Quantifying aerosol mixing state with entropy and diversity measures

This paper presents the first quantitative metric for aerosol population mixing state, defined as the distribution of per-particle chemical species composition. This new metric, the mixing state index χ , is an affine ratio of the average per-particle species diversity Dα and the bulk population species diversityDγ , both of which are based on information-theoretic entropy measures. The mixing state index χ enables the first rigorous definition of the spectrum of mixing states from so-called external mixture to internal mixture, which is significant for aerosol climate impacts, including aerosol optical properties and cloud condensation nuclei activity. We illustrate the usefulness of this new mixing state framework with model results from the stochastic particle-resolved model PartMC-MOSAIC. These results demonstrate how the mixing state metrics evolve with time for several archetypal cases, each of which isolates a specific process such as coagulation, emission, or condensation. Further, we present an analysis of the mixing state evolution for a complex urban plume case, for which these processes occur simultaneously. We additionally derive theoretical properties of the mixing state index and present a family of generalized mixing state indexes that vary in the importance assigned to low-mass-fraction species.


Introduction
Our quantitative understanding of the aerosol impact on climate still has large gaps and hence introduces large uncertainties in climate predictions (IPCC, 2007).One of the challenges is the inherently multi-scale nature of the problem: the macro-scale impacts of aerosol particles are governed by processes that occur on the particle-scale, and these microscale processes are difficult to represent in large-scale models (Ghan and Schwartz, 2007).
An important quantity in this context is the so-called mixing state of the aerosol population, which we define as the distribution of the per-particle chemical species compositions.Recent observations made in the laboratory and in the field using single-particle measurement techniques have revealed that the mixing states of ambient aerosol populations are complex.Even freshly emitted particles can have complex compositions by the time they enter the atmosphere.For example, the mixing state of particles originating from vehicle engines depends strongly on fuel type and operating conditions (Toner et al., 2006).The initial particle composition is further modified in the atmosphere as a result of aging processes including coagulation, condensation of secondary aerosol species, and heterogeneous reactions (Weingartner et al., 1997).
While the extent to which mixing state needs to be represented in models is still an open research question, there is evidence that mixing state matters for adequately modeling aerosol properties such as optical properties (Jacobson, 2001;Chung and Seinfeld, 2005;Zaveri et al., 2010), cloud condensation nuclei activity (Zaveri et al., 2010), and wet removal (Koch et al., 2009;Stier et al., 2006;Liu et al., 2012).Therefore, in recent years, efforts have been made to represent mixing state in models to some extent.This is the case for models on the regional scale (Riemer et al., 2003) as well as on the global scale (Jacobson, 2002;Stier et al., 2005;Bauer et al., 2008;Wilson et al., 2001).
In discussions about mixing state, the terms "external mixture" and "internal mixture" are frequently used to describe Published by Copernicus Publications on behalf of the European Geosciences Union.
how different chemical species are distributed over the particle population.An external mixture consists of particles that each contain only one pure species (which may be different for different particles), whereas an internal mixture describes a particle population where different species are present within one particle.If all particles consist of the same species mixture and the relative abundances are identical, the term "fully internal mixture" is commonly used.
While these terms may be appropriate for idealized cases, observational evidence shows that ambient aerosol populations rarely fall in these two simple categories.In this paper we present the first quantitative measure of aerosol mixing state, the mixing state index χ, based on diversity measures derived from the information-theoretic entropy of the chemical species distribution among particles.
The measurement of species diversity and distribution using information-theoretic entropy measures has a long history in many scientific fields.In ecology, the study of animal and plant species diversity within an environment dates back to Good (1953) and MacArthur (1955), but rose to prominence with the work of Whittaker (1960Whittaker ( , 1965Whittaker ( , 1972)).Whittaker proposed measuring species diversity by the species richness (number of species), the Shannon entropy, and the Simpson index, which are now referred to as generalized diversities of order 0, 1, and 2, respectively (see Appendix A for details).
Whittaker also introduced the fundamental concepts of alpha, beta, and gamma diversity, where alpha diversity D α measures the average species diversity within a local area, beta diversity D β measures the diversity between local areas, and D γ measures the overall species diversity within the environment, given as the product of alpha and beta diversities.In the context of aerosols, we regard alpha diversity as measuring the average species diversity within a single particle, beta diversity as quantifying diversity between particles, and gamma diversity as describing the overall diversity in bulk population (see Section 2 for details).From these measures we construct the mixing state index χ as an affine ratio of alpha and gamma diversity.
The ecology literature in the 1960s and 1970s contains much work on species diversity and distribution, although there was also significant confusion about the underlying mathematical framework (Hurlbert, 1971;Hill, 1973).Within the last decade, the profusion of diversity measures have been largely categorized (Tuomisto, 2013(Tuomisto, , 2012(Tuomisto, , 2010)), although disagreement in the literature is still present (Tuomisto, 2011;Gorelick, 2011;Jurasinski and Koch, 2011;Moreno and Rodríguez, 2011).Despite the current controversies, certain principles are now well-established, such as the use of the effective number of species as the fundamentally correct way to measure diversity (Hill, 1973;Jost, 2006;Chao et al., 2008Chao et al., , 2010;;Jost et al., 2010).
This paper is organized as follows.In Sect. 2 we define the well-established entropy and diversity measures, adapted to the aerosol context, and use these to define our new mixing state index χ.This section also contains examples of diversity and mixing state and a summary of the properties of these measures.Section 3 presents a suite of simulations for archetypal cases using the stochastic particle-resolved model PartMC-MOSAIC (Riemer et al., 2009;Zaveri et al., 2008).These simulations show how the diversity and mixing state measures evolve under common atmospheric processes, including emissions, dilution, coagulation, and gas-to-particle conversion.A more complex urban plume simulation is then considered in Sect.4, for which the above processes occur simultaneously.Appendix A presents a generalization of the diversity and mixing state measures to ascribe different levels of importance to low-mass-fraction species, while Appendix B contains mathematical proofs for the results summarized in Sect. 2.

