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References: lognormal

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Sylwester Arabas edited this page Jan 20, 2023 · 27 revisions

in reference books:

  • Jacobson 2005
    "Particle components, size distributions, and size structures" (in "Fundamentals of Atmospheric Modeling")

1920-ties:

  • Hatch & and Choate 1929 (J. Franklin Inst. 207)
    "Statistical description of the size properties of non uniform particulate substances"

    Recent investigations by Drinker (1925) and by Loveland and Trivelli (1927), however, have shown that the asymmetrical or "skewed" frequency curves of non-uniform particulate materials can, in general, be transformed into symmetrical curves following the ordinary "law of chance," or normal probability curve, when the logarithms of the sizes are substituted for the sizes themselves.`

1930-ties:

  • Hatch 1933
    "Determination of ``average particle size'' from the screen-analysis of non-uniform particulate substances"

    It has been pointed out by Drinker 1925 and more recently by Loveland and Trevelli 1927 that the size-frequency curves of non-uniform particulate substances produced by crushing, chemical precipitation and other means have the shape of the logarithmic probability curve.

1940-ties:

  • Epstein 1947
    "The mathematical description of certain breakage mechanisms leading to the logarithmico-normal distribution"

    A number of writers have observed that the particle size distributions obtained from some breakage processes appear to be logarithmico-normal. There have been virtually no attempts to explain this phenomenon. In this paper a statistical model is constructed for a breakage mechanism which will generate size distributions which are asymptotically logarithmico-normal.

    The only mathematical papers on the subject appear to be one by Kolmogorov_1941 and another by Halmos_1944.

  • Horton 1948
    "STATISTICAL DISTRIBUTION OF DROP SIZES AND THE OCCURRENCE OF DOMINANT DROP SIZES IN RAIN"

    ... frequency curves are slightly skewed but closely resemble the curve of the normal or Gaussian law of error ...

  • Howell 1949
    "The shape of the distribution curve is virtually unknown. Tha gaussian distribution would be a fovored guess except that the unilateral limit at zero radius becomes serious when the distribution is so broad. The picture is more reasonable if a Gaussian distribution of the logarithm of the radius is assumed"

1960-ties:

  • Fay & Ashford
    "Size distribution of airborne dust samples from British coalmines"

    plotting the logarithm of the observed number of particles per unit interval of particle size against (in turn) particle size, the square root of particle size and the logarithm of particle size

  • Fletcher 1962
    "The Physics of Rainclouds"

    A somewhat different approach has been adopted by Levin (1954b) who remarks that in fact all observed drop size-distributions are very close to log-normal distribution typical of most collections of particles having a common origin (see chapter 4). He further pointed out that both the Marshall-Palmer distribution (7.19) and that used by Khrgian, Mazin and Cao (1952), namely ... are close approximations to the log-normal distribution at values of x rather greater than the median value. The log-normal distribution has the further advantage of limiting the number of very small drops in the distribution, a feture found in the observations ... but not shown by either of the distributions.

    It is physically reasonable that the distribution should be log-normal, but as yet no more detailed description can be given.

    As with most dispersion processes, the size-distribution is, to a first approximation, log-normal (see fig. 4.1.).

  • Mitchell 1968
    "Permanence of the Log-Normal Distribution"

    the log-normal distribution is a better representation than the normal for the distribution of the sum of log-normal variates

  • Kovetz 1969
    "An Analytical Solution for the Change of Cloud and Fog Droplet Spectra Due to Condensation"

    There are both theoretical and experimental resons to believe that the distribution of condensation nuclei, from which the newly created droplets are formed, is a log-normal one (e.g., Fletcher, 1962, chap. 4).

1970-ties:

  • Bencala & Seinfeld 1976 (Atmos. Env. 10)
    "On frequency distributions of air pollutant concentrations"

    Air pollutant concentration frequency distributions are the result of complex phenomena. The direct prediction of these distributions does not appear to be possible.Observed data are generally represented as log-normal, although other common statistical distributions are capable of representing the data as well as or better than the log-normal. The log-normal is convenient because the mean and variance can be easily determined from a log-probability plot of the data. The fundamental question of why concentration distributions tend to be approximately log-normal cannot be answered unequivocally. The persistence of log-normality for all averaging times can be explained if the raw data are themselves log-normally distributed. The log-normality of the raw data, i.e. the instantaneous concentrations, can be shown to result if wind speed is log-normally distributed. Conventional models for mean concentrations, such as the Gaussian plume and eddy diffusion, can also lead to log-normality for concentration distributions if the wind speeds are log-normal.

1980-ties

1990-ties:

  • Szwed, Wrochna & Wróblewski 1990
    "Genesis of the lognormal multiplicity distribution in the e+ e- collisions and other stochastic processes"

    ... scale invariant branching is assumed as a mechanism within which all these properties could be derived ... It is also shown that many other phenomena encountered in nature have the similar statistical properties.

