Stochastic differential equations and their corresponding Fokker-Planck equations 

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The phases of the ISM

Observations show that the ISM presents a wide variety of local properties, which are usually grouped in five phases based mostly on the ionization state of the gas and on its temperature (Ferrière 2001). The physical conditions of each phase are given in Table 2. From the densest to the most tenuous, those phases are:
• The molecular gas: neutral gas in which hydrogen is mainly in the form of dihydrogen molecules (H2). It usually lies in the inner part of clouds. This phase is actually rather a sub-phase of the cold neutral medium (next in the list) with roughly the same order of temperatures and higher densities, but is usually included in the five-phases description of the ISM. Most of the work of this thesis concerns this phase of the ISM. Dense molecular cores are often subject to self-gravity, contrary to the other phases.
• The cold neutral medium (CNM): dense and cold neutral gas. Interstellar clouds are regions where the ISM is in this phase (or in the molecular phase). If the molecular gas is distinguished, then this phase refers to the cold atomic medium only. It is usually less dense than the molec-ular gas, as a dense gas quickly absorbs the dissociative UV photons and becomes molecular. When molecular gas is present, it is always surrounded by a layer of atomic medium. But clouds can be entirely atomic if they are not large or dense enough to protect their core from UV photons, in which case they are usually called diffuse clouds.
• The warm neutral medium (WNM): neutral gas is also found in a state of much lower density with high temperature, in rough pressure equilib-rium with the CNM. The origin of this coexistence of two phases in the same pressure range and conditions is explained below. This phase usu-ally surrounds an ensemble of clouds, forming giant cloud complexes.
• The warm ionized medium (WIM): the gas surrounding stars is usually ionized by the stellar radiation, forming a roughly spherical region of ionized gas called an HI I region. But these localized structures are not the only ionized parts of the ISM. It was more recently recognized that the UV radiations of bright O stars of the galaxy somehow manage to escape their immediate vicinity and collectively create extremely ex-tended regions of ionized gas (with physical conditions similar to the WNM), mainly located in the halo of the galaxy. For a very detailed discussion of this phase, see the review by Haffner et al. [2009].
• The hot inter-cloud medium (HIM): Finally, supernova explosions create bubbles (or super-bubbles when several supernovae successively ex-plode in the same region) of extremely low density and extremely high temperatures. These expanding bubbles often connect to each other and to the galactic halo, creating tunnels. It is probably this network of al-most empty tunnels which allows the UV radiation of O stars to travel far across the galaxy and out of the disk, and to ionize large portions of gas, creating the WIM phase.
As we are more interested in the dense parts of the ISM, we now discuss in more details the neutral phases (CNM and WNM). We first discuss the mechanism that causes the coexistence of two different phases in the same pressure conditions, before giving a finer classification for the different types of CNM regions.
The cold atomic gas of the ISM has been shown to be thermally unstable. For a given gas density, the temperature corresponding to thermal balance be-tween the heating terms (the photoelectric effect largely dominates the heat-ing in the neutral medium) and the cooling terms (radiative cooling mostly caused by C+ and O) can be computed, and the corresponding pressure de-rived. The result is an equilibrium curve as shown on Fig. 3 (taken from a recent version of this analysis: Wolfire et al. 2003). The parts where pressure increases with density are stable (the gas “resists” when it is compressed by external forces), while the region where pressure decreases when density in-creases is unstable and tends to collapse to the high density stable branch. As the average density of the neutral medium (slightly higher than 1 cm 3 at the radius where the Sun is located) falls into the unstable regime.