Entropy, diversity, and mixing state index
We consider a population of N aerosol particles, each consisting of some amounts of A distinct aerosol species.The mass of species a in particle i is denoted µ a i , for i = 1, . . ., N and a = 1, . . ., A. From this basic description of the aerosol particles we can construct all other masses and mass fractions, as detailed in Table 1.Using the distribution of aerosol species within the aerosol particles and within the population, we can now define mixing entropies, species diversities, and the mixing state index, as shown in Table 2.Note that entropy and diversity are equivalent concepts, and that either could be taken as fundamental.We retain both in this paper to enable connections with the historical and current literature.
The entropy H i or diversity D i of a single particle i measures how uniformly distributed the constituent species are within the particle.This ranges from the minimum value  (H i = 0, D i = 1) when the particle is a single pure species, to the maximum value (H i = ln A, D i = A) when the particle is composed of equal amounts of all A species.As shown in Fig. 1, the diversity D i of a particle measures the effective number of equally distributed species in the particle.If the particle is composed of equal amounts of 3 species then the number of effective species is 3, for example, while 3 species unequally distributed will result in an effective number of species somewhat less than 3. Extending the single-particle diversity D i to an entire population of particles gives three different measures of population diversity.Alpha diversity D α measures the average perparticle diversity in the population, beta diversity D β measures the inter-particle diversity, and gamma diversity D γ measures the bulk population diversity.The bulk population diversity (D γ ) is the product of diversity on the per-particle level (D α ) and diversity between the particles (D β ), giving

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bulk population diversity (1) Alpha diversity D α measures the average per-particle effective number of species in the population, and ranges from 1 when all particles are pure (each composed of just one species, not necessarily all the same), to a maximum when all particles have identical mass fractions.Gamma diversity D γ measures the effective number of species in the bulk population, ranging from 1 if the entire population contains just one species, to a maximum when there are equal bulk mass fractions of all species.Beta diversity D β is defined by an affine ratio of gamma to alpha diversity, so it measures interparticle diversity and ranges from 1 when all particles have identical mass fractions, to a maximum when every particle is pure but the bulk mass fractions are all equal.
Fig. 3: Particle diversities D i of representative particles.The particle diversity measures the effective number of species within a particle, so a pure single-species particle has D i = 1 and a particle consisting of 2 or 3 species in even proportion will have D i = 2 or D i = 3, respectively.A particle with unequal amounts of 2 species will have an effective number of species somewhat less than 2, while a particle with unequal amounts of 3 species will have effective species below 3, and possibly even below 2 if the distribution is very unequal.(MacArthur, 1955;Goodman, 1975;McCann, 2000;Ives and Carpenter, 2007).Other important research questions include 120 the sources of diversity (Tsimring et al., 1996;De'ath, 2012), extensions of diversity to include a concept of species distance Chao et al. (2010); Leinster and Cobbold (2012); Feoli (2012); Scheiner (2012), and techniques for measuring diversity (Chao and Shen, 2003;Schmera and Podani, 2013; 125 Gotelli and Chao, 2013), despite the well-known difficulties in estimating entropy in an unbiased fashion (Harris, 1975;Paninski, 2003).Beyond ecology, the study of diversity is also important in economics (Garrison and Paulson, 1973;Hannah and Kay, 1977;Attaran and Zwick, 1989;Malizia 130 and Ke, 1993;Drucker, 2013), immunology (Tsimring et al., 1996), neuroscience (Panzeri and Treves, 1996;Strong et al., 1998), and genetics (Innan et al., 1999;Rosenberg et al., 2002;Falush et al., 2007).This paper is organized as follows.In Section 2 we define 135 the well-established entropy and diversity measures, adapted to the aerosol context, and use these to define our new mixing state index .This section also contains examples of diversity and mixing state and a summary of the properties of these measures.Section 3 presents a suite of simulations for 140 archetypal cases using the stochastic particle-resolved model PartMC-MOSAIC (Riemer et al., 2009;Zaveri et al., 2008).These simulations show how the diversity and mixing state measures evolve under common atmospheric processes, including emissions, dilution, coagulation, and gas-to-particle 145 conversion.A more complex urban plume simulation is then considered in Section 4, for which the above processes occur simultaneously.Appendix A presents a generalization of the diversity and mixing state measures to ascribe different levels of importance to low-mass-fraction species, while Ap-150 pendix B contains mathematical proofs for the results summarized in Section 2.  The particle diversity measures the effective number of species within a particle, so a pure single-species particle has D i = 1 and a particle consisting of 2 or 3 species in even proportion will have D i = 2 or D i = 3, respectively.A particle with unequal amounts of 2 species will have an effective number of species somewhat less than 2, while a particle with unequal amounts of 3 species will have effective species below 3, and possibly even below 2 if the distribution is very unequal.marizes the conditions under which the diversity measures attain their maximum and minimum values.
The two population diversities D α (per-particle) and D γ (bulk) can be combined to give the single mixing state index χ, which measures the homogeneity or heterogeneity of the population.It ranges from χ = 0 when all particles are pure (a fully externally mixed population) to χ = 1 when all particles have identical mass fractions (a fully internally mixed population).For example, a population with a mixing state index of χ = 0.3 (equivalently, χ = 30 %) can be interpreted as being 30 % internally mixed, and thus 70 % externally mixed.
Examples for different population diversities and mixing states are shown graphically in Fig. 2. Because the population diversity D γ cannot be less than the per-particle diversity D α , only a triangular region is accessible on the mixing state diagram.Representative populations and their diversities are indicated on this diagram, as listed in Table 4.
The diversity measures and mixing state index behave in characteristic ways when the particle population undergoes coagulation or when two particle populations are mixed, as is the case when particles are emitted into a pre-existing population.This is summarized in Table 5.The population mixing results (Table 5 and Theorem 3) show that the diversities and entropies are intensive quantities.For example, doubling the size of particle i leaves H i unchanged, and doubling the population leaves H α unchanged.Extensive versions of these quantities can be defined by mass-weighting, so that the total mass-extensive entropy is H = i µ i H i , for example.