  • Lee et al. 1997 (J. Colloid Interface Sci.)
    "The log-normal size distribution theory for Brownian coagulation in the low Knudsen number regime"

    TODO

  • Söderlund et al 1998 (Phys. Rev. Lett. 80)
    "Lognormal Size Distributions in Particle Growth Processes without Coagulation"

    It should be noted that once initial nucleation has taken place, the proposed model fully accounts for the lognormal size distribution found in numerous experiments. Growth by vapor absorption is sufficient, and the model does not involve coagulation. However, if subsequent growth by Brownian coagulation would occur in regions where absorption cannot take place, this is known to be able to preserve the initial lognormal distribution (Lee et al 1997)

  • Hinds 1999
    "Aerosol Technology: Properties, Behaviour, and Measurements of Airbirne Particles"

    There is no fundamental theoretical reason why particle size data should approximate the lognormal distribution, but it has been found to apply to most single-source aerosols”

    Appendix 2 (added in 2nd edition?): Theoretical basis for aerosol particle size distributions: Section 4.4 states that there is no fundamental theoretical basis for aerosols having a lognormal size distribution. While this is true in general, several authors have noted that when the particle formation process follows the "law of proportionate effect," a lognormal size distribution results. This law requires changes in size to occur in steps, with the particle size for each step being a random multiplicative factor of the size of the previous step.

2000-ties:

  • Limpert et al. 2000
    "Life is log normal: Keys and clues to understand patterns of multiplicative interactions from the disciplinary to the transdisciplinary level"

  • Limpert et al. 2001
    "_ Log-normal Distributions across the Sciences: Keys and Clues: On the charms of statistics, and how mechanical models resembling gambling machines offer a link to a handy way to characterize log-normal distributions, which can provide deeper insight into variability and probability—normal or log-normal: That is the question _"

    Among other examples are size distributions of aerosols and clouds and parameters of turbulent processes. In the context of turbulence, the size of which is distributed log-normally (Limpert et al. 2000b)

  • Maul et al. 2005 (Phys. Rev. B 72)
    "Lognormal mass distributions of nanodiamonds from proportionate vapor growth"

    lognormal distribution arises from a theory of elementary errors combined with a multiplicative process, just as the underlying change of growth is multiplicative rather than additive

    lognormal size distributions have long been explained on the basis of Brownian coagulation models Friedlander and Wang 1966, Lee et al 1997, but it turned out later that essentially any process where the basic mechanism is particle diffusion and drifts through a finite region exhibits lognormal growth Soederlund et al 1998. Here, the time available for particles to grow determines their peculiar size distribution

    process constant k j needs to be small in order to obtain lognormal growth behavior

    lognormal mass distributions which are indicative for proportionate growth processes in the vapor phase

  • Espiau de Lamaëstre & Bernas 2006 (Phys. Rev. B 73)
    "Significance of lognormal nanocrystal size distributions"

    he diversity of processes leading to lognormal size distributions is the first thing to emphasize. Aggregation of colloids (Gmachowski 2001,Gmachowski 2002) or of aerosols (Graham_and_Robinson_1976), some crystallization processes (Yatsuya et al 1973), (Granqvist and Buhrman 1976), some complex nucleation and growth processes (Miotello et al 2001, Borsella et al 2001, Andersen and Johnson 1995, Hagege and Dahmen_1996), all may lead to limiting size distributions that display apparently lognormal shapes. The fact that these processes involve very different microscopic mechanisms is clearly indicative of a very general property that we aimed to discern. This involved other questions. Where does the generality of lognormal shapes come from? Or conversely, what information on the physical processes occurring in an evolutionary system is contained in the lognormal distribution?

    A. When are size distributions lognormal? A brief survey

    C. How does a size distribution become lognormal?

  • Limpert et al. 2008 (Aerobiologia 24)
    "Data, not only in aerobiology: how normal is the normal distribution?"

    In aerosol science, the measured number distribution of physical properties of particles (e.g., size, surface area or volume/mass) is commonly highly skewed. Thus, in preference to the normal distribution, the data is routinely fitted to a log-normal distribution and corresponding statistics included as standard in instrument software. More than this, for modeling the formation of aerosol particles, the log-normal probability density function is justified from first principles. Specifically, any proportional size-modifying process (e.g., condensational growth or agglomeration) gives rise to a log-normal particle size distribution, similar to fragmentation which has long been shown to fit log-normality (Kolmogoroff1941).

2010-ties:

  • Kok 2011 (PNAS 108)
    "A scaling theory for the size distribution of emitted dust aerosols suggests climate models underestimate the size of the global dust cycle"

    Mineral dust aerosols impact Earth’s radiation budget through interactions with clouds, ecosystems, and radiation, which constitutes a substantial uncertainty in understanding past and predicting future climate changes. One of the causes of this large uncertainty is that the size distribution of emitted dust aerosols is poorly understood.

    I therefore take a different approach here and utilize the closest analog problem to dust emission that is quantitatively understood: the fragmentation of brittle materials Astrom 2006, Kolmogorov 1941

  • Boucher 2015 (In: Atmospheric Aerosols)
    "Atmospheric Aerosols: Physical, Chemical and Optical Aerosol Properties"

    Finally, the log-normal law is the most used distribution law because of its many advantages. It appears to be very universal and can describe observed size distributions fairly well especially those spanning many orders of magnitude. This distribution is convenient because $r_0$ happens to equal the mean geometric radius and $\sigma_g=\exp(\sigma_0)$ is the geometric standard deviation as defined above. Moreover, the lognormal law has interesting mathematical properties: the distribution of the various moments also follow lognormal laws, while both the median and the mode are equal to $r_0$. The effective radius, $r_e$, is equal to $r_0 \exp(\frac{5}{2}\sigma_0^2)$ and the effective variance, $\nu_e$, is $\exp(\sigma_0^2)-1$.

2020-ties:

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