An equation which is used to study the nonequilibrium behavior of a collection of particles. In a state of equilibrium a gas of particles has uniform composition and constant temperature and density. If the gas is subjected to a temperature difference or disturbed by externally applied electric, magnetic, or mechanical forces, it will be set in motion and the temperature, density, and composition may become functions of position and time; in other words, the gas moves out of equilibrium. The Boltzmann equation applies to a quantity known as the distribution function, which describes this nonequilibriium state mathematically and specifies how quickly and in what manner the state of the gas changes when the disturbing forces are varied. See also: Kinetic theory of matter; Statistical mechanics
is the Boltzmann transport equation, where f is the unknown distribution function which, in its most general form, depends on a position vector r , a velocity vector v , and the time t. The quantity ∂f/∂t on the left side of Eq. (1) is the rate of change of f at fixed values of r and v. The equation expresses this rate of change as the sum of three contributions: first, (∂f/∂t)force arises when the velocities of the particles change with time as a result of external driving forces; second, (∂f/∂t)diff is the effect of the diffusion of the particles from one region in space to the other; and third, (∂f/∂t)coll is the effect collisions of the particles with each other or with other kinds of particles.
The distribution function carries information about the positions and velocities of the particles at any time. The probable number of particles N at the time t within the spatial element dxdydz located at (x,y,z) and with velocities in the element dυx dυy dυz at the point (υx,υy,υz is given by Eq. (2)
or, in vector notation, by Eq. (3).
It is assumed that the particles are identical; a different distribution function must be used for each species if several kinds of particles are present.
Specific expressions can be found for the terms on the right side of Eq. (1). Suppose that an external force Fx acts on each particle, producing the acceleration ax = and hence changing the velocity by Δυx = axΔt in the time interval Δt. If a group of particles has the velocity υx at time t, the same particles will have the velocity υx + Δvx at the later time t + Δt. Therefore the distribution functions f(υx,t) and f(υx + Δυx,t + Δt) satisfy the equality of Eq. (4),
The generalization of this result for acceleration in an arbitrary direction is then given by Eq. (8).
The quantity (∂f/∂t)force therefore depends on both a, the rate of change of velocity of the particles, and ∂f/∂v the variation of the distribution function with velocity.
In a similar way (∂f/∂t)diff depends on both v, the rate of change of position of the particles, and (∂f/∂r), the variation of the distribution function with position. One writes Δx = υxΔt in place of Δυx = axΔt. Then the form of Eqs. (4)–(7) is unchanged, except that υx is replaced by x and ax by υx. The final expression is given by Eq. (9).
which is the usual form of the Boltzmann equation. Before it can be solved for f, a specific expression must be found for (∂f/∂t)coll, the rate of change of f(r,v, t) due to collisions. The calculation of (∂f/∂t)coll begins with a mathematical description of the forces acting between particles. Knowing the type of statistics obeyed by the particles (that is, Fermi-Dirac, Bose-Einstein, or Maxwell-Boltzmann), the manner in which the velocities of the particles are changed by collisions can then be determined. The term (∂f/∂t)coll is expressed as the difference between the rate of scattering from all possible velocities v′ to the velocity v, and the rate of scattering from v to all possible v′. For example, if the particles are electrons in a metal or semiconductor and if they are scattered elastically by imperfections in the solid, it is found that (∂f/∂t)coll obeys Eq. (11),
where Wvv ′ is the rate of scattering of one particle from v to v′ or the reverse. A basic requirement for obtaining a meaningful expression for (∂f/∂t)coll in this way is that the duration of a collision be small compared to the time between collisions. For gas of atoms or molecules this condition implies that the gas be dilute, and for electrons in a solid it implies that the concentration of imperfections must not be too high.
The Boltzmann expression, Eq. (10), with a collision term of the form in Eq. (11), is irreversible in time in the sense that if f(r,v, t) is a solution then f(r,−v,−t) is not a solution. Thus if an isolated system is initially not in equilibrium, it approaches equilibrium as time advances; the time-reversed performance, in which the system departs farther from equilibrium, does not occur. The Boltzmann equation therefore admits of solutions proceeding toward equilibrium but not of time-reversed solutions departing from equilibrium. This is paradoxical because actual physical systems are reversible in time when looked at on an atomic scale. For example, in a classical system the time enters the equations of motion only in the acceleration d2 r/dt2, so if t is replaced by −t in a solution, a new solution is obtained. If the velocities of all particles were suddenly reversed, the system would retrace its previous behavior.
