Boltzmann equation explained

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.^[1] 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.

d ^{3} r ) centered at the position r , and has momentum nearly equal to a given momentum vector p (thus occupying a very small region of momentum space d ^{3} p ), 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). See also convection–diffusion equation.

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.^{[2] [3]}

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. The small volume ("differential volume element") is written $$d^3\backslash mathbf\; \backslash ,\; d^3\backslash mathbf\; =\; dx\; \backslash ,\; dy\; \backslash ,\; dz\; \backslash ,\; dp\_x\; \backslash ,\; dp\_y\; \backslash ,\; dp\_z.$$

Since the probability of molecules, which all have and within

d ^{3r} d ^{3p} , 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,$$dN\; =\; f\; (\backslash mathbf,\backslash mathbf,t)\; \backslash ,\; d^3\backslash mathbf\; \backslash ,\; d^3\backslash mathbf$$is the number of molecules which all have positions lying within a volume element d ^{3r} about and momenta lying within a momentum space element d ^{3p} about, at time .^[4] Integrating over a region of position space and momentum space gives the total number of particles which have positions and momenta in that region:

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^[5] $$\backslash frac\; =\; \backslash left(\backslash frac\backslash right)\_\backslash text\; +\; \backslash left(\backslash frac\backslash right)\_\backslash text\; +\; \backslash left(\backslash frac\backslash right)\_\backslash text,$$

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.

Note that some authors use the particle velocity instead of momentum ; they are related in the definition of momentum by .

The force and diffusion terms

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

Suppose at time some number of particles all have position within element

d ^{3r} and momentum within d ^{3p} . If a force instantly acts on each particle, then at time their position will be r + \Delta r = r + (p)/(m) \Delta t and momentum . Then, in the absence of collisions, must satisfy

$$f\; \backslash left\; (\backslash mathbf+\backslash frac\; \backslash ,\; \backslash Delta\; t,\backslash mathbf+\backslash mathbf\; \backslash ,\; \backslash Delta\; t,\; t+\backslash Delta\; t\; \backslash right)\backslash ,d^3\backslash mathbf\backslash ,d^3\backslash mathbf\; =\; f(\backslash mathbf,\; \backslash mathbf,t)\; \backslash ,\; d^3\backslash mathbf\; \backslash ,\; d^3\backslash mathbf$$

Note that we have used the fact that the phase space volume element

d ^{3r} d ^{3p} 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 d ^{3r} d ^{3p} changes, so

where is the total change in . Dividing by

d ^{3r} d ^{3p} \Delta t and taking the limits and, we have

The total differential of is:where is the gradient operator, is the dot product,$$\backslash frac\; =\; \backslash mathbf\_x\backslash frac\; +\; \backslash mathbf\_y\backslash frac\; +\; \backslash mathbf\_z\; \backslash frac=\; \backslash nabla\_\backslash mathbff$$is a shorthand for the momentum analogue of, and,, are Cartesian unit vectors.

Final statement

Dividing by and substituting into gives:

$$\backslash frac\; +\; \backslash frac\backslash cdot\backslash nabla\; f\; +\; \backslash mathbf\; \backslash cdot\; \backslash frac\; =\; \backslash left(\backslash frac\; \backslash right)\_\backslash mathrm$$

In this context, is the force field acting on the particles in the fluid, and 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, where individual collisions are replaced with long-range aggregated interactions, e.g. Coulomb interactions, is often called the Vlasov equation.

This equation is more useful than the principal one above, yet still incomplete, since cannot be solved unless the collision term in 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 (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 "German: 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:$$\backslash left(\backslash frac\backslash right)\_\backslash text\; =\; \backslash iint\; g\; I(g,\; \backslash Omega)[f(\backslash mathbf\{r\},\backslash mathbf\{p\text{'}\}\_A,\; t)\; f(\backslash mathbf\{r\},\backslash mathbf\{p\text{'}\}\_B,t)\; -\; f(\backslash mathbf\{r\},\backslash mathbf\{p\}\_A,t)\; f(\backslash mathbf\{r\},\backslash mathbf\{p\}\_B,t)]\; \backslash ,d\backslash Omega\; \backslash ,d^3\backslash mathbf\_B,$$where and are the momenta of any two particles (labeled as A and B for convenience) before a collision, and are the momenta after the collision,$$g\; =\; |\backslash mathbf\_B\; -\; \backslash mathbf\_A|\; =\; |\backslash mathbf\_B\; -\; \backslash mathbf\_A|$$is the magnitude of the relative momenta (see relative velocity for more on this concept), and is the differential cross section of the collision, in which the relative momenta of the colliding particles turns through an angle into the element of the solid angle, due to the collision.

