domain: phys math sde

text:
Alain Mailliat

Finely Intricate Random Multiphase Systems 0 A MODEL FOR FINELY, INTRICATE, RANDOM MULTIPHASE SYSTEMS

Keywords:

A multiphase system is said fine for a given scale of observation if its phases are scattered in a large number of individual space domains within a unit of volume corresponding to the observation scale. In the same way, the system is said fine with regard to time if, at a given point, phases alternate very often during the time unit corresponding to the observation time scale. In addition, when time and space distributions of the phases does not present repetitive or periodic patterns, the system is said a finely intricate one. Finally, if intricacy results of the random distributions both in time and space of the phase domains, such systems are defined as Finely Intricate Random Multiphase (FIRM) systems. Canopies are simple and common ordinary FIRM multiphase systems. In forests, the air flows through canopies where the leaves are a second phase. An important practical question is the airborne particle deposition rates by time and volume units (let us say a few minutes and a cubic meter) on several hundred leaves which are ruffled by coupling with the turbulent air flow. Let us assume that all the le local instantaneous quantities required for describing a FIRM system are available. Let us assume that they obey the basic law of physics, and then both balance equations for each phase and transfer equations for their exchanges are available. Assuming we are able to solve such a set of equations, we obtain the behaviour of the system. But, the solutions will be intractable and useless because the amount of data required to describe the local instantaneous values of these quantities will be so huge that it will prevents any one from deriving understandable information directly from them. Then, how to proceed? First, we retain the idea to write down the set of basic balance and transfer equations for the local and instantaneous quantities. Two, instead to handle directly this set, we modify it using weighting operators in such a way to obtain new balance equations for weighted quantities. The reason for such a procedure is the following. Local equations are defined only where and when a phase is present and we know that such space time domains can be highly variable for FIRM systems. On the contrary, balance equations for the weighted quantities are defined at any point and instant. They provide a way to describe the behavior of these systems. Here after, starting from the balance equations of the local and instantaneous quantities, the FIRM model development provides balance equations for global quantities which are valid everywhere at the scale of interest for applications. A following paper will provide applications of this model to forest canopy and sea.

FIGURES

INTRODUCTION

Air quality studies, radioactive hazards and, more generally, atmosphere dispersal of polluting airborne particles, their transfer to the environment and the associated consequence for heath require determining particle deposits. So to say, it is necessary to estimate the amount of airborne particles extracted from the atmosphere boundary layer which settle in the environment, this one being regarded as a set of surfaces which collect the particles called here after environmental surfaces (ES).

On one hand, ES can have rather simple geometries as water surfaces of lakes, pounds, grounds with grass or small size vegetables. For such ES aerosol deposit estimates are rather correctly estimated from usual models. Geometries can be more complex in cases of large forests or surfaces found in urban areas. These situations are more difficult to take into account when looking for deposit estimates. Nevertheless they are still "rather" tractable because only one kind of ground cover has to be considered, even if it is made of complex surface geometries. Many studies have been devoted to these two situations and we will not deal with them here after.

On the other hand, a lot of more complex environmental situations exist for which not enough attention has been brought in terms of model development. For such situations we find rather confused space distributions of the phases, sometime random distributions of surfaces with various geometrical characteristics. For example, soil where several ground covers alternate; wooded surfaces and others ones with small size vegetables and/or urban areas with buildings, streets, small houses. In some others situations the complexity is not related to the space distributions but to time evolution of the surface distributions. For illustration, let us consider aerosol deposits in case of atmosphere and ocean exchanges during strong wind conditions. We find rather "smooth" liquid surfaces between wave crests. But, we have also "rough" complex surfaces at the wave crests or when waves break producing water lumps and droplets which stay in the atmosphere for some instants. In addition, such smooth and rough area distributions occur again and again.

These are the environmental conditions we intend to explore here after. We suggest calling such systems, Finely Intricate Random Multiphase (FIRM) systems according to their main characteristics.

To summarize the development here after, a model for FIRM system model is developed through four steps.

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Finely Intricate Random Multiphase Systems 2

• Firstly, we use some math to describe the complex phase domains distributions both in space and time which are specific of a FIRM system and their local instantaneous non-continuous stochastic functions providing its physical properties. • Secondly, global continuous quantities, at the scale of interest for practical applications are derived from local instantaneous non-continuous stochastic functions. They are deduced from weighting operations both with respect to time and space and, in case of stochastic systems, with the addition of a probability averaging. • Thirdly, time and space derivatives and differential operators of such global quantities are obtained. These operations produce additional fluxes resulting of weighting and averaging of local exchanges between phases through their interfaces.

• Finally, we write down balance equations for global quantities with the benefit for applications to be applicable everywhere at the scale of interest whereas the balance equations of the local and instantaneous quantities are valid only in their various phase domains.

But, before entering in the "maths" parts of this paper, we need to define the meaning of each of the words Finely, Intricate, Random and Multiphase in our context.

FINELY, INTRICATE, RANDOM, MULTIPHASE

Now, we provide the specific meaning of the previous words in the context of FIRM systems. We start by the last one.

MULTIPHASE

First, what the word phase means? It is more or less obvious to define a phase according to an environmental situation.

figure 1 : Air ocean system

Considering the system of the atmosphere and the ocean. We have clearly two phases, one is the part of the space where the liquid is found, and this is the liquid phase. The other one is the place where we find air, it is the gas phase.

There are a lot of such obvious two phase systems. Collecting a full bucket on the seashore we have a water phase in which the sand particles form the solid phase. In this case the solid phase is a sum of small space domains of sand material.

Environmental situations can be less obvious. Two phases can be made of the same material, so to say air but with different conditions such as temperature, turbulence intensity or particle concentration. Considering airborne particles flowing through an olive trees plantation as we see on the figure against, we can define three phases. The first one is the solid phase corresponding to the space domains where of the vegetables elements are leaves, branches and trunks. The second one is the part of the air flow which goes between the tree rows without encounter any vegetables elements. We call it the smooth air phase. Then, the last phase is the part of the air flow which goes inside the canopies and the vertical space above them. We call it the air rough phase. The raisons for this choice are the following. The turbulent characteristics of the flow in the smooth and rough phases are quite different due to dynamic interactions of the wind and leaves within the canopies. In addition, the particle concentrations are different in the smooth and rough phases because the airborne particles are collected on the vegetable surfaces. In this case the three phase localisations are rather permanent, but situations can be more complex when phases space distribution changes with time. The figure below sketches ocean with strong wind conditions. The liquid phase is made of water volumes and spray drops of various sizes which are thrown up and stay a few seconds in the air where they form water phase domains before falling down back to the ocean. We have also two phases above the ocean. The first is the rough phase above the breaking wave crests where water phase elements are included. The second one is the smooth phase above wave troughs. Having some practical examples of what has to be understood by the word phase, now, we can give a more formal definition.

Phases are defined by space domains and, for each point of theses domains, various physical quantities like density, velocity internal energy etc. from which they can identified. These quantities are called local instantaneous quantities. They vary according to both the phase domain movements and the different physical and chemical processes which modify their properties.

FINELY

In our multiphase system context, finely word is related with the concept of plurality. Before any formal description of the associated idea, a practical understanding of finely is given through two examples.

Let us imagine we collected a shovelful of sand from a humid beach shore and filled a 10 centimetres side cube. This length is called our observation scale. Inside this cube we have a liquid phase -the water-and a solid phase made of several hundreds of fine sand particles distributed in the water. This is a finely distributed two phase system. It has to be well underlined that the idea of finely is associated with both a specific system and its observation scale. If the previous cube side is reduced to a few millimetres instead of the previous 10 centimetres, our second cube with this new size includes just one or two sand particles (or may be, not only one) and cannot be regarded as a finely distributed system.

As a second example, we look at a forest canopy. Such vegetable systems can be considered as a two phase systems where air is one of the phases and all the leaves, branches and trunks the other one. Then, for a scale of observation of a few meters, so to say 2 meters, a cube of 8 cubic meters includes thousands of leaves of various sides. But, once more, if the scale of observation is reduced, the system might not be considered as finely distributed one. A 10 centimetres scale of observations conducts to a very few number of leaves, if not, in the cube. In a theoretical way, a multiphase system is said finely distributed in space for a given scale of observation L if, at a given time, its phases are divided in a very large number of individual space domains within a unit of volume corresponding to 3 L , the observation scale. This can be express in another way. Let us consider a given time and a point. The system is said finely distributed in a given space direction if when shifting this point over a length L equal to the scale of observation through the system, it goes through one phase to another one a very large number of times.

A system can be also finely distributed with regard to time. In this case, for a given time scale of observation T and at a given point in space the phase found at this location changes very often for the duration of the observation T . This is can be also applied to a specific physical property associated to a given point of the phase if the property changes often.

For the forest example, when wind blows through the canopy, branches and leaves are put in motion and a given location inside the canopy is swept by the vegetables elements. If the wind is such that we observe at this point a frequent change of the phases for the duration T , the system is said finely time distributed.

For this forest example, the two phase system is finely distributed both in space and time as compared with the beach example where the water-sand system is only finely distributed in space.

INTRICATE.

This word is related with the notion of complexity. Let us imagine a volume in the multiphase system based on the scale of observation. If the phase arrangement is so complex that it cannot be formally given or even described by a repetitive pattern in the considered volume, the multiphase system is said intricate. Consider a volume inside a canopy forest. There are thousands leaves of several shapes, different sizes and with various orientations. Therefore, asking a person who looks at this volume to provide a description in such a precise way that someone else would be able to reproduce exactly the observation is probably impossible.

figure 6: Leaves intricateness

For a second example we consider the ocean atmosphere system. When the wind is strong enough, the air-water interface is rather complex and change continuously with time. In addition, water volumes and drops of various sizes are thrown up and stay a few seconds in the air where they form new water phase domains before falling down back to the sea.

figure 7 Water spray intricateness

Therefore, the phase distribution for this sea-water system has an intricate structure due its space complexity.

The notion of an intricate system can be also related to time. Consider a given point of a volume based on the scale of observation inside a multiphase system. During an observation time T there is a series of time periods for which one phase is observed at the given point or a phase property found inside a given interval of values. If this series of periods cannot be easily described nor exhibit specific patterns, as, for example, a periodic repetition, the multiphase system is defined as intricate in time. This is obviously the case of this windy air-sea system we just considered above.

It has to be underlined that intricate does not means random. Consider an intricate physical system produce in a laboratory. Performing the experiment several times, the intricate distribution of the phase domains is identically reproduced from one experiment to the next one as if the same movie was replayed several times.

RANDOM figure 8 Random phase ditribution

Consider a lottery glass sphere of one meter diameter with its 49 five centimeter diameter balls. We regard the balls as a solid phase distributed in the air of the sphere domain. Using the sphere diameter as the observation scale, clearly, this system is not finely distributed because there are only 49 elements inside the observation volume. It is neither an intricate one because a description of any phase distributions can be exactly given through the list of the 49 positions of the ball centers. This two phase system is rather simple, neither intricate nor finely distributed. Then, let us run the lottery sphere and take a picture of the ball distribution after exactly 20 seconds. This is what we call an experiment or an occurrence or a realization.

