Boltzmann equation

3059: 2559: 3054:{\displaystyle {\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} }{m}}dt+{\frac {\partial f}{\partial \mathbf {p} }}\cdot \mathbf {F} \,dt\end{aligned}}} 2318: 1955: 8644:. 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 1125: 8878: 2313:{\displaystyle {\begin{aligned}dN_{\mathrm {coll} }&=\left({\frac {\partial f}{\partial t}}\right)_{\mathrm {coll} }\Delta t\,d^{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 f\,d^{3}\mathbf {r} \,d^{3}\mathbf {p} \end{aligned}}} 42: 761: 3298: 4466: 1838: 3513: 1394: 7521: 3099: 1120:{\displaystyle {\begin{aligned}N&=\int \limits _{\mathrm {momenta} }d^{3}\mathbf {p} \int \limits _{\mathrm {positions} }d^{3}\mathbf {r} \,f(\mathbf {r} ,\mathbf {p} ,t)\\&=\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{aligned}}} 6816: 7269: 4287: 4084:

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:

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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. The assumption in the BGK approximation is that the

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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.

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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.

5412: 256:, 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. 3293:{\displaystyle {\frac {\partial f}{\partial \mathbf {p} }}=\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}}}=\nabla _{\mathbf {p} }f} 5418: 5704: 4089: 4889: 4509: 6052: 4040: 5570: 1618: 4461:{\displaystyle {\frac {\partial f_{i}}{\partial t}}+{\frac {\mathbf {p} _{i}}{m_{i}}}\cdot \nabla f_{i}+\mathbf {F} \cdot {\frac {\partial f_{i}}{\partial \mathbf {p} _{i}}}=\left({\frac {\partial f_{i}}{\partial t}}\right)_{\text{coll}},} 2521: 7061: 7669:. 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 1833:{\displaystyle 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} } 7681:

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

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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

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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.

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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

6920: 5063: 6208: 3612: 3508:{\displaystyle {\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} }} 1389:{\displaystyle {\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}},} 6451: 5319: 6562: 7516:{\displaystyle {\hat {\mathbf {L} }}_{\mathrm {GR} }=p^{\alpha }{\frac {\partial f}{\partial x^{\alpha }}}-\Gamma ^{\alpha }{}_{\beta \gamma }p^{\beta }p^{\gamma }{\frac {\partial f}{\partial p^{\alpha }}}=C,} 3598:

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

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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

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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}

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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. These must be derived by using the

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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

6811:{\displaystyle {\frac {\partial }{\partial t}}(u+{\tfrac {1}{2}}\rho V_{i}V_{i})+{\frac {\partial }{\partial x_{j}}}(uV_{j}+{\tfrac {1}{2}}\rho V_{i}V_{i}V_{j}+J_{qj}+P_{ij}V_{i})-nF_{i}V_{i}=0,} 1944: 1888: 543: 7133: 5151: 5969: 7264:{\displaystyle {\hat {\mathbf {L} }}_{\mathrm {NR} }={\frac {\partial }{\partial t}}+{\frac {\mathbf {p} }{m}}\cdot \nabla +\mathbf {F} \cdot {\frac {\partial }{\partial \mathbf {p} }}\,.} 4988: 6127: 5100: 4771: 2440: 7322:

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

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

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The Boltzmann equation can be used to derive the fluid dynamic conservation laws for mass, charge, momentum, and energy. For a fluid consisting of only one kind of particle, the

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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)

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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

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Bhatnagar, P. L.; Gross, E. P.; Krook, M. (1954-05-01). "A Model for Collision Processes in Gases. I. Small Amplitude Processes in Charged and Neutral One-Component Systems".

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that carry internal degrees of freedom, the corresponding generalized Boltzmann equations must have inelastic collision integrals. Such equations can describe invasions of

7145:(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 5559:{\displaystyle \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 Ap_{j}\rangle ),} 182: 160: 5820:{\displaystyle \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,} 6081: 5214: 1560: 4204:{\displaystyle {\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),} 1398:

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

6925: 5897: 5870: 5268: 5241: 5178: 4733:{\displaystyle \left({\frac {\partial f_{i}}{\partial t}}\right)_{\mathrm {coll} }=\sum _{j=1}^{n}\iint g_{ij}I_{ij}(g_{ij},\Omega )\,d\Omega \,d^{3}\mathbf {p'} ,} 4258: 4231: 5001: 8839: 7762: 3609:

assumption". Under this assumption the collision term can be written as a momentum-space integral over the product of one-particle distribution functions:

5693:{\displaystyle \int AF_{j}{\frac {\partial f}{\partial p_{j}}}\,d^{3}\mathbf {p} =-nF_{j}\left\langle {\frac {\partial A}{\partial p_{j}}}\right\rangle ,} 8649: 8373:

Pareschi, L.; Russo, G. (2000-01-01). "Numerical Solution of the Boltzmann Equation I: Spectrally Accurate Approximation of the Collision Operator".

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is the local Maxwellian distribution function given the gas temperature at this point in space. This is also called "relaxation time approximation".

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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

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the fully covariant approach has been used to study the cosmic microwave background radiation. More generically the study of processes in the

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which gives this probability per unit phase-space volume, or probability per unit length cubed per unit momentum cubed, at an instant of time

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term – accounting for the forces acting between particles in collisions. Expressions for each term on the right side are provided below.

8334:"Aerodynamic optimisation of a hypersonic reentry vehicle based on solution of the Boltzmann–BGK equation and evolutionary optimisation" 756:

over a region of position space and momentum space gives the total number of particles which have positions and momenta in that region:

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The mathematical theory of non-uniform gases: an account of the kinetic theory of viscosity, thermal conduction, and diffusion in gases

8301:"A discontinuous finite element solution of the Boltzmann kinetic equation in collisionless and BGK forms for macroscopic gas flows" 8244:

Maartens R, Gebbie T, Ellis GFR (1999). "Cosmic microwave background anisotropies: Nonlinear dynamics". Phys. Rev. D. 59 (8): 083506

2363: 7637:) are generally used to find approximate solutions to the various forms of the Boltzmann equation. Example applications range from 4945: 9194: 9189: 7686:. The collision term is modified in Enskog equations such that particles have a finite size, for example they can be modelled as 6344:{\displaystyle {\frac {\partial }{\partial t}}(\rho V_{i})+{\frac {\partial }{\partial x_{j}}}(\rho V_{i}V_{j}+P_{ij})-nF_{i}=0,} 3873:{\displaystyle \left({\frac {\partial f}{\partial t}}\right)_{\text{coll}}=\iint gI(g,\Omega )\,d\Omega \,d^{3}\mathbf {p} _{B},} 1139:

is associated with a number of particles, the phase space is for one-particle (not all of them, which is usually the case with

5407:{\displaystyle \int A{\frac {\partial f}{\partial t}}\,d^{3}\mathbf {p} ={\frac {\partial }{\partial t}}(n\langle A\rangle ),} 8825: 8637: 8422: 8254:

Philip T. Gressman; Robert M. Strain (2011). "Global Classical Solutions of the Boltzmann Equation without Angular Cut-off".

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Debbasch, Fabrice; Willem van Leeuwen (2009). "General relativistic Boltzmann equation II: Manifestly covariant treatment".

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DiPerna, R. J.; Lions, P.-L. (1989). "On the Cauchy problem for Boltzmann equations: global existence and weak stability".

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DiPerna, R. J.; Lions, P.-L. (1989). "On the Cauchy problem for Boltzmann equations: global existence and weak stability".

7626: 1895: 1901: 1845: 500: 3570: 28: 7302:. It is not a priori clear that the state of a quantum system can be characterized by a classical phase space density 5105: 8443: 8002: 243: 7767: 9224: 7662: 7286:

quantum systems in which the number of particles is not conserved in collisions. This has several applications in

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Debbasch, Fabrice; Willem van Leeuwen (2009). "General relativistic Boltzmann equation I: Covariant treatment".

9199: 9093: 5070: 3578: 20: 7845:, G. L. Trigg, VHC publishers, 1991, ISBN (Verlagsgesellschaft) 3-527-26954-1, ISBN (VHC Inc.) 0-89573-752-3. 7747: 7310:

exists which is the solution of an effective Boltzmann equation that can be derived from first principles of

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in space causing heat to flow from hotter regions to colder ones, by the random but biased transport of the

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M. Drewes; C. Weniger; S. Mendizabal (8 January 2013). "The Boltzmann equation from quantum field theory".