Dust emission

Grains are heated by the light they absorb. They also radiate energy away by emitting photons, and this emission carries precious information about the dust population. Big grains ( 100 nm) emit thermally in the far-infrared with a modified black-body law, and their temperature can be directly de-duced from this emission. For very small grains (. 10 nm), the picture is more complex. As small grains have small heat capacities, they are sensitive to the energy of the individual UV photons, which cause spikes in the grain temperature. Small grains thus have constantly fluctuating temperatures, and their emission is dominated by the transient high-temperature phases, result-ing in near-infrared emission. This effect is discussed and modeled in details
in Chap. 5 and 6. This emission in the near- and mid-infrared is mainly constituted of emission bands. The smallest grains are thought to be large molecules of the PAH family (see Tielens 2008 for a review), and the emission bands correspond to C C stretching (6.2 mm and 7.7 mm), C H stretching (3.3 mm) and bending (8.6 and 11.3 mm) modes in these molecules. A typi-cal dust emission spectrum for the diffuse intergalactic medium is shown on Fig. 10.
The mechanism of stochastic heating of very small grains and PAHs that gives rise to the near- and mid-infrared spectrum is discussed and modeled in detail in Draine and Li [2001]. Li [2004] also gives a discussion of the transient excitation and re-emission of large molecules such as PAHs following the absorption of an UV photon.
Dust is also observed to emit in the microwave domain (see Lazarian and Finkbeiner [2003] for a review), in excess to the predictions from thermal (vibrational) emission. This could be due to rotational emission from very small asymmetrical grains, or to magnetic dipole emission from magnetic grains. Recent works favor the hypothesis of rotating very small grains (e.g. Ysard et al. 2010).

Elemental depletions and abundance constraints

Dust models trying to fit the observational constraints discussed above will make hypotheses about the composition of dust grains. As a result, they require some amounts of the heavy elements (C, Si, Mg, Fe, etc.) to be locked up in the dust grains. For a model to be realistic, these amounts must be available in the ISM.
The elemental abundances in the ISM are not directly measurable, and we have to extrapolate from solar and stellar abundances. Once some hypoth-esis is made about the ISM abundances, the gas phase abundances can be measured, and the difference constitutes the available material for dust.
The interstellar abundances are usually inferred from either the solar (Hol-weger 2001) or stellar abundances for either B stars or F and G stars (Sofia and Meyer 2001). Observations suggest that the interstellar abundances can-not be substantially lower than the solar or F and G stellar abundances. As discussed by Draine [2004a], possible processes of segregation between gas and dust during star formation probably induce differences between ISM and stellar abundances, possibly resulting in lower abundances of grain con-stituents in stars.
The gas phase abundances are reviewed by Jenkins [2009] over a large number of lines of sight. The observed depletion of oxygen in the gas phase seems to be problematic, as the missing quantity is much more than what is accounted for by current dust models (see Whittet [2010]). This could suggest that an additional oxygen-rich dust component is present.

dust models

As the observations described in the previous section result from the com-bined effects of all grain types and sizes, with the dust possibly experiencing a variety of environments along the line of sight, neither the composition and size distribution of dust grains nor the number of distinct dust components can be directly deduced from the data. Nevertheless, the wealth of observa-tional evidence gives tight constraints on hypothetical dust models that can be proposed.
A dust model consists in specifying the different dust components (types of grains), their composition, their size-dependent optical and thermal prop-erties (absorption coefficient, albedo, scattering g parameter, heat capacity, etc.), and a size distribution for each component.
Several aspects of the observations hint at possible compositions and dust components, so that models are not built on pure speculations. Absorption features are often signatures of specific families of materials. For instance, the 9.7 and 18 mm features are characteristic of silicates. Moreover, the lack of structure in those absorption features mostly rules out crystalline silicates. In a similar way, the near- and mid-infrared emission features are quite strongly similar to PAH spectra that are theoretically predicted or measured in the lab-oratory. In addition, theories of dust formation and processing can provide hypothetical compositions.
As a result, most dust models agree on broad categories of grains (see for instance the review by Draine 2003c):
• a silicate family, mostly amorphous, which might be of composition close to olivine MgFeSiO4 (Sofia and Meyer 2001).
• a carbonaceous family, with an appreciable fraction of aromatic car-bon (from the 2175 Å absorption bump), and some aliphatic carbon (to explain the 3.4 mm extinction feature). This might be graphite, hydro-genated amorphous carbon, PAHs and PAH clusters.
The precise composition and number of distinct components depend on the specific grain model, but those two categories most likely constitute two dis-tinct components (as silicate features are polarized, while carbon features seem not to be). Dust models then have to be compared to the observational constraints as discussed in Draine 2004a.

