic approach to triplet exciton dynamics in anthracene. Physical Review, 165(3), 974.
Brown, D. W., & Kenkre, V. M. (1983). Quasielastic neutron scattering in metal hydrides: Effects of the quantum mechanical motion of interstitial hydrogen atoms. In Electronic structure and properties of hydrogen in metals (pp. 177–182). New York: Springer.
Brown, D. W., & Kenkre, V. M. (1985). Coupling of tunneling and hopping transport interactions in neutron scattering lineshapes. Journal of Physics and Chemistry of Solids, 46(5), 579–583.
Brown, D. W., & Kenkre, V. M. (1986). Neutron scattering lineshapes for nearly-incoherent transport on non-bravais lattices. Journal of Physics and Chemistry of Solids, 47(3), 289–293.
Brown, D. W., & Kenkre, V. M. (1987). Neutron scattering lineshapes for hydrogen trapped near impurities in metals. Journal of Physics and Chemistry of Solids, 48(9), 869–876.
Casella, R. C. (1983). Theory of excitation bands of hydrogen in bcc metals and of their observation by neutron scattering. Physical Review B, 27(10), 5943.
Fayer, M. D., & Harris, C. B. (1974). Coherent energy migration in solids. I. Band-trap equilibria at Boltzmann and non-Boltzmann temperatures. Physical Review B, 9(2), 748.
Fitchen, D. B. (1968). Zero-phonon transitions. Physics of color centers (pp. 293–350). New York, NY: Academic.
Giuggioli, L., Sevilla, F. J., & Kenkre, V. M. (2009). A generalized master equation approach to modelling anomalous transport in animal movement. Journal of Physics A: Mathematical and Theoretical, 42, 434004.
Grover, M., & Silbey, R. (1971). Exciton migration in molecular crystals. The Journal of Chemical Physics, 54(11), 4843–4851.
Haken, H., & Reineker, P. (1972). The coupled coherent and incoherent motion of excitons and its influence on the line shape of optical absorption. Zeitschrift für Physik, 249(3), 253–268.
Haken, H., & Strobl, G. (1973). An exactly solvable model for coherent and incoherent exciton motion. Zeitschrift für Physik A Hadrons and nuclei, 262(2), 135–148.
Harris, C. B., & Zwemer, D. A. (1978). Coherent energy transfer in solids. Annual Review of Physical Chemistry, 29(1), 473–495.
Hochstrasser, R. M. (1966). Electronic spectra of organic molecules. Annual Review of Physical Chemistry, 17(1), 457–480.
Hochstrasser, R. M., & Prasad, P. N. (1972). Phonon sidebands of electronic transitions in molecular crystals and mixed crystals. The Journal of Chemical Physics, 56(6), 2814–2823.
Holstein, T. (1959a). Studies of polaron motion: Part I. The molecular-crystal model. Annals of Physics, 8(3), 325–342.
Holstein, T. (1959b). Studies of polaron motion: Part II. The “small” polaron. Annals of Physics, 8(3), 343–389.
Ierides, A. A., & Kenkre, V. M. (2018). Reservoir effects on the temperature dependence of the relaxation to equilibrium of three simple quantum systems. Physica A: Statistical Mechanics and Its Applications, 503, 9–25.
Kenkre, V. M. (1974). Coupled wave-like and diffusive motion of excitons. Physics Letters A, 47, 119–120.
Kenkre, V. M. (1975b). Relations among theories of excitation transfer. Physical Review B, 11(4), 1741.
Kenkre, V. M. (1975c). Relations among theories of excitation transfer. II. Influence of spectral features on exciton motion. Physical Review B, 12(6), 2150.
Kenkre, V. M. (1977a). The generalized master equation and its applications. In Landman, U. (Ed.), Statistical mechanics and statistical methods in theory and application (pp. 441–461). New York: Plenum.
Kenkre, V. M. (1978a). Generalization to spatially extended systems of the relation between stochastic Liouville equations and generalized master equations. Physics Letters A, 65(5–6), 391–392.
Kenkre, V. M. (1978d). Theory of exciton transport in the limit of strong intersite coupling. I. Emergence of long-range transfer rates. Physical Review B, 18(8), 4064.