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and integrating over momentum yields four terms, which, using integration by parts, can be expressed as

8970: 8886: 8645: 7752: 7670: 7279: 6047:{\displaystyle {\frac {\partial }{\partial t}}\rho +{\frac {\partial }{\partial x_{j}}}(\rho V_{j})=0,} 4058: 2541: 1512: 1474: 713: 671: 190: 111: 8568:"Multi-Cue Kinetic Model with Non-Local Sensing for Cell Migration on a Fiber Network with Chemotaxis" 8417:

H.J.W. Müller-Kirsten, Basics of Statistical Mechanics, Chapter 13, 2nd ed., World Scientific (2013),

5910: 230:, when a fluid is in transport. One may also derive other properties characteristic to fluids such as 9209: 8807: 8514:

van Noije, T. P. C.; Ernst, M. H. (1997-06-03). "Ring Kinetic Theory for an Idealized Granular Gas".

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of the particle, the integrated Boltzmann equation becomes the conservation of momentum equation:

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of the collision, in which the relative momenta of the colliding particles turns through an angle

165: 143: 8993: 8983: 8909: 8536: 7757: 4884:{\displaystyle g_{ij}=|\mathbf {p} _{i}-\mathbf {p} _{j}|=|\mathbf {p'} _{i}-\mathbf {p'} _{j}|,} 3533: 1891: 1613:{\displaystyle \mathbf {r} +\Delta \mathbf {r} =\mathbf {r} +{\frac {\mathbf {p} }{m}}\,\Delta t} 239: 6568:

of the particle, the integrated Boltzmann equation becomes the conservation of energy equation:

2516:{\displaystyle {\frac {df}{dt}}=\left({\frac {\partial f}{\partial t}}\right)_{\mathrm {coll} }} 9040: 8382: 7593:

can affect the dynamics, making it questionable whether the classical phase space distribution

7056:{\textstyle J_{qi}={\tfrac {1}{2}}\rho \langle (v_{i}-V_{i})(v_{k}-V_{k})(v_{k}-V_{k})\rangle } 753: 6057: 5966:

of the particle, the integrated Boltzmann equation becomes the conservation of mass equation:

5183: 8925: 8848: 7072: 6454: 3338: 7963: 9068: 8744: 8707: 8667: 8219: 8184: 8132: 8024: 7916: 7650: 7599: 7311: 7283: 5875: 5848: 5842: 5246: 5219: 5156: 4236: 4035:{\displaystyle g=|\mathbf {p} _{B}-\mathbf {p} _{A}|=|\mathbf {p'} _{B}-\mathbf {p'} _{A}|} 235: 78: 62: 35: 4216: 8: 9134: 9113: 7964: 7694: 7331: 3546: 348: 8748: 8711: 8671: 8223: 8188: 8136: 8028: 7920: 468:{\displaystyle d^{3}\mathbf {r} \,d^{3}\mathbf {p} =dx\,dy\,dz\,dp_{x}\,dp_{y}\,dp_{z}.} 222:

The Boltzmann equation can be used to determine how physical quantities change, such as

9161: 9073: 9050: 9031: 8975: 8789: 8760: 8723: 8683: 8600: 8567: 8515: 8281: 8263: 8148: 8122: 7939: 7906: 7894: 7872: 7772: 7717: 7578: 7566: 7531: 7338: 7287: 7142: 4910:. The integration is over the momentum components in the integrand (which are labelled 3556:

This equation is more useful than the principal one above, yet still incomplete, since

657:{\displaystyle dN=f(\mathbf {r} ,\mathbf {p} ,t)\,d^{3}\mathbf {r} \,d^{3}\mathbf {p} } 7326:; in galaxies, physical collisions between the stars are very rare, and the effect of 1898:). However, since collisions do occur, the particle density in the phase-space volume 9166: 9063: 9036: 8947: 8937: 8930: 8904: 8764: 8727: 8687: 8633: 8605: 8587: 8439: 8418: 8400: 8355: 8285: 8152: 8090: 8062: 7998: 7975: 7944: 7823: 7812: 7586: 7574: 4043: 1143: 1130: 8277: 9058: 8781: 8752: 8715: 8675: 8595: 8579: 8548: 8467: 8392: 8345: 8312: 8273: 8227: 8192: 8144: 8140: 8032: 7934: 7924: 7895:"Global classical solutions of the Boltzmann equation with long-range interactions" 7864: 7804: 7638: 3595: 1231: 250: 70: 9108: 1223:

It is assumed the particles in the system are identical (so each has an identical

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only, whose total value is conserved in a collision. Assume also that the force

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not due to other particles (see the collision term for the latter treatment).

9183: 9139: 8698:(1972). "On the Boltzmann equation part II: The full initial value problem". 8591: 8404: 8359: 7725: 7585:, particles are continuously created and annihilated. In such an environment 7306:. However, for a wide class of applications a well-defined generalization of 6915:{\textstyle u={\tfrac {1}{2}}\rho \langle (v_{i}-V_{i})(v_{i}-V_{i})\rangle } 4918:). The sum of integrals describes the entry and exit of particles of species 1140: 102: 7929: 9156: 8695: 8609: 8552: 8036: 7948: 7611: 7590: 7299: 5058:{\displaystyle \langle A\rangle ={\frac {1}{n}}\int Af\,d^{3}\mathbf {p} .} 3584: 7803:. Lecture Notes in Physics (LNP, vol. 660). Berlin, Heidelberg: Springer. 8965: 8960: 8537:"Modeling cell movement in anisotropic and heterogeneous network tissues" 8520: 7676: 7607: 7295: 4066: 3341: 3093: 282: 278: 98: 8817: 7997:

McGraw Hill Encyclopaedia of Physics (2nd Edition), S. P. Parker, 1993,

8955: 8793: 8756: 8719: 8679: 7876: 7729: 6446:{\displaystyle P_{ij}=\rho \langle (v_{i}-V_{i})(v_{j}-V_{j})\rangle } 9016: 8735:

Arkeryd, Leif (1972). "On the Boltzmann equation part I: Existence".

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Arkeryd, Leif (1972). "On the Boltzmann equation part I: Existence".

7777: 7581:. In the very dense medium formed by the primordial plasma after the 1403: 1399: 317: 231: 8785: 7868: 7798: 6557:{\displaystyle A={\frac {m(v_{i})^{2}}{2}}={\frac {p_{i}p_{i}}{2m}}} 1842:

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

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is the differential cross-section, as before, between particles

7721: 7701: 7687: 7649:, solution of the Boltzmann equation can be represented by an 2408:{\displaystyle d^{3}\mathbf {r} \,d^{3}\mathbf {p} \,\Delta t} 545:, is in question, at the heart of the equation is a quantity 74: 7274:

Quantum theory and violation of particle number conservation

7075:, the Boltzmann equation is often written more generally as 7712:

Boltzmann-like equations are also used for the movement of

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expansion). The first two terms of this expansion give the

5963: 3541: 1224: 223: 162:, and has momentum nearly equal to a given momentum vector 8209: 8174: 8084: 7151:

is the collision operator. The non-relativistic form of

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The collision term (Stosszahlansatz) and molecular chaos

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An introduction to the theory of the Boltzmann equation

7602:. This includes the formation of the light elements in 264: 8491:"Enskog Equation - an overview | ScienceDirect Topics" 7800:

Invariant Manifolds for Physical and Chemical Kinetics

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Limitations and further uses of the Boltzmann equation

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For a mixture of chemical species labelled by indices

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is in question. It is not part of the analysis to use

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Evans, Ben; Morgan, Ken; Hassan, Oubay (2011-03-01).

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The equation arises not by analyzing the individual