The Mathis, Rumpl, and Nordsieck 1977 (MRN) model

Mathis et al. [1977] gave one of the first attempt to solve the inversion problem of deducing the dust size distribution from the observations. They considered only the extinction curve and the abundance constraints, and tried to find the best fitting size distributions for various combination of components (with-out using a prescribed functional form for the size distributions). They find that the choice of the components is not settled by the comparison: in addi-tion to the fact that only a limited number of hypotheses were tested, several combinations gave a similar quality of fit. All involved graphite (which was the only carbon component tested) supplemented by silicates (olivine and en-statite gave similar results). For reasons that have been discussed much later by Zubko et al. [2004] (and inherent to the mathematical inversion problem), the resulting size distributions are discontinuous (with an alternation of peak bins and zero bins). However, once those discontinuities are smoothed out , a clear power-law shape emerges for all components with exponent a ’ 3.5, extending between a = 5 nm and 250 nm for the graphite component, and between 25 and 250 nm for the silicate component (olivine and enstatite give similar results). The authors note that the lower limit of the graphite distri-bution is not well constrained as for the range of wavelength used (0.11 to 1 mm), 5 nm grains are in the Rayleigh regime which is not sensitive to size. The biggest grains also approach the grey extinction regime, but abundance constraints limit the maximum size.

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The Draine model

One of the currently most used models has been built and refined by Bruce Draine and collaborators in a series of papers (Weingartner and Draine 2001a, Draine and Li 2001, Li and Draine 2001, 2002, Draine and Li 2007). This model follows the MRN model by taking a carbonaceous (graphite) compo-nent and a (amorphous) silicate component, but extends the carbon com-ponent to down to molecular sizes by using a continuous transition to PAH properties for small sizes (below 5 nm) in order to reproduce the near-infrared emission caused by the stochastic heating of very small grains (this mecha-nism is modeled in Draine and Li [2001]). Noting that the size distribution of the very small grains necessary for the emission has little effect on the extinction, they separately adjust the very small grain part and the bigger grains. The size distribution of the large graphite and silicate components are fitted to the extinction curves (using the parametrization of Fitzpatrick 1999 as a function of RV ) and to the abundance constraints simultaneously in Weingartner and Draine [2001a] by assuming a given functional form (modi-fied power laws with additional curvature and exponential cut-off), while the size distribution of very small carbonaceous grains is composed of two man-ually adjusted lognormal distributions that fit the emission spectrum (Li and Draine 2001). This model is revisited in Draine and Li [2007], with updated PAH properties and the new emission data provided by Spitzer. The resulting distribution is described in Chap. 5 (Sect. 5.3.3). This model fits most of the observational constraints very well, but requires too much heavy elements (especially twice the solar abundance of Fe)

The Zubko et al. 2004 models

Zubko et al. [2004] took an in depth look at the inversion problem consisting in deducing the size distribution directly from the observational constraints (given a set of predefined dust components with given properties). They show that this problem leads to a Fredholm integral equation of the first kind, an ill-posed problem. However, with the addition of a few assumptions on the smoothness of the size distribution, a regularization method can be used to find the solution.
The problem is solved under the simultaneous constraints of the extinc-tion curve, the infrared emission spectrum, and the heavy element abun-dances (using the average observations of the diffuse ISM). As possible dust components, they consider: PAHs, graphite, amorphous carbon, silicates (MgFeSiO4) and composite grains made of a mix of silicates, organic refrac-tory material, water ice and voids. They apply their solution method to different combinations of these components, and show that good fits can be obtained for all of the proposed dust models:
• PAH – graphite – silicate, same components as the Draine model, for which a better fit is obtained, which verifies the abundance constraints.
• PAH – amorphous carbon – silicates, amorphous carbon being a more realistic interstellar material as graphite is unlikely to form and survive in the ISM.
• PAH – graphite – silicate – composites.
• PAH – amorphous carbon – silicates – composites.
• PAH – silicates – composites for a low carbon model in the hypothesis of the ISM abundances being the B stars abundances.
Their results seem to favor slightly the models with composite grains, which are also found to match the albedo and g constraints (which were not in-cluded in the resolution procedure). But Dwek et al. [2004] further compared the models to small-angle X-ray scattering data and found them to favor mod-els without composites.