When the experiment is reproduced several times, the pictures provide solid phase distributions in the observation volume at a given time 20 seconds through the precise positions of each ball. But, all these distributions are different from one picture to another one and it is observed that the space distributions of the spheres -our solid phase locationare not predictable from available information. Such a system is called a random system. Each picture provides an observation of one possible state of the system. Each possible state of a random system is call one occurrence, or one realization, of its possible states. According to the system, it can present a finite number of states or an uncountable number, or even a continuous number of possible states.

Obviously, a random system can have also finely and intricate phase characteristics.

REPRESENTATION OF FIRM SYSTEMS

In the previous chapters we illustrated some examples of FIRM systems and underlined their main aspects. Now, we intend to use some math to describe the phase domains in space and time, their local and instantaneous physical properties through random continuous by pieces functions and the ways to reduce them to averaged or weighted quantities.

REPRESENTATIONS OF DETERMINISTIC SYSTEMS

PHASE DISTRIBUTION

Let us assume that we have a large space domain Ω where a system of N phases exists during a duration D . Each phase, -e.g. water, air, vegetables-is identified by an index k .

Inside a unit of space volume based on a given observation scale several phases are distributed in a finely intricate pattern. Generally, a phase consists of a large number of separated subdomains, cf.

( ) ( ) { } [ [ 3 0 0 k k M ,t : f M ,t , Ω = > ⊂ × ∞   
.

A section of k Ω for a given time t -or for a given location M  -provides the space distribution or -the time distribution-of the phase k : Finely Intricate Random Multiphase Systems 10 At a given time t the boundary surface between a phase and the other part of the system which is called the phase interface

( ) ( ) { } 3 0 k k t M : f M ,t Ω = > ⊂    or ( ) ( ) { } [ [ 0 0 k k M t : f M ,t , Ω = > ⊂ ∞  
( ) k t Ω ∂
is the union of the disjointed phase domain surfaces inside Ω :

(2) ( ) ( ) { } 0 k k t M : f M ,t Ω ∂ = =  
.

Regarding interphase surfaces, we ignore their points where three phases coexist and we assume that such circumstances have no notable impact on the system behavior. Therefore, for a point on an interphase surface there are only two collinear normals which are opposite.

A noter que ceci réduit les configurations physiques. Considérant les parois physiques du milieu multiphase comme une phase, on exclut, par exemple, les écoulements slug flow, lesébullitions en parois.

A phase domain Ω includes all the multiphase system.

The number of phase subdomains inside the space domain Ω is

k m t Ω ∂ . (3) ( ) ( ) ( ) ( ) 1 1 k k N N k k m k k m m m t t t t Ω Ω Ω Ω = = = ∂ = ∂   . A subdomain ( ) k m t Ω can
( ) { } 0 i i k m k m;k m t M : f M ,t Ω ′ ′ ′ ′ = >  
. The indexes { } N subdomains, we obtain the phase k interface number as :

0 i i k m;k m k m;k m t M : f M ,t Ω ′ ′ ′ ′ ∂ = =   . The k m i k m ′ ′ arrangement
1 1 k N N k m;k m k N ′ ′ = = ∑∑ . 12
The interfaces inside one subdomain k m Ω with another phase k′ (with k k ′ ≠ ) are obtained by adding those of the k m;k N ′ inclusions.

(4) ( )

1 k m;k i N k m;k m i t Ω ′ ′ ′ = ∂  .
Then, its external interface has to be added to the internal ones. To derive the full interface ( ) ( ) ( )

1 1 k m;k k m k m i N N k m l p ;k m k m;k m k i k k t t t Ω Ω Ω ′ ′ ′ ′ = = ′≠     ∂ = ∂ ∂          . For convenience, for k k ′ = 1 k m;k N = and 1 1 m′ = the quantity 1 k m;k Ω ∂ has to be replaced by k m k m l p ;k m Ω ∂
in such a way that relation just above can be reduced as:

(5)

( ) ( ) 1 1 k m;k i N N k m k m;k m k i t t Ω Ω ′ ′ ′ ′ = = ∂ = ∂   .
Finally, the surface limiting the phase domain ( )

k t
Ω is obtained by adding the interfaces of the various subdomains m :

(6) ( ) ( ) 1 1 1 k m;k k i N N N k k m;k m m k i t t Ω Ω ′ ′ ′ ′ = = = ∂ = ∂    .
In this relation we have to keep in mind that the quantities N , k N , k m;k N ′ can change with time when both phase subdomains and their inclusions disappear or are created.

; N = { } { } 3 i m′ = , with interface 3 2 2 3 ; Ω ∂ ... 3 2 Ω external surface is 2 2 3 2 ; Ω ∂ . 14 PHASE SURFACE UNIT VECTOR At a point M  belonging to the phase surface ( ) 0 i k m;k m f M ,t ′ ′ = 
there is an outward unit

normal vector i k m;k m n ′ ′  (see A-7) directed outside the subdomain ( ) 0 i k m;k m f M ,t ′ ′ >  : i i i k m;k m k m;k m k m;k m f n f ′ ′ ′ ′ ′ ′ -∇ = ∇   
This one is the outward unit normal vector

i i ek k m ; k m k m;k m n n ′ ′ ′ ′ ′ =   of the space domain i k m Ω ′ ′ .
But, the surface

i k m;k m Ω ′ ′ ∂ being a surface inside the subdomain k m Ω and i k m;k m n ′ ′
 pointing inside this one, the outward unit vector of k m Ω on this internal interface is the opposite vector:

i i i ek ek k m;k m k m;k m k m ; k m n n n ′ ′ ′ ′ ′ ′ ′ = - = -    . Now, considering the external surface of k m Ω given by ( ) ( ) { } 0 k m k m k m k m l p ;k m l p ;k m t M : f M ,t Ω ∂ = =   its outward unit vector is l p ;k m k m k m l p ;k m k m k m l p ;k m i k m k m ek f n f -∇ = ∇    PHASE INDEX A phase function ( ) k M ,t α 
is associated to each phase. Its value is one if the phase exists at point M  at time t and is zero otherwise. This function is defined with a Heaviside distribution:

( ) ( ) k k M ,t H f M ,t α   =     . [ ] 1 k k H f α = = if 0 k f > and [ ] 0 k k H f α = = if 0 k f ≤ . For a given point ( ) M ,t  the phase
function is a discrete quantity having two values 0 and 1. For a phase domain

( ) k t Ω with k N different subdomains ( ) k m t Ω its phase function is a sum of ( ) k m M ,t α  each subdomain: (7) ( ) ( ) ( ) ( ) 1 k N k k m k m k m m M ,t M ,t M ,t H f M ,t α α α =   = =   ∑     .
The question is to define

( ) k m M ,t α  taking into account the various subdomains included in ( ) k m t Ω
. Considering the quantity:

( ) ( ) k m k m i l p ;k m k m;k m H f M ,t H f M ,t α ′ ′     = -      
, where the domain Ω can be built as:

k m Ω is included itself in a subdomain k m k m l p Ω identified by k m l and k m p with k m l k ≠ .
α = . • Two, M  is inside k m Ω but outside i k m;k m Ω ′ ′ so
( ) ( ) ( ) 1 k m ,k k m k m i N N k ,m l p ;k m k m;k m k k i M ,t H f M ,t H f M ,t α ′ ′ ′ ′≠ =     = -     ∑ ∑   
. Quantities such

( ) k g M ,t 
, defined on space pieces changing with time, are said continuous by pieces functions. Their definition domain is ( )

k t Ω .
We assume that inside their definition domain they have the required properties for deriving the usual math properties (sums; derivatives etc.) and balance equations of physics. Hereafter, functions such as

( ) k g M ,t 
are called local instantaneous discontinuous quantities.

REPRESENTATIONS OF RANDOM MULTIPHASE SYSTEMS

Above, we defined a deterministic multiphase system through local, instantaneous, discontinuous functions such as

( ) k g M ,t 
. When handling random systems we need the same kind of quantities but, in addition, we have to specify their random characteristics and their associated probability density functions.

The space domain of a phase k , its properties -e.g. density, energy, etc. -can exhibit random behaviors so they will change according to space and time but also with a given realization of the system. To identify the realizations of the system we introduce a set of indexes

{ } Λ λ =
. So, phase distributions in space are given by functions

( ) k f ; M ,t λ  and
physical properties are provided by quantities such as ( )

k g ; M ,t λ  .
For a given point ( )

M ,t  the phase function ( ) k ;M ,t α λ  is a discrete random quantity { } 0 1 , Λ →
having two values 0 and 1. For writing ease and if there is no chance of errors, we use the following writing shortcut M ,t ≡ •  so we set ( )

k f ; λ • for ( ) k f ;M ,t λ  .
For a given occurrence of the physical system, the physical quantities ( )

k g ; λ • are defined in a 4D domain: (8) ( ) ( ) ( ) { } [ [ 3 0 0 k k M ,t : f ; , Ω λ λ = • > ⊂ × ∞  
, which is limited by a phase interface:

(9) ( ) ( ) ( ) { } [ [ 3 0 0 k k M ,t : f ; , Ω λ λ ∂ = • ⊂ × ∞ =  
.

The section of k Ω for a given time or a given location provides the random space phase distribution ( )

3 k ;t Ω λ ⊂  or the random time distribution ( ) [ [ 0 k , ;M Ω λ ⊂ ∞  of the phase k .
The previous ideas are illustrated by figure 12 for a simplified case where space has only one dimension.

The gray areas are the space time distributions ( )

k Ω λ where a one dimension phase k is found for a given realization λ .

The red horizontal line at t on the time axis cuts the grey areas in several points which define

( ) k ;t Ω λ
as the union of several space pieces in yellow on the space axis. This is the fraction of the space where the phase k is present at t for this realization λ of the system. For a random multiphase system the grey areas change with each realization.

figure 12 Phase space and time distributions

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Finely Intricate Random Multiphase Systems 18

In the same way, we consider a given point M  on the space axis: the vertical red line in the figure above. The time period

( ) [ [ 0 k , ;M Ω λ ⊂ ∞ 
for which the phase k is observed at this point M  is the union of several time sections in yellow on the time axis.

The lines on the figure (but surfaces for real cases) limiting the grey areas are the space time distributions of the phase k interfaces. For a given time t , ( )

3 k ;t Ω λ ⊂ ∂
 is the union of several surfaces which limit the phase distribution in space. It is the interface of the phase k at this time for the realization λ .

PROBABILITY AND WEIGHTED MEASURES FOR FIRM SYSTEMS

PROBABILITY MEASURE

Often, the index set

{ } Λ λ =
is considered as finite or as  , so to say countable. Herewith, { } Λ λ = is an uncountable abstract set having the power 1 of  . The main reason for such a choice is the fact that the physical quantities being defined as real numbers, the set of their possible values has the same power than the set of the real numbers  : It is uncountable. So, the set of the realization index λ requires having the same power than the set of the possible values of the physical quantities, so Λ ⊂  .

To handle such random systems we provide Λ with a σ-algebra 2 Θ of subsets and a probability measure [ ]

0 1 P : , Θ →
. (French readers could find simplified elements of probability theory in 3 , similar information are available in 4 ). Probability measure ( ) P ε of any subset ε of Λ belonging to Θ is given by a Lebesgue integral:

1 Two sets have the same power if their elements can be put into one-to-one correspondence each other.

2 A family Θ of subsets of Λ including Λ itself is a σ-algebra if any finite or countable intersections n  or unions n  of its elements belongs to Θ . A probability measure is defined on the abstract set Λ . But what we are looking for is a probability measures associated to physical conditions. For example, the probability to find a density value

( ) k ; M ,t ρ λ  for phase k within a given interval [ ] a, a ρ ρ - + .
It means that we need to connect such intervals of  with elements of the σ-algebra Θ for which a probability measure exits. The way to connect these real quantity intervals to abstract sets of Θ is to use measurable 5 functions. Technically, any random function ( )

g ; M ,t λ  from [ [ 3 0, Λ × × ∞ 
has to be defined as a measurable function using the σ -algebra product of Θ and Borel σ -algebra in 4  .