Single-process studies
Having established the key quantities to characterize mixing state, and having explored their properties and their physical interpretation, we illustrate in this section their behavior with a suite of simulation scenarios.The cases presented in this section are "single-process" simulations.They are designed to isolate the impacts of emission, coagulation, and condensation on the aerosol mixing state, and exemplify how each mixing state index -0 to 100 % degree to which population is externally mixed (χ = 0) versus internally mixed (χ = 100 %) A when all bulk mass fractions are equal χ 0 % when all particles are pure 100 % when all particles have identical mass fractions process impacts the quantities D α , D γ and χ.Expanding on this, in Sect. 4 we analyze a more complex urban plume case with emission, dilution, coagulation and condensation occurring simultaneously.We used the particle-resolved model PartMC-MOSAIC (Particle Monte Carlo Model for Simulating Aerosol Interactions and Chemistry) (Riemer et al., 2009;Zaveri et al., 2008) for this study (PartMC version 2.2.0).This stochastic particle-resolved model explicitly resolves the composition of individual aerosol particles in a population of differ-ent particle types in a Lagrangian air parcel.PartMC simulates particle emissions, dilution with the background, and Brownian coagulation stochastically by generating a realization of a Poisson process.Gas-and aerosol-phase chemistry are treated deterministically by coupling with the MOSAIC chemistry code.The governing model equations and the numerical algorithms are described in detail in Riemer et al. (2009).Since the model tracks the per-particle composition as the population evolves over time, we can calculate the mixing state quantities as detailed in Sect. 2. We excluded Atmos.Chem.Phys., 13, 11423-11439, 2013 www.atmos-chem-phys.net/13/11423/2013/ aerosol water from calculating total particle masses of particles (i.e., we use dry mass to define the mass fractions in Table 1).Note that from the information on per-particle composition, it is straightforward to calculate per-particle properties, such as hygroscopicity (Riemer et al., 2010;Zaveri et al., 2010;Ching et al., 2012;Tian et al., 2013), optical properties (Zaveri et al., 2010), or particle reactivity (Kaiser et al., 2011).

Single-process case descriptions
The following model setup applies to the cases listed in Table 6.The simulation time was 24 h, and 10 5 computational particles were used to initialize the simulations.To simplify the interpretation of the results, we applied a flat weighting function in the sense of DeVille et al. (2011).The temperature was 288.15 K, the pressure was 10 5 Pa, the mixing height of the box was 300 m, and the relative humidity (RH) was 0.7.Dilution with background air was not simulated.Each initial monodisperse mode was defined by an initial  total number concentration of N tot = 3 × 10 7 m −3 and by an initial diameter of D = 0.1 µm.
For the cases that included particle emissions (Cases 1, 2, and 3), the diameter of the emitted particles was D = 0.1 µm.The emitted particle flux was E = 5 × 10 7 m −2 s −1 for Case 1, E = 1.6 × 10 10 m −2 s −1 for Case 2, and E = 1.6 × 10 8 m −2 s −1 for Case 3.For the cases that included chemistry (Cases 5, 6, 7, and 8), the initial conditions for the gas phase were 50 ppb O 3 , 4 ppb NH 3 and 1 ppb HNO 3 .The gas phase emissions of NO and NH 3 were prescribed at a constant rate of 60 nmol m −2 s −1 and 9 nmol m −2 s −1 , respectively.Photolysis rates were constant, corresponding to a solar zenith angle of 0 • .Coagulation was not simulated except for Case 4.
The results from the single-process studies are summarized in Figs. 3, 4, and 5.The left column of Figs. 3 and  4 shows time series of the per-particle species diversity distribution, n(t, D i ).The right column in these figures shows the time series of the average per-particle diversity D α , the population diversity D γ , and the corresponding mixing state index χ .Each simulation is also depicted in the mixing state diagram in Fig. 5.The next sections discuss the main features of these results.

