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Memory Functions from Static Disorder: Effective Medium Theory

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Memory Functions, Projection Operators, and the Defect Technique

Part of the book series: Lecture Notes in Physics ((LNP,volume 982))

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Abstract

Whereas the rest of the treatment in the book has dealt with ordered systems such as crystals, and with disorder that is dynamic, i.e. arising from the movement of the constituents of the crystals, this chapter showed how memory functions arise in statically disordered systems exemplified by amorphous molecular aggregates. We addressed the concept behind the effective medium approach. It was shown how it arises from a judicious and noteworthy application of the defect technique to replace a statically disordered time-local Master equation by an ordered counterpart which is non-local in time. The memory associated with this non-local description was not merely postulated but calculated: an explicit prescription to obtain the memory functions from given probability distributions of the randomness of the disorder was provided and used to examine the extent of validity of the procedure. The typical shape of the EMT memory was determined and an exponential approximation was introduced. Numerical calculations were used to obtain the memory in general as well as in specific cases and a number of effects were discovered with its help. Among others, they included behavior in finite size systems and the appearance of spatially long range memories. A brief conceptual discussion of the method was also provided.

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Notes

  1. 1.

    The correctness of your solution of the exercise should be, thus, verified at least through the mighty power of democracy. No reason to worry, as some say, that it is teetering these days on the brink of a precipice.

  2. 2.

    The usefulness of the form of the basic equation we have presented, Eq. (13.4), should be already clear by comparison to the well-known asymptotic result for the effective rate Eq. (13.8).

  3. 3.

    When the EMT memory is calculated in the Laplace domain via our prescription based on Eq. (13.4), the derived quantities D(t) and 〈m 2〉 can be obtained very simply in the Laplace domain by dividing \(\tilde {\mathcal {F}}(\epsilon )\) by 𝜖 and 𝜖 2 (except for proportionality constants) respectively.

  4. 4.

    No ensemble average was involved in that context.

  5. 5.

    It has always seemed to me surprising that there were investigators (Pollak 1977; Silver et al. 1979) who had to be persuaded with the help of a formal exercise (Klafter and Silbey 1980) that disorder could be treated with memory functions.

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  1. University of New Mexico, Albuquerque, NM, USA

    V. M. (Nitant) Kenkre

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Kenkre, V.M.(. (2021). Memory Functions from Static Disorder: Effective Medium Theory. 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_13

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