A first example. The set of the realizations k ε for which a given phase k is observed at a given ( ) M ,t  is written with some math as the relation:

( ) ( ) { } 0 k k M ,t : f ; M ,t ε λ Λ λ Θ = ∈ > ⊂  
. This set k ε can be also defined with

( ) k ; M ,t α λ  the phase function: ( ) ( ) { } 1 k k M ,t : ; M ,t ε λ Λ α λ = ∈ =  
.

The probability to observe the phase k at ( )

M ,t  is ( ) ( ) ( ) ( ) k k M ,t P M ,t dP ε ε λ = ∫   .
A second example.

For ease we use the following writing shortcut

M ,t ≡ •  . The set ( ) k δ • of the realizations
for which the density of the phase k at ( )

M ,t  is within the interval 2 2 a a , ρ ρ   - +    
with a ∈  is given by relation:

( ) ( ) 2 2 k k a a : ; , δ λ Λ ρ λ ρ ρ Θ     • = ∈ • ∈ - + ⊂         . Its probability is ( ) ( ) k P δ • .
For any function

( ) k g ; λ • which is limited to ( ) k Ω λ we can construct a quantity k G defined for any ( ) [ [ 3 0 M ,t , ∈ × ∞   using the phase function: k k k G g α = 5 A,B being two sets ; A B , Θ Θ their σ-algebras f : A B → is measurable if ( ) 1 B A f ε Θ ε Θ - ∀ ∈ ∈ April 2023
Finely Intricate Random Multiphase Systems 20

Having assumed that any function in the system is measurable, the product

k k k G g α = is also measurable. The subset k G
φ of the occurrences Λ for which this product is below a given value x ∈  is given by:

(11) ( ) ( ) { } k G k x, : G , x φ λ λ • = • ≤ . According to k G measurability, the subset k G φ is also an element of Θ : k G φ Θ ⊂ . Its probability measure is ( ) ( ) k G P x, φ • . This probability measure is called the distribution function k G Φ of the property k G (12) ( ) ( ) ( ) k k G G x, P x, Φ φ • = • . If ( ) k G x, Φ
• is derivable with respect to x , we obtain the probability density function (pdf)

G k ϕ defined for any phase property at any point and time ( )

M ,t 
, even where or when the phase is not present:

(13) ( ) ( ) G k k G x, x, x ϕ Φ ∂ • = • ∂ .
So, the mean value also called average value of any phase quantity is defined and can be written down as:

(14) ( ) ( ) ( ) ( ) ( ) G k G k k k k k k D G g g , dP x x, dx Λ α α λ λ ϕ • = • = • = • ∫ ∫ , where k G D is the set of k G possible values, generally k G D ⊂  .
The previous development, starting from a rather abstract concept of a probability conceived using abstract sets and Lebesgue integral, provides the well-known classic method for averaging physical quantities with a probability density function.

At this stage, the average in probability of any multi-phase property can be determined, at least formally, and any quantity can be represented as a sum of a mean and a fluctuation:

(15) ( ) ( ) ( ) k k k G , G G , λ λ ′ • = • + • with ( ) 0 k G , λ ′ • = .

WEIGHTED MEASURE

Let us assume that we know all the le local instantaneous quantities required for describing a FIRM system. The amount of data will be so huge that it will prevent anyone from deriving understandable information directly from it. Instead, to be able to handle this set, we will modify it using weighting operators in such a way to obtain new quantities defined from the local and instantaneous quantities. function is defined as:

(16) ( ) ( ) ( ) 3 V g M ,t du w M ,t u g ,u d ξ ξ ξ ∞ -∞ = - - ∫ ∫    
, where w is a weighting function w with dimensions Note that weighting for space or time only can be deduced from ( 16) using weighting functions with a Dirac distribution:

3 1 L T - -     with ( ) 0 w M ,t u ξ - -≥   whatever is the space time location ( ) 4 M ,t u ξ - -∈    .
( ) ( ) ( ) w M ,t w M t ξ τ ξ δ τ - -= - -     or ( ) ( ) ( ) w M ,t w t M ξ τ τ δ ξ - -= - -     .
In the same way, the weighting operation can reduced to a surface using a Dirac function for a given direction.

The weighting function has to verify three requirements. First, w is normalized:

(17) ( ) 3 1 V w M ,t u d du ξ ξ ∞ -∞ - - = ∫ ∫   .
So, if a quantity is constant on the weighting domain then its weighted value is identical to this constant.

Two, w is a function of both distance and time intervals without reference to any specific position or time. Therefore their derivatives have to verify the following equalities

(18) 1 2 3 j j w w j , , x w w t u ξ ∂ ∂ = - = ∂ ∂ ∂ ∂ = - ∂ ∂
.

Three: The objective of the third condition is to decrease the importance of local values with the increase of their distance -in terms of space and time-from the location ( )

M ,t 
under consideration. By this way, the weighted quantity is mainly related to the local values in a certain vicinity of ( )

M ,t 
which is defined by the "shape" of w :

( ) ( )

0 0 t u M lim w lim w ξ -→∞ -→∞ → →   .
Using the previous three properties, it can be demonstrated (see Math-complement chapter 6.1) that the derivative and weighting operations commute so:

(19) i i g g g g t t x x ∂ ∂ ∂ ∂ = = ∂ ∂ ∂ ∂
.

For a random function ( )

g ; M ,t λ 
, the weighted quantity is also a random function defined by:

(20) ( ) ( ) ( ) 3 V g ; M ,t w M ,t u g ; ,u d du λ ξ λ ξ ξ ∞ -∞ = - - ∫ ∫     ,
It is worth to note that probability averaging, as defined by relation ( 14), does not imply space or time coordinates, therefore, it commutes with the weighting operation and we can write:

(21)

g g = .
Often in science papers, we find the words mean or average in space or in time for the weighting operations. But, the reader will take care that we do not use such words for a weighted quantity. Hereafter, the words mean and average are used only for operations related with probability operations.

It is important to note that such weighting operations make sense under two conditions when used for physics. On one hand, the physical process which produces these values does not change of nature within the weighting space and time domains. On the other hand, these weighting space and time domains are small fractions of the FIRM system both in terms of their typical space and time scales of variations. But, nevertheless, they have to be large enough for including many individual elements most often.

April 2023

Finely Intricate Random Multiphase Systems

24

A weighting function can be chosen null outside a space time domain in such a way that local instantaneous quantities outside this domain do not contribute to the weighted values.

Let υ and τ the measures of such space and time domains located at ( )

M ,t  : ( ) ( ) M t υ τ  .
The following function is a weighting function:

( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 0 M ,t u M t w M ,t u M ,t u M t ξ υ τ υτ ξ ξ υ τ  ∀ - -∈ ×  - -=   ∀ - -∉ ×          ,
A double weighting operation is defined as:

(22) ( ) ( ) ( ) 3 V g M ,t w M ,t g , d d ξ τ ξ τ ξ τ ∞ -∞ = - - ∫ ∫     .
According to ( 16) and ( 22), ( 24), it has been underlined that simple and double weightings are not identical: g g ≠ . Nevertheless, for some multiphase systems it is possible to adjust adequately the weighting function parameters to obtain g g ≅ .

The intuitive reason for such an approximation is the following. If a physical process can be depicted using weighted quantities it means that their weighted values do not exhibit large variations within the effective weighting domain. Then, on one hand, we have

( ) ( ) g X , g M ,t θ ≅  
for any point ( )

X ,θ  within ( ) ( ) M t υ τ × 
On the other hand, the function g being continued on ( ) ( )

M t υ τ × 
, we know, according to the integration properties, that there is a point ( )

X ,θ ′ ′  within the integration domain ( ) ( ) M t υ τ × 
for which (22) can be written

( ) ( ) g M ,t g X ,θ ′ ′ =  
. Using these two previous points, we can assume that it is possible to have

( ) ( ) g M ,t g M ,t ≅  
if the weighting function is carefully chosen according to the multiphase system under consideration.

BALANCE EQUATIONS FOR FIRM SYSTEMS

Let us assume that • all the le local instantaneous quantities required for describing a FIRM system are known; • they obey the basic law of physics;

• both balance equations for each phase and transfer equations for their exchanges through their interfaces are available. Then, imagining being able to solve such a set of equations, the behaviour of the system will be obtained, in theory.

But, the solutions will be intractable and useless because the amount of data required to describe the local instantaneous values of these quantities will be so huge that it will prevents any one from deriving understandable information directly from them.

Then, how to proceed?

• First, we retain the idea to write down the set of basic balance and transfer equations for the local and instantaneous quantities. • Two, instead to handle directly this set, we modify it using weighting operators in such a way to obtain new balance equations for weighted quantities.

The reason for such a procedure is the following. Local equations are defined only where and when a phase is present and, from chapter one, we know that such space time domains can be highly variable for FIRM systems. On the contrary, balance equations for the weighted quantities are defined at any point and instant. They provide a way to describe the behavior of these systems even if they give an approximate view of them.

PHASE WEIGHTED QUANTITIES

VOLUME WEIGHTED PHASE INDEXES

Let us consider the weighting operation ( 16) with weighting volume V inside a multiphase system. With a constant quantity 1 g = according we find:

( ) 3 1 V du w M ,t u d ξ ξ ∞ -∞ = - - ∫ ∫   , April 2023
Finely Intricate Random Multiphase Systems 26 Generally, the weighting volume V is shared between various phase domains

1 k N k k V V Ω = = =  
and the left hand previous integral can be written with the phase indexes k α so:

( ) ( )

3 3 1 1 1 k k N k N k k k V V du w d du w M ,t u ,u d Ω ξ ξ α ξ ξ ∞ ∞ = = = = -∞ -∞ = = - - ∑ ∑ ∫ ∫ ∫ ∫     .
In most right hand part of the previous relation we recognize the weighting of the phase indexes. Finally we obtain

1 1 k N k k α = =
= ∑ and using (21) if the system has random features we have also

1 1 k N k k α = = = ∑ .
For short writing we define a weighted phase index

k k k α α α ≡ ≡  : (23) 1 1 k N k k α = = = ∑  .
The weighted phase index k α  can be understood as the fraction of a given volume where this phase k is found, according to the weighting process. For instance, in a water steam or liquid air two phase flow, the weighted phase index related to steam or air is often call the void fraction.

VOLUME WEIGHTED PHASE QUANTITIES

In the same way, we define the weighted quantity:

k k k k k g g g α α = =  see relation (16). ( ) ( ) ( ) ( ) 3 k k k V g M ,t du w M ,t u ,u g ,u d ξ α ξ ξ ξ ∞ -∞ = - - ∫ ∫      
, It has to be keeping in mind that a weighted quantity such k g  is defined at any space location and time whatever is the actual phase existing at this position at this specific time.

It noteworthy to mention that k g  obtained from the previous weighting operation refers to a volume which is occupied by various phases, so k g  value is representative of the value of k g as if the phase k is alone in the weighting volume. In the same way that gas partial pressure is defined.