Emission cases (Cases 1, 2, and 3)
Cases 1, 2, and 3 explore the impact of particle emissions into a pre-existing aerosol population.In these cases coagulation    and condensation were not simulated.Depending on the relative magnitudes of the per-particle diversities of the emitted versus the pre-exisiting particles, emissions can have different impacts on the aerosol mixing state.6.
-Case 1: we considered an initial particle population that contained ammonium sulfate (D i = 1.8 effective species) combined with emissions of pure BC particles (D i = 1 effective species).This process is shown in Fig. 3a with the number concentration of the ammonium sulfate particles remaining constant, and the number concentration of the emitted BC particles increasing over time.Due to the emission of particles with lower D i than the initial population, the average per-particle diversity D α decreased (Fig. 3b).On the other hand, adding particles of a different species than the initial particles increased the population species diversity D γ .This results in a decreasing mixing state index χ and is consistent with the particles becoming on average more simple, and the population more inhomogeneous.In this particular case the population evolved from 100 % internally mixed (χ = 1) to 30 % internally mixed (χ = 0.3).The blue solid line in Fig. 5 shows this process on the mixing state diagram.
-Case 2: we prescribed an initial particle population of two monodisperse modes, with mode 1 consisting of mineral dust (model species OIN, "other inorganics") and SO 4 , and mode 2 consisting of BC and OC.Since all particles contained two species in equal amounts, D i = 2 for all particles.The emissions consisted of mode-2 particles.Since the D i -values of all particles were identical, we only observe one line in Fig. 3c, and the average per-particle species diversity D α in Fig. 3d was constant with time.However, since the emitted particles had the same composition as one of the initial modes, D γ decreased in this simulation, hence the mixing state index χ increased from 33 % to 90 % internally mixed.This is an example of a process where the average diversity on a particle-level did not change, but on a population-level diversity decreased.The blue dashed line in Fig. 5 shows this process on the mixing state diagram.
-Case 3: we considered an initial particle population of pure BC particles (D i = 1 effective species) combined with emissions of particles containing ammonium sulfate (D i = 1.8 effective species) (Fig. 3e).This case represents the opposite of Case  6. mixed particles with a higher per-particle diversity than that of the initial particles increased D α , as the particles became more diverse on average.At the same time population diversity D γ also increased.As a result the mixing state index χ increased, indicating that the population became more homogeneous (Fig. 3e).The cyan line in Fig. 5 shows this process on the mixing state diagram.

Coagulation case (Case 4)
Case 4 explores the impact of coagulation.Emissions and condensation were not simulated.We considered an initial particle population that contained a subpopulation of pure BC particles and another subpopulation of pure SO 4 , giving D i = 1 effective species for all particles at the start of the simulation.Coagulation of the particles produced mixed particles with 1 ≤ D i ≤ 2, as Fig. 3g shows.The largest possible value for D i was 2 effective species, resulting from coagulation events that led to equal amounts of SO 4 and BC in the particles.Values of D i smaller than 2 developed as a result of multiple coagulation events, when one species dominated the composition of the constituent particles.Since coagulation produced mixed particles, D α increased, indicating that particles became more complex on average.In contrast, as stated in Theorem 2, the population diversity D γ remained constant, as shown in Fig. 3h.As a result χ increased from 0 % to about 75 % internally mixed, indicating that the population became more homogeneous.The red line in Fig. 5 shows this process on the mixing state diagram.Since in this scenario the bulk amounts of SO 4 and BC were equal, this line traces the upper edge of the triangle in the mixing state diagram.

Condensation cases (Cases 5-8)
Cases 5-8 explore the impact of condensation.In these cases particle emissions and coagulation were not simulated.Since we only prescribed gas phase emissions of NO and NH 3 , only ammonium nitrate formed as a secondary species.Similar to the emission cases, the condensation cases illustrate that the same process (here condensation) can lead to different outcomes in terms of mixing state, depending on the conditions of the scenario.As we will demonstrate below, on a Atmos.Chem.Phys., 13, 11423-11439, 2013 www.atmos-chem-phys.net/13/11423/2013/6 and shown in Figs. 3 and 4. population level, condensation can produce either more homogeneous or less homogeneous populations, and on a particle level, it can produce either less diverse or more diverse particles.
-Case 5: we considered an initial monodisperse particle population of pure BC particles, hence D i was initially 1 (Fig. 4a).Over the course of the simulation, secondary ammonium nitrate formed on the particles, with the same amount on each particle.Therefore D α increased and was at all times equal to D γ .This resulted in a constant value χ = 1, as shown in Fig. 4b.While the population was always 100 % internally mixed, the increase of D α and D γ can be interpreted as an increase in diversity both on a per-particle level and on a population level.This process is represented by the black solid line in the mixing state diagram (Fig. 5).
-Case 6: here we initialized each particle with a mixture of BC, ammonium sulfate and ammonium nitrate, so the particles started out with D i = 3.7 effective species (Fig. 4c).The formation of secondary ammonium nitrate led to a decrease of D α , again with χ = 1 at all times for a 100 % internally mixed population (Fig. 4d).The decrease of D α and D γ can be interpreted as a decrease in the complexity of the particles.This is consistent with the particle composition becoming more dominated by the condensing species.This process is represented by the black dashed line in the mixing state diagram (Fig. 5).
-Case 7: we initialized the population with two monodisperse modes.One consisted predominantly of BC with some ammonium sulfate (D i = 1.5 effective species).The other was mainly ammonium sulfate with a small amount of BC (D i = 2.35 effective species).The condensation of ammonium nitrate on all particles led to increasing D i for each subpopulation (Fig. 4e).Ammonium nitrate condensed on all particles, hence the overall population became more homogeneous, indicated by increasing χ from 50 % to about 75 % internally mixed (Fig. 4f).This process is represented by the green solid line in the mixing state diagram (Fig. 5).
-Case 8: we initialized the population with two monodisperse modes.They both consisted predominantly of BC, but differed in their composition of inorganic species (see Table 6) and contained 1.4 and 1.6 effective species, respectively.This case was designed so that differences would occur in the ammonium nitrate formation on the two modes based on differences in aerosol water content (Fig. 4g).The result was that the two subpopulations diverged from each other in composition.While faster, hence χ decreased (Fig. 4h) from 90 % to 78 % internally mixed.In this case ammonium nitrate condensed preferentially on one of the two subpopulations, hence condensation caused the overall population to be more inhomogeneous.This process is represented by the green dashed line in the mixing state diagram (Fig. 5).