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.^[6] 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:

$$\backslash frac\; +\; \backslash frac\backslash cdot\backslash nabla\; f\; +\; \backslash mathbf\; \backslash cdot\; \backslash frac\; =\; \backslash nu\; (f\_0\; -\; f),$$

where

\nu is the molecular collision frequency, and f _{0} 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 the equation for species is

$$\backslash frac\; +\; \backslash frac\; \backslash cdot\; \backslash nabla\; f\_i\; +\; \backslash mathbf\; \backslash cdot\; \backslash frac\; =\; \backslash left(\backslash frac\; \backslash right)\_\backslash text,$$

where, and the collision term is

$$\backslash left(\backslash frac\; \backslash right)\_\; =\; \backslash sum\_^n\; \backslash iint\; g\_\; I\_(g\_,\; \backslash Omega)[f\text{'}\_i\; f\text{'}\_j\; -\; f\_i\; f\_j]\; \backslash ,d\backslash Omega\backslash ,d^3\backslash mathbf,$$

where, the magnitude of the relative momenta is

$$g\_\; =\; |\backslash mathbf\_i\; -\; \backslash mathbf\_j|\; =\; |\backslash mathbf\_i\; -\; \backslash mathbf\_j|,$$

and 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.^[7] For a fluid consisting of only one kind of particle, the number density is given by$$n\; =\; \backslash int\; f\; \backslash ,d^3\backslash mathbf.$$

The average value of any function is$$\backslash langle\; A\; \backslash rangle\; =\; \backslash frac\; 1\; n\; \backslash int\; A\; f\; \backslash ,d^3\backslash mathbf.$$

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

x \mapsto x _{i} and p \mapsto p _{i} = m v _{i} , where v _{i} is the particle velocity vector. Define A(p _{i)} as some function of momentum p _{i} only, which is conserved in a collision. Assume also that the force F _{i} is a function of position only, and that f is zero for p _{i} \to \pm inf ty . Multiplying the Boltzmann equation by A and integrating over momentum yields four terms, which, using integration by parts, can be expressed as

$$\backslash int\; A\; \backslash frac\; \backslash ,d^3\backslash mathbf\; =\; \backslash frac\; (n\; \backslash langle\; A\; \backslash rangle),$$

$$\backslash int\; \backslash frac\backslash frac\; \backslash ,d^3\backslash mathbf\; =\; \backslash frac\backslash frac(n\backslash langle\; A\; p\_j\; \backslash rangle),$$

$$\backslash int\; A\; F\_j\; \backslash frac\; \backslash ,d^3\backslash mathbf\; =\; -n\; F\_j\backslash left\backslash langle\; \backslash frac\backslash right\backslash rangle,$$

$$\backslash int\; A\; \backslash left(\backslash frac\backslash right)\_\backslash text\; \backslash ,d^3\backslash mathbf\; =\; 0,$$

where the last term is zero, since is conserved in a collision. The values of correspond to moments of velocity

v _{i} (and momentum p _{i} , as they are linearly dependent).

Zeroth moment

Letting

A = m(v ^{0} _{i)} = m , the mass of the particle, the integrated Boltzmann equation becomes the conservation of mass equation:$$\backslash frac\backslash rho\; +\; \backslash frac(\backslash rho\; V\_j)\; =\; 0,$$where \rho = mn is the mass density, and V _{i} = \langle v _{i\rangle} is the average fluid velocity.

First moment

Letting

A = m(v ^{1} _{i)} = p _{i} , the momentum of the particle, the integrated Boltzmann equation becomes the conservation of momentum equation:

$$\backslash frac(\backslash rho\; V\_i)\; +\; \backslash frac(\backslash rho\; V\_i\; V\_j+P\_)\; -\; n\; F\_i\; =\; 0,$$

where

P _{ij} = \rho \langle (v _{i-V} _{i)} (v _{j-V} _{j)} \rangle is the pressure tensor (the viscous stress tensor plus the hydrostatic pressure).

Second moment

Letting

A = (m(v ^{2} _{i)})/(2) = (p _{i} p _{i})/(2 m) , the kinetic energy of the particle, the integrated Boltzmann equation becomes the conservation of energy equation:

$$\backslash frac(u\; +\; \backslash tfrac\backslash rho\; V\_i\; V\_i)\; +\; \backslash frac\; (uV\_j\; +\; \backslash tfrac\backslash rho\; V\_i\; V\_i\; V\_j\; +\; J\_\; +\; P\_V\_i)\; -\; nF\_iV\_i\; =\; 0,$$

where $u\; =\; \backslash tfrac\; \backslash rho\; \backslash langle\; (v\_i-V\_i)\; (v\_i-V\_i)\; \backslash rangle$ is the kinetic thermal energy density, and $J\_\; =\; \backslash tfrac\; \backslash rho\; \backslash langle(v\_i\; -\; V\_i)(v\_k\; -\; V\_k)(v\_k\; -\; V\_k)\backslash rangle$ is the heat flux vector.