If a quantity closer of the instantaneous local quantity is needed the following quantity can be used :

k k k k k k g g c α α α = =   .
To underline the difference between k c and k g  let us assume that the instantaneous local quantity is constant: k g γ = . So, k g  is proportional to the volume fraction of the phase k in the weighting volume:

k k g α γ =  
. Now, regarding k c , from the previous relation we have k c γ = which is not related to volume fraction of the considered phase. For general phase conditions, the difference between k c and k g  is not so obvious but, anyway, k c will be closer of the he instantaneous local quantity value than k g  .

BALANCE EQUATIONS FOR LOCAL QUANTITIES

Local balance equations for multi-phase systems are built using local instantaneous quantities k g by unit of mass associated to a phase k within its phase domain.

BALANCE EQUATIONS FOR PHASES

The general form of such balance equations is the following:

(24) ( ) 0 m m k k k k k gk gk m g g u j t x ρ ρ Γ ∂ ∂ + + - = ∂ ∂ ,
where k g stands for mass, momentum or energy of a given phase k , k ρ is its mass density, with dimensions [ ]

3 k ML ρ - = .
The left hand first term is a rate of variation by unit of volume of the quantity g . The second one is resulting of flux of g due to both a macroscopic transport Γ is a generation rate by unit of volume of the quantity g in the phase domain k . For each phase we have five balance equations; one for mass, three for momentum components and one for energy. For a FIRM system with N phases, we have 5 N × such balance equations.

If we write [ ]

g for k g dimensions, we have:

[ ][ ] 1 gk g T Γ ρ -   =   and [ ][ ] 1 m gk j g LT ρ -   =   .
Table 1 below provides the terms required for equation ( 24) according to mass, momentum or energy balance: Regarding gk Γ the generation rates given in the previous table, we make the choice to have no mass generated in the phase domains. For momentum, we need the sum of the volume forces acting on the phase and we took just a gravity force with acceleration vector components being

j γ 2 j LT γ -   =  
. For energy we considered only the power of the volume forces so the corresponding generation rate is

3 1 i i k k i u ρ γ = ∑ .

BALANCE EQUATIONS ON INTERFACES

In this study we consider an interface between two phases as a surface with no thickness. In reality, when two phases are in contact, their molecules interact on a given zone, generally very small, on both sides of the interface. Its real thickness depends of various parameters such as pressure, temperature, and so on. The zone properties might be quite different from the two phases which are in contact, especially if chemical reactions take place there.

For the present study of FIRM systems the interphase zones between phases will be always considered as negligible by comparison with the smallest of its phase domains. So, their volumes, masses, momentum are ignored in regard of the phase quantities. Even, if interface energy is present, it is quite often negligible in comparison with the phase energy.

Nevertheless, even if the interface mass, momentum and energy are neglected by themselves, they are locations of the interactions between the phases, places where exchanges of mass, momentum and energy take place. So, we need balance equations for such exchanges through interfaces because they modify the amount of the corresponding quantities in the phase domains. The previous relations (24) provide balance equations at any point inside the various phases. In addition, we represent hereafter both the interactions between phases themselves which take place at their interfaces and those which might be produce by the material surfaces which limit the system and are external surfaces of some phases. On these surfaces the system must satisfy the principles of conservation of mass, momentum and energy.

April 2023

Finely Intricate Random Multiphase Systems 30 Such conditions provide balance equations for phase interfaces. We write down these interactions for the instantaneous values because they are needed for deriving weighted quantities balance equations. Previously, we established that a phase k can have



provide the physical and chemical processes acting on system interfaces. Such quantities have to be known to describe the interactions between phases and, therefore, to study the system behavior. We can also define normal generation rates of g by unit of surface )

i i i s s ek gk / k m;k m k m;k m gk / k m;k m .n Γ Γ ⊥ ′ ′ ′ ′ ′ ′ =   and i i i s s ek gk / k m ;k m k m ;k m gk / k m ;k m .n Γ Γ ′ ⊥ ′ ′ ′ ′ ′ ′ ′ ′ =   .
( ) i k k k k m;k m g u V ρ ′ ′ -  
0 i i i s ek k k k k m;k m gk k m;k m gk / k m;k m g u V j .n ρ Γ ⊥ ′ ′ ′ ′ ′ ′   - + - =         .
A positive value of

0 i i i i i i s ek k k k k m;k m gk k m;k m gk / k m;k m s ek k k k k m ;k m gk k m ;k m gk / k m ;k m g u V j .n g u V j .n ρ Γ ρ Γ ⊥ ′ ′ ′ ′ ′ ′ ′ ⊥ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′   - + -       + - + - =             .
Taking into account that the interface velocity is unique

i i k m;k m k m ;k m V V ′ ′ ′ ′ ≡  
; and phase outward normal vectors are in opposite directions

i i i ek ek k m ;k m k m;k m k m;k m n n n ′ ′ ′ ′ ′ ′ ′ = - =   
, the previous relation is written as:

0 i i i i i i k k k k m;k m gk k k k k m ;k m gk k m;k m s s k m;k m gk / k m;k m gk / k m ;k m g u V j g u V j . n n ρ ρ Γ Γ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′   - + - - +     - - =           . (26) ( ) ( ) ( ) ( ) ( )
This equality can be understood as following. If generation rates s gk / ....

Γ

 for a quantity are identical or zero on both sides of an interface, this quantity k g is just transferred from one subdomain to the embedded one through the interface. But, if the rates differ, there is a production or absorption of the quantity at the interface and its flux presents a variation there.

When considering the surface 11 11

;

Ω ∂ of 11
Ω which encloses all the system, the previous relation ( 26) cannot be applied. For this surface the condition (25) can be used involving 1 11 11

s g / ;
Γ ⊥ the normal generation rate by unit of surface of the physical quantity k g on this surface, but, on its external side, boundary conditions of the full system have to be known in order to predict the system behavior.

BALANCE EQUATIONS FOR WEIGHTED AND MEAN QUANTITIES

BALANCE FOR WEIGHTED QUANTITIES

We are looking for balance equations for weighted quantities for each phase valid everywhere in the space domain where is located the FIRM system at any instant.

These equations are derived by multiplying equation ( 24) by the phase function k α and re arranging the terms to introduce the quantity k k g α defined everywhere and at any moment and finally applying a weighting operation. So we have:

(27) ( ) 0 k k k k k k k gk k gk g g u j t ρ α α ρ α Γ ∂ + ∇. + - = ∂    .
Using the derivatives (see A-17 A-18 A-19) of weighted quantities k α 8 we can rewrite the previous relation as following:

(28) ( ) ( ) 1 1 1 k m ,k k i i k k k k k k k gk k gk N N N k k k k m;k m gk k m;k m m k i g . g u j t g u V j α ρ α ρ α Γ ρ Π ′ ′ ′ ′ ′ ′ = = = ∂ + ∇ + - = ∂   - - + ⋅     ∑∑ ∑        .
The right hand side of ( 28) is modified using the conditions (25) on the phase interfaces and knowing that

i i ek k m;k m k m;k m n n ′ ′ ′ ′ = -   and i k m;k m i i s k m;k m k m;k m n Π σ ′ ′ ′ ′ ′ ′ =  
. Finally the balance equations for the weighted quantities are :

(29) ( )

0 sV k k k k k k k gk k gk gk g . g u j t α ρ α ρ α Γ Γ ∂ + ∇ + - - = ∂    ,
where appear two volume rates. Γ is the generation of k g on the phase k interfaces with all the other phases included in the volume used to define the weighted quantity k k g α :

(30)

1 1 1 k m ,k k i i N N N s sV s gk k m;k m gk / k m;k m m k i Γ σ Γ ′ ⊥ ′ ′ ′ ′ ′ = = = = ∑∑ ∑ .
For example, applying (30) for phase 1 k = and the first term of the previous relation for 1 k′ = is obtained with 11 1 1 ; N = so: This sum (30) can be made first for all the subdomains of phase k and a given phase and then for all the phases:

11 1 1 1 1 1 11 11 11 1 1 11 11 11 1 2 1 2 1 1 ,k m ,k i i i i N N N N N s s s sV s s s g ; ;k m m;k m g / ; g / ;k m g / m;k m k i m k i Γ σ Γ σ Γ σ Γ ′ ′ ⊥ ⊥ ⊥ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ = = = = = = + + ∑ ∑ ∑∑ ∑
(31) 1 N sV sV gk gk / k k k Γ Γ ′ ′= = ∑ with 1 1 k k i i N N m;k s sV s gk / k k k m;k m gk / k m;k m m i Γ σ Γ ′ ⊥ ′ ′ ′ ′ ′ = = = ∑ ∑ . BALANCE FOR AVERAGED QUANTITIES
Instead of weighting by time and space integrations, a probability averaging operation can be used if the random aspects are dominant. It will be seen that the same form of balance equations for the mean quantities is obtained. Relation ( 24) is multiplied by k α and averaged:

( )

0 k k k k k k k gk k gk g g u j t ρ α α ρ α Γ ∂ + ∇ + - = ∂    .
According to the possibility to permute averaging and derivation operators, the previous equality can be written:

( ) ( ) k k k k k k k k gk k gk k k k k k gk k g g u j g g u j t t α ρ α α ρ α Γ ρ ρ α ∂ ∂ + ∇ + - = + + ⋅∇ ∂ ∂       .
Using the derivatives (see A-27, A-28) of weighted quantities k α , performing the same transformations previously we arrive to:

(32) ( ) (
)

0 sV k k k k k k k k gk k gk gk g g u j t α ρ α ρ α α Γ Γ ∂ + ∇. + - - = ∂    . (33) 1 1 1 k m,k k i i N N N s sV s gk k m;k m gk / k m;k m m k i Γ σ Γ ′ ⊥ ′ ′ ′ ′ ′ = = = = ∑∑ ∑

BALANCE FOR WEIGHTED AND AVERAGED QUANTITIES

It can appear that averaged quantities continue to be too complex in terms of time or space variations or alternatively, the weighted quantities may continue to exhibit random characteristics. For such conditions, both weighting and averaging operators can be applied and the useful quantity will be G G = , keeping in mind that the two kinds of operations can be permuted:

(34) ( ) 0 sV k k k k k k k gk k gk gk g . g u j t α ρ α ρ α Γ Γ ∂ + ∇ + - - = ∂    , April 2023
Finely Intricate Random Multiphase Systems

BALANCE FOR THE WHOLE SYSTEM

When considering a given point and time ( )

, M t 
in the sytem the value of any quantities u g ρ  as specific mass, velocity or any parameters for the whole system are those of the specific phase being at ( ) ,

M t 

. A general relation for the whole sytem properties can be written as following because at a given point and time only one of the

( ) k M ,t α  is egal to one: 1 1 1 N N N k k k k k k k k k k k k g g g u g u ρ α ρ ρ α ρ ρ α ρ = = = = = = ∑ ∑ ∑   ,
These quantities being continuous by pieces functions they need to be weighted or averaged, therefore the whole sytem will be represented by

