Boltzmann equation
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.[2] 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
The Boltzmann equation can be used to determine how physical quantities change, such as
The equation is a nonlinear integro-differential equation, and the unknown function in the equation is a probability density function in six-dimensional space of a particle position and momentum. The problem of existence and uniqueness of solutions is still not fully resolved, but some recent results are quite promising.[3][4]
Overview
The phase space and density function
The set of all possible positions r and momenta p is called the
Since the probability of N molecules, which all have r and p within , is in question, at the heart of the equation is a quantity f which gives this probability per unit phase-space volume, or probability per unit length cubed per unit momentum cubed, at an instant of time t. This is a probability density function: f(r, p, t), defined so that,
which is a
It is assumed the particles in the system are identical (so each has an identical mass m). 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[6]
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.[6]
Note that some authors use the particle velocity v instead of momentum p; they are related in the definition of momentum by p = mv.
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 the latter treatment).
Suppose at time t some number of particles all have position r within element and momentum p within . If a force F instantly acts on each particle, then at time t + Δt their position will be and momentum p + Δp = p + FΔt. Then, in the absence of collisions, f must satisfy
Note that we have used the fact that the phase space volume element is constant, which can be shown using
|
(1)
|
where Δf is the total change in f. Dividing (1) by and taking the limits Δt → 0 and Δf → 0, we have
|
(2)
|
The total differential of f is:
|
(3)
|
where ∇ is the gradient operator, · is the dot product,
Final statement
Dividing (3) by dt and substituting into (2) gives:
In this context, F(r, t) is the
This equation is more useful than the principal one above, yet still incomplete, since f cannot be solved 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
The collision term (Stosszahlansatz) and molecular chaos
Two-body collision term
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 "Stosszahlansatz" 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:[2]
Simplifications to the collision term
Since much of the challenge in solving the Boltzmann equation originates with the complex collision term, attempts have been made to "model" and simplify the collision term. The best known model equation is due to Bhatnagar, Gross and Krook.[7] 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:
where is the molecular collision frequency, and 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 i = 1, 2, 3, ..., n the equation for species i is[2]
where fi = fi(r, pi, t), and the collision term is
where f′ = f′(p′i, t), the magnitude of the relative momenta is
and Iij is the differential cross-section, as before, between particles i and j. The integration is over the momentum components in the integrand (which are labelled i and j). The sum of integrals describes the entry and exit of particles of species i in or out of the phase-space element.
Applications and extensions
Conservation equations
The Boltzmann equation can be used to derive the fluid dynamic conservation laws for mass, charge, momentum, and energy.[8]: 163 For a fluid consisting of only one kind of particle, the number density n is given by
The average value of any function A is
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 and , where is the particle velocity vector. Define as some function of momentum only, which is conserved in a collision. Assume also that the force is a function of position only, and that f is zero for . Multiplying the Boltzmann equation by A and integrating over momentum yields four terms, which, using integration by parts, can be expressed as
where the last term is zero, since A is conserved in a collision. The values of A correspond to moments of velocity (and momentum , as they are linearly dependent).
Zeroth moment
Letting , the mass of the particle, the integrated Boltzmann equation becomes the conservation of mass equation:[8]: 12, 168
First moment
Letting , the momentum of the particle, the integrated Boltzmann equation becomes the conservation of momentum equation:[8]: 15, 169
where is the pressure tensor (the viscous stress tensor plus the hydrostatic pressure).
Second moment
Letting , the kinetic energy of the particle, the integrated Boltzmann equation becomes the conservation of energy equation:[8]: 19, 169
where is the kinetic thermal energy density, and is the heat flux vector.
Hamiltonian mechanics
In Hamiltonian mechanics, the Boltzmann equation is often written more generally as
Quantum theory and violation of particle number conservation
It is possible to write down relativistic quantum Boltzmann equations for relativistic quantum systems in which the number of particles is not conserved in collisions. This has several applications in physical cosmology,[9] 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.[10]
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[11]
In
Solving the equation
Exact solutions to the Boltzmann equations have been proven to exist in some cases;[15] this analytical approach provides insight, but is not generally usable in practical problems.
Instead,
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[20]). 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.[21]
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.[22] 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.[22]
Many real fluids like liquids or dense gases have besides the features mentioned above more complex forms of collisions, there will be not only binary, but also ternary and higher order collisions.[23] These must be derived by using the BBGKY hierarchy.
Boltzmann-like equations are also used for the movement of
See also
Notes
- ^
Gorban, Alexander N.; Karlin, Ilya V. (2005). Invariant Manifolds for Physical and Chemical Kinetics. Lecture Notes in Physics (LNP, vol. 660). Berlin, Heidelberg: Springer. ISBN 978-3-540-22684-0. Alt URL
- ^ a b c d Encyclopaedia of Physics (2nd Edition), R. G. Lerner, G. L. Trigg, VHC publishers, 1991, ISBN (Verlagsgesellschaft) 3-527-26954-1, ISBN (VHC Inc.) 0-89573-752-3.
- JSTOR 1971423.
- PMID 20231489.
- ISBN 978-0-471-81518-1.
- ^ ISBN 0-07-051400-3.
- .
- ^ ISBN 978-0-486-64741-8.
- ^ ISBN 978-0-201-62674-2.
- ^ S2CID 119253828.
- ^ Ehlers J (1971) General Relativity and Cosmology (Varenna), R K Sachs (Academic Press NY);Thorne K S (1980) Rev. Mod. Phys., 52, 299; Ellis G F R, Treciokas R, Matravers D R, (1983) Ann. Phys., 150, 487}
- .
- .
- ^ Maartens R, Gebbie T, Ellis GFR (1999). "Cosmic microwave background anisotropies: Nonlinear dynamics". Phys. Rev. D. 59 (8): 083506
- S2CID 115167686.
- .
- ISSN 0307-904X.
- ISSN 0036-1429.
- ISBN 978-981-4449-53-3.
- ISBN 0-521-40844-X
- .
- ^ a b "Enskog Equation - an overview | ScienceDirect Topics". www.sciencedirect.com. Retrieved 2022-05-10.
- arXiv:cond-mat/9706020.
- .
- PMID 35150333.
References
- Harris, Stewart (1971). An introduction to the theory of the Boltzmann equation. Dover Books. p. 221. ISBN 978-0-486-43831-3.. 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.
- Arkeryd, Leif (1972). "On the Boltzmann equation part I: Existence". Arch. Rational Mech. Anal. 45 (1): 1–16. S2CID 117877311.
- S2CID 119481100.
- Arkeryd, Leif (1972). "On the Boltzmann equation part I: Existence". Arch. Rational Mech. Anal. 45 (1): 1–16. S2CID 117877311.
- DiPerna, R. J.; Lions, P.-L. (1989). "On the Cauchy problem for Boltzmann equations: global existence and weak stability". Ann. of Math. 2. 130 (2): 321–366. JSTOR 1971423.