Hamiltonian mechanics

In Hamiltonian mechanics, the Boltzmann equation is often written more generally as$$\backslash hat[f]=\backslash mathbf[f],$$where is the Liouville operator (there is an inconsistent definition between the Liouville operator as defined here and the one in the article linked) describing the evolution of a phase space volume and is the collision operator. The non-relativistic form of is$$\backslash hat\_\backslash mathrm\; =\; \backslash frac\; +\; \backslash frac\; \backslash cdot\; \backslash nabla\; +\; \backslash mathbf\backslash cdot\backslash frac\backslash ,.$$

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,^[8] 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.^[9]

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^[10] $$\backslash hat\_\backslash mathrm[f]\; =\; p^\backslash alpha\backslash frac\; -\; \backslash Gamma^\backslash alpha\_\; p^\backslash beta\; p^\backslash gamma\; \backslash frac\; =\; C[f],$$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.^{[11] [12]}

In physical cosmology the fully covariant approach has been used to study the cosmic microwave background radiation.^[13] More generically the study of processes in the early universe often attempt to take into account the effects of quantum mechanics and general relativity. In the very dense medium formed by the primordial plasma after the Big Bang, particles are continuously created and annihilated. In such an environment quantum coherence and the spatial extension of the wavefunction can affect the dynamics, making it questionable whether the classical phase space distribution f that appears in the Boltzmann equation is suitable to describe the system. In many cases it is, however, possible to derive an effective Boltzmann equation for a generalized distribution function from first principles of quantum field theory. This includes the formation of the light elements in Big Bang nucleosynthesis, the production of dark matter and baryogenesis.

Solving the equation

Exact solutions to the Boltzmann equations have been proven to exist in some cases;^[14] this analytical approach provides insight, but is not generally usable in practical problems.

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^{[15] [16]} to plasma flows.^[17] 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.^[18]

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^[19]). 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.^[20]

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.^[21] 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.^[21]

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.^[22] These must be derived by using the BBGKY hierarchy.

Boltzmann-like equations are also used for the movement of cells.^{[23] [24]} 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.

See also

• Vlasov equation
• The Vlasov–Poisson equation
• Fokker - Planck equation
• Williams–Boltzmann equation
  • Derivation of Navier–Stokes equation from LBE Derivation of Jeans equation from BE
• Jeans's theorem
• H-theorem

References

• Book: Harris . Stewart . An introduction to the theory of the Boltzmann equation . Dover Books. 221 . 1971 . 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 . On the Boltzmann equation part I: Existence . Arch. Rational Mech. Anal. . 45 . 1 . 1–16 . 1972 . 10.1007/BF00253392 . 1972ArRMA..45....1A . 117877311 .
• Arkeryd . Leif . Leif Arkeryd . On the Boltzmann equation part II: The full initial value problem . Arch. Rational Mech. Anal. . 45 . 1 . 17–34 . 1972 . 10.1007/BF00253393 . 1972ArRMA..45...17A . 119481100 .
• Arkeryd . Leif . On the Boltzmann equation part I: Existence . Arch. Rational Mech. Anal. . 45 . 1 . 1–16 . 1972 . 10.1007/BF00253392 . 1972ArRMA..45....1A . 117877311 .
• DiPerna . R. J. . Lions . P.-L. . On the Cauchy problem for Boltzmann equations: global existence and weak stability . Ann. of Math. . 2 . 130 . 2 . 321–366 . 1989 . 10.2307/1971423. 1971423 .

External links

• The Boltzmann Transport Equation by Franz Vesely
• Boltzmann gaseous behaviors solved

