domain: phys math

text:
Alain Mailliat

Forward and Backward Lagrangian Probability Density Properties 0 Forward and Backward Lagrangian Probability Density Properties

INTRODUCTION

A basic relationship between forward and backward lagrangian probability density functions (pdfs.) is derived for any continuous medium if mass conservation is postulated. For an incompressible medium these pdfs are identical: see equation ( 45). This identity is used to generalize some previous results obtained under more restrictive hypotheses. It is shown:

1. Identity of forward and backward variances for dispersion was established under the condition of unstrained stationary flow [1]. The same identity is obtained hereafter under homogeneity condition only.

2. Identity of Eulerian and Lagrangian velocity moments was established under homogeneous and stationary flow conditions [START_REF] Tennekes | A first Course in Turbulence[END_REF]. This identity and the equality of the lagrangian and eulerian velocity pdfs are established here for homogeneous and unsteady conditions.

These results were published first year 1981 in French in the "comptes-rendus" of the French Science Academy [START_REF] Mailliat | Sur les propriétés des densités de probabilité lagrangienne[END_REF]. Such an issue imposes a very concise paper limiting additional developments and explanations. We propose herewith such additions providing an opportunity for longer explanations of how such results are obtained.

Chapter two is devoted to the description of deterministic random displacement of a continuous medium, e.g. a turbulent fluid flow. In addition, elementary elements of probability tools are provided in connection with the developments of the chapter three. This one addresses the question of forward and backward displacements and their respective lagrangian pdfs and proves their equality. Finally, chapter four establishes the results 1 and 2 mentioned above in this introduction using forward and backward pdfs. equality.

Forward and Backward Lagrangian Probability Density Properties 1 2 DETERMINISTIC CONTINUOUS MEDIUM DISPLACEMENTS.

DEFINITIONS

Displacements

In order to represent the movements of continuous systems, e. g. fluid flows, we follow the proposal of P. Germain and P.

Muller [START_REF] Germain | Introduction à la mécanique des milieux continus[END_REF]. They state that a set of space domains ( )

t Σ
which vary according to time defines the configurations of a cinematic system if and only if, for any times 1 t and 2 t there is a bijective mapping f of ( )

1 t Σ on ( ) 2 t Σ Figure 1 : Cinematic Sytem Mapping
This mapping is written as ( )

2 1 f t x,t 
where the right part of vertical bar gives the location and the time of the point under consideration -here x  at 1 t . The location at time 2 t of this point ( )

1
x,t  is ( )

2 1 y f t x,t =  
. The mapping is assumed having the following properties:

Identical application:

(1) ( )

f t x,t x =   .
Transitivity of the application:

(2) ( ) ( ) ( )

3 1 3 2 1 2 f t x,t f t f t x,t ,t =  
.

So to say, the application related to two displacements of a same point is obtained by a composition of the applications related to each displacement, see figure (1) above.

The inverse application 1 fis deduced from [START_REF] Tennekes | A first Course in Turbulence[END_REF] where we put ( )

2 1 y f t x,t =  
and 3 1 t t = .

So ( ) ( )

1 1 1 2 f t x,t f t , y,t =  
and applying (1) ( )

1 1
f t x,t x =   we conclude that:

(3) If ( )

2 1 y f t x,t =   then ( ) ( )
1 1 2 2 1
x f t , y,t f t y,t

- = ≡   
.

The locations y  given by the function ( )

2 1 y f t x,t =  
when 2 t changes can be regarded as the successive positions of a punctual particle trajectory identified by its position x  at time 1 t taken as a reference time.

Lagrangian and eulerian velocities

Having identified a particle through its location x  at a reference time 1 t , we define its velocity as the time derivative of its displacement.

(4) ( ) ( )

1 1 L f t x,t V t x,t t ∂ = ∂   .
For various times t ( )

1 L V t x,t 
provides the velocities of a same particle along its trajectory and is called its lagrangian velocity. Now, considering the reference time

1 t t = , ( )
1 1 L V t x,t 
gives the velocity of a particle located at x  when it is identified, before any future displacements. We define a velocity field at each point of the space domain ( )

1 t Σ by ( ) ( ) 1 1 1 E L U x,t V t x,t =  
. This velocity is called the eulerian velocity at point x  at time 1 t . Such an eulerian velocity field is no longer related to a specific particle but is regarded as a velocity field defined on ( )

1 t Σ .
The question is to know if such an eulerian velocity field ( )