8014: 7317: 7290:, including the formation of the light elements in 7128:{\displaystyle {\hat {\mathbf {L} }}=\mathbf {C} ,} 4263: 73:in 1872. The classic example of such a system is a 7573:often attempt to take into account the effects of 7515: 7263: 7127: 7055: 6914: 6810: 6556: 6445: 6343: 6191: 6121: 6075: 6046: 5954: 5891: 5864: 5819: 5692: 5558: 5406: 5302: 5262: 5235: 5208: 5172: 5145: 5094: 5057: 4982: 4883: 4732: 4460: 4252: 4225: 4203: 4034: 3872: 3507: 3292: 3053: 2515: 2407: 2312: 1938: 1882: 1832: 1612: 1533: 1495: 1388: 1234:, one distribution is needed for each, see below. 1119: 734: 692: 656: 537: 467: 211: 176: 154: 132: 8480: 8460:Philosophical Transactions of the Royal Society A 8298: 7837: 7835: 7833: 7831: 3902:are the momenta of any two particles (labeled as 3233: 3184: 3135: 1409:Note that some authors use the particle velocity 9181: 8808:The Boltzmann Transport Equation by Franz Vesely 8535:Chauviere, A.; Hillen, T.; Preziosi, L. (2007). 5146:{\displaystyle \mathbf {p} \mapsto p_{i}=mv_{i}} 1437: 7899:Proceedings of the National Academy of Sciences 7330:can be neglected for times far longer than the 8513: 7828: 7797:Gorban, Alexander N.; Karlin, Ilya V. (2005). 7550:phase space as opposed to fully contravariant 4925: 4042:is the magnitude of the relative momenta (see 3562:cannot be solved unless the collision term in 1547:instantly acts on each particle, then at time 8833: 8372: 8056: 7763:Derivation of Navier–Stokes equation from LBE 668:have positions lying within a volume element 289:, and three more for each momentum component 281:of the system; in other words a set of three 8771: 8256:Journal of the American Mathematical Society 8108: 8106: 8080: 8078: 7993: 7991: 7970:(Second ed.). New York: Wiley. p. 7854: 7796: 7050: 6960: 6909: 6848: 6440: 6379: 6116: 6103: 5802: 5796: 5547: 5531: 5395: 5389: 5011: 5005: 4983:{\displaystyle n=\int f\,d^{3}\mathbf {p} .} 3600: 3300:is a shorthand for the momentum analogue of 1242:The general equation can then be written as 8541:American Institute of Mathematical Sciences 8331: 6922:is the kinetic thermal energy density, and 6122:{\displaystyle V_{i}=\langle v_{i}\rangle } 5835:is conserved in a collision. The values of 4766:, the magnitude of the relative momenta is 1465:some number of particles all have position 61:) describes the statistical behaviour of a 8840: 8826: 7278:It is possible to write down relativistic 4233:is the molecular collision frequency, and 3589: 3536:acting on the particles in the fluid, and 34:"BTE" redirects here. For other uses, see 8847: 8599: 8566:Conte, Martina; Loy, Nadia (2022-02-12). 8519: 8471: 8386: 8349: 8332:Evans, B.; Walton, S.P. (December 2017). 8316: 8267: 8126: 8103: 8085:Edward Kolb & Michael Turner (1990). 8075: 7988: 7938: 7928: 7910: 7257: 5749: 5617: 5474: 5349: 5270:is a function of position only, and that 5095:{\displaystyle \mathbf {x} \mapsto x_{i}} 5036: 4961: 4930: 4706: 4699: 3844: 3837: 3040: 2830: 2786: 2742: 2695: 2665: 2635: 2600: 2398: 2382: 2290: 2274: 2242: 2226: 2179: 2062: 2046: 1920: 1864: 1814: 1798: 1751: 1735: 1708: 1685: 1603: 1099: 1085: 1071: 1064: 1057: 1050: 875: 638: 622: 519: 448: 434: 420: 413: 406: 381: 8565: 8052: 8050: 8048: 8046: 7841:Encyclopaedia of Physics (2nd Edition), 7066: 5310:. Multiplying the Boltzmann equation by 5180:is the particle velocity vector. Define 40: 8734: 8694: 8657: 8456:"Theme issue 'Hilbert's sixth problem'" 7617: 184:(thus occupying a very small region of 9182: 8625: 8434:Sydney Chapman; Thomas George Cowling 4922:in or out of the phase-space element. 1890:is constant, which can be shown using 1642:. Then, in the absence of collisions, 1237: 8821: 8043: 7961: 4079:Simplifications to the collision term 3932:are the momenta after the collision, 3910:for convenience) before a collision, 8438:, Cambridge University Press, 1970. 8061:. New York: Dover Publications Inc. 7768:Derivation of Jeans equation from BE 6192:{\displaystyle A=m(v_{i})^{1}=p_{i}} 5303:{\displaystyle p_{i}\to \pm \infty } 2552: 2433: 1948: 265:The phase space and density function 105:region of space (mathematically the 8057:de Groot, S. R.; Mazur, P. (1984). 5899:, as they are linearly dependent). 5829:where the last term is zero, since 101:that the particle occupies a given 13: 8813:Boltzmann gaseous behaviors solved 8375:SIAM Journal on Numerical Analysis 7479: 7471: 7427: 7407: 7399: 7368: 7365: 7246: 7242: 7226: 7199: 7195: 7184: 7181: 6648: 6644: 6583: 6579: 6255: 6251: 6218: 6214: 6000: 5996: 5979: 5975: 5775: 5771: 5730: 5722: 5667: 5659: 5601: 5593: 5509: 5505: 5458: 5450: 5374: 5370: 5340: 5332: 5297: 4992:The average value of any function 4703: 4638: 4560: 4557: 4554: 4551: 4536: 4521: 4436: 4421: 4392: 4377: 4350: 4309: 4294: 4159: 4151: 4131: 4104: 4096: 3841: 3672: 3632: 3624: 3499: 3496: 3493: 3490: 3475: 3467: 3445: 3437: 3417: 3390: 3382: 3347: 3276: 3256: 3248: 3207: 3199: 3158: 3150: 3114: 3106: 3021: 3013: 2980: 2962: 2954: 2919: 2911: 2888: 2870: 2862: 2814: 2806: 2770: 2762: 2726: 2718: 2686: 2678: 2656: 2648: 2626: 2618: 2591: 2583: 2507: 2504: 2501: 2498: 2483: 2475: 2399: 2268: 2153: 2138: 2116: 2040: 2034: 2031: 2028: 2025: 2010: 2002: 1981: 1978: 1975: 1972: 1724: 1709: 1686: 1604: 1572: 1364: 1356: 1326: 1318: 1288: 1280: 1230:). For a mixture of more than one 975: 972: 969: 966: 963: 960: 957: 954: 951: 938: 935: 932: 929: 926: 923: 920: 854: 851: 848: 845: 842: 839: 836: 833: 830: 803: 800: 797: 794: 791: 788: 785: 355:. The small volume ("differential 269:The set of all possible positions 14: 9236: 8801: 7589:and the spatial extension of the 1534:{\displaystyle d^{3}\mathbf {p} } 1496:{\displaystyle d^{3}\mathbf {r} } 735:{\displaystyle d^{3}\mathbf {p} } 693:{\displaystyle d^{3}\mathbf {r} } 664:is the number of molecules which 212:{\displaystyle d^{3}\mathbf {p} } 133:{\displaystyle d^{3}\mathbf {r} } 16:Equation of statistical mechanics 8876: 8572:Bulletin of Mathematical Biology 7353: 7318:General relativity and astronomy 7250: 7233: 7214: 7169: 7109: 7086: 6464: 5955:{\displaystyle A=m(v_{i})^{0}=m} 5902: 5761: 5629: 5486: 5361: 5110: 5075: 5048: 4973: 4859: 4839: 4813: 4798: 4719: 4397: 4367: 4325: 4264:General equation (for a mixture) 4163: 4141: 4119: 4013: 3993: 3967: 3952: 3857: 3815: 3806: 3780: 3771: 3738: 3728: 3698: 3688: 3449: 3427: 3405: 3281: 3230: 3181: 3132: 3118: 3036: 3025: 2992: 2937: 2923: 2901: 2394: 2378: 2302: 2286: 2254: 2238: 2213: 2205: 2191: 2175: 2134: 2126: 2107: 2097: 2074: 2058: 1932: 1916: 1876: 1860: 1826: 1810: 1785: 1777: 1763: 1747: 1704: 1696: 1676: 1666: 1594: 1584: 1576: 1565: 1527: 1489: 1442:Consider particles described by 1402:of particles, and "coll" is the 891: 883: 871: 820: 728: 686: 650: 634: 609: 601: 531: 515: 393: 377: 205: 170: 148: 126: 8559: 8528: 8507: 8448: 8428: 8411: 8366: 8325: 8292: 8278:10.1090/S0894-0347-2011-00697-8 8247: 8238: 8203: 8168: 8159: 7893:& Robert M. Strain (2010). 