1 1 1 N N N m k k m m k k k m m m k k k k k k k g g g u g u ρ α ρ ρ α ρ ρ α ρ = = = = = = ∑ ∑ ∑   , or 1 1 1 N N N m k k m m k k k m m m k k k k k k k g g g u g u ρ α ρ ρ α ρ ρ α ρ = = = = = = ∑ ∑ ∑   .
The interfaces inside the system being without mass they do not affect the corresponding balance equations for mass and momentum. For energy, we have to assume that interface movements do not produce or absorb energy. If it is not the case, capillarity effect has to be added. Therefore, balance equations for the whole system are obtained adding (29) or (32) for all the phases:

(35) 0 sV m m m m m k gk k gk gk k k k g . g u . j t ρ ρ α α Γ Γ ∂ + ∇ + ∇ - - = ∂ ∑ ∑ ∑     , (36) 0 sV m m m m m k gk k gk gk k k k g . g u . j t ρ ρ α α Γ Γ ∂ + ∇ + ∇ - - = ∂ ∑ ∑ ∑     , .
Within these relations the right hand last term is related to the generation of the property of k g on all the interfaces in the FIRM system, see relation ( 30) , ( 31)

1 1 1 N N N sV sV gk gk / k k k k k Γ Γ ′ ′ = = = = ∑ ∑∑ ,
where the double sum over N in can be rearranged as ( )

1 1 1 N N N sV sV sV gk gk / k k gk / k k k k k Γ Γ Γ ′ ′ ′ ′ = = > = + ∑ ∑∑ .
With the exception of the limiting surface 11 11 ; Ω ∂ of the full system which is taken into account only for its internal side, all the other surfaces of the system are involved for their both sides. The term sV gk / k k Γ ′ stands for production of the quantity g on all the interfaces between the phase k and k k ′ > on their sides which are in contact with phase k . In the same way, sV gk / k k Γ ′ ′ stands for the production of on these same interfaces but on their k′ side, so it can be written as:

1 k i i N s sV s gk / k k k m ;k m gk / k m ;k m m Γ σ Γ ⊥ ′ ′ ′ ′ ′ ′ ′ = = ∑ .
Taking into account that the interface densities are related to the same surface ( )

k m;k i t Ω ′ ∂
seen from its k or k′ side, we have obviously

i i s s k m;k m k m ;k m σ σ ′ ′ ′ ′ ≡
. The phase outward normal vectors being in opposite directions

i i ek ek k m;k m k m ;k m n n ′ ′ ′ ′ ′ = -  
we have:

( )

1 1 k m ,k k i i i i N N s s sV sV s ek gk / k k gk / k k k m;k m k m;k m gk / k m;k m gk / k m ;k m m i n Γ Γ σ Γ Γ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ = = + = - ∑ ∑   
where appears the difference of production of the quantity g on both sides of the interfaces between phases. Finally, after having added the rate on the limiting surface 11 11 ; Ω ∂ of the full system which is taken into account only for its internal side, we can write for the term which stands for the interaction through the interfaces of the FIRM system:

April 2023

Finely Intricate Random Multiphase Systems 38

( )

11 11 1 11 11 1 1 1 1 1 k m ,k k N N N N N s s s sV s s gk ; k m;k i k i;k m g / ; gk / k m;k i gk / k i;k m k k k m i n Γ σ Γ σ Γ Γ ′ ⊥ ′ ′ ′ ′ ′ ′ = = > = = = - - ∑ ∑∑ ∑ ∑    ,
If the number of subdomains does not change with time :

( )

11 11 1 11 11 1 1 1 1 1 k m ,k k N N N N N s s s sV s s gk ; k m;k i k i;k m g / ; gk / k m;k i gk / k i;k m k k k m i n Γ σ Γ σ Γ Γ ′ ⊥ ′ ′ ′ ′ ′ ′ = = > = = = - - ∑ ∑∑∑ ∑    .
For probability averaged balance equations, the corresponding terms is

(38) ( ) 11 11 1 11 11 1 1 1 1 1 k m ,k k N N N N N s s s sV s s gk ; k m;k i k i;k m g / ; gk / k m;k i gk / k i;k m k k k m i n Γ σ Γ σ Γ Γ ′ ⊥ ′ ′ ′ ′ ′ ′ = = > = = = - - ∑ ∑∑∑ ∑    ,

SUMMARY

Finally, we derived balance equations for the weighted or averaged or simultaneously weighted and averaged quantities. Such equations are valid at any point in the space domain where the system is found, without any reference to the real locations of the various phase domains. This is a great advantage in terms of system description because, at any point and time, we have continuous functions involved in the differential problem to solve. But, as usually, one hand advantages have to be paid by some disadvantages for the other hand. For us, the difficulties are the following.

Relations as (26) above will be useful only if the terms which provide the interactions and exchanges between phases such as sV gk Γ can be determined. Even if analytical relations for the instantaneous generation rates on interfaces are known, the possibility to determine correctly sV gk Γ depends of the knowledge of the exact distribution of the interfaces and their time variations inside the weighting volume. For a FIRM system it is clear that such knowledge is rarely or even never available. In case of a probability approach, the corresponding term is sV gk Γ in equation ( 27). In this case we need to know the probability density function -pdf-of the parameters e.g. temperature, velocity, concentration, which control the generation rates

s gk / k m;k i Γ ⊥ ′ 
and the surface area, orientations etc. For FIRM system it is obvious that such pdf are unknown at can be just estimated.

To complete the 3 N × balance differential equations written down previously we need two sets of additional conditions for the phase quantities. The first set provides the initial state of phases and the second one are conditions given on surfaces which enclose the FIRM system.

The initial state means that phase quantities have to be provided at any point of the FIRM system at a given time taken as at time zero.

The second set, called boundary conditions, provides the interactions between phases and the surfaces which enclose the FIRM system. If we call ( )

out t Ω ∂
this surface, phase quantity such k g have to be known on the surfaces ( ) ( )

out k t t Ω Ω ∂ ∂  .
Regarding the initial conditions, it could be though that having the initial states of the various phases at any points we could derive initial conditions for the weighted quantities using the relations such as (15). In fact, this is roughly unreachable because for FIRM system it is generally impossible to know the physical quantities values at any point of the system at a given time chosen as an initial instant. Consider for example the olive tree field in Figure 2 above. If the question is to determine the particle deposits on the leaves, theoretically we need to know their movements, orientations, etc. Such information should be obtained from their interaction with the air flow through the canopy which required the knowledge of the initial arrangement of the leaves at a given instant. Obviously, such knowledge is out of access. Considering now the boundary conditions for the weighted quantities the situation is also out of access. As previously explain, FIRM systems are too complex for allowing knowledge of the phase quantities at any points belonging to the boundary surfaces. In addition, the weighted quantities being defined through volume integration their boundary conditions cannot be deduced from the instantaneous boundary conditions which are known only on boundary surfaces and not everywhere in the averaging volume.

Therefore, both the initial conditions and boundary conditions for the weighted balance conditions are known approximately only. A way to correct this incertitude will be to research how solutions of the balance equations are modified by variations of these conditions. → . So, we have:

ξ τ ξ τ ξ τ ξ τ ξ τ ξ τ τ ξ τ ξ τ ξ τ ξ τ ξ τ ξ τ τ τ ξ τ ξ τ ξ τ ξ ξ τ ξ τ τ ξ τ ξ τ ξ τ ∞ ∞ -∞ -∞ ∞ ∞ -∞ -∞ ∞ +∞ -∞ -∞ ∂ ∂ ∂     = - - - -     ∂ ∂ ∂ ∂ ∂   - - + - -   ∂ ∂ ∂ = - - + - - ∂ ∂ = - - ∂ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫  ( ) V g d M ,t t τ ∞ -∞ ∂ = ∂ ∫ ∫  1
A-5 ( ) ( ) g g M ,t M ,t t t ∂ ∂ = ∂ ∂  
For the space derivatibles we have the same kind of relations:

A-6 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 3 3 3 3 =- - - i i i i i i i V V i i V V x j k x g M ,t w M ,t g , d d w M ,t g , d d x x g , w M ,t g , d d w M ,t d d g d w M ,t g , d d w M ,t ξ ξ ξ τ ξ τ ξ τ ξ τ ξ τ ξ τ ξ ξ τ ξ τ ξ τ ξ τ ξ τ ξ τ ξ ξ τ ξ τ ξ τ ξ ξ ξ τ ∞ ∞ -∞ -∞ ∞ ∞ -∞ -∞ +∞ +∞ +∞ -→+∞ -→-∞ -∞ -∞ -∞ ∂ ∂ ∂     = - - - -     ∂ ∂ ∂ ∂ ∂   - - + - -   ∂ ∂ ∂ = - - + - - ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫  ( ) ( ) ( ) ( ) 3 3 i V i i V , d d g , g w M ,t d d M ,t x ξ τ ξ τ ξ ξ τ ξ τ ξ τ ξ ∞ -∞ ∞ -∞ ∂ ∂ ∂ = - - = ∂ ∂ ∫ ∫ ∫ ∫  With the property i i w w x ξ ∂ ∂ = - ∂ ∂ and ( ) 0 M lim w ξ -→∞
→ . So, we have:

( ) ( ) i i g g M ,t M ,t x x ∂ ∂ = ∂ ∂   .
Applying this relation to the various component of a vectorial function we have :

div g divg =   .
finally, we can write:

g g ∂ ∂ ≡ ∂ • ∂ • . April 2023
Finely Intricate Random Multiphase Systems 2

SURFACE PROPERTIES

A phase domain

( ) k t Ω in 3
 is a bounded region of space having an inside and an outside limited by a surface  belonging to the phase surface there is an outward unit normal vector with components:

A-7 k k f n f -∇ = ∇    k i i k f x n f ∂ ∂ = - ∇  .
Let us consider a point P  , located on the normal at a given point M  , close to the surface

( ) 0 k f M ,t =  : P M n β = +    with 0 β > .
According to the choice made for the normal unit vector defined by A-7, the question is to know if the point P  is inside or outside the phase k .The sign of ( ) k f P,t  can be obtained through the development:

( ) ( ) ( )

0 k k k k k k k k f f P,t f M n,t f M ,t f .n f . f f β β β β ∇ = + = + ∇ = -∇ = -∇ < ∇           , So, ( ) 0 k f P,t < 
, P  is outside the phase and we conclude that the normal defined by A-7 is pointing outwards. This is the outward unit normal vector of phase k at point M  and we write it e n n =   .

An explicit equation of the surface can be derived from a solution of :

( ) ( ) 0 k k f x, y,z x, y,t ,t = .
Then the normal components can be written down as :

1 2 2 2 1 1 k / k k k x y z z z z z y x n n n n x y n n n ∂ ∂     ∂ ∂ ∂   ∂   = + + = = = -       ∂ ∂          
The measure of the elementary dσ of ( )

k t Ω ∂
having the area dxdy in the , x y plan is:

A-8 k k z f dxdy d dxdy f n z σ ∇ = = ∂ ∂



The elementary surface vector dσ   is:

A-9 k i k i k z i f x z n d nd dxdy d dxdy dxdy f n x z σ σ σ ∂ ∂ ∂ = = = = - ∂ ∂ ∂     ,
where we take advantage of the relations:

A-10 k k k k k k f f z z y x f f x y z z ∂ ∂ ∂ ∂ ∂ ∂ = - = - ∂ ∂ ∂ ∂ ∂ ∂ .