E U y,t 
can be defined at any other instant t in the domain ( )

t Σ
in the same way, so to say, through the velocity of the particle which is located at this point y  at time t . Precisely, using the inverse function defined by (3), the location x  at the reference time 1 t of the particle which is at y  at time Forward and Backward Lagrangian Probability Density Properties 3 t is ( ) ( )

1 1 1 x f t y,t f t , y,t - = ≡  
 . Then its lagrangian velocity when it arrives at y  is ( )

1 L V t x,t 
. We states that the eulerian velocity ( )

E U y,t 
is defined by the relation:

(

( )

1 1 1 1 E L x f t y ,t f t x,t U y,t V t f t y,t ,t t = ∂ = = ∂      .
But, the previous equality makes sense only if this definition of E U is independent of the reference time. The derivative of L V with respect to 1 t is:

( ) ( )

3 3 1 1 1 1 1 1 1 1 1 1 1 L L L L L E j j j j j X x X x j V t X ,t V t X ,t V t x,t V t x,t V t x,t U x,t t t X X = = = = ∂ ∂ ∂ ∂ + = + ∂ ∂ ∂ ∂ ∑ ∑          
, where we use the definition ( ) ( )

1 1 1 E L U x,t V t x,t =   .
To demonstrate that the derivative is zero, and relation ( 5) makes sense, we return to (3) with ( )

1 x f t y,t =   then ( ) ( ) 1 1 y f t f t y,t ,t =  
and we derive this last equality with respect to 1 t :

3 1 1 1 1 1 0 L j j j X x X x f t X ,t f t X ,t V t y,t t X = = = ∂ ∂ = + ∂ ∂ ∑        . (7) ( ) ( ) ( )
The quantity ( )

1 L j V t y,t 
is the velocity at 1 t of the particle which is at y  at t . This one was at x  at 1 t according to ( )

1 y f t x,t =   , then ( ) ( ) ( ) 1 1 1 1 L L E j j j V t y,t V t x,t U x,t = =    and (7) becomes:
(8) ( ) ( ) ( )

3 1 1 1 1 1 0 E j j X x f t X ,t f t x,t U x,t t X = ∂ ∂ = + ∂ ∂ ∑     
, Taking the derivative of the previous relation with respect to t we arrive to Forward and Backward Lagrangian Probability Density Properties 4 (9) ( ) ( ) ( )

3 1 1 1 1 1 0 L L E j j X x V t X ,t V t x,t U x,t t X = ∂ ∂ = + ∂ ∂ ∑     
, which demonstrates that the relation ( 6) is null and gives the possibility to define a eulerian velocity field at any time. This possibility is based on the transitivity condition (3) of the displacements.

RANDOM DISPLACEMENTS OF A CONTINUOUS MEDIUM.

The application f with the previous properties describes one occurrence or realization of a cinematic system. Having in mind the study of systems which exhibit random displacements, e.g. turbulent flows, we need to represent not only one occurrence of such systems but a large number of its realizations from which mean properties can be derived from the theory of probabilities.

Let us assume that the collection of occurrences of the same physical system is a continuous set

{ } Λ λ =
. For each realization λ there is a bijective application ( )

2 1
f t x,t ;λ  having the properties (1), (2), (3) which describes the system through a mapping of the space domains ( )

1 t ; Σ λ on ( ) 2 t ; Σ λ
. Each application f is call a stochastic process (this way of handle random functions can be found in [START_REF] Wiener | Non Linear Problems in Random Theory[END_REF]).

To handle such stochastic processes we provide Λ with a σ-algebra 1 Θ of subsets and a probability measure Π (French readers could find a simplified theory of probability theory in [START_REF] Mailliat | Les milieux aérosols et leurs représentations[END_REF], similar information are available in [START_REF] Lumley | Stochastic Tools in Turbulence[END_REF]). The probability measure ( ) Π ε of any subset ε of Λ belonging to Θ is given by a Lebesgue integral:

(10) ( ) ( ) d ε Π ε Π λ = ∫
The physicist needs to associate probability measures defined for subsets of Λ with events in the physical domain of the continuous medium. For example, after having identified a particle by its location x  and time 1 t , an event will be to find this particle later at 2 t in a given space domain Ω of 3  , see figure [START_REF] Tennekes | A first Course in Turbulence[END_REF] below. The physicist wants to determine the subset ε of the realizations for which such an event is observed and, finally, to obtain its probability measure. The way for answering this question is to force a measurability property on the stochastic processes f . Such a property provides a link between subsets of the abstract set Λ of the realizations -having a probability measure-with the events of the physical system, so to say with the subsets of 3  . The measurability is stated as:

(11) The function ( )

( ) 5
3 2 1 f : y f t x,t ; Λ λ → =   
gives the position at time 2 t of the particle which is located at x  at time 1 t for the realization λ of the physical system. This is mainly this application we will consider hereafter.

