Let us now proceed to the derivation of the main equation kinetic theory gases - the equation that determines the distribution function.

If collisions of molecules could be neglected altogether, then each gas molecule would represent a closed subsystem and the Liouville theorem would be valid for the distribution function of molecules, according to which

(see V, § 3). Total derivative here means differentiation along the phase trajectory of the molecule, determined by its equations of motion. Recall that Liouville's theorem holds for the distribution function defined precisely as the density in phase space(i.e. in the space of variables that are canonically conjugate generalized coordinates and momenta).

This circumstance does not interfere. Of course, the fact that the function f itself can then be expressed through any other variables.

In the absence of external field the quantities Γ of a freely moving molecule remain constant and only its coordinates change; wherein

If the gas is, for example, in an external field acting on the coordinates of the center of inertia of the molecule (say, in the gravitational field), then

where is the force acting on the molecule from the field.

Taking collisions into account violates equality (3.1); the distribution function ceases to be constant along phase trajectories. Instead of (3.1) we must write

where the symbol means the rate of change of the distribution function due to collisions: there is a change per unit time due to collisions in the number of molecules in the phase volume Written in the form

equation (3.4) (from (3.2)) determines the total change in the distribution function in given point phase space; the term is a decrease (in 1 s) in the number of molecules in a given element of phase space, associated with their free movement.

The quantity is called the collision integral, and equations of the form (3.4) are generally called kinetic equations. Of course kinetic equation acquires real meaning only after establishing the form of the collision integral. We will now turn to this issue.

When two molecules collide, the values ​​of their values ​​Γ change. Therefore, every collision experienced by a molecule takes it out of a given interval; such collisions are spoken of as acts of escape.

Full number collisions with transitions with everyone possible values; for a given Γ occurring per unit time in a volume dV is equal to the integral

However, such collisions (“arrival”) also occur, as a result of which molecules that initially had values ​​of the values ​​of Γ lying outside the given interval fall into this interval. These are collisions with transitions again with all possible for a given G. The total number of such collisions (per unit time in volume dV) is equal to

Subtracting the number of acts of departure from the number of acts of arrival, we find that as a result of all collisions, the number of molecules in question increases by 1 s

where for brevity we denote

Thus, we find the following expression for the collision integral:

In the second term in the integrand, integration over applies only to the function w; the factors do not depend on these variables. Therefore, this part of the integral can be transformed using the unitarity relation (2.9). As a result, the collision integral takes the form

in which both terms enter with the same coefficient.

Having established the form of the collision integral, we thereby got the opportunity to write the kinetic equation

This integro-differential equation is also called the Boltzmann equation. It was first established by the founder of kinetic theory, Ludwig Boltzmann, in 1872.

Equilibrium statistical distribution must satisfy the kinetic equation identically. This condition is indeed met. The equilibrium distribution is stationary and (in the absence of an external field) homogeneous; That's why left-hand side equation (3.8) vanishes identically. Equal to zero also the collision integral: due to equality (2.5), it vanishes integrand. Of course, the equilibrium distribution for a gas in an external field also satisfies the kinetic equation. It is enough to remember that the left side of the kinetic equation is the total derivative df/dt, which identically vanishes for any function depending only on the integrals of motion; the equilibrium distribution is expressed only through the integral of motion - full energy molecules

In the derivation of the kinetic equation presented, collisions of molecules were considered essentially as instantaneous events occurring at one point in space. It is therefore clear that the kinetic equation allows, in principle, to monitor the change in the distribution function only over time intervals that are large compared to the duration of the collisions, and at distances that are large compared to the size of the collision region. The last order of magnitude of the radius of action molecular forces d (for neutral molecules matching their sizes); the collision time is of the order of magnitude. These values ​​​​set the lower limit of distances and durations, the consideration of which is allowed by the kinetic equation (we will return to the origin of these restrictions in § 16). But in fact, there is usually no need (or even possibility) for such detailed description system behavior; this would require, in particular, specifying the initial conditions (spatial distribution of gas molecules) with the same accuracy, which is practically impossible. In real physical issues there are characteristic parameters of length L and time T imposed on the system by the conditions of the problem (characteristic lengths of gradients of macroscopic gas quantities, lengths and periods of propagating in it sound waves and so on.). In such problems, it is enough to monitor the behavior of the system at distances and times that are small only in comparison with these L and T. In other words, physically infinitesimal elements of volume and time should be small only in comparison with L and T. Averaged over such elements are given and initial conditions tasks.