6132: 6129:is the average fluid velocity. 4046:for more on this concept), and 944: 9195:Partial differential equations 9190:Eponymous equations of physics 8338:Applied Mathematical Modelling 8305:Applied Mathematical Modelling 8145:10.1016/j.physletb.2012.11.046 8059:Non-Equilibrium Thermodynamics 8008: 7955: 7883: 7848: 7790: 7507: 7501: 7380: 7374: 7357: 7173: 7119: 7113: 7102: 7096: 7090: 7047: 7021: 7018: 6992: 6989: 6963: 6906: 6880: 6877: 6851: 6770: 6664: 6636: 6592: 6502: 6488: 6437: 6411: 6408: 6382: 6313: 6271: 6243: 6227: 6167: 6153: 6032: 6016: 5937: 5923: 5805: 5790: 5550: 5525: 5398: 5383: 5291: 5203: 5190: 5114: 5079: 4874: 4832: 4824: 4792: 4696: 4644: 4641: 4619: 4195: 4176: 4028: 3986: 3978: 3946: 3834: 3831: 3802: 3796: 3767: 3758: 3724: 3718: 3684: 3678: 3675: 3663: 2223: 2201: 1795: 1773: 1047: 984: 901: 879: 619: 597: 29:Maxwell–Boltzmann distribution 1: 8619: 5216:as some function of momentum 1438:The force and diffusion terms 285:for each position coordinate 254:integro-differential equation 244:convection–diffusion equation 8863:Principle of maximum entropy 8632:. Dover Books. p. 221. 7753:Fokker–Planck equation 6453:is the pressure tensor (the 4504:, and the collision term is 3605:" and is also known as the " 557:probability density function 177:{\displaystyle \mathbf {p} } 155:{\displaystyle \mathbf {r} } 55:Boltzmann transport equation 7: 8232:10.1016/j.physa.2009.01.009 8197:10.1016/j.physa.2008.12.023 7758:Williams–Boltzmann equation 7748:The Vlasov–Poisson equation 7735: 7280:quantum Boltzmann equations 4926:Applications and extensions 3366: 3354: 3068: 2530: 2356: 2327: 1194:for particle 2, etc. up to 706:and momenta lying within a 320:: a point in this space is 259: 140:) centered at the position 21:Boltzmann's entropy formula 10: 9241: 8887:Statistical thermodynamics 8584:10.1007/s11538-021-00978-1 4059:differential cross section 1894:(see the discussion under 219:), at an instant of time. 33: 18: 9122: 9084: 9049: 9004: 8946: 8885: 8874: 8855: 8737:Arch. Rational Mech. Anal 8700:Arch. Rational Mech. Anal 8660:Arch. Rational Mech. Anal 8397:10.1137/S0036142998343300 8351:10.1016/j.apm.2017.07.024 8318:10.1016/j.apm.2010.07.027 7635:lattice Boltzmann methods 7063:is the heat flux vector. 6083:is the mass density, and 4278:the equation for species 3594:A key insight applied by 1146:systems), since only one 477:Since the probability of 347:, and each coordinate is 9147:Condensed matter physics 9130:Statistical field theory 8626:Harris, Stewart (1971). 7784: 7604:Big Bang nucleosynthesis 7328:gravitational collisions 7292:Big Bang nucleosynthesis 6076:{\displaystyle \rho =mn} 5209:{\displaystyle A(p_{i})} 4075:, due to the collision. 4065:into the element of the 316:. The entire space is 6- 9225:Thermodynamic equations 9005:Mathematical approaches 8994:Lennard-Jones potential 8910:thermodynamic potential 7930:10.1073/pnas.1001185107 7690:having a fixed radius. 7671:Hilbert's sixth problem 7667:Navier–Stokes equations 7639:hypersonic aerodynamics 3590:Two-body collision term 3364:and substituting into ( 1557:their position will be 1448:, each experiencing an 240:electrical conductivity 9041:conformal field theory 8553:10.3934/nhm.2007.2.333 8037:10.1103/PhysRev.94.511 7962:Huang, Kerson (1987). 7700:Many real fluids like 7517: 7337:Its generalization in 7265: 7129: 7057: 6916: 6812: 6558: 6447: 6345: 6193: 6123: 6077: 6048: 5956: 5893: 5866: 5821: 5694: 5560: 5408: 5304: 5264: 5237: 5210: 5174: 5147: 5096: 5059: 4984: 4931:Conservation equations 4885: 4734: 4589: 4462: 4254: 4227: 4205: 4036: 3874: 3601: 3549:, is often called the 3509: 3294: 3055: 2517: 2415:and taking the limits 2409: 2314: 1940: 1884: 1834: 1614: 1535: 1497: 1390: 1121: 736: 694: 658: 539: 469: 213: 178: 156: 134: 46: 9200:Statistical mechanics 8956:Ferromagnetism models 8849:Statistical mechanics 8495:www.sciencedirect.com 8473:10.1098/rsta/376/2118 7966:Statistical Mechanics 7518: 7266: 7130: 7073:Hamiltonian mechanics 7067:Hamiltonian mechanics 7058: 6917: 6813: 6559: 6457:plus the hydrostatic 6455:viscous stress tensor 6448: 6346: 6194: 6124: 6078: 6049: 5957: 5894: 5892:{\displaystyle p_{i}} 5867: 5865:{\displaystyle v_{i}} 5822: 5695: 5561: 5409: 5305: 5265: 5263:{\displaystyle F_{i}} 5238: 5236:{\displaystyle p_{i}} 5211: 5175: 5173:{\displaystyle v_{i}} 5148: 5097: 5060: 4985: 4886: 4735: 4569: 4463: 4255: 4253:{\displaystyle f_{0}} 4228: 4206: 4037: 3875: 3510: 3295: 3056: 2518: 2410: 2315: 1941: 1885: 1835: 1615: 1536: 1498: 1391: 1122: 737: 695: 659: 540: 470: 214: 179: 157: 135: 79:temperature gradients 44: 7695:inelastic collisions 7651:asymptotic expansion 7618:Solving the equation 7606:, the production of 7600:quantum field theory 7345: 7312:quantum field theory 7294:, the production of 7161: 7079: 6926: 6824: 6574: 6473: 6357: 6209: 6141: 6087: 6058: 5970: 5911: 5876: 5849: 5705: 5571: 5419: 5320: 5278: 5247: 5220: 5184: 5157: 5106: 5071: 5002: 4946: 4772: 4510: 4288: 4237: 4226:{\displaystyle \nu } 4217: 4090: 3936: 3613: 3547:Coulomb interactions 3376: 3100: 2560: 2441: 2364: 1956: 1902: 1892:Hamilton's equations 1846: 1654: 1561: 1513: 1475: 1415:instead of momentum 1246: 762: 714: 672: 582: 501: 363: 236:thermal conductivity 191: 166: 144: 112: 69:; it was devised by 63:thermodynamic system 36:BTE (disambiguation) 25:Stefan–Boltzmann law 19:For other uses, see 9205:Transport phenomena 9135:elementary particle 8900:partition functions 8749:1972ArRMA..45....1A 8712:1972ArRMA..45...17A 8672:1972ArRMA..45....1A 8224:2009PhyA..388.1818D 8189:2009PhyA..388.1079D 8137:2013PhLB..718.1119D 8029:1954PhRv...94..511B 7921:2010PNAS..107.5744G 7718:composite particles 7332:age of the universe 4672: 4659: 1896:Liouville's theorem 1238:Principal statement 578:, defined so that, 9162:information theory 9069:correlation length 9064:Critical exponents 9051:Critical phenomena 9032:stochastic process 9012:Boltzmann equation 8905:equations of state 8757:10.1007/BF00253392 8720:10.1007/BF00253393 8680:10.1007/BF00253392 8089:. Westview Press. 