An actual system does not necessarily move toward equilibrium, although it is overwhelmingly probable that it does so. From a mathematical point of view it is puzzling that one can begin with the exact equations of motion, reversible in time, and by making reasonable approximations arrive at the irreversible Boltzmann equation. The resolution of this paradox lies in the statistical nature of the Boltzmann equation. It does not describe the behavior of a single system, but the average behavior of a large number of systems. Mathematically, the irreversibility arises from the collision term, Eq. (11), where the approximation has been made that the distribution function f(r,v, t) or f(r,v′,t) applies to particles both immediately before and immediately after a collision.
A number of equations closely related to the Boltzmann equation are often useful for particular applications. If collisions between particles are disregarded, the right side of the Boltzmann equation, Eq. (10), is zero. The equation is then called the collisionless Boltzmann equation or Vlasov equation. This equation has been applied to a gas of charged particles, also known as a plasma. The coulomb forces between particles have such a long range that it is incorrect to consider the particles as free except when colliding. The Vlasov equation can be used by including the forces between particles in the term a ċ (∂f/∂v). One takes a = q E/m, q being the charge of a particle and m the mass. The electric field E includes the field produced by the particles themselves, in addition to any externally produced field.
can be derived. The particles are now considered as a continuous fluid with density ρ and mean velocity v. F is the external force per unit volume and p is the pressure tensor. Equation (12a) states mathematically that the mass of the fluid is conserved, while Eq. (12b) equates the rate of change of momentum of an element of fluid to the force on it. An energy conservation equation can also be derived. See also: Navier-Stokes equation
The Boltzmann equation can be used to calculate the electronic transport properties of metals and semiconductors. For example, if an electric field is applied to a solid, one must solve the Boltzmann equation for the distribution function f(r,v, t) of the E electrons, taking the acceleration in Eq. (10) as a = q E/m and setting ∂f/∂r = 0, corresponding to spatial uniformity of the electrons. If the electric field is constant, the distribution function is also constant and is displaced in velocity space in such a way that fewer electrons are moving in the direction of the field than in the opposite direction. This corresponds to a current flow in the direction of the field. The relationship between the current density J and the field E is given by Eq. (13),
where σ, the electrical conductivity, is the final quantity of interest. See also: Free-electron theory of metals
With the Boltzmann equation one can also calculate the heat current flowing in a solid as the result of a temperature difference, the constant of proportionality between the heat current per unit area and the temperature gradient being the thermal conductivity. In still more generality, both an electric field E and a temperature gradient ∂T/∂r can be applied, where T is the temperature. Expressions are obtained for the electrical current density J and the heat current density U in the form of Eqs. (14a) and (14b).
L11 is the electrical conductivity and −L22 the thermal conductivity. Equations (14a) and (14b) also describe thermoelectric phenomena, such as the Peltier and Seebeck effects. For example, if a thermal gradient is applied to an electrically insulated solid so that no electric current can flow (J = 0), Eq. (14a) shows that an electric field given by Eq. (15)
will appear. The quantity (L12/L11) is called the Seebeck coefficient.
Finally, if a constant magnetic field B is also applied, the coefficients Lij in Eqs. (14a) and (14b) become functions of B. It is found that the electrical conductivity usually decreases with increasing B, a behavior known as magnetoresistance. These equations also describe the Hall effect, the appearance of an electric field in the y direction if there is an electric current in the x direction and a magnetic field in the z direction, as well as more complex thermomagnetic phenomena, such as the Ettingshausen and Nernst effects. The conductivity σ in Eq. (13) and the Lij in Eqs. (14a) and (14b) are tensor quantities in many materials. If the field E and the temperature gradient ∂T/∂r are in a given direction, the currents U and J are not necessarily in the same direction. See also: Conduction (electricity); Conduction (heat); Galvanomagnetic effects; Hall effect; Magnetoresistance; Thermoelectricity; Thermomagnetic effects
Nonequilibrium properties of atomic or molecular gases such as viscosity, thermal conduction, and diffusion have been treated with the Boltzmann equation. Although many useful results, such as the independence of the viscosity of a gas on pressure, can be obtained by simple approximate methods, the Boltzmann equation must be used in order to obtain quantitatively correct results. See also: Diffusion; Viscosity
If one proceeds from a neutral gas to a charged gas or plasma, with the electrons partially removed from the atoms, a number of new phenomena appear. As a consequence of the long-range Coulomb forces between the charges, the plasma can exhibit oscillations in which the free electrons move back and forth with respect to the relatively stationary heavy positive ions at the characteristic frequency known as the plasma frequency. This frequency is proportional to the square root of the particle density. If the propagation of electromagnetic waves through a plasma is studied, it is found that a plasma reflects an electromagnetic wave at a frequency lower than the plasma frequency, but transmits the wave at a higher frequency. This fact explains many characteristics of long-distance radio transmission, made possible by reflection of radio waves by the ionosphere, a low-density plasma surrounding the Earth at altitudes greater than 40 mi (65 km). A plasma also exhibits properties such as electrical conductivity, thermal conductivity, viscosity, and diffusion. See also: Plasma (physics)
If a magnetic field is applied to the plasma, its motion can become complex. A type of wave known as an Alfvén wave propagates in the direction of the magnetic field with a velocity proportional to the field strength. The magnetic field lines are not straight, however, but oscillate like stretched strings as the wave passes through the plasma. Waves that propagate in a direction perpendicular to the magnetic field have quite different properties.
