Mechanisms for log normal concentration distributions in the environment

Log normal-like concentration distributions are ubiquitously observed in the environment. However, the mechanistic origins are not well understood. In this paper, we show that first order exponential kinetics onsets log-normal concentration distributions, under certain assumptions. Given the ubiquity of exponential kinetics, e.g., source and sink processes, this model suggests an explanation for the frequent observation in the environment, and elsewhere. We compare this model to other mechanisms affecting concentration distributions, e.g., source mixing. Finally, we discuss possible implications for data analysis and modelling, e.g., log-normal rates and fluxes.


S1. A stochastic differential equation for sink kinetics
In this section we show that the concentration distribution, i.e., the probability density distribution 14 (pdf), that describes the variability of a first order sink process with a randomly varying rate is the 15 log-normal distribution. Mathematically, we represent this sink process by a stochastic differential 16 equation (SDE): Eq. (2) in the main manuscript (here reproduced as Eq. (S1) for convenience). 17 The derivation of the log-normal distribution from this SDE is composed of 5 main steps: 18 i) Perform a variable change of the SDE, where the multiplicative noise term is replaced by an 19 expression with additive noise, which is more convenient for further analysis. term. The FPE is a partial differential equation the describe the time-evolution of the pdf.    There is no mathematical novelty in this derivation, as solutions have been presented for 26 mathematically similar models 1,2 . However, we here express the model for the specific system of 27 pysico/chemical sink kinetics. We aim for an explicit and hopefully transparent presentation, 28 allowing for readers that are not familiar with SDEs to follow the argument. This includes a 29 simplified notation, akin to regular calculus. The resulting partial differential equations are for 30 clarity also solved explicitly. Eq. (S1) represents a SDE where the random term ((t)) is multiplied with the function we want  We assume that (t)) is identical and independent normal distributed with mean equal to zero and 37 standard deviation equal to one.

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In Eq. (S1) we have what is called multiplicative noise: the noise term is multiplied with the 39 variable X. However, the standard formulation of SDEs is based on additive noise. We therefore 40 need to do a variable transformation of the form: In regular calculus, we would then compute the differential: For stochastic calculus, this relation does not apply. Instead, we calculate the differential using 45 Itô's formula.

Itô's formula and variable transformation in SDEs 48
Consider a general stochastic differential equation of the form:  Itô's formula gives the following general expression for the differential: Now we want to relate this stochastic differential equation to an equation describing the time-68 evolution of the concentration distribution of Z, i.e., the probability density distribution (P(Z,t)).

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The equation that describes the evolution of P(Z,t)) is a partial differential equation called the  76 To solve the partial differential equation (S8), we use separation of variables, setting:

iii). Solving the Fokker Planck equation: Separation of variables
The corresponding partial derivatives are: (S11) 85 We can then define q and s as: Through these definitions, the direct dependencies on b(Z,t) and ( , ) in Eq. (S11) are removed, Which is formally equivalent to the diffusion/heat equation, which we do not solve here, see, e.g., 94 Zwanzig 3 . The solution is a normal distribution: Combing Eqs (S9), (S12-S13) and (S15), we have: Which is a normal distribution with mean − − 1 2 2 and variance 2 .

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In the main manuscript, the formulation of the kinetics model is focused on sink or loss processes.

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But what about processes involving formation or production?

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Consider Eq. (2) from the main manuscript (Eq (S1) reprinted for convenience: The formation kinetics for Y may be written as (removing the minus sign in Eq. (S22)): Since X is a log-normally distributed stochastic variable we conclude that Y may also be a log 131 normally distributed stochastic variable.