Boltzmann equation

thumb|The place of the Boltzmann kinetic equation on the stairs of model reduction from microscopic dynamics to macroscopic continuum dynamics (illustration to the content of the book)

The Boltzmann equation or Boltzmann transport equation (BTE) describes the statistical behaviour of a thermodynamic system not in a state of equilibrium; it was devised by Ludwig Boltzmann in 1872.

The classic example of such a system is a fluid with temperature gradients in space causing heat to flow from hotter regions to colder ones, by the random but biased transport of the particles making up that fluid. In the modern literature the term Boltzmann equation is often used in a more general sense, referring to any kinetic equation that describes the change of a macroscopic quantity in a thermodynamic system, such as energy, charge or particle number.

The equation arises not by analyzing the individual positions and momenta of each particle in the fluid but rather by considering a probability distribution for the position and momentum of a typical particle—that is, the probability that the particle occupies a given very small region of space (mathematically the volume element <math>d^3 \mathbf{r}</math>) centered at the position <math>\mathbf{r}</math>, and has momentum nearly equal to a given momentum vector <math> \mathbf{p}</math> (thus occupying a very small region of momentum space <math>d^3 \mathbf{p}</math>), at an instant of time.

The Boltzmann equation can be used to determine how physical quantities change, such as heat energy and momentum, when a fluid is in transport. One may also derive other properties characteristic to fluids such as viscosity, thermal conductivity, and electrical conductivity (by treating the charge carriers in a material as a gas).

Overview

The phase space and density function

The set of all possible positions r and momenta p is called the phase space of the system; in other words a set of three coordinates for each position coordinate x, y, z, and three more for each momentum component , , . The entire space is 6-dimensional: a point in this space is , and each coordinate is parameterized by time t. A relevant differential element is written

<math display="block"> d^3\mathbf{r} \, d^3\mathbf{p} = dx \, dy \, dz \, dp_x \, dp_y \, dp_z. </math>

Since the probability of molecules, which all have and within <math> d^3\mathbf{r} \, d^3\mathbf{p}</math>, is in question, at the heart of the equation is a quantity which gives this probability per unit phase-space volume, or probability per unit length cubed per unit momentum cubed, at an instant of time . This is a probability density function: , defined so that,

<math display="block">dN = f (\mathbf{r},\mathbf{p},t) \, d^3\mathbf{r} \, d^3\mathbf{p}</math>

is the number of molecules which all have positions lying within a volume element <math> d^3\mathbf{r}</math> about and momenta lying within a momentum space element <math> d^3\mathbf{p}</math> about , at time . Integrating over a region of position space and momentum space gives the total number of particles which have positions and momenta in that region:

<math display="block">\begin{align}

N & = \int\limits_\mathrm{momenta} d^3\mathbf{p} \int\limits_\mathrm{positions} d^3\mathbf{r}\,f (\mathbf{r},\mathbf{p},t) \\[5pt]

& = \iiint\limits_\mathrm{momenta} \quad \iiint\limits_\mathrm{positions} f(x,y,z, p_x,p_y,p_z, t) \, dx \, dy \, dz \, dp_x \, dp_y \, dp_z

\end{align}</math>

which is a 6-fold integral. While 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 only one and is in question. It is not part of the analysis to use , for particle 1, , for particle 2, etc. up to , for particle N.

It is assumed the particles in the system are identical (so each has an identical mass ). For a mixture of more than one chemical species, one distribution is needed for each, see below.

Principal statement

The general equation can then be written as

\frac{df}{dt} =

\left(\frac{\partial f}{\partial t}\right)_\text{force} +

\left(\frac{\partial f}{\partial t}\right)_\text{diff} +

\left(\frac{\partial f}{\partial t}\right)_\text{coll},

</math>

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 assumption in the BGK approximation is that the effect of molecular collisions is to force a non-equilibrium distribution function at a point in physical space back to a Maxwellian equilibrium distribution function and that the rate at which this occurs is proportional to the molecular collision frequency. The Boltzmann equation is therefore modified to the BGK form:

<math display="block">\frac{\partial f}{\partial t} + \frac{\mathbf{p{m}\cdot\nabla f + \mathbf{F} \cdot \frac{\partial f}{\partial \mathbf{p = \nu (f_0 - f),</math>

where <math>\nu</math> is the molecular collision frequency, and <math>f_0</math> is the local Maxwellian distribution function given the gas temperature at this point in space. This is also called "relaxation time approximation".

General equation (for a mixture)

For a mixture of chemical species labelled by indices <math>i=1,2,3,\dots,n</math> the equation for species is For a fluid consisting of only one kind of particle, the number density is given by

<math display="block">n = \int f \,d^3\mathbf{p}.</math>

The average value of any function is

<math display="block">\langle A \rangle = \frac 1 n \int A f \,d^3\mathbf{p}.</math>

Since the conservation equations involve tensors, the Einstein summation convention will be used where repeated indices in a product indicate summation over those indices. Thus <math>\mathbf{x} \mapsto x_i</math> and <math>\mathbf{p} \mapsto p_i = m v_i</math>, where <math>v_i</math> is the particle velocity vector. Define <math>A(p_i)</math> as some function of momentum <math>p_i</math> only, whose total value is conserved in a collision. Assume also that the force <math>F_i</math> is a function of position only, and that f is zero for <math>p_i \to \pm\infty</math>. Multiplying the Boltzmann equation by A and integrating over momentum yields four terms, which, using integration by parts, can be expressed as