LAGRANGIAN PROBABILITY DENSITY FUNCTIONS.

FORWARD AND BACKWARD LAGRANGIAN PROBABILITY DENSITY FUNCTIONS.

Let us define a domain ( )

D p  of 3
 by: ( ) { }

3 1 2 3 i i D p m : m p , i , , = ∈ < ∀ =    . ( ) D p
 is an element of the Borel σ-algebra on 3  . We identify the subset ε of realizations for which a point located at x  at time 1 t arrives inside the domain ( )

D p  at time 2 t : (12) ( ) ( ) ( ) ( ) { } 2 1 2 1 D p ,t ,x,t : f t x,t ; D p ε λ λ = ∈     .
The function f being measurable, this subset ε is an element of Θ , the σ-algebra on Λ and, therefore, its probability measure exists:

2 Let be 1 E and 2 E two sets; 1 Θ and 2 Θ their respective σ-algebra;

1 2 f : E E → a function. This function is measurable if ( ) 1 2 2 2 1 A f A Θ Θ - ∀ ∈ ∈ . (13) ( ) ( ) ( ) ( ) ( )
( )

2 1 2 1 D p ,t ,x ,t D p ,t ,x,t d ε Π ε Π λ = ∫    
, and can be used to define a probability measure P on 3  through the relation:

(

( )

2 1 2 1 P D p ,t ,x,t D p ,t ,x,t Π ε =    
.

We call distribution function the quantity

2 1 L X X F (15) ( ) ( ) ( ) 2 1 2 1 2 1 L X X F p,t ,x,t P D p ,t ,x,t =     . Assuming 2 1 L X X F derivable, the associated probability density function (pdf) 2 L X X W is: (16) ( ) ( ) 2 1 2 1 2 1 2 1 1 2 3 L L X X X X F p,t ,x,t W p,t x,t p p p ∂ = ∂ ∂ ∂    
.

With this writing the event on left hand of the vertical bar is the random event X -here to be located at p  at 2 t .On the right side of the bar we have the condition event X -here to be located at x  at 1 t . So, we read

2 1 L X X F or 2 1 L X X W the distribution function or pdf of event 2 X if the event 1 X is realized.
This pdf ( ) It means that we follow the particle located at this point 1 M  , therefore, we can call it a forward lagrangian pdf. The Figure 2 above illustrates this question for 3 realizations of the system. Using this pdf we can write:

2 1 2 2 1 1 L X X W M ,t M ,t   provides the probability ( ) ( ) 2
(17) ( ) ( ) ( ) ( ) 2 1 2 3 2 2 1 1 2 1 1 L X X M P M ,t ,M ,t W z ,t M ,t d z Ω Ω = ∫      .
The mean value f  of the random location ( )

2 1 1 f t M ,t ;λ  at 2 t of a point started from 1 M  at 1
t can be obtained both from the Lebesgue integral on Λ or an integration on 3  :

(18)

( ) ( ) ( ) ( ) 2 1 3 3 2 1 1 2 1 1 2 1 1 L X X f t M ,t f t M ,t ; d z W z ,t M ,t d z Λ λ Π λ = = ∫ ∫       
.

The average value of any function ϕ of the random position of a particle located at 1 1 M ,t  is obtained as:

(19) ( ) ( ) ( ) ( ) ( ) ( ) 2 1 3 3 2 1 1 2 1 1 2 1 1 L X X t ,M ,t f t M ,t ; d z W z ,t M ,t d z Λ ϕ ϕ λ Π λ ϕ = = ∫ ∫       . If the point 1 M 
is chosen inside the space domain where the continuous medium is never found at time 1 t : so to say ( )

1 1 M t , λ Λ Σ λ ∀ ∈ ∉   , the quantity 2 1 L X X W cannot be defined. Then, we impose 2 1 0 L X X W
= to extend its definition domain to 3  . Now, we consider what is called the backward problem. We look for the random locations at time 1 2

t t < of a point if this one is y  at time 2 t .
Such locations are given by the function ( )

1 2 1 f t y,t ;λ - 
which is also ( )

1 2
f t y,t ;λ  according to relation [START_REF] Mailliat | Sur les propriétés des densités de probabilité lagrangienne[END_REF], see also figure (3) below.