Kenkre, V. M., Andersen, J. D., Dunlap, D. H., & Duke, C. B. (1989). Unified theory of the mobilities of photoinjected electrons in naphthalene. Physical Review Letters, 62(10), 1165.
Kenkre, V. M., & Brown, D. W. (1985). Exact solution of the stochastic Liouville equation and application to an evaluation of the neutron scattering function. Physical Review B, 31(4), 2479.
Kenkre, V. M., & Dresden, M. (1971). Exact transport parameters for driving forces of arbitrary magnitude. Physical Review Letters, 27(1), 9.
Kenkre, V. M., & Dresden, M. (1972). Theory of electrical resistivity. Physical Review A, 6(2), 769.
Kenkre, V. M., Endicott, M. R., Glass, S. J., & Hurd, A. J. (1996). A theoretical model for compaction of granular materials. Journal of the American Ceramic Society, 79(12), 3045–3054.
Kenkre, V. M. (Nitant), & Giuggioli, L. (2020). Theory of the spread of epidemics and movement ecology of animals: An interdisciplinary approach using methodologies of physics and mathematics. Cambridge: Cambridge University Press.
Kenkre, V. M., & Knox, R. S. (1974a). Generalized-master-equation theory of excitation transfer. Physical Review B, 9, 5279–5290.
Kenkre, V. M., Kühne, R., & Reineker, P. (1981). Connection of the velocity autocorrelation function to the mean-square-displacement and to the memory function of generalized master equations. Zeitschrift für Physik B Condensed Matter, 41(2), 177–180.
Kenkre, V. M., Montroll, E. W., & Shlesinger, M. F. (1973). Generalized master equations for continuous-time random walks. Journal of Statistical Physics, 9, 45–50.
Kenkre, V. M., & Phatak, S. M. (1984). Exact probability propagators for motion with arbitrary degree of transport coherence. Physics Letters A, 100(2), 101–104.
Kenkre, V. M., & Rahman, T. S. (1974). Model calculations in the theory of excitation transfer. Physics Letters A, 50(3), 170–172.
Kenkre, V. M., & Reineker, P. (1982). Exciton dynamics in molecular crystals and aggregates. In Springer tracts in modern physics (Vol. 94). Berlin: Springer.
Kenkre, V. M., & Sevilla, F. J. (2007). Thoughts about anomalous diffusion: Time-dependent coefficients versus memory functions. In T. S. Ali & K. B. Sinha (Eds.), Contributions to mathematical physics: A tribute to Gerard G. Emch (pp. 147–160). New Delhi: Hindustani Book Agency.
Kopelman, R., Monberg, E. M., Ochs, F. W., & Prasad, P. N. (1975). Exciton percolation: Isotopic-mixed 1B2u naphthalene. Physical Review Letters, 34(24), 1506.
Kubo, R. (1957). Statistical-mechanical theory of irreversible processes. I. General theory and simple applications to magnetic and conduction problems. Journal of the Physical Society of Japan, 12(6), 570–586.
Landman, U., Montroll, E. W., & Shlesinger, M. F. (1977). Random walks and generalized master equations with internal degrees of freedom. Proceedings of the National Academy of Sciences USA, 74(2), 430–433.
Mainardi, F. (1997). Fractional calculus, some basic problems in continuum and statistical mechanics. In A. Carpinteri & F. Mainardi (Eds.), Fractals and fractional calculus in continuum mechanics. Wien: Springer-Verlag.
Mannella, R., Grigolini, P., & West, B. J. (1994). A dynamical approach to fractional Brownian motion. Fractals, 2(01), 81–94.
Montroll, E. W., & Weiss, G. H. (1965). Random walks on Lattices II. Journal of Mathematical Physics, 6(2), 167–181.
Montroll, E. W., & West, B. J. (1979). On an enriched collection of stochastic processes. In E. W. Montroll & J. J. Lebowitz (Eds.), Studies in statistical mechanics: Vol. VII. Fluctuation phenomena (pp. 61–175). Amsterdam: North Holland Publishing.
Munn, R. W. (1973). Direct calculation of exciton diffusion coefficient in molecular crystals. The Journal of Chemical Physics, 58(8), 3230–3232.