Overview of urban plume case
In this section we discuss the case of a more complex urban plume scenario.The details of this scenario are described in Zaveri et al. (2010) and Ching et al. (2012).We assumed that a Lagrangian air parcel containing background air was advected within the mixed layer across a large urban area.The start of the simulation was at 06:00 LT in the morning.During the first 12 h of simulation, while the air parcel traveled over the urban area, we prescribed continuous gas emissions NO x , SO 2 , CO, and volatile organic compounds (VOCs), as well as emissions of three different particle types, which originated from gasoline engines, diesel engines, and meat-cooking activities.The specifics of the particle emissions and the initial particle distributions, here initialized as log-normal distributions, are listed in Table 7.We slightly modified two details of the original urban plume case presented in Zaveri et al. (2010).The initial and background particles of the urban plume case in Zaveri et al. (2010) contained small amounts of BC.Here we changed this, so that these particle types only contained ammonium sulfate and SOA, while BC is exclusively associated with particle emissions.We also set the initial concentration of HCl to zero.
Both of these modifications simplify the discussion in this section.
Unlike the single-process cases presented in Sect.3, this urban plume case included diurnal variations of the meteorological variables (temperature, relative humidity, mixing height and solar zenith angle).Dilution with background air occured during the entire simulation period, and gas and aerosol chemistry as well as coagulation amongst the particles modified the aerosol population further.For reference we show the bulk time series of the aerosol species in Fig. 6.The BC and POA mass concentration increased during the emission phase, and decreased thereafter due to dilution with the background.The time series of the secondary aerosol species sulfate and SOA were determined by the interplay between loss by dilution and photochemical production.The ammonium nitrate mass concentration depended on the gas concentrations of its precursors, HNO 3 and NH 3 .When the two gas precursors were abundant during the emission phase, ammonium nitrate formed rapidly.After emissions and photochemistry ceased, HNO 3 and NH 3 decreased due to dilution, and the ammonium nitrate evaporated.

Evolution of mixing state for urban plume case
To analyze the mixing state evolution for the urban plume case, we graph the same quantities as for the single-process cases.We show two versions of this analysis, first we only include the subpopulation of BC-containing particles in Fig. 7, and then we include the whole population in Fig. 8.
Focusing on the BC-containing particles, Fig. 7a shows that the particle diversity values D i covered a wide range at any point in time during the simulation, and this range changed over the course of the simulation.To explain this, we refer to Table 7, which lists the initial particle diversity values of the different particle types.Particles originating from diesel engine emissions and gasoline engine emissions entered the simulation with D i = 1.8 effective species and D i = 1.7 effective species, respectively.However, coagulation and, more importantly, condensation altered these initial values quickly.Given the particular mix of gas precursor emissions, the number of secondary species in this simulation was 8, and adding the primary species BC and POA, the total number of species is 10, which is the maximum number of effective species for this simulation.Indeed, as shown in Fig. 7a, many particles during the first 12 h of simulation acquired diversity values of up to 9 effective species.At the same time, due to fresh emissions, particles with lower particle diversity values were replenished, which maintained the spread of D i values during the emission period.During the emission phase we also observe BC-containing particles with D i values lower than their initial value of 1.7 or 1.8 effective species.These arose due to coagulation with meat cooking aerosol particles with D i = 1 (see Table 7).After the emission period, and especially on the second day, the majority of particles resided in a narrow range of D i values between 6  and 7 effective species.This is consistent with our notion of "aging", which results in a less diverse population.However it is interesting to note that at all times a range of particles still existed with lower number of effective species.