As was done previously, we define the subset ϖ of the realizations of Λ for which a point identified by its position y  at time 2 t was inside the domain ( )

D p  at time 1 t :
Forward and Backward Lagrangian Probability Density Properties 8 (20)

( ) ( ) ( ) ( ) { } 1 1 1 2 D p ,t , y,t : f t y,t ; D p ϖ λ λ = ∈     .
The function ( )

1 2
f t y,t ;λ  being measurable, the probability measure of this subset ϖ defines a probability measure on 3  , its distribution function and its probability density through the following relations:

( ) ( ) ( ) ( ) ( ) 1 2 1 2 P D p ,t , y,t D p ,t , y,t Π ϖ =     (21) ( ) ( ) ( ) 1 2 1 2 1 2 L X X F p,t y,t P D p ,t , y,t =     ( ) ( ) 1 2 1 2 1 2 1 2 1 2 3 L L X X X X F p,t y,t W p,t y,t p p p ∂ = ∂ ∂ ∂     This pdf ( ) 1 2 1 1 2 2 L X / X W M ,t M ,t  
provides the probability to find at time 1 t inside a space domain ( )

1 2 1 3 1 1 2 2 1 2 2 L X X M P M ,t ,M ,t W z ,t M ,t d z Ω Ω = ∫      .
It means that we follow backward the particle located at this point 2 M  , therefore, we can call backward lagrangian this pdf:

The mean value of any function ϕ of the random departure location of a particle arrived at

( ) ( ) ( ) ( ) ( ) ( ) 1 2 3 3 1 2 1 1 2 2 1 2 2 L X X t M ,t f t M ,t ; d z W z ,t M ,t d z Λ ϕ ϕ λ Π λ ϕ = = ∫ ∫       . If the point 2 M 
is chosen inside the space domain where the continuous medium is never found at time 2 t : so to say ( )

2 2 M t , λ Λ Σ λ ∀ ∈ ∉   , the quantity 1 2 L X X W cannot be defined. Then, we impose 1 2 0 L X X W
= to extend its definition domain to 3



For fluid flow studies, the density of the fluid at a given point and time ( )

x,t ρ
 is required. This quantity being considered as a random variable which changes for the various realizations of the flow, so we write it ( )

x,t; ρ λ 
. This function is assumed continuous, derivable and measurable:

(24) the function ( ) 4

x,t; :

ρ λ Λ + × →  
 is measurable with regard to the product B Θ × where B is a Borel σ-algebra on 4  .

Random density and displacements being defined on the same set of occurrences Λ their joint event probability can be defined as previously through the probability on Λ and its probability image on 3 + ×   .

For example; considering a given time 2 t and a point 2 M  , a first event is to find there the density value within a given 2 δ × interval of 0

ρ : ( ) [ ] 2 2 0 0 M ,t ; , ρ λ φ ρ δ ρ δ + ∈ = - + ⊂  
and a second event: to find inside a space domain ( )

2 M Ω  centered on 2 M  one particle if
this one is identified previously by its location 1 M  at a given time 1 t .

The probability of these joint events is obtained from its pdf as:

0 2 1 0 2 3 2 2 1 1 2 1 1 L R, X X M P , M ,t ,M ,t W r,z ,t M ,t drd z ρ δ ρ δ Ω φ Ω + - = ∫ ∫      (25) ( ) ( ) ( ) ( )
. This is the forward lagrangian join pdf. The corresponding backward pdf can be constructed in the same way. We look for the joint events considering a given time 1 t and a point 1 M  , a first event is to find there the density value within a given interval φ and a second event being to find inside a space domain ( )

1 M Ω  centered on 1 M  one particle if
this one is identified by its location 2 M  at a given time 2 t . The backward pdf is written :

( )

1 2 1 2 2 L R, X X W r,z ,t M ,t   .
It has to be understood that densities and displacements are not independent random variables if mass conservation is assumed during the displacements of the continuous medium. Since the mass included in a small volume 3 d x centered on point x  at 1 t has to be equal to the mass of fluid included in the volume 3 d z centered on point z  at 2 t if x  is the location where was the point arrived at z  , the mass conservation implies:

(26) ( ) ( ) ( ) 3 1 3 2 1 z ,t ; d z f z ,t ; d x ρ λ ρ λ - =  
, where ( )

1 f z
- is a short writing for ( )

1 2 1 f t z ,t ;λ -  .
Furthermore, the points x  and z  being related by the function ( )

2 1 f t x,t ;λ 
, the volume 3 d z is given by the Jacobian of the transformation f :

)

1 1 1 1 1 2 3 3 3 2 2 2 1 2 3 3 3 3 1 2 3 x f z f f f x x x J f J f f f f d z d x J x J x x x x f f f x x x - = ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = = ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂  
.