The Compiègne et al. 2011 model

Compiègne et al. [2011] proposed a model matching the extinction and emis-sion properties of the diffuse ISM at high galactic latitudes. The model is quite similar to the Draine model, but graphite has been replaced by amor-phous carbon. As said before, amorphous carbon is a more likely material in the ISM harsh conditions. Moreover, shocks observations show that carbon grains are almost completely destroyed (see for instance Slavin 2009, Welty et al. 2002), a fact that favors amorphous carbon as it is more easily destroyed in shocks (see Serra Díaz-Cano and Jones 2008).
As a result, a larger fraction of PAHs than in the Draine model (7.7% in mass versus 4.6%) is required to explain the 2175 Å absorption bump. The size distributions of the different components are described in Chap. 5 (Sect. 5.3.3).

The Jones et al. 2013 model

Jones et al. [2013] propose a model based on the carefully derived (Jones 2012a,b,c) size-dependent properties of hydrogenated amorphous carbon grains (HAC), including the effect of photo-processing converting H-rich aliphatic carbon into H-poor aromatic carbon up to a depth of 20 nm. As a result a sin-gle carbonaceous population can exhibit PAH-like feature due to very small mostly aromatic grains, and aliphatic features from the big grains. They add a silicate component coated with a 5 nm layer of aromatic carbon.
A standard model is adjusted to the observations, and the complete set of data makes it possible to study the evolution of the observable properties as photo-processing, accretion and coagulation occur. It is thus the first model to take into account an hypothetical formation and evolution scenario. However, the HAC optical properties had to be adjusted to the observa-tions, in particular to reduce the strength of the 2175 Å absorption bump.

Table of contents :