For a monatomic gas, the quantities Γ are reduced to three components of the atomic momentum , and according to (2.8) the function w in the collision integral can be replaced by the function

Having then expressed this function through the differential collision cross section according to see (2.2)), we obtain

Its function and the cross section determined according to (2.2) contain -functional factors expressing the laws of conservation of momentum and energy, due to which the variables (for a given ) are in fact not independent. But after the collision integral is expressed in the form (3.9), we can assume that these -functions have already been eliminated by the corresponding integrations; then will be the usual scattering cross section, depending (for a given ir) only on the scattering angle.

Which describes systems that are far from thermodynamic equilibrium, for example, in the presence of temperature gradients and electric fields). Boltzmann's equation is used to study the transport of heat and electrical charge in liquids and gases, and from it transport properties such as electrical conductivity, Hall effect, viscosity and thermal conductivity are derived. The equation is applicable for rarefied systems, where the interaction time between particles is short (molecular chaos hypothesis).

Formulation

The Boltzmann equation describes the evolution over time ( t) density distribution functions f(x, p, t) in single-particle phase space, where x And p- coordinate and momentum, respectively. The distribution is defined so that

proportional to the number of particles in the phase volume d³x d³p at a point in time t. Boltzmann equation

Here F(x, t) is the field of forces acting on particles in a liquid or gas, and m- mass of particles. A term on the right side of the equation has been added to account for collisions between particles. If it is zero, then the particles do not collide at all. This case is often called the Liouville equation. If the field of forces F(x, t) replace with a suitable self-consistent field depending on the distribution function f, then we obtain the Vlasov equation, which describes the dynamics of charged plasma particles in a self-consistent field. The classical Boltzmann equation is used in plasma physics, as well as in the physics of semiconductors and metals (to describe kinetic phenomena, i.e. charge or heat transfer, in e-liquid).

Derivation of Boltzmann's equation

The microscopic derivation of the Boltzmann equation from first principles (based on the exact Liouville equation for all particles of the medium) is carried out by breaking the chain of Bogolyubov equations at the level of the pair correlation function for classical and quantum systems. Accounting for kinetic equations in the chain correlation functions more high order allows you to find corrections to the Boltzmann equation.

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See what the “Boltzmann Equation” is in other dictionaries:

Boltzmann equation- - [A.S. Goldberg. English-Russian energy dictionary. 2006] Energy topics in general EN Boltzmann equation ... Technical Translator's Guide

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Equation for the distribution function f (ν, r, t) of gas molecules over velocities ν and coordinates r (depending on time t), describing nonequilibrium processes in low-density gases. The function f determines the average number of particles with velocities... ... Great Soviet Encyclopedia

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MOSCOW ENERGY INSTITUTE

(Technical University)

FACULTY OF ELECTRONIC ENGINEERING

ABSTRACT ON THE TOPIC

TO INETIC EQUATION B OLTZMAN.

COMPLETED:

Korkin S.V.

TEACHER

Sherkunov Yu.B.

The second half of the work is quite packed complex mathematics . Author ( [email protected], [email protected])does not consider this coursework ideal; it can only serve as a starting point for writing a more perfect (and understandable) work. The text is not a copy of the book. See end for supporting literature.

The coursework was accepted with the mark "Excellent". (The final version of the work is a little lost. I suggest using the penultimate “version”).