Using the previous equalities, it appears that the random variables density and displacement satisfy the relation:

( )

1 1 1 2 x f z f z ,t ; J f z ,t ; J x ρ λ ρ λ - - = =     ,
and, therefore, f and ρ are not independent quantities. We can deduce also from (27) that for an incompressible fluid, the elementary volumes have a same measure 3 3 d z d x = .

FORWARD AND BACKWARD LAGRANGIAN PDF IDENTITY.

Let us consider the space domain ( )

1
t , Σ λ where the fluid is at time 1 t for a realization λ .

Its transformation ( )

2 t , Σ
λ for the same realization at a subsequent moment 2 t is obtained through the mapping ( )

2 1 f t ,x,t ;λ  .
We intend to express the integral:

(29) ( ) ( ) ( ) ( ) ( ) 1 3 2 1 2
( ) 2 t , Σ λ then ( ) ( ) 2 1 2 1 I t , y,t ; t , y,t λ ϕ =   .
To determine I we can perform the integration in the domain ( )

2 t , Σ λ using a new
variable define by : ( )

2 1 z f t x,t ;λ =  
. The previous variable x  is given by the relation (

( )

1 2 1 x f t ,z ,t ;λ - =  
. Hereafter, for convenience, we will only write down the variables used for the transformations. We see from (26) that the elementary volumes are related through the mass conservation:

(30) ( ) ( ) ( ) 3 1 3 2 1 z ,t ; d z f z ,t ; d x ρ λ ρ λ - =  
, Using this equality and (28) in (29) we have

( )

2 2 1 3 1 1 t , z ,t I y z y f z d z f z ,t Σ λ ρ δ ϕ ρ - - = - ∫∫∫       .
If the point y  belongs to the space domain ( )

2 t , Σ
λ the result is:

Forward and Backward Lagrangian Probability Density Properties 12 (32)

( ) ( ) ( ) ( ) ( ) ( ) 2 1 1 1 y,t I y f y f y ,t ρ ϕ ρ - - =     .
Using a Dirac distribution, the previous result can be re-written as an integral in ( )

1 t , Σ λ : (33) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 1 3 1 t , y,t I y x f y x d x x,t Σ λ ρ δ ϕ ρ - = - ∫∫∫      
. , Returning to (29), the first form of the integral we can write:

(34) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 1 3 1 0 t , y,t f x y x f y x d x x,t Σ λ ρ δ δ ϕ ρ -     -- - =       ∫∫∫       
. Such an equality having to be verified for any function ( )

x ϕ
 , we deduced the following relationship:

(

x f y f x y y,t

x,t

δ δ ρ ρ - - - =       , or ( ) ( ) ( ) ( ) ( ) ( )
1 2 1 y,t f x y x,t x f y σ δ σ δ - - = -       where 1 σ ρ =
is the fluid specific volume. Writing down all the variables, we have:

(36) ( ) ( ) ( ) ( ) ( ) ( ) 1
σ λ δ λ σ λ δ λ - - = -       .
Then, using ( ) ( )

1 2 1 1 2 f t y,t ; f t y,t ; λ λ - =  
from (11) and the property ( ) ( )

X X δ δ - =  
of Dirac distributions we arrive to:

σ λ δ λ σ λ δ λ - = -       .
On the left side of (37) the Dirac is related to the event to find the random arrival location at time 2 t of the particle which is departed from x  at time 1 t and, on the right side, the Dirac is related to the event to find the random departure location at time 1 t of the particle which is arrived at y  at time 2 t .

For a given fluid flow realization λ , (37) is verified if the two points y  and x  belong to ( ) ( )

2 1 t , t , Σ λ Σ λ ∩
. If we need (37) for all the realizations, so to say, all the values of Λ , we have to choose y  and x  in the space domain where the fluid is always found whatever is its realization, so to say ( ) ( )

1 2 t , t , λ Λ Σ λ Σ λ ∀ ∈



. This condition is immediately verified if the fluid flow remains inside a given space domain within permanent physical boundaries or if the fluid domain is the same ( ) ( )

t, t Σ λ Σ =
whatever the realization is. We take the mean of both sides of (37):

(38) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
Λ Λ σ λ δ λ Π λ σ λ δ λ Π λ - = - ∫ ∫      
.

The left and right sides can be obtained through integration in 3

+ ×   : (39) ( ) ( ) ( ) ( ) 2 1
3

1 2 3 3 2 1 3 1 2 L ,X X L ,X X s y z W s,z ,t x,t dsd z s x z W s,z ,t y,t dsd z Σ Σ δ δ + + - = - ∫ ∫ ∫ ∫            
.