8087:The Early Universe 7891:Philip T. Gressman 7732:-related effects. 7716:. Since cells are 7579:general relativity 7567:physical cosmology 7532:Christoffel symbol 7513: 7339:general relativity 7288:physical cosmology 7261: 7143:Liouville operator 7125: 7053: 6955: 6912: 6843: 6808: 6693: 6611: 6554: 6443: 6341: 6189: 6119: 6073: 6044: 5952: 5889: 5862: 5817: 5690: 5556: 5404: 5300: 5260: 5233: 5206: 5170: 5143: 5092: 5055: 4980: 4881: 4730: 4660: 4647: 4458: 4250: 4223: 4201: 4032: 3870: 3505: 3290: 3051: 3049: 2513: 2405: 2310: 2308: 1936: 1880: 1830: 1610: 1531: 1493: 1386: 1117: 1115: 980: 943: 859: 808: 732: 690: 654: 535: 465: 249:The equation is a 209: 174: 152: 130: 65:not in a state of 51:Boltzmann equation 47: 9177: 9176: 9167:Boltzmann machine 9037:mean-field theory 8938:Maxwell relations 8639:978-0-486-43831-3 8423:978-981-4449-53-3 8096:978-0-201-62674-2 8068:978-0-486-64741-8 7981:978-0-471-81518-1 7905:(13): 5744–5749. 7818:978-3-540-22684-0 7724:cells in tissue, 7647:local equilibrium 7627:numerical methods 7587:quantum coherence 7575:quantum mechanics 7493: 7421: 7360: 7255: 7221: 7206: 7176: 7093: 6954: 6842: 6692: 6662: 6610: 6590: 6552: 6515: 6269: 6225: 6014: 5986: 5787: 5782: 5746: 5737: 5681: 5615: 5523: 5501: 5472: 5445: 5381: 5347: 5025: 4543: 4452: 4443: 4408: 4345: 4316: 4168: 4126: 4111: 4044:relative velocity 3648: 3639: 3571:Maxwell–Boltzmann 3517:In this context, 3482: 3454: 3412: 3397: 3270: 3236: 3221: 3187: 3172: 3138: 3123: 3076: 3075: 3030: 2999: 2969: 2928: 2877: 2828: 2784: 2740: 2693: 2663: 2633: 2598: 2538: 2537: 2490: 2462: 2335: 2334: 2114: 2017: 1683: 1601: 1380: 1371: 1342: 1333: 1304: 1295: 1267: 945: 914: 824: 779: 481:molecules, which 9232: 9210:Ludwig Boltzmann 9059:Phase transition 8880: 8879: 8842: 8835: 8828: 8819: 8818: 8797: 8768: 8731: 8691: 8650:Landau equations 8643: 8614: 8613: 8603: 8563: 8557: 8556: 8532: 8526: 8525: 8523: 8521:cond-mat/9706020 8511: 8505: 8504: 8502: 8501: 8487: 8478: 8477: 8475: 8452: 8446: 8432: 8426: 8415: 8409: 8408: 8390: 8381:(4): 1217–1245. 8370: 8364: 8363: 8353: 8329: 8323: 8322: 8320: 8296: 8290: 8289: 8271: 8251: 8245: 8242: 8236: 8235: 8207: 8201: 8200: 8183:(7): 1079–1104. 8172: 8166: 8163: 8157: 8156: 8130: 8121:(3): 1119–1124. 8110: 8101: 8100: 8082: 8073: 8072: 8054: 8041: 8040: 8012: 8006: 7995: 7986: 7985: 7969: 7959: 7953: 7952: 7942: 7932: 7914: 7887: 7881: 7880: 7852: 7846: 7839: 7826: 7822: 7794: 7561: 7549: 7529: 7522: 7520: 7519: 7514: 7494: 7492: 7491: 7490: 7477: 7469: 7467: 7466: 7457: 7456: 7447: 7446: 7438: 7435: 7434: 7422: 7420: 7419: 7418: 7405: 7397: 7395: 7394: 7373: 7372: 7371: 7362: 7361: 7356: 7351: 7270: 7268: 7267: 7262: 7256: 7254: 7253: 7241: 7236: 7222: 7217: 7212: 7207: 7205: 7194: 7189: 7188: 7187: 7178: 7177: 7172: 7167: 7156: 7150: 7140: 7134: 7132: 7131: 7126: 7112: 7095: 7094: 7089: 7084: 7062: 7060: 7059: 7054: 7046: 7045: 7033: 7032: 7017: 7016: 7004: 7003: 6988: 6987: 6975: 6974: 6956: 6947: 6941: 6940: 6921: 6919: 6918: 6913: 6905: 6904: 6892: 6891: 6876: 6875: 6863: 6862: 6844: 6835: 6817: 6815: 6814: 6809: 6798: 6797: 6788: 6787: 6769: 6768: 6759: 6758: 6743: 6742: 6727: 6726: 6717: 6716: 6707: 6706: 6694: 6685: 6679: 6678: 6663: 6661: 6660: 6659: 6643: 6635: 6634: 6625: 6624: 6612: 6603: 6591: 6589: 6578: 6563: 6561: 6560: 6555: 6553: 6551: 6543: 6542: 6541: 6532: 6531: 6521: 6516: 6511: 6510: 6509: 6500: 6499: 6483: 6452: 6450: 6449: 6444: 6436: 6435: 6423: 6422: 6407: 6406: 6394: 6393: 6372: 6371: 6350: 6348: 6347: 6342: 6331: 6330: 6312: 6311: 6296: 6295: 6286: 6285: 6270: 6268: 6267: 6266: 6250: 6242: 6241: 6226: 6224: 6213: 6198: 6196: 6195: 6190: 6188: 6187: 6175: 6174: 6165: 6164: 6128: 6126: 6125: 6120: 6115: 6114: 6099: 6098: 6082: 6080: 6079: 6074: 6053: 6051: 6050: 6045: 6031: 6030: 6015: 6013: 6012: 6011: 5995: 5987: 5985: 5974: 5961: 5959: 5958: 5953: 5945: 5944: 5935: 5934: 5898: 5896: 5895: 5890: 5888: 5887: 5871: 5869: 5868: 5863: 5861: 5860: 5840: 5834: 5826: 5824: 5823: 5818: 5789: 5788: 5785: 5783: 5781: 5770: 5764: 5759: 5758: 5748: 5747: 5744: 5742: 5738: 5736: 5728: 5720: 5699: 5697: 5696: 5691: 5686: 5682: 5680: 5679: 5678: 5665: 5657: 5651: 5650: 5632: 5627: 5626: 5616: 5614: 5613: 5612: 5599: 5591: 5589: 5588: 5565: 5563: 5562: 5557: 5546: 5545: 5524: 5522: 5521: 5520: 5504: 5502: 5494: 5489: 5484: 5483: 5473: 5471: 5470: 5469: 5456: 5448: 5446: 5441: 5437: 5436: 5426: 5413: 5411: 5410: 5405: 5382: 5380: 5369: 5364: 5359: 5358: 5348: 5346: 5338: 5330: 5309: 5307: 5306: 5301: 5290: 5289: 5269: 5267: 5266: 5261: 5259: 5258: 5242: 5240: 5239: 5234: 5232: 5231: 5215: 5213: 5212: 5207: 5202: 5201: 5179: 5177: 5176: 5171: 5169: 5168: 5152: 5150: 5149: 5144: 5142: 5141: 5126: 5125: 5113: 5101: 5099: 5098: 5093: 5091: 5090: 5078: 5064: 5062: 5061: 5056: 5051: 5046: 5045: 5026: 5018: 4997: 4989: 4987: 4986: 4981: 4976: 4971: 4970: 4941: 4901: 4890: 4888: 4887: 4882: 4877: 4872: 4871: 4866: 4865: 4852: 4851: 4846: 4845: 4835: 4827: 4822: 4821: 4816: 4807: 4806: 4801: 4795: 4787: 4786: 4765: 4739: 4737: 4736: 4731: 4726: 4725: 4716: 4715: 4695: 4694: 4685: 4684: 4668: 4655: 4634: 4633: 4618: 4617: 4605: 4604: 4588: 4583: 4565: 4564: 4563: 4548: 4544: 4542: 