i introduction 
1 the interstellar medium 
1.1 The life cycle of galactic matter
1.2 Constituents and organization
1.2.1 Constituents
1.2.2 The phases of the ISM
1.2.3 The ISM structure
1.3 Molecules in the ISM
1.3.1 Molecular richness
1.3.2 Observational diagnostics
2 dust and surface chemistry 
2.1 Observational evidences
2.1.1 Extinction of light
2.1.2 Scattering
2.1.3 Polarization
2.1.4 Dust emission
2.1.5 Elemental depletions and abundance constraints
2.2 Dust models
2.2.1 The Mathis, Rumpl, and Nordsieck 1977 (MRN) model
2.2.2 The Draine model
2.2.3 The Zubko et al. 2004 models
2.2.4 The Compiègne et al. 2011 model
2.2.5 The Jones et al. 2013 model
2.3 Physical processes
2.3.1 Energy balance
2.3.2 Charge balance
2.4 Surface chemistry
2.4.1 H2 formation
2.4.2 Other reactions and ice formation
2.5 Formation and evolution
2.5.1 Formation processes
2.5.2 Processing and destruction
3 turbulence 
3.1 Turbulence
3.1.1 Definition
3.1.2 Kolmogorov theory and scalings
3.1.3 Intermittency
3.1.4 Compressible turbulence and MHD turbulence
3.2 Interstellar turbulence
3.2.1 Observational evidences
3.2.2 Energy sources
3.2.3 Roles
3.3 Turbulent chemistry
3.3.1 CH+, SH+ and H2 in the diffuse ISM
3.3.2 Modeling the interstellar turbulent chemistry
4 stochastic processes 
4.1 General definitions
4.1.1 Stochastic processes
4.1.2 The several meanings of “continuous”
4.1.3 Markov processes
4.2 The Chapman-Kolmogorov equation and its differential forms
4.2.1 The Chapman-Kolmogorov equation
4.2.2 General differential form of the Chapman-Kolmogorov equation
4.2.3 The Liouville equation
4.2.4 The Fokker-Planck equation
4.2.5 The Master equation
4.2.6 Important examples
4.3 Stochastic differential equations
4.3.1 Langevin equations and definition problems
4.3.2 It ¯o integral and It ¯o stochastic differential equations
4.3.3 It ¯ o’s lemma for variable changes
4.3.4 Stochastic differential equations and their corresponding Fokker-Planck equations
ii dust : temperature fluctuations and surface chemistry
5 a simple dust model 
5.1 Dust-radiation interactions
5.1.1 Heat capacity
5.1.2 Radiative processes for a spherical dust grain
5.1.3 External radiation field
5.1.4 Equilibrium temperature
5.1.5 PAHs
5.2 H2 formation on dust grains
5.2.1 Binding sites
5.2.2 Surface processes
5.2.3 Langmuir-Hinshelwood mechanism
5.2.4 Eley-Rideal mechanism
5.3 Dust components and size distribution
5.3.1 A simplistic model
5.3.2 The dust model of the Meudon PDR Code
5.3.3 Comparison with more complete dust models
6 dust temperature fluctuations and dust emissivity 
6.1 A Master Equation approach
6.1.1 Master Equation
6.1.2 Properties of the eigenvalues
6.1.3 Resolution method
6.2 Numerical resolution
6.2.1 Discretization of the problem
6.2.2 Numerical tests
6.2.3 Results
6.3 Limit cases
6.3.1 Kramers-Moyal expansion in the big grain limit
6.3.2 Continuous cooling approximation
6.4 Dust emissivity
6.4.1 Computation of the dust emissivity
6.4.2 Comparison with DustEM
6.5 A fast treatment of dust emissivity
6.5.1 Single photon approximation for small grains
6.5.2 A complete emissivity approximation
6.5.3 Implementation in the Meudon PDR Code
7 chemistry on fluctuating dust grains: the case of h2 
7.1 A Master Equation approach
7.1.1 General Master Equation
7.1.2 Langmuir-Hinshelwood: general resolution method
7.1.3 Eley-Rideal: a simplified equation
7.2 Numerical resolution
7.2.1 Langmuir-Hinshelwood
7.2.2 Eley-Rideal
7.2.3 Numerical tests
7.3 Approximations of the formation rate
7.3.1 Langmuir-Hinshelwood
7.3.2 Eley-Rideal
7.3.3 Accuracy assessment
8 results : bron et al. 2014 
8.1 The article
8.2 Conclusions and perspectives
iii turbulence-driven chemistry in the diffuse ism 
9 a stochastic model for turbulent chemistry 
9.1 Moment approaches and turbulent chemistry
9.1.1 The moment approaches
9.1.2 Strengths and limitations
9.2 A PDF method for turbulence-driven chemistry
9.2.1 The one-point PDF transport equation
9.2.2 Closed and unclosed terms
9.2.3 A stochastic Lagrangian model of turbulent dissipation
9.3 Application to homogeneous turbulence in the interstellar medium
9.3.1 Marginal PDF in the chemical composition-dissipation space only
9.3.2 Simplification for homogeneous turbulence
9.4 A Lagrangian Monte Carlo method
10 application to the excitation of h2 in the interstellar medium 
10.1 Simulation and test of the stochastic diffusion process
10.1.1 Physics and algorithm
10.1.2 Numerical test
10.2 Diffuse atomic ISM: temperature distribution
10.2.1 The model
10.2.2 Net cooling function
10.2.3 Gas heat capacity
10.2.4 Results
10.3 Diffuse molecular gas: H2 excitation
10.3.1 The model
10.3.2 Results
10.4 Conclusions and perspectives
iv simulations of the ngc7023 north-west pdr 
11 models of the ngc7023 north-west pdr 
11.1 The object
11.1.1 Characteristics
11.1.2 Previous works
11.1.3 Constraining the model from the literature
11.2 The observations
11.3 The models
11.3.1 Constant density vs. constant pressures
11.3.2 Best model
11.3.3 Additional aspects
v conclusions and perspectives 
vi appendix
a tools and notations in probability 
a.1 Single-variable case
a.1.1 Random variables and probabilities
a.1.2 Cumulative distribution functions and probability distribution functions
a.1.3 Mean and moments
a.2 Multi-variable case
a.2.1 Independence
a.2.2 Probability density function and moments
a.2.3 Marginal PDF
a.2.4 Conditional PDF and moments
a.3 Stochastic processes and fields
a.3.1 Definitions
a.3.2 Probability density functions
a.3.3 Autocorrelation function
b interstellar cloud models: the meudon pdr code 
b.1 Overview
b.1.1 Principle and hypotheses
b.1.2 Model parameters
b.1.3 Code structure
b.2 Physico-chemistry
b.2.1 Radiative transfer and excitation of species
b.2.2 Chemistry
b.2.3 Thermal balance
b.2.4 Dust physics
c the fredholm code 
c.1 Description of the Fredholm code
c.1.1 Code structure
c.1.2 Configuration and use
c.2 Coupling between Fredholm and the Meudon PDR Code
c.2.1 Implementation
c.2.2 Configuration and use


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