In this relation we used the pdf for the specific volume The joined event pdf can be rewritten as the product of the specific volume conditional pdf if the particle has made the displacement between x

 to z  z  ( ) ( ) 2 1 2 1 X X W s z ,x,t ,t Σ -   by the pdf to be in z  if the particle is started from x  ( ) 2 1 2 1 L X X W z ,t x,t   so: (40) ( ) ( ) ( ) ( ) 2 1 2 1 2 1 2 1 2 1 2 1 L L ,X X X X X X W s,z ,t x,t W s z ,x,t ,t W z ,t x,t Σ Σ - =      
.

Doing the same for the backward pdf:

Forward and Backward Lagrangian Probability Density Properties 14 (41)

( ) ( ) ( ) ( ) 1 2 1 2 1 2 1 2 1 2 1 2 L L ,X X X X X X W s,z ,t y,t W s z , y,t ,t W z ,t y,t Σ Σ- =      
.

Introducing these pdfs in (39) we have: for its left side

(42) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 1 2 1 3 2 1 2 1 2 1 2 1 3 2 1 2 1 2 1 2 1 2 1 2 1 L X X X X L X X X X L X X X X s y z W s z ,x
Σ Σ δ σ + + - - - - = = ∫ ∫ ∫                 
.

And for the right one:

( )

1 2 1 2 3 1 2 1 2 1 2 1 2 3 1 2 1 2 1 2 1 2 1 2 1 2 L X X X X L X X X X L X X X X s x z W s z ,
Σ Σ δ σ + + - - - - = = ∫ ∫ ∫                 
.

So, we obtain the relationship between forward and backward lagrangian pdfs as:

(44)

( ) ( ) ( ) ( ) ( ) ( ) 2 1 1 2 2 1 1 2 2 1 2 1 1 2 1 2 L L X X X X X X X X y,x,t ,t W y,t x,t x, y,t ,t W x,t y,t σ σ - - =         ,
where the mean values of the specific volumes are obtained using conditional pdfs.

This relation is simplified if the continuous medium has a constant specific volume (a constant density):

(45)

( ) ( ) 2 1 1 2 2 1 1 2 L L X X X X W y,t x,t W x,t y,t =     .
This relationship states that the pdf to find a particle at y  if it started from x  is identical to the pdf to find a particle at a starting point x  if it arrived at y  .

Integrating the previous relation on 3  with respect of y  the random quantity of

2 1 L X X
W the result is one, then for the left side we obtain:

(46) ( )

1 2 3 3 1 2 1 L X X W x,t y,t d y = ∫   
.

Doing the same for

1 2 L X X W
with respect of x  we have:

(47) ( )

2 1 3 3 2 1 1 L X X W y,t x,t d x = ∫   
.

The main reasons of such equalities (44), ( 45) are

• The mapping transitivity property which allows defining a reciprocal function of the forward displacement providing a backward location whatever the later position is. This hypothesis states an infinite memory of the continuous medium. • The existence of a conservative measure for the elementary volumes of the continuous medium through its mass conservation. • The hypothesis that the mapping is a measurable function which allows using probabilities tools.

CONSEQUENCES.

Hereafter, we consider two consequences of the forward and backward pdf identity: the equalities of forward and backward dispersion average properties and the lagragian and eulerian velocity pdfs.

FORWARD AND BACKWARD DISPERSIONS.

We identify a point x  somewhere in the fluid domain. This point can be regarded as a particle departure location for a forward displacement. Taking t as starting time, after a flight of duration τ the random arrival point of the particle is ( )

f t x,t; τ λ + 
and its random forward displacement is :

( ) ( ) F ,x,t; f t x,t; x ∆ τ λ τ λ =+ -     .
Let us consider a second kind of displacement we call backward. Now, the point x  is a fixed arrival point for a particle arriving at time t τ + . Having considered the forward displacements during the period τ starting à t , to compare to backward characteristics in case of unsteady system we need to consider the same period of time. Therefore, we look for the random particle departure location which arrives at this fixed point after a flight of duration τ . The random departure point of the particle is ( )

f t x,t ; τ λ + 
and its random backward displacement is:

( ) ( ) B ,x,t; x f t x,t ; ∆ τ λ τ λ = - +    
. This displacement B ∆  points to the departure locations while F ∆  points to the arrival locations instead.

Hereafter, we replace 2 X and 1 X respectively by D and A which stand for departure at time t and arrival at time t τ + . According to the case, arrival or departure location is the random variable.