When the plasma moves as a fluid, it tends to carry the magnetic field lines with it. The plasma becomes partially trapped by the magnetic field in such a way that it can move easily along the magnetic field lines, but only very slowly perpendicular to them. The outstanding problem in the attainment of a controlled thermonuclear reaction is to design a magnetic field configuration that can contain an extremely hot plasma long enough to allow nuclear reactions to take place. See also: Magnetohydrodynamics
Plasmas in association with magnetic fields occur in many astronomical phenomena. Many of the events occurring on the surface of the Sun, such as sunspots and flares, as well as the properties of the solar wind, a dilute plasma streaming out from the Sun in all directions, are manifestations of the motion of a plasma in a magnetic field. It is believed that a plasma streaming out from a rapidly rotating neutron star will radiate electromagnetic energy; this is a possible explanation of pulsars, stars emitting regularly pulsating radio signals that have been detected with radio telescopes.
Many properties of plasmas can be calculated by studying the motion of individual particles in electric and magnetic fields, or by using the hydrodynamic equations, Eqs. (12a) and (12b), or the Vlasov equation, together with Maxwell's equations. However, subtle properties of plasmas, such as diffusion processes and the damping of waves, can best be understood by starting with the Boltzmann equation or the closely related Fokker-Planck equation. See also: Maxwell's equations
It is usually difficult to solve the Boltzmann equation, Eq. (10), for the distribution function f(r,v,t) because of the complicated form of (∂f/∂t)coll. This term is simplified if one makes the relaxation time approximation shown in Eq. (16),
where f0(υ) is the equilibrium distribution function and π(υ), the relaxation time, is a characteristic time describing the return of the distribution function to equilibrium when external forces are removed. For some systems Eq. (16) follows rigorously from Eq. (11), but in general it is an approximation. If the relaxation time approximation is not made, the Boltzmann equation usually cannot be solved exactly, and approximate techniques such as variational, perturbation, and expansion methods must be used. An important situation in which an exact solution is possible occurs if the particles exert forces on each other that vary as the inverse fifth power of the distance between them. See also: Relaxation time of electrons
As an illustration of how the Boltzmann equation can be solved, consider a spatially uniform electron gas in a metal or a semiconductor. Let the scattering of electrons be described by Eq. (16). One can calculate the electric current which flows when a small, constant electric field Ex is applied. The spatial uniformity of the electron gas implies that ∂f/∂r = 0; since the electric field is constant, one requires also that the distribution function be constant (∂f/∂ t = 0) so that the current does not depend on time. If one writes the acceleration of the electron as ax = Fx−m = −eEx−m, where (- e) is the charge and m is the mass of the electron, Eq. (10) becomes Eq. (17).
The difference between f and f0 is small; therefore f can be replaced by f0 in the term containing Ex, since Ex itself is small. Thus Eq. (18)
Here f0(v) is the equilibrium distribution function describing the state of the electrons in the absence of an electric field. For a high electron density, as in metals, f0(v) is the Fermi-Dirac distribution function given by Eq. (20),
where ζ is the Fermi energy, k is the Boltzmann constant, and T is the absolute temperature. In writing f0(v) in the form of Eq. (20), the simplifying assumption that the energy of an electron is mυ 2/2 has been made. If the electron density is low, as it often is in semiconductors, f0(v) can be approximated by the Maxwell-Boltzmann distribution given in Eq. (21).
Therefore f0(v) is centered at v = 0 in such a way that the net flow of electrons in every direction is zero. The nonequilibrium distribution function f(v) similar to f0(v), except that is shifted in the (−υx) direction in velocity space. Consequently more electrons move in the (−x) direction than in the (+x) direction, producing a net current flow in the (+x) direction. The current density Jx is calculated from f v by Eq. (22),
where the spatial integral is to be taken over a unit volume, so that ∫d3 x = 1. f v as given by Eq. (19) is substituted into Eq. (22), the term involving the equilibrium distribution function f0(v) by itself gives zero current, since no current flows at equilibrium. Hence one obtains Eq. (23).
is an average relaxation time. Equation (25) is the final expression for the electrical conductivity σ.
The above treatment is applicable only if the electrons in the solid behave as if they were nearly free. In general, the wave vector k, rather than the velocity v, must be used as a variable in the distribution function, and the final expression for σ depends on the electronic band structure of the solid.