VELOCITY OF A POINT ON A SURFACE

Let assume that at time t a point M  stays on a surface during its displacements or its deformations. At time t dt + , the point moving with the surface is located at M dM

+
( ) 0 k f M ,t = 
is defined by a:

A-12 ( ) ( ) ( ) s k k k M ,t f f M ,t σ δ = ∇    ,
In this relation δ is a Dirac distribution which is zero everywhere except on the surface

( ) 0 k f M ,t = 
where it is infinite. An integral of ( )

s k M ,t σ 
in a given space domain provides the measure of the area of the surface which is found inside this domain. This property is shown here after.

SURFACE DENSITY PROPERTIES

The surface density s σ is defined in 3

 as a distribution from a surface S ( )

0 f M =  by: A-13 ( ) ( ) ( ) s M f f M σ δ = ∇    ,
where δ is a Dirac distribution. So, s σ is zero everywhere except for the points which belong to the surface S . This quantity s σ can be understood as the surface by unit of volume, this idea is explained below. For the moment we demonstrate the relation:

A-14 ( ) ( ) s S g M d g P d Ω Ω σ ω σ = ∫ ∫    ,
where Ω is a space domain of 3  .

Let us call S Ω  the part(s) of S which belongs to Ω . In order to determine the volume integral A-14 we consider a new variable u

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) i i s i i D x,y D u x,y,z x,y,u g x, y,z x, y,z dxdydz f g x, y,z x, y,u u dxdydu f z Ω σ δ       =     ∇     ∂   ∂   ∫ ∑ ∫∫ ∫  .
According to Dirac property ( ) u δ , the term inside the left integrals is not zero only for 0 u = , so to say, for points belonging to the surfaces which are inside Ω :

( ) ( ) ( ) ( ) ( )
( ) 0 0 i i D x,y x,y,z x,y, f g x, y,z x, y, dxdy f z           ∇     ∂   ∂   ∫∫  .
For such a relation, when a couple ( )

x, y covers ( ) D x, y a point So to say, the integral of s σ in a volume provides the measure of the part of the surface which is enclosed in this volume. This result explains why s σ is called a surface density.

The relation A-14 can be applied to the flux of a vector A  on a surface. In that case g is a scalar product of a vector by a normal unit vector of the surface s

k k k s k k k d d d Ω Ω Ω Ω Ω σ ω σ σ ∂ ∂ = = ∑ ∑ ∫ ∫ ∫    ,
provides the sum of the measures of the surface parts which are included in Ω .

PHASE FUNCTION DERIVATIVES

According to Heaviside distribution properties and A-11, their derivatives can be written down as:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) dH f M ,t dH f M ,t f M ,t V M ,t . f M ,t f M ,t t df t dH f M ,t H f M ,t f M ,t f M ,t f M ,t df δ δ     ∂     = = - ∇ ∂ ∂       ∇ = ∇ = ∇                   . The condition ( ) 0 k f M ,t = 
provides the surface of the space domain

( ) 0 k f M ,t > 
with its unit outward vector n  and its surface density s σ , see 6.2 and 6.3

( ) ( ) ( ) ( ) ( ) ( ) s f M ,t n f M ,t M ,t f M ,t f M ,t σ δ ∇ = -∇ =∇          .
Using such relations for Heaviside derivatives we arrive to:

A-16 ( ) ( ) ( ) ( ) ( ) s s dH f M ,t M ,t V M ,t .n t H f M ,t n M ,t σ σ     = ∂   ∇ = -            .
Let us now consider relation A-16 applied to the phase index of a phase subdomain k m Ω defined in chapter 3.1:

April 2023

Finely Intricate Random Multiphase Systems 8 ( ) ( ) ( )

1 k m ,k k m k m i N N k ,m l p ;k m k m;k m k k i M ,t H f M ,t H f M ,t α ′ ′ ′ ′≠ =     = -     ∑ ∑   
.

For the derivative of the first Heaviside the unit normal vector is A-17

1 1 1 1 k m ,k k m;k m k m;k m i i i k m ,k k m;k m i i N N k ,m s k m;k m k i N N s k ,m k m;k m k i V .n t .n α σ α σ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ = = ′ ′ ′ = = ∂ = ∂ ∇ = - ∑ ∑ ∑ ∑    
where we set for writing ease. Previous relations can be also written:

A-18 ( ) ( ) 1 1 1 1 k m ,k i i k m ,k i k m;i j N N k m k m;k m k m;k m k i N N k m k m;k m k i s k m;i j k m;i j M ,t V t M ,t n α Π α Π Π σ ′ ′ ′ ′ ′ ′ ′ = = ′ ′ ′ = = ∂ = ⋅ ∂ ∇ = - = ∑ ∑ ∑ ∑         .
Finally, using (7) and summing for the k N sub domains of a phase k we derive the following relations:

A-19 ( ) ( ) 1 1 1 1 1 1 k m,k k i i k m,k k i N N N k k m;k m k m;k m m k i N N N k k m;k m m k i M ,t V t M ,t α Π α Π ′ ′ ′ ′ ′ ′ ′ = = = ′ ′ ′ = = = ∂ = ⋅ ∂ ∇ = - ∑∑ ∑ ∑∑ ∑      
We have to keep in mind that both k N and k m;k N ′ , the number of phase subdomains and the number of inclusion, might vary with time, for example in case of coalescence or fragmentation. Nevertheless, within the previous sums only, only one term k m α is different of zero at a given time and its derivative is not affected. When a phase does not enclose other phase inclusions, there is only one term k m k m l p ;k m in the previous relations. It means that its phase function variations are only due to the variations of the external interface of the phase, so to say, its volume or shape variations. But, in general, relations A-19 contain several terms which, in addition to the previous variations, provide the phase function variations due to the behaviors of the volume or the shapes of the inclusions inside the phase domain.

PHASE FUNCTION INTEGRALS

Here after, we consider integrals within a space domain V where derivatives of the phase function are involved. These integrals are useful when weighted quantities are evaluated, for example

k k j g x α ∂ ∂ : A-20 ( ) ( ) ( ) ( ) ( ) 3 k k k k j j S k k k k j j S V g g M ,t ,u du x x ,u g M ,t ,u w M ,t u g ,u d x α α α ξ α ξ ξ ξ ξ +∞ -∞ ∂ ∂ ≡ ∂ ∂ ∂ ∂ = - - ∂ ∂ ∫ ∫       .
The volume integral is modified using the second relations of A-19 for the derivative of k α and its j component. Furthermore, we switch the integration and the sums : Regarding the weighting of a phase function time derivative, we proceed in the same way than previously. Using the first relation of A-19, an interface velocity appears and we have:

A-24 ( )

3 1 1 1 k m ,k k i i N N N k k k k m;k m k m;k m V S m k i g M ,t,u wg V d t α Π ξ ′ ′ ′ ′ ′ ′ = = = ∂ = + ⋅ ∂ ∑∑ ∑ ∫    .
Then, applying the time weighting operation with the same comment then previously regarding the variations of N k N and km,k N ′ we obtain:

A-25 ( )

1 1 1 k m ,k k i i N N N k k k k m;k m k m;k m m k i g M ,t g V t α Π ′ ′ ′ ′ ′ ′ = = = ∂ = + ⋅ ∂ ∑∑ ∑   
. CETTE PARTIE EST A CORRIGER As we mentioned above the equalities A-23 A-24 are valid only if the numbers k N and km,k N ′ do not vary with time or their time variations are small enough during the weighting duration. If such conditions are not satisfied, the sums on k N and km,k N ′ and time integration cannot be permuted and the corresponding relations are the following:

A-26 1 1 1 1 1 1 k m ,k k k m ,k k N N N s k k m;k i k m;k i k m i k k N N N s k k k k m;k i k m;k i k m;k i k m i k k A A n g g V n t α σ α σ ′ ′ ′ ′ ′ = = = ′≠ ′ ′ ′ ′ = = = ′≠ ⋅∇ = - ⋅ ∂ = ⋅ ∂ ∑ ∑ ∑ ∑ ∑ ∑       .
To conclude this section, if k g is a quantity continuous by pieces associated to a phase k with phase function k α , we have:

A-27

1 1 1 1 1 1 k m,k k i i k m,k k i N N N k k k k k k m;k m k m;k m m k i N N N j k k k k k k m;k m j j m k i g g g V t t

FIGURE 1 :

 1 FIGURE 1 : AIR OCEAN SYSTEM .................................................................................................................... FIGURE 2 :OLIVE TREE FIELD IN SAINTE VICTOIRE COUNTY ............................................................... FIGURE 3 : ROUGH AND SMOOTH PHASE ELEMENTS ............................................................................. FIGURE 4 ; ROUGH AND SMOOTH PHASE ELEMENTS FOR THE OCEAN ............................................. FIGURE 5: LEAVES FINELY DISTRIBUTED .................................................................................................. FIGURE 6: LEAVES INTRICATENESS ............................................................................................................ FIGURE 7 WATER SPRAY INTRICATENESS ................................................................................................. FIGURE 8 RANDOM PHASE DITRIBUTION ..................................................................................................

FIGURE 9 :

 9 FIGURE 9 : A DISTRIBUTED PHASE k ........................................................................................................

FIGURE 10 :

 10 FIGURE 10: INTERFACE ILLUSTRATION.................................................................................................... FIGURE 11: CONTINUITY BY PIECES. ......................................................................................................... FIGURE 12 PHASE SPACE AND TIME DISTRIBUTIONS ........................................................................... FIGURE 13 WEIGHTING CONCEPT .............................................................................................................. FIGURE 14 INTERFACE BALANCE .............................................................................................................. FIGURE 6-1 : PHASE DOMAIN ......................................................................................................................... FIGURE 6-2 SURFACE DENSITY .....................................................................................................................

Figure 2 :

 2 Figure 2 :Olive tree field in Sainte Victoire county

figure 3 :

 3 figure 3 : Rough and smooth phase elements

figure 4 ;

 4 figure 4 ; Rough and smooth phase elements for the ocean

  figure 5: leaves finely distributed

  figure 9 : A distributed phase k

A

  figure 10: interface illustration

  At a given time t , a point M  can have three possibilities.• One, the point is outside k m Ω and also outside an included subdomain such

  the first Heaviside is one but the second is zero, therefore 1 α = . The point is located inside the phase included in the space domain k m Ω . so both Heaviside are equal to one and 0 α = . The point is inside the space domain k m Ω but it belongs to a phase k′ located in the space subdomain inclusion i k m;k m Ω ′ ′ . In summary, a phase function for the subdomain k m

  , e.g. density, concentration, impulsion, energy, etc. are defined only where and when the considered phase exists.The figure 11 depicts this situation for a one dimension domain. The red zones are separated space domains, called pieces, where a quantity phaseis defined at a given time t .

figure 11 :

 11 figure 11: continuity by pieces.

3

  Les milieux aérosols et leurs représentations. Alain Mailliat. EDP Sciences, Paris 2010 ISBN 978-2-7598-0553-2. 4 Stochastic tools in turbulence. John L. Lumley. Academic Press 1970, Lib.Congress Catalog Card Number 70-117080.

figure 13

 13 figure 13 Weighting conceptBut, in return, they are tractable and they correctly depict FIRM system behaviour if adequately chosen.

  weighted value has the same dimensions that the corresponding local instantaneous one. The weighted quantity ( ) g M ,t  is defined at any point of space and time where the local instantaneous ( ) g M ,t  is defined. The quantity ( ) g M ,t  is continuous and derivable. Its value at a given point is typical of the properties in the vicinity of this point.