<math display="block">\int A \frac{\partial f}{\partial t} \,d^3\mathbf{p} = \frac{\partial }{\partial t} (n \langle A \rangle),</math>

<math display="block">\int \frac{p_j A}{m}\frac{\partial f}{\partial x_j} \,d^3\mathbf{p} = \frac{1}{m}\frac{\partial}{\partial x_j}(n\langle A p_j \rangle),</math>

<math display="block">\int A F_j \frac{\partial f}{\partial p_j} \,d^3\mathbf{p} = -n F_j\left\langle \frac{\partial A}{\partial p_j}\right\rangle,</math>

<math display="block">\int A \left(\frac{\partial f}{\partial t}\right)_\text{coll} \,d^3\mathbf{p} = \frac{\partial }{\partial t}_\text{coll} (n \langle A \rangle) = 0,</math>

where the last term is zero, since is conserved in a collision. The values of correspond to moments of velocity <math>v_i</math> (and momentum <math>p_i</math>, as they are linearly dependent).

Zeroth moment

Letting <math>A = m(v_i)^0 = m</math>, the mass of the particle, the integrated Boltzmann equation becomes the conservation of mass equation: including the formation of the light elements in Big Bang nucleosynthesis, the production of dark matter and baryogenesis. It is not a priori clear that the state of a quantum system can be characterized by a classical phase space density f. However, for a wide class of applications a well-defined generalization of f exists which is the solution of an effective Boltzmann equation that can be derived from first principles of quantum field theory.

General relativity and astronomy

The Boltzmann equation is of use in 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.

Its generalization in general relativity is

<math display="block">\hat{\mathbf{L_\mathrm{GR}[f] = p^\alpha\frac{\partial f}{\partial x^\alpha} - \Gamma_{\beta\gamma}^\alpha p^\beta p^\gamma \frac{\partial f}{\partial p^\alpha} = C[f],</math>

where 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 phase space as opposed to fully contravariant phase space.

In physical cosmology the fully covariant approach has been used to study the cosmic microwave background radiation. More generically the study of processes in the early universe often attempt to take into account the effects of quantum mechanics and general relativity. this analytical approach provides insight, but is not generally usable in practical problems. Assuming a specific inverse power law between interacting particles, fully closed-form solutions to boundary value problems involving gross, spatially homogenous rates of deformation can be found and studied analytically.

Instead, numerical methods (including finite elements and lattice Boltzmann methods) are generally used to find approximate solutions to the various forms of the Boltzmann equation. Example applications range from hypersonic aerodynamics in rarefied gas flows to plasma flows. An application of the Boltzmann equation in electrodynamics is the calculation of the electrical conductivity - the result is in leading order identical with the semiclassical result.

Close to local equilibrium, solution of the Boltzmann equation can be represented by an asymptotic expansion in powers of Knudsen number (the Chapman–Enskog expansion). The first two terms of this expansion give the Euler equations and the Navier–Stokes equations. The higher terms have singularities. The problem of developing mathematically the limiting processes, which lead from the atomistic view (represented by Boltzmann's equation) to the laws of motion of continua, is an important part of Hilbert's sixth problem.

Limitations and further uses of the Boltzmann equation

The Boltzmann equation is valid only under several assumptions. For instance, the particles are assumed to be pointlike, i.e. without having a finite size. There exists a generalization of the Boltzmann equation that is called the Enskog equation. The collision term is modified in Enskog equations such that particles have a finite size, for example they can be modelled as spheres having a fixed radius.

No further degrees of freedom besides translational motion are assumed for the particles. If there are internal degrees of freedom, the Boltzmann equation has to be generalized and might possess inelastic collisions. These must be derived by using the BBGKY hierarchy.

Boltzmann-like equations are also used for the movement of cells. Since cells are composite particles that carry internal degrees of freedom, the corresponding generalized Boltzmann equations must have inelastic collision integrals. Such equations can describe invasions of cancer cells in tissue, morphogenesis, and chemotaxis-related effects.

Long-time derivation of the Boltzmann equation from Newtonian mechanics

In 2024, Yu Deng, Zaher Hani, and Xiao Ma established a major extension of Lanford's theorem by proving that the Boltzmann equation can be rigorously derived from the Newtonian mechanics of a dilute gas of hard spheres for practically unlimited time intervals. This result extends the 1975 breakthrough by Oscar Lanford, which was restricted to very short times, and provides a complete microscopic-to-mesoscopic derivation valid as long as the corresponding Boltzmann equation admits a classical solution. This work addresses a key aspect of Hilbert's sixth problem by rigorously linking particle-level Newtonian dynamics with the mesoscopic kinetic description, and contributes to understanding how macroscopic irreversibility emerges from time-reversible microscopic laws.

Notes

References

. Very inexpensive introduction to the modern framework (starting from a formal deduction from Liouville and the Bogoliubov–Born–Green–Kirkwood–Yvon hierarchy (BBGKY) in which the Boltzmann equation is placed). Most statistical mechanics textbooks like Huang still treat the topic using Boltzmann's original arguments. To derive the equation, these books use a heuristic explanation that does not bring out the range of validity and the characteristic assumptions that distinguish Boltzmann's from other transport equations like Fokker–Planck or Landau equations.

External links

The Boltzmann Transport Equation by Franz Vesely
Boltzmann gaseous behaviors solved