4534: 4533: 4532: 4519: 4503: 4467: 4465: 4464: 4459: 4454: 4453: 4450: 4448: 4444: 4442: 4434: 4433: 4432: 4419: 4409: 4407: 4406: 4405: 4400: 4390: 4389: 4388: 4375: 4370: 4362: 4361: 4346: 4344: 4343: 4334: 4333: 4328: 4322: 4317: 4315: 4307: 4306: 4305: 4292: 4281: 4277: 4273:= 1, 2, 3, ..., 4259: 4257: 4256: 4251: 4249: 4248: 4232: 4230: 4229: 4224: 4210: 4208: 4207: 4202: 4188: 4187: 4169: 4167: 4166: 4157: 4149: 4144: 4127: 4122: 4117: 4112: 4110: 4102: 4094: 4074: 4064: 4056: 4041: 4039: 4038: 4033: 4031: 4026: 4025: 4020: 4019: 4006: 4005: 4000: 3999: 3989: 3981: 3976: 3975: 3970: 3961: 3960: 3955: 3949: 3931: 3920: 3901: 3890: 3879: 3877: 3876: 3871: 3866: 3865: 3860: 3854: 3853: 3824: 3823: 3818: 3809: 3789: 3788: 3783: 3774: 3751: 3750: 3745: 3744: 3731: 3711: 3710: 3705: 3704: 3691: 3650: 3649: 3646: 3644: 3640: 3638: 3630: 3622: 3604: 3567: 3561: 3539: 3531: 3514: 3512: 3511: 3506: 3504: 3503: 3502: 3487: 3483: 3481: 3473: 3465: 3455: 3453: 3452: 3443: 3435: 3430: 3413: 3408: 3403: 3398: 3396: 3388: 3380: 3363: 3336: 3325: 3314: 3303: 3299: 3297: 3296: 3291: 3286: 3285: 3284: 3271: 3269: 3268: 3267: 3254: 3246: 3244: 3243: 3238: 3237: 3229: 3222: 3220: 3219: 3218: 3205: 3197: 3195: 3194: 3189: 3188: 3180: 3173: 3171: 3170: 3169: 3156: 3148: 3146: 3145: 3140: 3139: 3131: 3124: 3122: 3121: 3112: 3104: 3091: 3081: 3070: 3060: 3058: 3057: 3052: 3050: 3039: 3031: 3029: 3028: 3019: 3011: 3000: 2995: 2990: 2970: 2968: 2960: 2952: 2944: 2940: 2929: 2927: 2926: 2917: 2909: 2904: 2878: 2876: 2868: 2860: 2852: 2848: 2844: 2843: 2842: 2829: 2827: 2826: 2825: 2812: 2804: 2799: 2798: 2785: 2783: 2782: 2781: 2768: 2760: 2755: 2754: 2741: 2739: 2738: 2737: 2724: 2716: 2706: 2702: 2694: 2692: 2684: 2676: 2664: 2662: 2654: 2646: 2634: 2632: 2624: 2616: 2599: 2597: 2589: 2581: 2553: 2549: 2532: 2522: 2520: 2519: 2514: 2512: 2511: 2510: 2495: 2491: 2489: 2481: 2473: 2463: 2461: 2453: 2445: 2434: 2430: 2422: 2414: 2412: 2411: 2406: 2397: 2392: 2391: 2381: 2376: 2375: 2353: 2343: 2329: 2319: 2317: 2316: 2311: 2309: 2305: 2300: 2299: 2289: 2284: 2283: 2261: 2257: 2252: 2251: 2241: 2236: 2235: 2216: 2208: 2194: 2189: 2188: 2178: 2173: 2172: 2163: 2159: 2137: 2129: 2115: 2110: 2105: 2100: 2081: 2077: 2072: 2071: 2061: 2056: 2055: 2039: 2038: 2037: 2022: 2018: 2016: 2008: 2000: 1986: 1985: 1984: 1949: 1945: 1943: 1942: 1937: 1935: 1930: 1929: 1919: 1914: 1913: 1889: 1887: 1886: 1881: 1879: 1874: 1873: 1863: 1858: 1857: 1839: 1837: 1836: 1831: 1829: 1824: 1823: 1813: 1808: 1807: 1788: 1780: 1766: 1761: 1760: 1750: 1745: 1744: 1734: 1730: 1707: 1699: 1684: 1679: 1674: 1669: 1647: 1641: 1619: 1617: 1616: 1611: 1602: 1597: 1592: 1587: 1579: 1568: 1556: 1546: 1540: 1538: 1537: 1532: 1530: 1525: 1524: 1508: 1502: 1500: 1499: 1494: 1492: 1487: 1486: 1470: 1464: 1461:Suppose at time 1457: 1447: 1433: 1420: 1414: 1395: 1393: 1392: 1387: 1382: 1381: 1378: 1376: 1372: 1370: 1362: 1354: 1344: 1343: 1340: 1338: 1334: 1332: 1324: 1316: 1306: 1305: 1302: 1300: 1296: 1294: 1286: 1278: 1268: 1266: 1258: 1250: 1232:chemical species 1229: 1215: 1204: 1193: 1184: 1176:for particle 1, 1175: 1166: 1157: 1151: 1138: 1126: 1124: 1123: 1118: 1116: 1112: 1111: 1098: 1097: 1084: 1083: 1040: 1039: 1027: 1026: 1014: 1013: 979: 978: 942: 941: 907: 894: 886: 874: 869: 868: 858: 857: 823: 818: 817: 807: 806: 751: 747: 741: 739: 738: 733: 731: 726: 725: 705: 699: 697: 696: 691: 689: 684: 683: 663: 661: 660: 655: 653: 648: 647: 637: 632: 631: 612: 604: 577: 554: 550: 544: 542: 541: 536: 534: 529: 528: 518: 513: 512: 496: 490: 480: 474: 472: 471: 466: 461: 460: 447: 446: 433: 432: 396: 391: 390: 380: 375: 374: 346: 315: 306: 297: 218: 216: 215: 210: 208: 203: 202: 183: 181: 180: 175: 173: 161: 159: 158: 153: 151: 139: 137: 136: 131: 129: 124: 123: 71:Ludwig Boltzmann 9240: 9239: 9235: 9234: 9233: 9231: 9230: 9229: 9220:1872 in Germany 9215:1872 in science 9180: 9179: 9178: 9173: 9118: 9080: 9045: 9027:BBGKY hierarchy 9022:Vlasov equation 9000: 8989:depletion force 8982:Particles with 8942: 8881: 8877: 8872: 8851: 8846: 8804: 8786:10.2307/1971423 8640: 8622: 8617: 8564: 8560: 8533: 8529: 8512: 8508: 8499: 8497: 8489: 8488: 8481: 8454: 8453: 8449: 8433: 8429: 8416: 8412: 8371: 8367: 8330: 8326: 8311:(3): 996–1015. 8297: 8293: 8252: 8248: 8243: 8239: 8208: 8204: 8173: 8169: 8164: 8160: 8111: 8104: 8097: 8083: 8076: 8069: 8055: 8044: 8017:Physical Review 8013: 8009: 7996: 7989: 7982: 7960: 7956: 7888: 7884: 7869:10.2307/1971423 7853: 7849: 7840: 7829: 7819: 7795: 7791: 7787: 7782: 7773:Jeans's theorem 7743:Vlasov equation 7738: 7707:BBGKY hierarchy 7684:Enskog equation 7679: 7663:Euler equations 7631:finite elements 7620: 7551: 7546: 7536: 7528: 7524: 7486: 7482: 7478: 7470: 7468: 7462: 7458: 7452: 7448: 7439: 7437: 7436: 7430: 7426: 7414: 7410: 7406: 7398: 7396: 7390: 7386: 7364: 7363: 7352: 7350: 7349: 7348: 7346: 7343: 7342: 7320: 7276: 7249: 7245: 7240: 7232: 7213: 7211: 7198: 7193: 7180: 7179: 7168: 7166: 7165: 7164: 7162: 7159: 7158: 7152: 7146: 7136: 7108: 7085: 7083: 7082: 7080: 7077: 7076: 7069: 7041: 7037: 7028: 7024: 7012: 7008: 6999: 6995: 6983: 6979: 6970: 6966: 6945: 6933: 6929: 6927: 6924: 6923: 6900: 6896: 6887: 6883: 6871: 6867: 6858: 6854: 6833: 6825: 6822: 6821: 6793: 6789: 6783: 6779: 6764: 6760: 6751: 6747: 6735: 6731: 6722: 6718: 6712: 6708: 6702: 6698: 6683: 6674: 6670: 6655: 6651: 6647: 6642: 6630: 6626: 6620: 6616: 6601: 6582: 6577: 6575: 6572: 6571: 6544: 6537: 6533: 6527: 6523: 6522: 6520: 6505: 6501: 6495: 6491: 6484: 6482: 6474: 6471: 6470: 6467: 6431: 6427: 6418: 6414: 6402: 6398: 6389: 6385: 6364: 6360: 6358: 6355: 6354: 6326: 6322: 6304: 6300: 6291: 6287: 6281: 6277: 6262: 6258: 6254: 6249: 6237: 6233: 6217: 6212: 6210: 6207: 6206: 6183: 6179: 6170: 6166: 6160: 6156: 6142: 6139: 6138: 6135: 6110: 6106: 6094: 6090: 6088: 6085: 6084: 6059: 6056: 6055: 6026: 6022: 6007: 6003: 5999: 5994: 5978: 5973: 5971: 5968: 5967: 5940: 5936: 5930: 5926: 5912: 5909: 5908: 5905: 5883: 5879: 5877: 5874: 5873: 5856: 5852: 5850: 5847: 5846: 5836: 5830: 5784: 5774: 5769: 5768: 5760: 5754: 