Introduction………………………………………………………………………………3

Legend………………………………………………………………. 4

§1 Distribution function.

§2 Particle collision.

§3 Determination of the form of the collision integral

and Boltzmann equations.

§4. Kinetic equation for a weakly inhomogeneous gas.

Thermal conductivity of gas.

Some conventions:

n - particle concentration;

d is the average distance between particles;

V is a certain volume of the system;

P is the probability of some event;

f - distribution function;

Introduction.

The branches of physics thermodynamics, statistical physics and physical kinetics study physical processes, occurring in macroscopic systems - bodies consisting of a large number of microparticles. Depending on the type of system, such microparticles can be atoms, molecules, ions, electrons, photons or other particles. Today, there are two main methods for studying the states of macroscopic systems - thermodynamic, which characterizes the state of the system through macroscopic easily measured parameters (for example, pressure, volume, temperature, number of moles or concentration of a substance) and, in fact, does not take into account atomic-molecular structure substances, and a statistical method based on the atomic-molecular model of the system under consideration. The thermodynamic method will not be discussed in this work. Based on the known laws of behavior of particles of the system, the statistical method allows us to establish the laws of behavior of the entire macrosystem as a whole. In order to simplify the problem being solved, the statistical approach makes a number of assumptions (assumptions) about the behavior of microparticles and, therefore, the results obtained by the statistical method are valid only within the limits of the assumptions made. Statistical method uses a probabilistic approach to solving problems; to use this method, the system must contain enough a large number of particles. One of the problems solved by the statistical method is the derivation of the equation of state of a macroscopic system. The state of the system can be constant over time (equilibrium system) or can change over time (non-equilibrium system). Physical kinetics deals with the study of nonequilibrium states of systems and processes occurring in such systems.

The equation of state of a system developing over time is a kinetic equation, the solution of which determines the state of the system at any time. Interest in kinetic equations is associated with the possibility of their application in various areas physics: in the kinetic theory of gas, in astrophysics, plasma physics, fluid mechanics. This paper examines the kinetic equation derived by one of the founders statistical physics And physical kinetics Austrian physicist Ludwig Boltzmann in 1872 and bearing his name.

§1 Distribution function.

Gas particles are indistinguishable (identical);

Particles collide only in pairs (we neglect the collision of three or more particles simultaneously);

Immediately before the collision, the particles move in one straight line towards each other;

The collision of molecules is a direct central elastic impact;

And. Let us denote by

The translational motion of a classical particle is described by the coordinates

Boltzmann equation

Ludwig Boltzmann, Austrian theoretical physicist, member of the Austrian Academy of Sciences, one of the founders of classical kinetic theory.

Let us now take into account that two differently heated gases having temperatures T 1 and T 2 behave in a similar way when they come into contact. > T 1 . One of them heats up, the other cools down, and after some time their temperatures become equal (T). Gases come to a state thermal equilibrium and heat exchange stops. Let's illustrate this with a diagram.

So, W And T behave in exactly the same way: when gases come into contact, both of these characteristics change in the same way and are then compared, which corresponds to states of energy or thermal equilibrium. As rigorous calculations show, these characteristics are interconnected proportional dependence: T ~ W.

It would even be possible to measure the temperature of a gas by the kinetic energy of its molecules. However, this would be inconvenient, since then it would be necessary to measure the temperature in joules, which, firstly, is unusual and, secondly, would express the temperature in very small numbers. For example, the melting temperature of ice equal to 273 K would be expressed as 5.7 10 -21 L. To maintain the temperature at the usual Kelvin (or °C), most convenient to accept

where is the dimensional factor k ([k] - J/K) provides temperature measurement in K units, and numerical coefficient 2/3 is introduced because it stands at W to in the Clausius equation. The temperature measured in this way will be denoted by T and call thermodynamic temperature:

From the last expression it follows Boltzmann equation:

Where k = 1.38 10 -23 J/K - Boltzmann constant(her numeric value later we will get it theoretically). From the Boltzmann equation it follows physical meaning zero thermodynamic temperature (0 K): at T= 0 will be W k = 0, those. at zero Kelvin, the movement of molecules stops (i.e. thermal movement).

