Brownian Movement

The Brownian movement is a mathematical description of the random movement of “a large” particle immersed in a fluid and which is subjected to no other interaction but Choc S with “the small” molecules of the surrounding fluid. It results a very irregular movement from it from the large particle, which was described for the first time in 1827 by the biologist Robert Brown whereas it observed pollen of Clarkia pulchella (a species of North-American wild flower), then various other plants, in suspension in water.

The physical description most elementary of the phenomenon is the following one:

• between two shocks, the large particle moves in straight line with a constant speed;
• the large particle is accelerated when it meets a molecule of fluid or a wall.

This description makes it possible to describe successfully the behavior Thermodynamique Gaz (kinetic Théorie of the gases), as well as the phenomenon of diffusion.

Historical aspects

Brown saw in the fluid located at the interior of the grains of pollen (the Brownian movement was not observed on the grains of pollen themselves like often mentioned), of very small agitated particles of apparently chaotic movements. Those could be explained by flows, nor by no other known physical phenomenon. Initially, Brown thus allotted them to a vital activity. The correct explanation of the phenomenon will come later.

Brown is not exactly the first to have made this observation. It announces itself that several authors had suggested the existence of such a movement (in bond with the vitalistic theories of the time). Among those, some had actually described it. One can mention in particular the abbot John Turberville Needham (1713-1781), famous at his time for his great control of the microscope.

The reality of the observations of Brown was discussed throughout the XXe century. Taking into account the poor quality of the optics of which it laid out, some disputed that it could see truly the Brownian movement, which interests of the particles of some micrometers at the maximum. The experiments were remade by the English Brian Ford with the beginning of the year 1990, with the material employed by Brown and under the most similar possible conditions. The movement was indeed observed under these conditions, which validates the observations of Brown.

Mathematical rudiments

Concept of process stochastic

The difficulty of modeling of the Brownian movement lies in the fact that this movement is Aléatoire and that statistically, displacement is null: there is no overall movement, contrary to a wind or a current. More precisely:

• at a given moment, the vectorial sum speeds of all the particles is cancelled (there is no overall movement);
• if one follows a particle given during time, the Barycentre of its trajectory is its starting point, it “circles” around the same point.

Difficult under these conditions of characterizing the movement… The solution was found by Louis Bachelier in 1902. It showed that what characterizes the movement, it is not the arithmetic Mean positions < X > but the quadratic average $\backslash \; sqrt\; \{\backslash \; langle\; \backslash ,\; X^2\; \backslash ,\; \backslash \; rangle\; \backslash \}$: if X ( T ) is the distance from the particle to its starting position at the moment T , then:

$\backslash \; langle\; \backslash ,\; X^2\; (T)\; \backslash \; \backslash \; rangle\; \backslash \; =\; \backslash \; \backslash \; frac\; \{1\}\; \{T\}\; \backslash \; int\_\; \{0\}\; ^\; \{T\}\; x^2\; (\backslash \; tau)\; \backslash \; D\; \backslash \; tau$

It is shown that average quadratic displacement is proportional to time :

where D is the dimension of the movement (linear, plane, space), D the coefficient of diffusion, and T past time.

Formulate of Einstein

The preceding formula makes it possible to calculate the coefficient of diffusion of a couple particle-fluid. By knowing the characteristics of the diffusing particle or the fluid, one can deduce the characteristics from them from the other. By knowing the characteristics of both, one can evaluate the Nombre of Avogadro using the formula of Einstein (1905):

$D\; \backslash \; =\; \backslash \; \backslash \; frac\; \{R\; T\}\; \{6\; \backslash \; pi\; \backslash \; eta\; \backslash \; mathcal\; \{NR\}\; \_\; \{AV\}\; R\}$

where R is the Constante perfect gases, T the Température, $\backslash \; eta$ the viscosity of the fluid, R the ray of the particle and $\backslash \; mathcal\; \{NR\}\; \_\; \{AV\}$ the Nombre of Avogadro. The physicist Jean Perrin evaluated this number in 1908 thanks to this formula.