5750: 5743: 5729: 5721: 5719: 5715: 5714: 5706: 5703: 5702: 5674: 5670: 5666: 5658: 5656: 5652: 5646: 5642: 5628: 5622: 5618: 5608: 5604: 5600: 5592: 5590: 5584: 5580: 5572: 5569: 5568: 5541: 5537: 5516: 5512: 5508: 5503: 5493: 5485: 5479: 5475: 5465: 5461: 5457: 5449: 5447: 5432: 5428: 5427: 5425: 5420: 5417: 5416: 5373: 5368: 5360: 5354: 5350: 5339: 5331: 5329: 5321: 5318: 5317: 5285: 5281: 5279: 5276: 5275: 5254: 5250: 5248: 5245: 5244: 5227: 5223: 5221: 5218: 5217: 5197: 5193: 5185: 5182: 5181: 5164: 5160: 5158: 5155: 5154: 5137: 5133: 5121: 5117: 5109: 5107: 5104: 5103: 5086: 5082: 5074: 5072: 5069: 5068: 5047: 5041: 5037: 5017: 5003: 5000: 4999: 4993: 4972: 4966: 4962: 4947: 4944: 4943: 4939: 4933: 4928: 4899: 4894: 4873: 4867: 4858: 4857: 4856: 4847: 4838: 4837: 4836: 4831: 4823: 4817: 4812: 4811: 4802: 4797: 4796: 4791: 4779: 4775: 4773: 4770: 4769: 4759: 4743: 4718: 4717: 4711: 4707: 4690: 4686: 4680: 4676: 4664: 4651: 4626: 4622: 4610: 4606: 4597: 4593: 4584: 4573: 4550: 4549: 4535: 4528: 4524: 4520: 4518: 4514: 4513: 4511: 4508: 4507: 4497: 4483: 4476: 4471: 4449: 4435: 4428: 4424: 4420: 4418: 4414: 4413: 4401: 4396: 4395: 4391: 4384: 4380: 4376: 4374: 4366: 4357: 4353: 4339: 4335: 4329: 4324: 4323: 4321: 4308: 4301: 4297: 4293: 4291: 4289: 4286: 4285: 4279: 4269: 4266: 4244: 4240: 4238: 4235: 4234: 4218: 4215: 4214: 4183: 4179: 4162: 4158: 4150: 4148: 4140: 4118: 4116: 4103: 4095: 4093: 4091: 4088: 4087: 4081: 4069: 4062: 4047: 4027: 4021: 4012: 4011: 4010: 4001: 3992: 3991: 3990: 3985: 3977: 3971: 3966: 3965: 3956: 3951: 3950: 3945: 3937: 3934: 3933: 3930: 3922: 3919: 3911: 3900: 3892: 3889: 3881: 3861: 3856: 3855: 3849: 3845: 3819: 3814: 3813: 3805: 3784: 3779: 3778: 3770: 3746: 3737: 3736: 3735: 3727: 3706: 3697: 3696: 3695: 3687: 3645: 3631: 3623: 3621: 3617: 3616: 3614: 3611: 3610: 3607:molecular chaos 3602:Stosszahlansatz 3592: 3587: 3581:distributions. 3563: 3557: 3551:Vlasov equation 3537: 3518: 3489: 3488: 3474: 3466: 3464: 3460: 3459: 3448: 3444: 3436: 3434: 3426: 3404: 3402: 3389: 3381: 3379: 3377: 3374: 3373: 3359: 3350: 3348:Final statement 3335: 3327: 3324: 3316: 3313: 3305: 3301: 3280: 3279: 3275: 3263: 3259: 3255: 3247: 3245: 3239: 3228: 3227: 3226: 3214: 3210: 3206: 3198: 3196: 3190: 3179: 3178: 3177: 3165: 3161: 3157: 3149: 3147: 3141: 3130: 3129: 3128: 3117: 3113: 3105: 3103: 3101: 3098: 3097: 3087: 3079: 3048: 3047: 3035: 3024: 3020: 3012: 3010: 2991: 2989: 2961: 2953: 2951: 2942: 2941: 2936: 2922: 2918: 2910: 2908: 2900: 2869: 2861: 2859: 2850: 2849: 2838: 2834: 2821: 2817: 2813: 2805: 2803: 2794: 2790: 2777: 2773: 2769: 2761: 2759: 2750: 2746: 2733: 2729: 2725: 2717: 2715: 2714: 2710: 2685: 2677: 2675: 2655: 2647: 2645: 2625: 2617: 2615: 2614: 2610: 2590: 2582: 2580: 2573: 2563: 2561: 2558: 2557: 2545: 2497: 2496: 2482: 2474: 2472: 2468: 2467: 2454: 2446: 2444: 2442: 2439: 2438: 2424: 2416: 2393: 2387: 2383: 2377: 2371: 2367: 2365: 2362: 2361: 2349: 2338: 2307: 2306: 2301: 2295: 2291: 2285: 2279: 2275: 2259: 2258: 2253: 2247: 2243: 2237: 2231: 2227: 2212: 2204: 2190: 2184: 2180: 2174: 2168: 2164: 2133: 2125: 2106: 2104: 2096: 2095: 2091: 2079: 2078: 2073: 2067: 2063: 2057: 2051: 2047: 2024: 2023: 2009: 2001: 1999: 1995: 1994: 1987: 1971: 1970: 1966: 1959: 1957: 1954: 1953: 1931: 1925: 1921: 1915: 1909: 1905: 1903: 1900: 1899: 1875: 1869: 1865: 1859: 1853: 1849: 1847: 1844: 1843: 1825: 1819: 1815: 1809: 1803: 1799: 1784: 1776: 1762: 1756: 1752: 1746: 1740: 1736: 1703: 1695: 1675: 1673: 1665: 1664: 1660: 1655: 1652: 1651: 1643: 1621: 1593: 1591: 1583: 1575: 1564: 1562: 1559: 1558: 1548: 1542: 1526: 1520: 1516: 1514: 1511: 1510: 1504: 1488: 1482: 1478: 1476: 1473: 1472: 1471:within element 1466: 1462: 1453: 1443: 1440: 1422: 1416: 1410: 1377: 1363: 1355: 1353: 1349: 1348: 1339: 1325: 1317: 1315: 1311: 1310: 1301: 1287: 1279: 1277: 1273: 1272: 1259: 1251: 1249: 1247: 1244: 1243: 1240: 1227: 1214: 1206: 1203: 1195: 1192: 1186: 1183: 1177: 1174: 1168: 1165: 1159: 1153: 1147: 1134: 1131:6-fold integral 1114: 1113: 1107: 1103: 1093: 1089: 1079: 1075: 1035: 1031: 1022: 1018: 1009: 1005: 950: 949: 919: 918: 905: 904: 890: 882: 870: 864: 860: 829: 828: 819: 813: 809: 784: 783: 772: 765: 763: 760: 759: 749: 743: 727: 721: 717: 715: 712: 711: 701: 685: 679: 675: 673: 670: 669: 649: 643: 639: 633: 627: 623: 608: 600: 583: 580: 579: 560: 552: 546: 530: 524: 520: 514: 508: 504: 502: 499: 498: 492: 486: 478: 456: 452: 442: 438: 428: 424: 392: 386: 382: 376: 370: 366: 364: 361: 360: 359:") is written 343: 339: 335: 321: 313: 308: 304: 299: 295: 290: 267: 262: 204: 198: 194: 192: 189: 188: 169: 167: 164: 163: 147: 145: 142: 141: 125: 119: 115: 113: 110: 109: 39: 32: 17: 12: 11: 5: 9238: 9228: 9227: 9222: 9217: 9212: 9207: 9202: 9197: 9192: 9175: 9174: 9172: 9171: 9170: 9169: 9164: 9159: 9152:Complex system 9149: 9144: 9143: 9142: 9137: 9126: 9124: 9120: 9119: 9117: 9116: 9111: 9106: 9101: 9096: 9090: 9088: 9082: 9081: 9079: 9078: 9077: 9076: 9071: 9061: 9055: 9053: 9047: 9046: 9044: 9043: 9034: 9029: 9024: 9019: 9014: 9008: 9006: 9002: 9001: 8999: 8998: 8997: 8996: 8991: 8980: 8979: 8978: 8973: 8968: 8963: 8952: 8950: 8944: 8943: 8941: 8940: 8935: 8934: 8933: 8928: 8923: 8918: 8907: 8902: 8897: 8891: 8889: 8883: 8882: 8875: 8873: 8871: 8870: 8868:ergodic theory 8865: 8859: 8857: 8853: 8852: 8845: 8844: 8837: 8830: 8822: 8816: 8815: 8810: 8803: 8802:External links 8800: 8799: 8798: 8780:(2): 321–366. 8769: 8732: 8692: 8654: 8653: 8638: 8621: 8618: 8616: 8615: 8558: 8547:(2): 333–357. 8527: 8506: 8479: 8466:(2118). 2018. 8447: 8427: 8410: 8388:10.1.1.46.2853 8365: 8324: 8291: 8246: 8237: 8218:(9): 1818–34. 8202: 8167: 8158: 8102: 8095: 8074: 8067: 8042: 8023:(3): 511–525. 8007: 7987: 7980: 7954: 7882: 7863:(2): 321–366. 