The Maxwell- Boltzmann Law - English version | La loi de Maxwell Boltzmann

In kinetic theory of gases ; the Maxwell-Boltzmann law of speed distribution quantifies the statistical distribution of particle speeds in a homogeneous gas at thermodynamic equilibrium.  The speed vectors of the particles follow a normal law.  This law was established by James Clerk Maxwell in 1860, and subsequently confirmed by Ludwig Boltzmann from the physical bases which founded statistical physics in 1872 and 1877.

 This distribution was first defined and used to describe ideal gas particle velocity, where particles move freely without interacting with each other, except for very brief collisions in which they exchange energy.  and the amount of movement.  The term "particle" in this context refers only to particles in the gaseous state (atoms or molecules), and the particle system considered is assumed to have reached thermodynamic equilibrium.  The energies of these particles follow what is known as the Maxwell-Boltzmann statistic, and the statistical distribution of velocities is obtained by equating the energies of the particles to their kinetic energy.

 Mathematically, the Maxwell - Boltzmann distribution is the Law of χ with three degrees of freedom (the components of the velocity vector in Euclidean space), with a scale parameter that measures velocities in units proportional to the square root of  (the ratio between the temperature and the mass of the particles).

 The Maxwell-Boltzmann distribution is a result of the kinetic theory of gases.  It provides a simplified explanation of many fundamental properties of gases, including pressure and diffusion.  The Maxwell - Boltzmann distribution applies to three-dimensional particle velocities, but it appears to depend only on the particle velocity norm.  A particle velocity probability distribution indicates the most probable velocities.  The kinetic theory of gases applies to the ideal gas.  For real gases, various effects (for example, the presence of van der waals interactions or vortices also called vertex), can make the velocity distribution different from Maxwell - Boltzmann law.  However, rarefied gases at ordinary temperatures behave almost like ideal gases, and Maxwell's law of velocity distribution is an excellent approximation for these gases.  Palsmas, which are low density ionized gases, often have partially or fully Maxwellian particle distributions.

 States

 Let be the probability density of the speed  in a medium at thermodynamic equilibrium.Its expression is:

 Where

  is the mass of each of the particles;

  is Boltzamann's constant;

  is the thermodynamic temperature,

 Properties

 Several properties can be stated.

 This is a probability density, so the norm of this distribution is equal to unity:

 .

 The distribution depends only on the norm of the speed, which implies its isotropy:

 .

 It is given in a medium at rest.  The flow of momentum and therefore the average speed:  at the macroscopic scale are zero there:

 .

 The probability density of the speed norm is obtained by calculating the probability density in the interval of speeds, volume Distribution of the norm of the speed of oxygen molecules, at −100 ° C, 20  ° C and 600 ° C

 .

 The most probable speed norm can be deduced: .

 The mean translational kinetic energy of a particle, related to the mean square speed:

 .This quantity is enough to determine the internal energy of a monoatomic ideal gas made up of  particles (atoms), for which only translational movements are possible, considered at the macroscopic scale: .  In more general cases, it is necessary to include the kinetic energies of rotations of the molecules as well as the energies (kinetic and potential) of vibrations in the case of high temperatures.

 It is not possible to deduce from this distribution that of the positions of the particles which, moreover, we know is uniformly distributed.

 Obtaining the distribution by statistical physics

 The distribution function

 In the framework of statistical physics, considering that the studied system:  is in thermal equilibrium with a reservoir , the whole constitutes a canonical whole.  The only energy taken into account in the case of the Maxwell distribution is the kinetic energy of the particles which constitute the system : it extends over a continuous domain from 0 to infinity.  The probability that the system  has an energy in the energy slice  is given by:

 ,

 where  is a probability density of energy E

  is the density of states and  is the canonical partition function.  The number of microstates which have an energy in the energy slice:  is .  Since the energy  depends only on the norm of the speed, one can count these microstates by integration in the space of the speed in spherical coordinates: they have a speed included in the field hence

 .

 The probability can then be expressed as follows by showing the probability density 

 .

 The link with thermodynamics

 The identification of the constant  is related to the Thermodynamic temperature via the entropy

 .

 where  is the heat which corresponds to a variation of entropy.

 Hence the internal energy:

 .

 Obtained by analyzing the function

 The distribution function

 The starting hypothesis is the isotropy of the probability density: it does not depend on the direction studied.  This hypothesis implies a maximum disorder and therefore a system which can be characterized by a minimum of information.  In terms of information theory this is a necessary condition for maximum entropy.  In addition the isotropy involves an overall speed (macroscopic mean speed) zero: .  The studied macroscopic system is at rest in the reference frame in which one places oneself.

 We therefore assume that:

 ,

 From where we get:

 .

 On the other hand, using the relation: 

 .

 This makes it possible to establish:

 .

 The term on the left includes a priori all  while the term on the right only includes one of them.  So, if we apply this observation to the first component, we conclude that the right term and therefore the left one exclude  and .  With the same reasoning on the other components one finally excludes any dependence on  and one thus concludes that each of the terms is constant:

 .

 After integration:

 .

 The constant  is necessarily negative to prevent infinite speeds.

 The expression is identical for all : there is a unique distribution  which applies to the three components of the speed:

 .

 By normalization we obtain:

 .

 Link with thermodynamics

 The identification of the constant will be obtained by making the link with the thermodynamic temperature via the internal energy.  So :

 

 and:

 .

 Obtaining the distribution from the Boltzmann equation

 Main article: Chapman-Enskog method

 We can find the distribution from the Boltzmann equation which describes a medium in thermodynamic unbalance.  The Maxwellian distribution is found as a zero-order solution of an asymptotic development.  It corresponds to Euler's Equations.

Problème de visibilité d'équations - Problématique googliéen ; Article présentée par Karam Ouharou