Energy considerations

The quantity of energy implemented by the Brownian movement is negligible on a macroscopic scale. One cannot draw some from energy to carry out a perpetual motion of second species, and to thus violate the Second principle of thermodynamics.

Some modelings in an Euclidean space

Equation of Langevin (1908)

See also: Equation of Langevin

In the approach of Langevin, the large Brownian particle of mass m animated at the moment T a speed $v\; (T)$ is subjected to two forces:

• a fluid force of friction of the type $v\; \backslash ,\; =\; \backslash ,\; -\; \backslash ,\; K\; \backslash ,\; v$, where K is a positive constant;
• a White vibration Gaussian $\backslash \; eta\; (T)$

Gaussian White vibration:

A white Bruit Gaussian $\backslash \; eta\; (T)$ is a stochastic Processus of null average:

$\backslash \; langle\; \backslash ,\; \backslash \; eta\; (T)\; \backslash ,\; \backslash \; rangle\; \backslash \; =\; \backslash \; 0$

and completely décorrélé in time; its function of correlation to two points is worth indeed:

$\backslash \; langle\; \backslash ,\; \backslash \; eta\; (t\_1)\; \backslash \; \backslash \; eta\; (t\_2)\; \backslash ,\; \backslash \; rangle\; \backslash \; =\; \backslash \; \backslash \; Gamma\; \backslash \; \backslash \; delta\; (t\_1-t\_2)$

In this formula, $\backslash \; Gamma$ is a positive constant, and $\backslash \; delta\; (T)$ is the distribution of Dirac.

In these two formulas, the average is taken on all the possible achievements of the Gaussian white vibration . One can formalize this by introducing a functional integral, still called Intégrale of way according to Feynman, defined for the measurement Gaussian known as “measurement of Wiener”. Thus, one writes:

$\backslash \; langle\; \backslash ,\; \backslash \; eta\; (t\_1)\; \backslash \; \backslash \; eta\; (t\_2)\; \backslash ,\; \backslash \; rangle\; \backslash \; =\; \backslash \; \backslash \; int\; \backslash \; left\; \backslash ,\; \backslash \; mathcal\; \{D\}\; \backslash \; eta\; (T)\; \backslash ,\; \backslash \; right\; \backslash \; \backslash \; eta\; (t\_1)\; \backslash \; \backslash \; eta\; (t\_2)\; \backslash \; \backslash \; textrm\; \{E\}\; ^\; \{-\; \backslash \; \backslash \; frac\; \{\backslash \; displaystyle\; \backslash \; dowry\; \{\backslash \; eta\}\; ^2\; (T)\}\{\backslash \; displaystyle\; 2\; \backslash \; Gamma\}\}$

where $\backslash \; dowry\; \{\backslash \; eta\}$ is the derivative of $\backslash \; eta$ compared to time T .

The basic principle of the dynamics of Newton led to the stochastic equation of Langevin:

Process of Orstein-Uhlenbeck

The process of Ornstein-Uhlenbeck is a stochastic process describing (inter alia) the speed of a particle in a fluid, in dimension 1.

One defines it as being the solution $X\_t$ of the following stochastic differential equation: $dX\_t=\; \backslash \; sqrt2dB\_t-X\_tdt$, where $B\_t$ is a standard Brownian movement, and with $X\_0$ a given random variable. The $dB\_t$ term translates the many random shocks undergone by the particle, whereas the term $-\; \{X\_t\}\; dt$ represents the force of friction undergone by the particle.

The formula of Itô applied to the process $\{e^t\}\; X\_t$ gives us: $d\; (\{e^t\}\; X\_t)\; =\; \{e^t\}\; \{X\_t\}\; dt+\; \{e^t\}\; (\backslash \; sqrt\; \{2\}\; \{dB\_t\}\; -\; \{X\_t\}\; dt)\; +\; \{e^t\}\; dt=\; \{e^t\}\; \backslash \; sqrt\; \{2\}\; \{dB\_t\}\; +\; \{e^t\}\; dt$, maybe, in integral form: $X\_t=\; \{X\_0\}\; e^\; \{-\; T\}\; +\; \backslash \; sqrt\; \{2\}\; e^\; \{-\; T\}\; \backslash \; int\_0^t\; \{e^s\}\; dB\_s$

For example, if $X\_0$ is worth Presque surely $x$, the law of $X\_t$ is a Gaussian law of average $xe^\; \{-\; T\}$ and of variance $1-e^\; \{-\; 2t\}$, what converges in law when $t$ tends towards the infinite one towards the reduced centered Gaussian law.