  Cauchy stress tensor, is a diffusion flux of momentum. If there is no viscous effect in the phase, the Cauchy tensor reduces to the pressure term mj k p δ . For a viscous medium the corresponding stress tensor energy including the power due to the stress forces and a heat flux Qm k j .

N

  figure 14 Interface balance

.

  For momentum balance, exchanges are due to forces on the interfaces and for energy balance heat can be transferred from one phase to another one.

  k g to (or from) this interface. Generally, this flux results of two phenomena which are convection and diffusion. The first one is a transport by the flow rate k k k g u ρ  of the phase material which has to be corrected by the interface velocity. So, this flux is

.

  The second one is a flux due to molecular movement gk j  which was introduced in the previous chapter.The ratei s gk / k m;k m Γ ⊥ ′ ′ on the internal side of k m Ω has to be balanced by the part of the previous fluxes arriving on the same unit of interface. So, with k k ′ ≠ , at any point M

  for interactions between a phase k and any one of the other phases. By convention defined previously, index sets such as 1 k ,m;k , have to be replaced by k m k m l , p ;k ,m .Note that k m Ω has only one external interface with its surrounding because it is included in only one other phase subdomain (due to the fact that we ignore interface points where three phases coexist).Now we apply (25) on both sides of

.Ω

  The two first terms of the right hand are related to the interfaces imbedded in the first subdomain 1 m = of phase 1 k = . The first of them being the effect of which encloses all the system. Third term takes into account the interfaces within the other phase 1 subdomains. Remember that when k k′ =

  on the figure against.

Figure

  Figure 6-1 : Phase Domain

  Two functions with their time and space derivatives are quite often used in balance equations for FIRM systems. Hereafter we provide properties of the interface density distribution

.

  figure 6-2 Surface density

.

  y,z x, y,  covers one of the surfaces ( ) i S of S Ω  . On one of them an elementary surface element dσ is given by: previous integral can be transform to a surface integral. Finally, summing the various integrals for each reciprocal functions we arrive to:This last relation provides a clear view of the meaning of the surface density s σ . Given a function ( ) g M  being equal to one everywhere inside the space domain Ω , we have from A

  external interface with its surrounding because it is included in only one other phase subdomain (due to the fact that we ignore interface points where three phases coexistdomain k m Ω . The derivatives of the phase index are:

,

  the previous relations, it can be assumed that the numbers N , k N and km,k N ′ do not change or have small enough variations during the weighting characteristic time, the sums and the time integral of This last relation provides a way for writing down weighting quantities where the phase function derivatives are involved. If the function g is a vector component j A , the previous relation is written:

  the three components m k j of a vector quantity k j  associated to a phase k , the relation between the weighting of the divergence .

FIGURESFIGURE 1 :

 1 FIGURES

FIGURE 9 :

 9 FIGURE 9 : A DISTRIBUTED PHASE k ........................................................................................................

FIGURE 10 :

 10 FIGURE 10: INTERFACE ILLUSTRATION.................................................................................................... FIGURE 11: CONTINUITY BY PIECES. ......................................................................................................... FIGURE 12 PHASE SPACE AND TIME DISTRIBUTIONS ........................................................................... FIGURE 13 WEIGHTING CONCEPT .............................................................................................................. FIGURE 14 INTERFACE BALANCE .............................................................................................................. FIGURE 6-1 : PHASE DOMAIN ......................................................................................................................... FIGURE 6-2 SURFACE DENSITY .....................................................................................................................

include some subdomains of the other phases k′ with k

	Each phase subdomain	i Ω ′ ′ k m	( ) t	of a phase k′ is defined by a function	f	i ′ ′ k m;k m	( ) M ,t 	and
	the condition	( )
									k ′ ≠ (see
	figure 10 below). Let us assume that we have k m;k N	′ inclusions of phase k′ inside k m Ω .
	They are labelled	i Ω ′ ′ where a sequence { } k m i m′ with	i	1 ,.....,N  ∈ 	k m;k	′	  identifies the k′
	phase subdomains which are inside	Ω	( ) k m t	.

Table 1 : balance eqution terms

 1

Symbol # stands for a scalar quantity without dimensions.

0

MATH COMPLEMENT

Here after the reader will find definitions and justifications of properties used above within the present paper. This annex allows lightening chapters which are devoted to physics main aspects.

WEIGHTED FUNCTIONS DERIVATIVES

Given g a function defined for each point of a weighting domain

Due to the integral properties, this weighting operation can be seen as the composition of two weighting one space and time:

Its time derivative is:

The integration limit being constant, integration and derivative can be permuted:


François Roubille

Nadia Bouabdallaoui

Simon Kouz

David D Waters

Rafael Diaz

Aldo P Maggioni

Fausto J Pinto

Jean C Grégoire

Habib Gamra

Ghassan S Kiwan

Colin Berry

José Lopez-Sendon

Wolfgang Koenig

Laurent Delorme

Meyer Elbaz

Pierre Coste

Mylène Provencher

Zohar Bassevitch

Lucie Blondeau

Philippe L L'allier

Marie-Claude Guertin

Jean-Claude Tardif
email: jeanclaude.tardif@icm-mhi.org

Low-Dose Colchicine in Patients With Type 2 Diabetes and Recent Myocardial Infarction in the COLchicine Cardiovascular Outcomes Trial (COLCOT)

OBJECTIVE

The cardiovascular benefits of low-dose colchicine have been demonstrated in patients with coronary disease. Its effects were evaluated in this prespecified analysis in patients with type 2 diabetes (T2D) from the COLchicine Cardiovascular Outcomes Trial (COLCOT).

RESEARCH DESIGN AND METHODS

COLCOT was a randomized, double-blinded trial of colchicine, 0.5 mg daily, versus placebo initiated within 30 days after a myocardial infarction.

RESULTS

There were 959 patients with T2D enrolled and monitored for a median of 22.6months. A primary end point event occurred in 8.7% of patients in the colchicine group and in 13.1% in the placebo group (hazard ratio 0.65; 95% CI 0.44-0.96; P = 0.03). Nausea was reported in 2.7% and 0.8% in the study groups (P = 0.03), and pneumonia occurred in 2.4% and 0.4% (P = 0.008).

CONCLUSIONS

Among patients with T2D and a recent myocardial infarction, colchicine, 0.5 mg daily, leads to a large reduction of cardiovascular events. These results support the conduct of the COLCOT-T2D trial in primary prevention.

Inflammation is involved in the initiation, progression, and destabilization of atherosclerosis [START_REF] Roubille | Emerging anti-inflammatory therapies for atherosclerosis[END_REF]. Basic studies and clinical trials have demonstrated the benefits of reducing inflammation in atherosclerosis [START_REF] Roubille | Colchicine reduces atherosclerotic plaque vulnerability in rabbits[END_REF][START_REF] Ridker | Antiinflammatory therapy with canakinumab for atherosclerotic disease[END_REF]. Colchicine has been shown to exert beneficial effects, both in the COLchicine Cardiovascular Outcomes Trial (COLCOT) of patients with a recent myocardial infarction (MI) (4) and in the Low-Dose Colchicine-2 (LoDoCo2) trial of patients with stable coronary artery disease (CAD) [START_REF] Nidorf | LoDoCo2 Trial I. Colchicine in patients with chronic coronary disease[END_REF]. These results have led to the regulatory approval of lowdose colchicine for prevention of cardiovascular events in patients with atherosclerotic cardiovascular disease. Type 2 diabetes (T2D) is associated with increased rates of both macrovascular and microvascular disease and cardiovascular events [START_REF] Berry | Coronary heart disease in patients with diabetes: part I: recent advances in prevention and noninvasive management[END_REF]. Accumulating evidence suggests that inflammation is an important bridging link between diabetes and atherosclerosis [START_REF] Donath | Type 2 diabetes as an inflammatory disease[END_REF]. At least one-third of patients with acute coronary syndrome have T2D or prediabetes [START_REF] Boyer | Trends in clinical, demographic, and biochemical characteristics of patients with acute myocardial infarction from 2003 to 2008: a report from the American Heart Association Get With the Guidelines Coronary Artery Disease program[END_REF]. Patients with T2D are at higher risk of presenting a cardiovascular event after an MI (9). This appears to be at least in part due to more pronounced vascular inflammation [START_REF] Donath | Type 2 diabetes as an inflammatory disease[END_REF]. We hypothesized that patients with both T2D and a recent MI could draw significant benefit from colchicine.

RESEARCH DESIGN AND METHODS

The COLCOT trial was reported in detail elsewhere (4). Briefly, COLCOT was a randomized, double-blind, placebo controlled, investigator-initiated trial comparing colchicine, 0.5 mg once daily, with placebo in a 1:1 ratio. The protocol was approved by the institutional review board at each of the 167 centers in the 12 countries participating. All study support activities, including project coordination, data management, site monitoring, and statistical analyses, were performed at the Montreal Health Innovations Coordinating Center. Potential study end points were adjudicated by an independent clinical end point committee composed of experienced cardiologists and neurologists who were unaware of the trial group assignments. The study medication and matching placebo were provided by Pharmascience (Montreal, Quebec, Canada), which had no role in the trial design or conduct.

Adult patients were eligible if an MI had occurred within 30 days of enrollment. Patients were excluded if they had class III or IV heart failure, left ventricular ejection fraction <35%; stroke within the past 3 months; type 2 index MI, coronary bypass surgery within the past 3 years or planned, history of non-cutaneous cancer within the last 3 years, inflammatory bowel disease or chronic diarrhea, neuromuscular disease, or non-transient creatine phosphokinase greater than three times the upper limit of normal (ULN), significant non-transient hematological abnormalities, severe renal disease with serum creatinine greater than twice the ULN, severe hepatic disease, drug or alcohol abuse, chronic systemic steroid therapy, or history of sensitivity to colchicine.

Written informed consent was obtained from all patients before enrollment. Clinical evaluations occurred at 1 and 3 months following randomization and every 3 months thereafter.

The primary efficacy end point was a composite of cardiovascular death, resuscitated cardiac arrest, MI, stroke, or urgent hospitalization for angina requiring coronary revascularization in a time-to-event analysis.

The statistical analysis plan of this event-driven trial was published elsewhere (4) and included a prespecified subgroup analysis of the primary end point according to diabetes. The subgroup analysis was conducted according to the statistical analysis plan using positively adjudicated data, according to the intention-to-treat (ITT) principle and using a Cox proportional hazards model with terms for group, diabetes, and group × diabetes interaction. The treatment effect in patients with diabetes, along with its 95% CI, was estimated from this model. Patients with no event were censored at the time last known to be event free. The subgroup analysis of the primary end point was repeated on the per-protocol population of patients without major protocol deviations. To account for the occurrence of multiple primary end point events within patients, a recurrent-event analysis was undertaken in the subgroup of patients with diabetes, with the use of a negative binomial regression model. All statistical tests were two-sided and conducted at the 0.05 significance level. Statistical analyses were performed using SAS 9.4 software.

RESULTS

COLCOT enrolled 4,745 patients between December 2015 and August 2018. Patients were monitored for a median of 22.6 months. As reported previously, the primary end point occurred less often in patients in the colchicine group than in those allocated to placebo (hazard ratio 0.77; 95% CI 0.61-0.96; P = 0.02).