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N. Riemer and M. West: Quantifying Aerosol Mixing State 3 Fig. 3: Particle diversities D i of representative particles.The particle diversity measures the effective number of species within a particle, so a pure single-species particle has D i = 1 and a particle consisting of 2 or 3 species in even proportion will have D i = 2 or D i = 3, respectively.A particle with unequal amounts of 2 species will have an effective number of species somewhat less than 2, while a particle with unequal amounts of 3 species will have effective species below 3, and possibly even below 2 if the distribution is very unequal.(MacArthur, 1955;Goodman, 1975;McCann, 2000;Ives and Carpenter, 2007).Other important research questions include the sources of diversity (Tsimring et al., 1996;De'ath, 2012), extensions of diversity to include a concept of species distance Chao et al. (2010); Leinster and Cobbold (2012); Feoli (2012); Scheiner (2012), and techniques for measuring diversity (Chao and Shen, 2003;Schmera and Podani, 2013;Gotelli and Chao, 2013), despite the well-known difficulties in estimating entropy in an unbiased fashion (Harris, 1975;Paninski, 2003).Beyond ecology, the study of diversity is also important in economics (Garrison and Paulson, 1973;Hannah and Kay, 1977;Attaran and Zwick, 1989;Malizia and Ke, 1993;Drucker, 2013), immunology (Tsimring et al., ord.-q div.q D i 0 1 2 3 4 5 order parameter q 0 1 2 3 4 5 Fig. 4: Generalized per-particle diversity q D i of order q for varying q, shown for three different particles (inset square plots).Left: a particle with equal amounts of two species.Center: a particle with two species in unequal amounts.Right: a particle with three species in unequal amounts.The order q controls the importance of species with small mass fraction.When q = 0 all species are taken to be equally present, irrespective of mass fraction, so 0 D i is simply the number of species present in the particle.When q = 1 the generalized diversity is equal to the regular diversity defined from the Shannon entropy in Section 2. When q = 2 the generalized diversity 2 D i is the inverse of the Simpson index i = P A a=1 (p a i ) 2 for particle i (Simpson, 1949), often used in the form of the Gini-Simpson index 1 i (Peet, 1974;Jost, 2006).

Entropy, diversity, and mixing state index
We consider a population of N aerosol particles, each con-155 sisting of some amounts of A distinct aerosol species.The mass of species a in particle i is denoted µ a i , for i = 1, . . ., N and a = 1, . . ., A. From this basic description of the aerosol Fig. 10.Generalized per-particle diversity q D i of order q for varying q, shown for three different particles (inset square plots).Left: a particle with equal amounts of two species.Center: a particle with two species in unequal amounts.Right: a particle with three species in unequal amounts.The order q controls the importance of species with small mass fraction.When q = 0 all species are taken to be equally present, irrespective of mass fraction, so 0 D i is simply the number of species present in the particle.When q = 1 the generalized diversity is equal to the regular diversity defined from the Shannon entropy in Sect. 2. When q = 2 the generalized diversity 2 D i is the inverse of the Simpson index λ i = A a=1 (p a i ) 2 for particle i (Simpson, 1949), often used in the form of the Gini-Simpson index 1 − λ i (Peet, 1974;Jost, 2006).
Figure 7b shows the corresponding evolution of D α , D γ , and χ .The average particle diversity D α displayed a rapid increase during the condensation period of the first 6 h of simulation, consistent with the BC-containing particles becoming more complex in composition as condensation and coagulation were at work.After this, D α stayed essentially www.atmos-chem-phys.net/13/11423/2013/Atmos.Chem.Phys., 13, 11423-11439, 2013 constant, with small modulations induced by the diurnal cycle of evaporation and condensation of secondary material.
The importance of condensation is also reflected by the increase in the bulk population diversity D γ .The mixing state index χ decreased from 70 % to 50 % internally mixed during the first two hours because of the effect of meat cooking aerosol emissions; some of these particles coagulated with the BC-containing particles, making the subpopulation of BC-containing particles more heterogeneous.During the period of secondary aerosol formation on the first day (t = 2-11 h), χ increased to 75 % internally mixed, which means that the population of BC-containing particles became more homogeneous during that time.After another day of processing the aerosol in the air parcel, without fresh emissions, the simulation period ended with the aerosol population being 82 % internally mixed.
The same analysis performed for all particles reveals a qualitatively similar pattern for the distributions of D i and the time series of D α , D γ , and χ, with some differences in the details, as shown in Fig. 8.For example, during the first 12 h, when emissions were present, the minimum value of the per-particle diversities was D i = 1 effective species, due to the meat cooking particle emissions (Fig. 8a).The red line at D i = 1.8 in Fig. 8a can be attributed to background particles from which SOA components had evaporated once they entered the air parcel, hence ammonium sulfate particles remained.The values for χ from 2 h of simulation onwards was lower when considering the whole population, compared to χ of the subpopulation of BC-containing particles.This makes sense, since including all particle types allows for a more heterogeneous population.Figure 9 shows the mixing state diagram that corresponds to the urban plume case shown in Fig. 8.At the end of the two-day simulation period the whole population was 75 % internally mixed.

