Boltzmann equation

For other uses, see Boltzmann's entropy formula, Stefan–Boltzmann law and Maxwell–Boltzmann distribution

The Boltzmann equation, also often known as the Boltzmann transport equation, devised by Ludwig Boltzmann, describes the statistical behaviour of a gas not in thermodynamic equilibrium. It is one of the most important equations of non-equilibrium statistical mechanics. The Boltzmann equation is used to determine how physical quantities change, such as heat and momentum, when a fluid is in transport, from which other properties such as viscosity, and thermal conductivity can be derived.

The equation is not simple: a linear stochastic partial differential equation, and the problem of existence and uniqueness of solutions is still not fully resolved, but some recent results are quite promising.

Overview

The density function

The Boltzmann equation gives the time evolution of the probability density function f(r, p, t), in one-particle phase space, where r is position in and p momentum. The density function is defined so that

$dN=f(\mathbf {r} ,\mathbf {p} ,t)\,d^{3}\mathbf {r} \,d^{3}\mathbf {p}$

is the number of molecules which have positions lying within a volume element d³r about r and momenta lying within a momentum-space element d³p about p, at time t. Integrating over a region of position space and momentum space gives the total number of particles which have positions and momenta in that region:

$N=\int \limits _{\mathrm {positions} }d^{3}\mathbf {r} \int \limits _{\mathrm {momenta} }d^{3}\mathbf {p} f(\mathbf {r} ,\mathbf {p} ,t)$

While the function is associated with a number of particles, the phase space is for one-particle (not all of them, which is usually the case with deterministic many-body systems), since it is the probability density of a number of particles having the same r, p simaltaneously at time t.

It is assumed the particles in the system are identical (so each has an identical mass m). The function f can only be used for one species of particle, for a mixture another distribution needs to be used.

General equation (principal form)

The general equation can then be written:

${\frac {\partial f}{\partial t}}=\left({\frac {\partial f}{\partial t}}\right)_{\mathrm {force} }+\left({\frac {\partial f}{\partial t}}\right)_{\mathrm {diff} }+\left({\frac {\partial f}{\partial t}}\right)_{\mathrm {coll} }$

where the "force" term corresponds to the forces exerted on the particles by an external influence (not by the particles themselves), the "diff" term represents the diffusion of particles, and "coll" is the collision term - accounting for the forces acting between particles in collisions. Expressions for each term on the right side are provided below.

The force and diffusion terms

Consider particles described by f, each experiencing an external force F, not due to other particles (see the collision term for this treatment).

Suppose at time t some number of particles all have position r and momentum p. If a force F instantly acts on each particle, then at time t + Δt their position will be r + Δr = r + pΔt/m and momentum p + Δp = p + FΔt. Then f must satisfy (in absence of collisions),

$f\left(\mathbf {r} +{\frac {\mathbf {p} }{m}}\Delta t,\mathbf {p} +\mathbf {F} \Delta t,t+\Delta t\right)\,d^{3}\mathbf {r} \,d^{3}\mathbf {p} =f(\mathbf {r} ,\mathbf {p} ,t)\,d^{3}\mathbf {r} \,d^{3}\mathbf {p} ,$

Note that we have used the fact that the phase space volume element d³rd³p is constant, which can be shown using Hamilton's equations (see the discussion under Liouville's theorem). However, since collisions do occur, the particle density in the phase-space volume d³rd³p changes, so

${\begin{aligned}dN_{\mathrm {coll} }&=\left({\frac {\partial f}{\partial t}}\right)_{\mathrm {coll} }\Delta td^{3}\mathbf {r} d^{3}\mathbf {p} \\&=f\left(\mathbf {r} +{\frac {\mathbf {p} }{m}}\Delta t,\mathbf {p} +\mathbf {F} \Delta t,t+\Delta t\right)d^{3}\mathbf {r} d^{3}\mathbf {p} -f(\mathbf {r} ,\mathbf {p} ,t)d^{3}\mathbf {r} d^{3}\mathbf {p} \\&=\Delta fd^{3}\mathbf {r} d^{3}\mathbf {p} \end{aligned}}$

(1)

where Δf is the total change in f. Dividing (1) by d³rd³pΔt and taking the limits Δt → 0 and Δf → 0, we have

${\frac {df}{dt}}=\left({\frac {\partial f}{\partial t}}\right)_{\mathrm {coll} }$

(2)

The total differential of f is:

${\begin{aligned}df&={\frac {\partial f}{\partial t}}dt+\left({\frac {\partial f}{\partial x}}dx+{\frac {\partial f}{\partial y}}dy+{\frac {\partial f}{\partial z}}dz\right)+\left({\frac {\partial f}{\partial p_{x}}}dp_{x}+{\frac {\partial f}{\partial p_{y}}}dp_{y}+{\frac {\partial f}{\partial p_{z}}}dp_{z}\right)\\&={\frac {\partial f}{\partial t}}dt+\nabla f\cdot d\mathbf {r} +{\frac {\partial f}{\partial \mathbf {p} }}\cdot d\mathbf {p} \\&={\frac {\partial f}{\partial t}}dt+\nabla f\cdot {\frac {\mathbf {p} dt}{m}}+{\frac {\partial f}{\partial \mathbf {p} }}\cdot \mathbf {F} dt\end{aligned}}$

(3)

where ∇ is the gradient operator,

${\frac {\partial f}{\partial \mathbf {p} }}=\left(\mathbf {\hat {e}} _{x}{\frac {\partial f}{\partial p_{x}}}+\mathbf {\hat {e}} _{y}{\frac {\partial f}{\partial p_{y}}}+\mathbf {\hat {e}} _{z}{\frac {\partial f}{\partial p_{z}}}\right)$

is a shorthand, and ê_x, ê_y, ê_z are cartesian unit vectors.