The mean value of a function g of a random displacement F ∆  is obtained using

L A D W because the random variable z  is the arrival point (48) ( ) ( ) ( ) 3 3
L A D F g ,x,t g z x W z ,t x,t d z τ τ = - + ∫       .
The corresponding backward quantity is:

L D A B g ,x,t g x z W z ,t x,t d z τ τ = - + ∫       (49) ( ) ( ) ( ) 3 3
, where we use

L A D B g ,x,t g x z W x,t z ,t d z τ τ = - + ∫       . (50) ( ) ( ) ( ) 3 3
In general (48) and (50) are different. But, under the condition of statistical homogeneity of the continuous medium, pdfs like

L A D W or L D A W
depends only of the difference of locations instead of the locations themselves. So, the mean values F g and B g do no depend of the point considered and are obtained according to:

L L A D A D F L L A D A D B F B g ,t g z x W z x,t t d z g u W u,t t d u g ,t g x z W x z ,t t d z g u W u,t t d u g t ,t g t ,t τ τ τ τ τ τ τ τ = - - + = + = - -+ = + + = + ∫ ∫ ∫ ∫                 . (51) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 3 3 3 3 3 3 3 3
So, mean values of forward and backward displacement functions are identical. Hereafter we consider two consequences of such equality.

First case: the function g stands for the displacement itself:

(52) ( ) ( ) ( ) F B t ,t t ,t t ,t ∆ τ ∆ τ ∆ τ + = + = +    .
It means that the average location of the departure points of particles arriving at a given point after flight duration falls in line with the mean arrival locations of particles starting to this same point for same flight duration.

Second Case: the function g stands for the squared displacement used for diffusion studies in turbulent flow :

1 2 3

F B Fii Bii ii t ,
= + = + + = + = + ∀ =    τ ∆ τ ∆ τ ∆ τ ∆ τ ∆ τ +
Relations (53) show that the dispersion of the point locations which arrive at a point is equal to the dispersion of particles starting from this same point, in the same time interval

Adding the hypothesis of statistically stationarity of the continuous medium, the pdfs do not depend of the time itself but only of the time differences. Then, for homogeneous and stationary turbulent flow

( ) ( ) ( ) 2 2 2 F B ∆ τ ∆ τ ∆ τ = =  .

EULERIAN END LAGRANGIAN VELOCITY PDFS. IDENTITY.

Considering a particle identified by its location x  at time 1 t , its location at time 2 t is a random variable given by ( ) ( )

2 E V X W y,t υ  
based on the velocity of the particles which pass by this point at this time whatever their previous locations are at 1 t .

We consider the quantity (55)

( ) ( ) ( ) 2 1 2 1 L p V t x,t ; y f t x,t ; δ λ δ λ   - -       
, Which allows defining the pdf ( )

2 1 2 1 , , , , L V X X y t x t W υ  
of the joint events of a displacement ( )

)

2 1 2 1 2 1 1 2 L L p V t x,t ; y f t x,t ; p V t x,t ; x f t y,t ; δ λ δ λ δ λ δ λ     - - = - -               .
On the right side of (56) we use the relation between le lagrangian and eulerian velocities:

(57) ( ) ( ) ( ) ( ) 2 1 2 1 2 1 2 L L E V t x
δ λ δ λ δ λ δ λ     - - = - -              
.

The last Dirac on right side ( ) ( )

1 2
x f t y,t ;

δ λ -  
being related to the event to find the random departure location at time 1 t of a particle which arrives at y  at time 2 t . This right side of (58) provides the pdf ( )

1 2 , 1 2
, , ,

E U X X W x t y t υ  
of the joint events of the Eulerian velocity at 2 y,t  and the random initial location of a particle. If this pdf is integrated on all the possible random initial positions x  we get ( )

2 2 E V X W y,t υ  
without reference to a specific particle nor a reference time.

For a given fluid flow realization λ (56) and ( 58) are verified if the two points y  and x  belong to ( ) ( )

Λ Λ δ λ δ λ Π λ δ λ δ λ Π λ   - -       = - -     ∫ ∫      
The left and right sides can be obtained through integration on 3  for the velocities and on ( )

2 t Σ
for the locations :

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 1 3 2 1 2 3 2 3 3 2 1 3 3 1 2 L V ,X X t E V ,X X t p y z W ,z ,t x,t d d z p x z W ,z ,t y,t d d z Σ Σ δ υ δ υ υ δ υ δ υ υ - - = - - ∫ ∫ ∫ ∫           .
The result is:

(61) ( ) ( ) 2 1 1 2 2 1 1 2 L E V ,X X U ,X X W p, y,t x,t W p,x,t y,t =     .
Such an equality can be also obtained with another analysis based on the definition of the conditional pdf. Given both the set ( )

2 1 y,t x,t ε  
of realizations for which a particle started from x  at time 1 t is at y  at time 2 t and its subset ω of realizations belonging to ε for which the lagrangian velocity L V is inside a given velocity domain ( )

D p  : (62) ( ) ( ) { } ( ) ( ) ( ) ( )
ε λ Λ λ ω λ ε λ = ∈ =   = ∈ ∈                , the measure ( ) ω Π
is the conditional probability to find the lagrangian velocity of a particle inside ( ) D p  if this particle goes from x  to y  . Then, this measure provides the conditional pdf

( ) ( ) 2 1 2 1 V X X L , W p y,t x,t -   
which is multiplied by the displacement pdf to give the velocity and displacement joint event pdf, according to:

( ) ( ) ( ) ( ) 2 1 2 1 2 1 2 1 2 1 2 1 V X X L L L V,X X X X , W p, y,t x,t W p, y,t x,t W y,t x,t - =       .
Returning to the subset ω , for any of its element λ the particle being in x

 at 1 t is at y  at time 2 t so ( ) ( ) 2 1 2 L E V t x,t ; U y,t ; λ λ =  
and we can re-write ω as:

(63) ( ) ( ) ( ) ( ) 2 1 2 1 2 E p y,t x,t y,t x,t :U y,t ; D p ω λ ε λ   = ∈ ∈            ,
Forward and Backward Lagrangian Probability Density Properties 21

So ω provides

( ) ( ) 2 1 2 1 U X X E , W p, y,t x,t -   which is such as ( ) ( ) 2 1 2 1 U X X V X X E L W W - - =
Considering the joint event pdf ( )

2 1 2 1 2 1 V X X L L L V,X X X X W W W - =
, we can rewrite it as

( ) 2 1 2 1 2 1 U X X L E L V,X X X X W W W - =
. Then, taking into account the equality (45) between direct and backward displacements

2 1 1 2 L L X X X X W W =
for an incompressible fluid we arrive to

( ) 2 1 2 1 1 2 1 2 U X X L E L E V ,X X X X U,X X W W W W - = × =
which is the result (61) previously obtained by another way.

The pdf of the lagrangian velocity is obtained through an integration of

= = - ∫ ∫    .
The integration on y  is equivalent to integration on the random variable 1 X which disapperaes and, finally, we arrive to :

(66) ( ) ( )

2 2 U V L E W p t W p t = .
The previous egality between the pdf of the lagrangian and eulerian velocity pdfs implies The previous equality of the lagrangian and eulerian velocity pdfs implies the identity of the velocity moments.

(67) ( ) ( )

n n L E V t U t     =        
This result was established under homogeneous and stationary flow conditions using another way [START_REF] Tennekes | A first Course in Turbulence[END_REF] for stationay conditions. The present work allows to extend the result to unstationary conditions.

Such an identity of the lagrangian and eulerian velocity pdfs can be understood easily. In fact, the homogeneity hypothesis implies that statistical properties are the same whatever is the observation point choosen. So, there is no difference between tracking a particle at its succesive locations to obtain its statistical lagrangian properties or staying at an unique point and using the properties of the different particles which go through this point.

Figure 2 Forward

 2 Figure 2 Forward Lagangian pdf.

Figure 3

 3 Figure 3 : Backward Lagrangian pdf

  one being deduced from the second one trough elementary operations.

  quantity z  is the departure point. Carrying the relation (45) in the previous expression we obtain:

Forward

CONTENTS 1 INTRODUCTION 3 . 2 4 . 2

 13242 ............................................................................................................................................................................................................ 2.1.1 Displacements .................................................................................................................... 2.1.2 Lagrangian and eulerian velocities ..................................................................................... 2.2 Random displacements of a continuous medium. .............................................................. 3 LAGRANGIAN PROBABILITY DENSITY FUNCTIONS. ...................... 3.1 Forward and Backward Lagrangian probability density functions. ................................ Forward and Backward Lagrangian pdf Identity. .......................................................... 4 CONSEQUENCES. ............................................................................. Dispersions. ................................................................................ Eulerian end lagrangian velocity pdfs. identity................................................................

FIGURE 1 :

 1 FIGURE 1 : CINEMATIC SYTEM MAPPING ................................................................................................... FIGURE 2 FORWARD LAGANGIAN PDF. ...................................................................................................... FIGURE 3 : BACKWARD LAGRANGIAN PDF ...............................................................................................