Conclusions
With the advent of sophisticated measurement techniques on a single-particle level, a wealth of information about the composition of an aerosol population has become available.The observations show that, on a particle level, aerosols are complex mixtures of many species, and different particle types can coexist within one population.This reflects the particles' sources as well as their history during the transport in the atmosphere.To describe this distribution of per-particle compositions the term "mixing state" has been coined; however, so far this term has not been rigorously defined and no concept existed to quantify it.
This paper, for the first time, presents a framework for quantifying the mixing state of aerosol particle populations.In developing this framework we borrowed the idea of entropy-derived "diversity parameters" from other disciplines, allowing us to define the per-particle species diversity D i , the average particle diversity D α , the population diversity D γ , and the mixing state index χ.
The average particle diversity D α is a measure for the number of effective species on a per-particle level, while D γ quantifies the number of effective species of the bulk population.An affine ratio of the two, represented by the mixing state index χ, measures how close the population is to an external or internal mixture.
Using particle-resolved simulations to illustrate the evolution of the mixing state metrics for selected test cases revealed the following results.Coagulation always increases the degree of internal mixture.The impact of emission and condensation on mixing state is not as straightforward, and in Sect. 3 we showed examples of scenarios where χ increased, decreased or stayed constant as a result of emissions or condensation.However, in the case of emissions, perhaps the most intuitive scenario is where "fresh" particles (low D i ) are emitted into an "aged" population (high D i ) which decreases the degree of internal mixture and is reflected by decreasing χ.Similarly, in the case of condensation, the most intuitive case is where the same species condenses on all particles of an initially low-χ population, increasing the mixing state index, consistent with our notion of "aging" as a process that increases the degree of internal mixing.
We expect that the mixing state index χ will prove useful in communicating, discussing, and categorizing the aerosol mixing state of both observed and modeled aerosol populations.This, in turn, will facilitate answering the key research questions: (1) what is the mixing state at emission and how does it evolve in the atmosphere; (2) what is the impact of mixing state on climate-related and health-relevant aerosol properties; and (3) to what extent do models need to account for mixing state to answer these questions?In this context, it would be very useful to obtain quantitative information on per-particle composition from field observations and laboratory experiments, together with measurements of application-relevant bulk properties.

Generalized entropy and generalized diversity
The mixing entropy can be generalized to give more or less importance to species with small mass fractions.This generalization was originally due to Havrda and Charvát (1967) and then was independently rediscovered at least three times: in information theory (Daróczy, 1970;Aczél and Daróczy, 1975), in ecology (Patil andTaillie, 1979, 1982), and in physics (Tsallis, 1988(Tsallis, , 2009)).In the physics literature this generalized entropy is frequently called the Tsallis entropy or the HCDT (Havrda-Charvát-Daróczy-Tsallis) (Knuth, 1998, p. 473) of particle mass vectors.A multiset is, roughly speaking, a set where identical elements can appear multiple times, and for which the set difference operator \ and union operator have been appropriately extended.
Theorem 1.The diversities q D i , q D α , and q D γ all lie in the interval [1, A] and q D α ≤ q D γ .Furthermore, for q > 0, the extreme values satisfy: q D i = 1 if and only if particle i is pure (consists of a single species); q D i = A if and only if particle i contains all species in equal mass fractions; q D α = 1 if and only if all particles are pure; q D γ = A if and only if all species are present with equal bulk mass fractions; and q D α = D γ if and only if all particles have identical species mass fractions.
Proof.Here we show the results for the entropies and this implies the corresponding results for the diversities because q f is strictly increasing on its domain.
It follows immediately from the definition that q H i ≥ 0, and similarly for H α and H γ .The domain of allowable mass fractions p a i is the probability simplex defined by p a i ≥ 0 and a p a i = 1, which has normal vector (1, 1, . . ., 1).For q > 0, q H i is a sum of identical functions of each component p a i and so the gradient of q H i is in the direction of the normal vector if all p a i are equal.The KKT conditions are thus satisfied for all mass fractions equal (Boyd and Vandenberghe, 2004, Sect. 5.5.3), and by strong convexity of q H i this is a unique maximum.Evaluating at p A i = 1/A gives the maximum q H i = ln A. For q = 0 the upper bound is immediate.
The condition for q D γ = A follows from the same argument as above applied to the bulk mass fractions p a .The condition for q D α = 1 is due to the fact that q H α is the weighted arithmetic mean of the particle entropies with positive weights, so q D α = 1 if and only if each q H i = 0.
The restriction q H α ≤ q H γ is simply Jensen's inequality (Boyd and Vandenberghe, 2004, Sect. 3.1.8)for the concave entropy function.This can be seen by checking that p a = i p i p a i , and so the weights p i for combining q H i into q H α are the same as for combining the per-particle mass fractions into the bulk mass fractions.For q > 0, strict convexity of the entropy implies that Jensen's inequality is an equality if and only if the per-particle mass fractions are all identical.
Proof.It is sufficient to consider a single coagulation event and then iterate the result, so without loss of generality we assume that 2 = 1 \{µ i , µ j } {µ c }, where µ c = µ i +µ j .We observe that the total mass µ is preserved by coagulation and p c = p i + p j , where the mass fractions are com-puted with respect to the relevant population 1 or 2 .Furthermore, p c p a c = p i p a i + p j p a j so concavity of the entropy gives p c q H c ≥ p i q H i +p j q H j , with equality for q > 0 if and only if p a i = p a j by strict convexity.All other per-particle entropies in the populations are identical, so p c = p i + p j implies q H α ( 2 ) ≥ q H α ( 1 ) with the same condition for equality.
Because the bulk mass fractions are unchanged by coagulation, q H γ ( 2 ) = q H γ ( 1 ) and the inequalities for q H β and χ follow immediately.The fact that q f is strictly increasing transfers all results to diversities.Theorem 3. If populations X and Y are combined to give population Z = X Y then min( q D X α , q D Y α ) ≤ q D Z α ≤ max( q D X α , q D Y α ), min( q D X β , q D Y β ) ≤ q D Z β , min( q D X γ , q D Y γ ) ≤ q D Z γ , and q χ Z ≤ q max(χ X , χ Y ), where superscripts denote the population.
Proof.As q f is strictly increasing it is sufficient to obtain bounds for entropies, which then imply bounds on the corresponding diversities.
Taking the total population mass fraction λ = µ Y /(µ X + µ Y ) ∈ (0, 1), then for particles µ x ∈ X and µ y ∈ Y the mass fractions in the combined population are p Z x = (1 − λ)p X x and p Z y = λp Y y .Thus q H Z α is the convex combination q H Z α = (1−λ) q H X α +λ q H Y α and so q H Z α lies strictly between q H X α and q H Y α .The bulk mass fractions satisfy p Za = (1 − λ)p Xa + λp Y a and so concavity of q H γ gives q H Z γ ≥ (1−λ) q H X γ +λ q H Y γ ≥ min( q H X γ , q H Y γ ).To obtain the lower bound for q H Z β = q f −1 ( q D Z β ), we observe that q H β = q H γ − q H α 1 − (q − 1) q H α .