General equation (stronger form)

Dividing (3) by dt and substituting into (2) gives the stronger form of the equation:

${\frac {\partial f}{\partial t}}+{\frac {\mathbf {p} }{m}}\cdot \nabla f+\mathbf {F} \cdot {\frac {\partial f}{\partial \mathbf {p} }}=\left({\frac {\partial f}{\partial t}}\right)_{\mathrm {coll} }$

In this context, F(r, t) is the force field acting on the particles in the fluid, and m is the mass of the particles. The term on the right hand side is added to describe the effect of collisions between particles; if it is zero then the particles do not collide. The collisionless Boltzmann equation is often mistakenly called the Liouville equation (the Liouville Equation is a many-particle equation).

This equation is more useful than the principal one above, yet still incomplete, since f cannot be solved for unless the collision term in f is known. This term cannot be found as easily or generally as the others - it is a statistical term representing the particle collisions, and requires knowledge of the statistics the particles obey, like the Maxwell-Boltzmann, Fermi-Dirac or Bose-Einstein distributions.

The collision term (Stosszahl Ansatz) and molecular chaos

A key insight applied by Boltzmann was to determine the collision term resulting solely from two-body collisions between particles that are assumed to be uncorrelated prior to the collision. This assumption was referred to by Boltzmann as the "Stosszahl Ansatz", and is also known as the "molecular chaos assumption". Under this assumption the collision term can be written as a momentum-space integral over the product of one-particle distribution functions:

$\left({\frac {\partial f}{\partial t}}\right)_{\mathrm {coll} }=\iint g(\mathbf {p-p'} ,\mathbf {q} )\left[f(\mathbf {r} ,\mathbf {p+q} ,t)f(\mathbf {r} ,\mathbf {p'-q} ,t)-f(\mathbf {r} ,\mathbf {p} ,t)f(\mathbf {r} ,\mathbf {p'} ,t)\right]\,d\mathbf {p'} \,d\mathbf {q} .$

Applications and extensions

It is also possible to write down relativistic Boltzmann equations for systems in which a number of particle species can collide and produce different species. This is how the formation of the light elements in big bang nucleosynthesis is calculated. The Boltzmann equation is also often used in dynamics, especially galactic dynamics. A galaxy, under certain assumptions, may be approximated as a continuous fluid; its mass distribution is then represented by f; in galaxies, physical collisions between the stars are very rare, and the effect of gravitational collisions can be neglected for times far longer than the age of the universe.

In Hamiltonian mechanics, the Boltzmann equation is often written more generally as

${\hat {\mathbf {L} }}[f]=\mathbf {C} [f],\,$

where L is the Liouville operator describing the evolution of a phase space volume and C is the collision operator. The non-relativistic form of L is

${\hat {\mathbf {L} }}_{\mathrm {NR} }={\frac {\partial }{\partial t}}+{\frac {\mathbf {p} }{m}}\cdot \nabla _{\mathbf {r} }+\mathbf {F} \cdot \nabla _{\mathbf {p} }.$

The generalization to (general) relativity is

${\hat {\mathbf {L} }}_{\mathrm {GR} }=p^{\alpha }{\frac {\partial }{\partial x^{\alpha }}}-\Gamma _{\beta \gamma }^{\alpha }p^{\beta }p^{\gamma }{\frac {\partial }{\partial p^{\alpha }}},$

where Γ^α_βγ is the Christoffel symbol of the second kind (this assumes there are no external forces, so that particles move along geodesics in the absence of collisions), with the important subtlety that the density is a function in mixed contravariant-covariant (xⁱ, p_i) phase space as opposed to fully contravariant (xⁱ, pⁱ) phase space.

See also

• H-theorem
• Fokker–Planck equation
• Navier–Stokes equations
• Vlasov equation
• Vlasov–Poisson equation
• Lattice Boltzmann methods

References

• Arkeryd, Leif (1972). "On the Boltzmann equation part II: The full initial value problem". Arch. Rational Mech. Anal. 45: 17–34. Bibcode:1972ArRMA..45...17A. doi:10.1007/BF00253393.
• Arkeryd, Leif (1972). "On the Boltzmann equation part I: Existence". Arch. Rational Mech. Anal. 45: 1–16. Bibcode:1972ArRMA..45....1A. doi:10.1007/BF00253392.
• DiPerna, R. J.; Lions, P.-L. (1989). "On the Cauchy problem for Boltzmann equations: global existence and weak stability". Ann. of Math. (2). 130: 321–366.

External links

• The Boltzmann Transport Equation by Franz Vesely