Munn, R. W. (1974). Exciton transport at short times. Chemical Physics, 6(3), 469–473.
Pope, M., & Swenberg, C. E. (1999). Electronic processes in organic crystals and polymers (2nd ed.) New York: Oxford University Press.
Rackovsky, S., & Silbey, R. (1973). Electronic energy transfer in impure solids: I. Two molecules embedded in a lattice. Molecular Physics, 25(1), 61–72.
Reineker, P., Kenkre, V. M., & Kühne, R. (1981). Drift mobility of photo-electrons in organic molecular crystals: Quantitative comparison between theory and experiment. Physics Letters A, 84(5), 294–296.
Robinson, G. W. (1970). Electronic and vibrational excitons in molecular crystals. Annual Review of Physical Chemistry, 21(1), 429–474.
Scher, H., & Lax, M. (1973). Stochastic transport in a disordered solid. I. Theory. Physical Review B, 7, 4491–4502.
Scher, H., & Montroll, E. W. (1975). Anomalous transit-time dispersion in amorphous solids. Physical Review B, 12, 2455–2477.
Shelby, R. M., Zewail, A. H., & Harris, C. B. (1976). Coherent energy migration in solids: Determination of the average coherence length in one-dimensional systems using tunable dye lasers. The Journal of Chemical Physics, 64(8), 3192–3203.
Shlesinger, M. F. (1974). Asymptotic solutions of continuous-time random walks. Journal of Statistical Physics, 10(5), 421–434.
Shugard, W., & Reiss, H. (1976). Derivation of the continuous-time random walk equation in non-homogeneous lattices. Journal of Chemical Physics, 65, 2827.
Silbey, R. (1976). Electronic energy transfer in molecular crystals. Annual Review of Physical Chemistry, 27(1), 203–223.
Silbey, R., & Munn, R. W. (1980). General theory of electronic transport in molecular crystals. I. Local linear electron–phonon coupling. The Journal of Chemical Physics, 72(4), 2763–2773.
Sköld, K. (1978). Quasielastic neutron scattering studies of metal hydrides. In G. Alefeld & J. Vlkl (Eds.), Hydrogen in metals I. Topics in applied physics (Vol. 28). Berlin, Heidelberg: Springer. https://doi.org/10.1007/3540087052_49
Soos, Z. G. (1974). Theory of π-molecular charge-transfer crystals. Annual Review of Physical Chemistry, 25(1), 121–153.
Soules, T. F., & Duke, C. B. (1971). Resonant energy transfer between localized electronic states in a crystal. Physical Review B, 3(2), 262.
Springer, T. (1972). Quasielastic neutron scattering for the investigation of diffusive motions in solids and liquids. In Springer Tracts in Modern Physics (Vol. 64, pp. 1–100). New York: Springer.
Van Hove, L. (1954b). Time-dependent correlations between spins and neutron scattering in ferromagnetic crystals. Physical Review, 95(6), 1374.
West, B., Bologna, M., & Grigolini, P. (2012). Physics of fractal operators. Berlin: Springer Science & Business Media.
Wong, Y. M., & Kenkre, V. M. (1980). Extension of exciton-transport theory for transient grating experiments into the intermediate coherence domain. Physical Review B, 22(6), 3072.
Wu, M. W., & Conwell, E. M. (1997). Transport in α-sexithiophene films. Chemical Physics Letters, 266(3–4), 363–367.
Zewail, A. H, & Harris, C. B. (1975a). Coherence in electronically excited dimers. II. Theory and its relationship to exciton dynamics. Physical Review B, 11(2), 935.
Zewail, A. H., & Harris, C. B. (1975b). Coherence in electronically excited dimers. III. The observation of coherence in dimers using optically detected electron spin resonance in zero field and its relationship to coherence in one-dimensional excitons. Physical Review B, 11(2), 952.
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Kenkre, V.M.(. (2021). Relations of Memories to Other Entities and GME Solutions for the Linear Chain. In: Memory Functions, Projection Operators, and the Defect Technique. Lecture Notes in Physics, vol 982. Springer, Cham. https://doi.org/10.1007/978-3-030-68667-3_4
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