Steps randomly

See also: Walk randomly

One can also use a model of Marche randomly (or goes random), where the movement is made by discrete jumps between definite positions (there are then movements in straight line between two positions), such as for example in the case of the diffusion in the solids. If the x_i are the successive positions of a particle, then one has after N jumps:

Go randomly to a dimension of space ( Exemple )

Randomly let us consider walk particle on the axis OX . It is supposed that this particle carries out jumps length has between two contiguous positions located on the network: $\backslash \; \{\backslash ,\; N\; \backslash ,\; has\; \backslash ,\; N\; \backslash \; in\; \backslash \; mathbb\; \{Z\}\; \backslash ,\; \backslash \}$ of mesh has on the axis, each jump having one duration $\backslash \; tau$.

It is still necessary to give oneself a number p such as: 0 < p < 1 . The physical interpretation of this parameter is the following one:

• p represents the probability that the particle at every moment makes a jump towards the line ;
• Q = 1 - p represents the probability that the particle at every moment makes a jump towards the left .

The case of the Brownian movement corresponds to make the assumption of space isotropy . All the directions of physical space being a priori equivalent, one poses the equiprobability:

The figure below watch a typical example of result: one traces the successive positions X (K) of the particle at the moments K , on the basis of the initial condition X (0) =0 .

Probabilities of conditional transition

One defines the Probabilité of conditional transition:

as being probability of finding the particle with the site my at the moment $s\; \backslash \; tau$ knowing that it was with the site Na at the initial moment 0 .

The assumption of isotropy results in writing the law of evolution of this probability of conditional transition:

One from of deduced the following relation:

Convergence towards the Brownian movement. Equation of Fokker-Planck

Let us take the limit continues preceding equation when parameters:

• $\backslash \; tau\; \backslash \; \backslash \; to\; \backslash \; 0$

• $a\; \backslash \; \backslash to\; \backslash \; 0$

One will see at the end of calculation that the combination $a^2/2\; \backslash \; tau$ must in makes remain constant within this continuous limit.

It comes, by reintroducing the adequate parameter to make a limited development:

In addition, one can write:

so that the hook is reduced to:

One from of deduced the equation from Fokker-Planck:

that one can rewrite:

by introducing the coefficient of diffusion:

Solution of the equation of Fokker-Planck

In addition to the equation of Fokker-Planck, density of probability of conditional transition $P\; (x\_0|X,\; T)$ must check the two following additional conditions:

• the standardization of the total probabilities:

• the initial condition:

where $\backslash \; delta\; (X)$ is the distribution of Dirac.

Density of probability of conditional transition $P\; (x\_0|X,\; T)$ is thus primarily a Fonction of Green of the equation of Fokker-Planck. One can show that she is written explicitly:

Moments of the distribution:

Let us pose $x\_0\; =\; 0$ to simplify. Density of probability of conditional transition $P\_0\; (X,\; T)\; =\; P\; (0|X,\; T)$ allows the calculation of the various moments:

$\backslash \; langle\; \backslash ,\; x^n\; (T)\; \backslash \; \backslash \; rangle\; \backslash \; =\; \backslash \; \backslash \; int\_\; \{-\; \backslash \; infty\}\; ^\; \{+\; \backslash \; infty\}\; dx\; \backslash \; x^n\; \backslash \; P\_0\; (X,\; T)$

The function $P\_0$ being even , every moment of an odd nature is null. One can easily calculate every moment of an even nature while posing:

$\backslash \; alpha\; \backslash \; =\; \backslash \; \backslash \; frac\; \{1\}\; \{4\; D\; T\}$

and by writing that:

$\backslash \; langle\; \backslash ,\; x^n\; (T)\; \backslash \; \backslash \; rangle\; \backslash \; =\; \backslash \; \backslash \; sqrt\; \{\backslash \; frac\; \{\backslash \; alpha\}\; \{\backslash \; pi\}\}\; \backslash \; \backslash \; int\_\; \{-\; \backslash \; infty\}\; ^\; \{+\; \backslash \; infty\}\; dx\; \backslash \; x^\; \{2n\}\; \backslash \; \backslash \; mathrm\; \{E\}\; ^\; \{-\; \backslash \; alpha\; x^2\}\; \backslash \; =\; \backslash \; (-\; 1)\; ^n\; \backslash \; \backslash \; sqrt\; \{\backslash \; frac\; \{\backslash \; alpha\}\; \{\backslash \; pi\}\}\; \backslash \; \backslash \; frac\; \{d^n~\}\; \{D\; \backslash \; alpha^n\}\; \backslash \; \backslash \; left\; \backslash ,\; \backslash \; int\_\; \{-\; \backslash \; infty\}\; ^\; \{+\; \backslash \; infty\}\; dx\; \backslash \; \backslash \; mathrm\; \{E\}\; ^\; \{-\; \backslash \; alpha\; x^2\}\; \backslash ,\; \backslash \; right$

One obtains explicitly:

$\backslash \; langle\; \backslash ,\; x^n\; (T)\; \backslash \; \backslash \; rangle\; \backslash \; =\; \backslash \; (-\; 1)\; ^n\; \backslash \; \backslash \; sqrt\; \{\backslash \; frac\; \{\backslash \; alpha\}\; \{\backslash \; pi\}\}\; \backslash \; \backslash \; frac\; \{d^n~\}\; \{D\; \backslash \; alpha^n\}\; \backslash \; \backslash \; left\; \backslash ,\; \backslash \; sqrt\; \{\backslash \; frac\; \{\backslash \; pi\}\; \{\backslash \; alpha\}\}\; \backslash ,\; \backslash \; right\; \backslash \; =\; \backslash \; (-\; 1)\; ^n\; \backslash \; \backslash \; sqrt\; \{\backslash \; alpha\}\; \backslash \; \backslash \; frac\; \{d^n~\}\; \{D\; \backslash \; alpha^n\}\; \backslash \; \backslash \; left\; \backslash ,\; \backslash \; frac\; \{1\}\; \{\backslash \; sqrt\; \{\backslash \; alpha\}\}\; \backslash ,\; \backslash \; right$

One finds in particular for the moment of order two:

$\backslash \; langle\; \backslash ,\; x^2\; (T)\; \backslash \; \backslash \; rangle\; \backslash \; =\; \backslash \; -\; \backslash ,\; \backslash \; sqrt\; \{\backslash \; alpha\}\; \backslash ,\; \backslash \; frac\; \{d~\}\; \{D\; \backslash \; alpha\}\; \backslash ,\; \backslash \; left\; \backslash ,\; \backslash \; frac\; \{1\}\; \{\backslash \; sqrt\; \{\backslash \; alpha\}\}\; \backslash ,\; \backslash \; right\; \backslash \; =\; \backslash \; (-\; \backslash ,\; \backslash \; sqrt\; \{\backslash \; alpha\})\; \backslash ,\; \backslash \; times\; \backslash ,\; \backslash \; left\; (-\; \backslash ,\; \backslash \; frac\; \{1\}\; \{2\; \backslash \; alpha^\; \{3/2\}\}\; \backslash \; right)\; \backslash \; =\; \backslash \; \backslash \; frac\; \{1\}\; \{2\; \backslash \; alpha\}\; \backslash \; =\; \backslash \; 2\; D\; t$

Process of Wiener

Brownian movement on a riemannienne variety

One calls Brownian movement on a Variété riemannienne V the Markovian continuous stochastic process from which the Semigroupe of transition to a parameter is generated by $1/2\; \backslash ,\; \backslash \; Delta\_V$, where $\backslash \; Delta\_V$ is the Opérateur of Laplace-Beltrami on the variety V .

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