Notes and References

1. 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.
2. DiPerna . R. J. . Lions . P.-L. . On the Cauchy problem for Boltzmann equations: global existence and weak stability . Ann. of Math. . 2 . 130 . 321–366 . 1989 . 10.2307/1971423 . 2. 1971423 .
3. Philip T. Gressman . Philip Gressman . Robert M. Strain . amp. 2010 . Global classical solutions of the Boltzmann equation with long-range interactions . Proceedings of the National Academy of Sciences . 107 . 5744–5749 . 10.1073/pnas.1001185107 . 2010PNAS..107.5744G . 1002.3639 . 13 . 20231489 . 2851887. free .
4. Book: Huang, Kerson . 1987 . Statistical Mechanics . limited . New York . Wiley . 978-0-471-81518-1 . 53 . Second .
5. McGraw Hill Encyclopaedia of Physics (2nd Edition), S. P. Parker, 1993, .
6. A Model for Collision Processes in Gases. I. Small Amplitude Processes in Charged and Neutral One-Component Systems . Physical Review. 1954-05-01. 511–525. 94. 3. 10.1103/PhysRev.94.511. P. L.. Bhatnagar. E. P.. Gross. M.. Krook. 1954PhRv...94..511B .
7. Book: de Groot . S. R.. Mazur. P.. Non-Equilibrium Thermodynamics . 1984 . Dover Publications Inc. . New York . 978-0-486-64741-8.
8. Book: Edward Kolb . Michael Turner . amp. The Early Universe. 1990. Westview Press. 978-0-201-62674-2.
9. M. Drewes . C. Weniger . S. Mendizabal . Phys. Lett. B. 8 January 2013. 718. 3. 1119–1124. 10.1016/j.physletb.2012.11.046. 1202.1301 . 2013PhLB..718.1119D . The Boltzmann equation from quantum field theory. 119253828 .
10. 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}
11. Debbasch . Fabrice . Willem van Leeuwen . General relativistic Boltzmann equation I: Covariant treatment . Physica A . 388 . 7 . 1079–1104 . 2009. 2009PhyA..388.1079D . 10.1016/j.physa.2008.12.023 .
12. Debbasch . Fabrice . Willem van Leeuwen . General relativistic Boltzmann equation II: Manifestly covariant treatment . Physica A . 388 . 9 . 1818–34 . 2009 . 2009PhyA..388.1818D . 10.1016/j.physa.2009.01.009 .
13. Maartens R, Gebbie T, Ellis GFR (1999). "Cosmic microwave background anisotropies: Nonlinear dynamics". Phys. Rev. D. 59 (8): 083506
14. 10.1090/S0894-0347-2011-00697-8 . 1011.5441 . Global Classical Solutions of the Boltzmann Equation without Angular Cut-off . Philip T. Gressman . Robert M. Strain . Journal of the American Mathematical Society . 24 . 3 . 2011 . 771. 115167686 .
15. A discontinuous finite element solution of the Boltzmann kinetic equation in collisionless and BGK forms for macroscopic gas flows. Applied Mathematical Modelling. 2011-03-01. 996–1015. 35. 3. 10.1016/j.apm.2010.07.027. Ben. Evans. Ken. Morgan. Oubay. Hassan. free.
16. Evans. B.. Walton. S.P.. December 2017. Aerodynamic optimisation of a hypersonic reentry vehicle based on solution of the Boltzmann–BGK equation and evolutionary optimisation . Applied Mathematical Modelling. 52. 215–240 . 10.1016/j.apm.2017.07.024. 0307-904X . free.
17. Numerical Solution of the Boltzmann Equation I: Spectrally Accurate Approximation of the Collision Operator. SIAM Journal on Numerical Analysis. 2000-01-01. 0036-1429. 1217–1245. 37. 4. 10.1137/S0036142998343300. L.. Pareschi. G.. Russo. 10.1.1.46.2853.
18. H.J.W. Müller-Kirsten, Basics of Statistical Mechanics, Chapter 13, 2nd ed., World Scientific (2013), .
19. Sydney Chapman; Thomas George Cowling The mathematical theory of non-uniform gases: an account of the kinetic theory of viscosity, thermal conduction, and diffusion in gases, Cambridge University Press, 1970.
20. 10.1098/rsta/376/2118 . 376 . 2018 . Philosophical Transactions of the Royal Society A . Theme issue 'Hilbert's sixth problem' . 2118 . free .
21. Web site: Enskog Equation - an overview ScienceDirect Topics . 2022-05-10 . www.sciencedirect.com.
22. van Noije . T. P. C. . Ernst . M. H. . 1997-06-03 . Ring Kinetic Theory for an Idealized Granular Gas . cond-mat/9706020 .
23. Chauviere, A. . Hillen, T. . Preziosi, L. . 2007 . Modeling cell movement in anisotropic and heterogeneous network tissues . American Institute of Mathematical Sciences . 2 . 2. 333–357. 10.3934/nhm.2007.2.333.
24. Conte . Martina . Loy . Nadia . 2022-02-12 . Multi-Cue Kinetic Model with Non-Local Sensing for Cell Migration on a Fiber Network with Chemotaxis . Bulletin of Mathematical Biology . en . 84 . 3 . 42 . 10.1007/s11538-021-00978-1 . 1522-9602 . 8840942 . 35150333.