A total of 959 patients (20.2%) had T2D, with 462 and 497 being assigned to the colchicine and placebo groups, respectively. The characteristics of patients with T2D at baseline are shown in Table 1. Patients were enrolled a mean of 14.0 days after the index MI. The mean age was 62.4 years, and 22.2% were women. The mean BMI was 29.9 kg/m2, and 74.9% of patients presented with hypertension.

Most patients (90.8%) underwent a percutaneous coronary intervention for their index MI. Aspirin, another antiplatelet agent, a statin, and a b-blocker were taken by 98.3%, 96.9%, 98.7%, and 91.6% of the patients, respectively. Metformin, insulin, a sodium-glucose cotransporter 2 inhibitor, a glucagon-like peptide 1 receptor agonist, and a dipeptidyl peptidase 4 inhibitor were used by 75.6%, 30.9%, 17.3%, 8.2%, and 21.0% of the patients, respectively.

The ITT analysis showed the primary end point occurred in 8.7% of the patients in the colchicine group and in 13.1% of those in the placebo group (hazard ratio, 0.65; 95% CI 0.44-0.96; P = 0.03), as presented in Fig. 1 and Table 2. This result is consistent with the perprotocol analysis (hazard ratio 0.63; 95% CI 0.42-0.95; P = 0.03).

In the ITT analysis, the total number of primary end point events (first and recurrent) was 50 in the colchicine group and 93 in the placebo group, over periods of 10,311 and 10,847 patient-months of follow-up, respectively. Thus, the primary end point event rates per 100 patient-months were 0.48 in the colchicine group and 0.86 in the placebo group (rate ratio, 0.53; 95% CI 0.33-0.87; P = 0.01).

The interaction between history of diabetes (presence or absence) and the study treatment group was not statistically significant (P = 0.27). The rate of the primary end point in patients with no diabetes was 4.8% and 5.6% in the colchicine and placebo groups, respectively (hazard ratio 0.85; 95% CI 0.64-1.13).

The incidence of adverse events considered to be related to the study drug in patients with T2D was 14.6% in the colchicine group and 12.8% in the placebo group (Table 3). At least one gastrointestinal adverse event occurred in 15.3% of the patients in the colchicine group compared with 16.1% of those in the placebo group. Diarrhea was reported in 8.0% of the patients in the colchicine group and in 9.8% of those in the placebo group (P = 0.34). Nausea was more common in the colchicine group than in the placebo group (2.7% vs. 0.8%; P = 0.04). Pneumonia was reported as a serious adverse event in 2.4% of the patients receiving colchicine compared with 0.4% of those receiving placebo (P = 0.008).

CONCLUSIONS

In this prespecified subgroup analysis of COLCOT, the risk of the primary efficacy end point, consisting of a composite of cardiovascular death, resuscitated cardiac arrest, MI, stroke, or urgent hospitalization for angina requiring coronary revascularization, was reduced by 35% in the patients with T2D in the colchicine group compared with those in the placebo group.

Approximately 30-40% of MI patients have T2D or metabolic syndrome [START_REF] Boyer | Trends in clinical, demographic, and biochemical characteristics of patients with acute myocardial infarction from 2003 to 2008: a report from the American Heart Association Get With the Guidelines Coronary Artery Disease program[END_REF]. Patients with T2D present a very high risk of cardiovascular events after an MI, with an event rate nearly twice as high as in those without T2D [START_REF] Goldberg | The Care Investigators. Cardiovascular events and their reduction with pravastatin in diabetic and glucose-intolerant myocardial infarction survivors with average cholesterol levels: subgroup analyses in the Cholesterol and Recurrent Events (CARE) trial[END_REF]. In COLCOT, patients with T2D presented a 1.86fold higher risk of a primary end point cardiovascular event. Inflammation has been shown to contribute to the increased risk of cardiovascular events in patients with T2D [START_REF] Donath | Type 2 diabetes as an inflammatory disease[END_REF].

The most common adverse events observed were gastrointestinal. Diarrhea was reported in 8.0% of the patients in the colchicine group and in 9.8% of those in the placebo group, and nausea occurred in 2.7% and 0.8%, respectively. Pneumonia, as a serious adverse event, was more frequent in the colchicine group than in the placebo group (2.4% vs. 0.4%). The latter observation could be due to a play of chance or might reflect altered immunologic responses. This difference in the incidence of infections was not observed in LoDoCo2 [START_REF] Nidorf | LoDoCo2 Trial I. Colchicine in patients with chronic coronary disease[END_REF]. The role of HbA1c or LDL-cholesterol could not be analyzed. The effects of different glucose-lowering medications or possible hypoglycemic episodes could also not be assessed. Finally, patients were not stratified at inclusion for the presence of diabetes.

Duality of Interest.

J.-C.T. reports research grants from Amarin, AstraZeneca, Ceapro, Dal-Cor Pharmaceuticals, Esperion, Ionis, Merck, Novartis, and Pfizer, honoraria from DalCor Pharmaceuticals, HLS Pharmaceuticals, Pendopharm, and Pfizer, minor equity interest in DalCor Pharmaceuticals, and authorship of a patent on the use of colchicine after myocardial infarction in which he waived his rights and does not stand to gain financially. W.K. reports advisory board fees from AstraZeneca, Novartis, Amgen, Pfizer, The Medicines Company, DalCor, Kowa, Corvidia, OMEICOS, Daiichi-Sankyo, Novo Nordisk, New Amsterdam Pharma, TenSixteen Bio, Esperion, and Genentech, lecture fees from Bristol-Myers Squibb, Novartis, Amgen, Berlin-Chemie, Sanofi, and AstraZeneca, and grants and nonfinancial support from Abbott, Roche Diagnostics, Beckmann, and Singulex. No other potential conflicts of interest relevant to this article were reported.

Funding.

COLCOT was funded by the Government of Quebec, the Canadian Institutes of Health Research, and philanthropic foundations.

Author Contributions. F.R., N.B., S.K., D.D.W., R.D., A.P.M., F.J.P., J.C.G., H.G., G.S.K., C.B., J.L.-S., W.K., L.D., M.E., P.C., M.P., Z.B., L.B., P.L.L'A., M.-C.G., and J.-C.T. were involved in the conception, design, and conduct of the study and the analysis and interpretation of the results. F.R. and J.-C.T. wrote the first draft of the manuscript. All authors edited, reviewed, and approved the final version of the manuscript. J.-C.T. is the guarantor of this work and, as such, had full access to all the data in the study and takes responsibility for the integrity of the data and the accuracy of the data analysis.


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• When a robot writes a play. https://www.theaitre.com, Prague, 2021.

DRAMATURGIES NUMÉRIQUES

• Certaines dramaturgies numériques actuelles consistent à mettre en scène des textes écrits à l'aide de l'IA.

• L'option inverse, d'un texte dramatique écrit par un auteur humain et mis en scène à l'aide de l'IA, est plus difficile.

• Mais plus intéressante ! DRAMATURGIES NUMÉRIQUES

• Ce serait une dramaturgie qui peut être jouée par des acteurs virtuels et éclairée par des machines autonomes, qui ouvrirait aux auteurs de théâtre des perspectives nouvelles pour représenter sur scène des mondes intérieurs ou imaginaires.

• Penser les dramaturgies numériques comme tissage algorithmique des actions humaines dans des mondes virtuels spectaculaires.

• C'est une perspective qui demeure lointaine et nécessite des nouvelles recherches en informatique. Il faut aussi que des dramaturges s'emparent de ces nouvelles possibilités.

• D'où la nécessité de créer une nouvelle discipline … • Traduction du terme anglais « computer theater » introduit par Claudio Pinhanez au cours de sa thèse au MIT en 1996.

• Analogie avec les couples « computer music / informatique musicale » et « computer graphics / informatique graphique » qui sont communément admis.

• Lorsque l'informatique s'empare de la musique, elle y ajoute de nouveaux instruments (synthétiseurs, séquenceurs) et de nouvelles techniques de composition (aléatoires, algorithmiques) sur la base de la partition musicale traditionnelle. • Mireille Losco et Rémi Ronfard.

• ENSATT, Lyon, Octobre 2022.

• https://hal.science/JIT/page/actes-2022

• Cyrielle Garson et Rémi Ronfard.

• Univ. Avignon, Octobre 2024.

• Appel à participation à venir !

Le métavers et la conquête de l'ubiquité

• Demande de Roselyne Bachelot, ministre de la culture, Bruno Lemaire, ministre de l'économie et Cédric 0, secrétaire d'état chargé de la transition numérique (14 février 2022)

• Rapport remis à Rima Abdul Malak, ministre de la culture, et Jean-Noël Barrot, ministre du numérique (24 octobre 2022).

Un métavers est un service en ligne donnant accès à des simulations d'espaces 3D temps réel, partagées et persistantes, dans lesquelles on peut vivre ensemble des expériences immersives. » 36 Tentative de définition « On peut y accéder avec ou sans visiocasque et y commercer avec ou sans blockchain » 37 Tentative de définition QUATRE GRANDES CATÉGORIES Sans casque Avec casque Sans blockchain Avec blockchain 38 5 GRANDS AXES POUR UNE STRATÉGIE MÉTAVERSIQUE 1 Des questions de sociétés brûlantes 2 Des enjeux et opportunités culturelles 3 Des freins structurels à dépasser 4 Soutenir l'innovation 5 Adapter les cadres de régulation

	INTERNET ET WEB JEUX MULTI-JOUEURS EN LIGNE MONDES VIRTUELS RÉALITÉ VIRTUELLE BLOCKCHAIN DES ENJEUX ET OPPORTUNITÉS CULTURELLES
	De nouveaux media immersifs et créatifs
	« 39

• https://mission-metavers.fr/ DE QUOI LE MÉTAVERS EST-IL LE NOM? -Enjeu de souveraineté culturelle : protéger l'indépendance de la création française -Valoriser notre patrimoine par des attelages hybrides entre structures françaises technologiques et institutions culturelles -Créer un "

institut de recherche et de coordination"

	«
	pour rapprocher recherche publique et création artistique Paul Valéry

On saura transporter ou reconstituer en tout lieu le système de sensations que dispense en un lieu quelconque un objet ou un événement quelconque. Les oeuvres acquerront une sorte d'ubiquité. » « Faire entendre en tout point du globe, dans l'instant même, une oeuvre musicale exécutée n'importe où. » « En tout point du globe, et à tout moment, restituer à volonté une oeuvre musicale. » 43 Ubiquité de la musique

	UBIQUITÉ DU THÉÂTRE ?

• Faire voir en tout point du globe, dans l'instant même, une oeuvre théâtrale exécutée n'importe où.

• En tout point du globe, et à tout moment, restituer à volonté une oeuvre théâtrale.

« Pourquoi

les films de théâtre ça n'intéresse pas ? Est-ce que c'est une question de prise de vue ? Ou est-ce qu'il y a un a priori comme quoi le théâtre est ennuyeux ? Est-ce qu'on ne pourrait pas inventer ensemble un théâtre numérique populaire? »

  Le théâtre va conquérir une forme d'ubiquité dans le metavers • Cette reproduction mécanique du théâtre va changer l'art du théâtre, pour le meilleur ou pour le pire Pour accompagner ces transformations, il sera utile de fédérer les chercheurs et artistes en informatique théâtrale dans un institut de recherche et coordination dédié aux arts immersifs

	EN GUISE DE CONCLUSION

• • Le théâtre privé pourra se développer dans les métavers privés • Le théâtre public aura besoin d'un métavers de service public •

• Les captations vidéo ne sont pas suffisantes