The statistical description of a gas is carried out by the distribution function of gas molecules in their phase space, where is the set of generalized coordinates of the molecule, is the set of generalized impulses corresponding to the coordinates, is time (the distribution function depends on time in a non-stationary state). Quite often, the symbol Г denotes the set of all variables on which the distribution function depends, with the exception of the coordinates of the molecule and time. The quantities have important property: These are movements that remain constant for each molecule during its free motion.

Thus, for a monatomic gas, the quantities are the three components of the atom. For diatomic molecule includes impulse and torque.

Basic kinetic equation

The basic equation of the kinetic theory of gases (or kinetic equation) is the equation defining the distribution function.

The equation:

where is the collision integral, equation (1) is called the kinetic equation. The symbol means the rate of change of the distribution function due to collisions of molecules. The kinetic equation acquires real meaning only after the collision integral is established. Then the kinetic equation takes the form (2). This integro-differential equation is also called the Boltzmann equation:

It is necessary to explain what is right part equation (2).

When two molecules collide, their values ​​change. Therefore, every collision experienced by a molecule takes it out of the given interval d. Total number of collisions with transitions with all possible values ​​for a given G, occurring per unit time in a volume dV, is equal to the integral:

(outgoing particles)

Some molecules, due to collisions, fall into the dG interval (collisions with transitions ). The total number of such collisions (per unit time in volume dV) is equal to:

(incoming particles).

If we subtract the number of acts of departure from the number of acts of arrival, it is clear that as a result of all collisions, the number of molecules in question increases by 1c

For a qualitative consideration of kinetic phenomena in a gas, a rough estimate of the collision integral is used using the concept of the mean free path l (a certain average distance traveled by a molecule between two successive collisions). The ratio is called the free run time. For a rough estimate of the collision integral, one assumes:

The difference in the numerator (3) takes into account that the collision integral turns to 0 for the equilibrium distribution function. The minus sign expresses the fact that collisions are a mechanism for establishing statistical equilibrium.

Boltzmann kinetic equation

The Boltzmann kinetic equation gives a microscopic description of the evolution of the state of a small gas. The kinetic equation is an equation of first order in time; it describes the irreversible transition of the system from some initial nonequilibrium state with a distribution function to the final equilibrium state with the most probable distribution function.

Solving the kinetic equation is very difficult with mathematical point vision. The difficulties in solving it are due to the multidimensionality of the function, which depends on seven scalar variables, and complex look the right side of the equation.

If the distribution function depends only on the x coordinate and the velocity component, the Boltzmann kinetic equation has the form:

where and are the distribution functions of molecules before and after the collision; – speed of molecules; is the differential effective scattering cross section per solid angle dW, depending on the interaction of molecules. — change in the distribution function as a result of collisions. -change in particle number density. is the force acting on the particle.

If the gas consists of particles of the same type, the kinetic equation can be written as:

Where – the average number of particles in an element of the phase volume near the point ( - change in the density of the number of particles near the point ( at time t per unit time.

Boltzmann's equation is valid if:

If the system is in a state of statistical equilibrium, then the collision integral vanishes and the solution to the Boltzmann equation is the distribution. Solving the Boltzmann equation for the appropriate conditions allows us to calculate the kinetic coefficients and obtain macroscopic equations for various processes transfer ( , viscosity, ). In the gravitational field of the earth, the solution to the Boltzmann equation is the well-known barometric formula.

Based on the solutions of the Boltzmann equation, the macroscopic behavior of gas, the calculation of viscosity and thermal conductivity coefficients are explained.

The kinematic equation is the basic equation for the dynamics of rarefied gases and is used for aerodynamic calculations aircraft on high altitudes flight.

Examples of problem solving

EXAMPLE 1