Fig. 1 .
Fig.1.Particle diversities D i of representative particles.The particle diversity measures the effective number of species within a particle, so a pure single-species particle has D i = 1 and a particle consisting of 2 or 3 species in even proportion will have D i = 2 or D i = 3, respectively.A particle with unequal amounts of 2 species will have an effective number of species somewhat less than 2, while a particle with unequal amounts of 3 species will have effective species below 3, and possibly even below 2 if the distribution is very unequal.

N
Fig. 24: Diversity and mixing state evolution for BCcontaining particles in the urban plume case.(a) Distribution of per-particle diversity D i as a function of time.(b) Time series of average particle diversity D ↵ , population diversity D , and the mixing state index .

Fig. 25 :
Fig. 25: Diversity and mixing state evolution for all particles in the urban plume case.(a) Distribution of per-particle diversity D i as a function of time.(b) Time series of average particle diversity D ↵ , population diversity D , and the mixing state index .

Fig. 26 :
Fig. 26: Mixing state diagram to illustrate the relationship between per-particle diversity D ↵ , bulk diversity D , and mixing state index for representative aerosol populations, as listed in Table??. See Section 2 and Table?? for more details.

Fig. 3 .
Fig. 3. Diversity and mixing state evolution for archetypal cases.Left column: Distributions of per-particle diversity D i as a function of time.Right column: time series of average particle diversity D α , population diversity D γ , and the mixing state index χ .Note that the left axis shows D α and D γ , and the right axis shows χ .The rows correspond to Cases 1 to 4 as defined in Table6.

Fig. 4 .
Fig. 4. Diversity and mixing state evolution for archetypal cases.Left column: distributions of per-particle diversity D i as a function of time.Right column: time series of average particle diversity D α , population diversity D γ , and the mixing state index χ .Note that the left axis shows D α and D γ , and the right axis shows χ .The rows correspond to Cases 5 to 8 as defined in Table6.

Fig. 5 .
Fig. 5. Mixing state diagram showing the normalized population species diversity D γ versus the normalized average particle species diversity D α for all single-process cases.The number labels refer to the cases defined in Table6and shown in Figs.3 and 4.

Fig. 6 .
Fig.6.Evolution of bulk aerosol species for the urban plume case.

Fig. 7 .Fig. 8 .
Fig. 7. Diversity and mixing state evolution for BC-containing particles in the urban plume case.(a) Distribution of per-particle diversity D i as a function of time.(b) Time series of average particle diversity D α , population diversity D γ , and the mixing state index χ .

Fig. 9 .
Fig. 9. Mixing state diagram for urban plume case showing the population species diversity D γ versus the average particle species diversity D α for the urban plume case presented in Fig. 8 (all particles included).

Table 1 .
Aerosol mass and mass fraction definitions and notations.The number of particles in the population is N, and the number of species is A.

Table 3 .
Conditions under which the maximum and minimum diversity values are reached.See Fig.2for a graphical representation of this information, and see Theorem 1 for precise statements.

Table 4 .
Representative particle populations shown on Fig.2, with the average per-particle diversity D α , the bulk population diversity D γ , and the mixing state index χ listed for each population.

Table 5 .
Change in population diversities and mixing state index due to change in the particle population.For population combinations the superscript indicates the population for which a quantity is evaluated.For coagulation, D α , D β , and χ stay constant when all particles have identical mass fractions.See Theorems 2 and 3 for precise statements.Quantity Change due to coagulation Combination of populations X and Y

Table 6 .
List of single-process case studies.Column "Chem."indicates if gas and aerosol phase chemistry were simulated, and column "Coag."indicates if coagulation was simulated.
Table ??.See Section 2 and Table ?? for more details.Mixing state diagram to illustrate the relationship between per-particle diversity D α , bulk diversity D γ , and mixing state index χ for representative aerosol populations, as listed in Table 4. See Section 2 and Table 3 for more details.

Table 7 .
Initial, background, and emission aerosol populations for the urban plume case, giving the number concentration N a or area rate of emission E a as appropriate.The aerosol population size distributions are log-normal and defined by the geometric mean diameter D g and the geometric standard deviation σ g .