7847: 7827: 7817: 7809:10.1007/b98103 7788: 7786: 7783: 7781: 7780: 7775: 7770: 7765: 7760: 7755: 7750: 7745: 7739: 7737: 7734: 7678: 7675: 7659:Chapman–Enskog 7655:Knudsen number 7619: 7616: 7571:early universe 7544: 7526: 7512: 7509: 7506: 7503: 7500: 7497: 7489: 7485: 7481: 7476: 7473: 7465: 7461: 7455: 7451: 7445: 7442: 7433: 7429: 7425: 7417: 7413: 7409: 7404: 7401: 7393: 7389: 7385: 7382: 7379: 7376: 7370: 7367: 7359: 7355: 7319: 7316: 7275: 7272: 7260: 7252: 7248: 7244: 7239: 7235: 7231: 7228: 7225: 7220: 7216: 7210: 7204: 7201: 7197: 7192: 7186: 7183: 7175: 7171: 7124: 7121: 7118: 7115: 7111: 7107: 7104: 7101: 7098: 7092: 7088: 7068: 7065: 7052: 7049: 7044: 7040: 7036: 7031: 7027: 7023: 7020: 7015: 7011: 7007: 7002: 6998: 6994: 6991: 6986: 6982: 6978: 6973: 6969: 6965: 6962: 6959: 6953: 6950: 6944: 6939: 6936: 6932: 6911: 6908: 6903: 6899: 6895: 6890: 6886: 6882: 6879: 6874: 6870: 6866: 6861: 6857: 6853: 6850: 6847: 6841: 6838: 6832: 6829: 6807: 6804: 6801: 6796: 6792: 6786: 6782: 6778: 6775: 6772: 6767: 6763: 6757: 6754: 6750: 6746: 6741: 6738: 6734: 6730: 6725: 6721: 6715: 6711: 6705: 6701: 6697: 6691: 6688: 6682: 6677: 6673: 6669: 6666: 6658: 6654: 6650: 6646: 6641: 6638: 6633: 6629: 6623: 6619: 6615: 6609: 6606: 6600: 6597: 6594: 6588: 6585: 6581: 6566:kinetic energy 6550: 6547: 6540: 6536: 6530: 6526: 6519: 6514: 6508: 6504: 6498: 6494: 6490: 6487: 6481: 6478: 6466: 6463: 6442: 6439: 6434: 6430: 6426: 6421: 6417: 6413: 6410: 6405: 6401: 6397: 6392: 6388: 6384: 6381: 6378: 6375: 6370: 6367: 6363: 6340: 6337: 6334: 6329: 6325: 6321: 6318: 6315: 6310: 6307: 6303: 6299: 6294: 6290: 6284: 6280: 6276: 6273: 6265: 6261: 6257: 6253: 6248: 6245: 6240: 6236: 6232: 6229: 6223: 6220: 6216: 6186: 6182: 6178: 6173: 6169: 6163: 6159: 6155: 6152: 6149: 6146: 6134: 6131: 6118: 6113: 6109: 6105: 6102: 6097: 6093: 6072: 6069: 6066: 6063: 6043: 6040: 6037: 6034: 6029: 6025: 6021: 6018: 6010: 6006: 6002: 5998: 5993: 5990: 5984: 5981: 5977: 5951: 5948: 5943: 5939: 5933: 5929: 5925: 5922: 5919: 5916: 5904: 5901: 5886: 5882: 5872:(and momentum 5859: 5855: 5841:correspond to 5816: 5813: 5810: 5807: 5804: 5801: 5798: 5795: 5792: 5780: 5777: 5773: 5767: 5763: 5757: 5753: 5741: 5735: 5732: 5727: 5724: 5718: 5713: 5710: 5689: 5685: 5677: 5673: 5669: 5664: 5661: 5655: 5649: 5645: 5641: 5638: 5635: 5631: 5625: 5621: 5611: 5607: 5603: 5598: 5595: 5587: 5583: 5579: 5576: 5555: 5552: 5549: 5544: 5540: 5536: 5533: 5530: 5527: 5519: 5515: 5511: 5507: 5500: 5497: 5492: 5488: 5482: 5478: 5468: 5464: 5460: 5455: 5452: 5444: 5440: 5435: 5431: 5424: 5403: 5400: 5397: 5394: 5391: 5388: 5385: 5379: 5376: 5372: 5367: 5363: 5357: 5353: 5345: 5342: 5337: 5334: 5328: 5325: 5299: 5296: 5293: 5288: 5284: 5257: 5253: 5230: 5226: 5205: 5200: 5196: 5192: 5189: 5167: 5163: 5140: 5136: 5132: 5129: 5124: 5120: 5116: 5112: 5089: 5085: 5081: 5077: 5054: 5050: 5044: 5040: 5035: 5032: 5029: 5024: 5021: 5016: 5013: 5010: 5007: 4979: 4975: 4969: 4965: 4960: 4957: 4954: 4951: 4937:number density 4932: 4929: 4927: 4924: 4897: 4880: 4876: 4870: 4864: 4861: 4855: 4850: 4844: 4841: 4834: 4830: 4826: 4820: 4815: 4810: 4805: 4800: 4794: 4790: 4785: 4782: 4778: 4755: 4729: 4724: 4721: 4714: 4710: 4705: 4702: 4698: 4693: 4689: 4683: 4679: 4675: 4671: 4667: 4663: 4658: 4654: 4650: 4646: 4643: 4640: 4637: 4632: 4629: 4625: 4621: 4616: 4613: 4609: 4603: 4600: 4596: 4592: 4587: 4582: 4579: 4576: 4572: 4568: 4562: 4559: 4556: 4553: 4547: 4541: 4538: 4531: 4527: 4523: 4517: 4493: 4481: 4474: 4457: 4447: 4441: 4438: 4431: 4427: 4423: 4417: 4412: 4404: 4399: 4394: 4387: 4383: 4379: 4373: 4369: 4365: 4360: 4356: 4352: 4349: 4342: 4338: 4332: 4327: 4320: 4314: 4311: 4304: 4300: 4296: 4265: 4262: 4247: 4243: 4222: 4200: 4197: 4194: 4191: 4186: 4182: 4178: 4175: 4172: 4165: 4161: 4156: 4153: 4147: 4143: 4139: 4136: 4133: 4130: 4125: 4121: 4115: 4109: 4106: 4101: 4098: 4080: 4077: 4030: 4024: 4018: 4015: 4009: 4004: 3998: 3995: 3988: 3984: 3980: 3974: 3969: 3964: 3959: 3954: 3948: 3944: 3941: 3926: 3915: 3896: 3885: 3869: 3864: 3859: 3852: 3848: 3843: 3840: 3836: 3833: 3830: 3827: 3822: 3817: 3812: 3808: 3804: 3801: 3798: 3795: 3792: 3787: 3782: 3777: 3773: 3769: 3766: 3763: 3760: 3757: 3754: 3749: 3743: 3740: 3734: 3730: 3726: 3723: 3720: 3717: 3714: 3709: 3703: 3700: 3694: 3690: 3686: 3683: 3680: 3677: 3674: 3671: 3668: 3665: 3662: 3659: 3656: 3653: 3643: 3637: 3634: 3629: 3626: 3620: 3591: 3588: 3586: 3583: 3501: 3498: 3495: 3492: 3486: 3480: 3477: 3472: 3469: 3463: 3458: 3451: 3447: 3442: 3439: 3433: 3429: 3425: 3422: 3419: 3416: 3411: 3407: 3401: 3395: 3392: 3387: 3384: 3349: 3346: 3331: 3320: 3309: 3289: 3283: 3278: 3274: 3266: 3262: 3258: 3253: 3250: 3242: 3235: 3232: 3225: 3217: 3213: 3209: 3204: 3201: 3193: 3186: 3183: 3176: 3168: 3164: 3160: 3155: 3152: 3144: 3137: 3134: 3127: 3120: 3116: 3111: 3108: 3074: 3073: 3064: 3062: 3046: 3043: 3038: 3034: 3027: 3023: 3018: 3015: 3009: 3006: 3003: 2998: 2994: 2988: 2985: 2982: 2979: 2976: 2973: 2967: 2964: 2959: 2956: 2950: 2947: 2945: 2943: 2939: 2935: 2932: 2925: 2921: 2916: 2913: 2907: 2903: 2899: 2896: 2893: 2890: 2887: 2884: 2881: 2875: 2872: 2867: 2864: 2858: 2855: 2853: 2851: 2847: 2841: 2837: 2833: 2824: 2820: 2816: 2811: 2808: 2802: 2797: 2793: 2789: 2780: 2776: 2772: 2767: 2764: 2758: 2753: 2749: 2745: 2736: 2732: 2728: 2723: 2720: 2713: 2709: 2705: 2701: 2698: 2691: 2688: 2683: 2680: 2674: 2671: 2668: 2661: 2658: 2653: 2650: 2644: 2641: 2638: 2631: 2628: 2623: 2620: 2613: 2609: 2606: 2603: 2596: 2593: 2588: 2585: 2579: 2576: 2574: 2572: 2569: 2566: 2565: 2536: 2535: 2526: 2524: 2509: 2506: 2503: 2500: 2494: 2488: 2485: 2480: 2477: 2471: 2466: 2460: 2457: 2452: 2449: 2404: 2401: 2396: 2390: 2386: 2380: 2374: 2370: 2333: 2332: 2323: 2321: 2304: 2298: 2294: 2288: 2282: 2278: 2273: 2270: 2267: 2264: 2262: 2260: 2256: 2250: 2246: 2240: 2234: 2230: 2225: 2222: 2219: 2215: 2211: 2207: 2203: 2200: 2197: 2193: 2187: 2183: 2177: 2171: 2167: 2162: 2158: 2155: 2152: 2149: 2146: 2143: 2140: 2136: 2132: 2128: 2124: 2121: 2118: 2113: 2109: 2103: 2099: 2094: 2090: 2087: 2084: 2082: 2080: 2076: 2070: 2066: 2060: 2054: 2050: 2045: 2042: 2036: 2033: 2030: 2027: 2021: 2015: 2012: 2007: 2004: 1998: 1993: 1990: 1988: 1983: 1980: 1977: 1974: 1969: 1965: 1962: 1961: 1934: 1928: 1924: 1918: 1912: 1908: 1878: 1872: 1868: 1862: 1856: 1852: 1828: 1822: 1818: 1812: 1806: 1802: 1797: 1794: 1791: 1787: 1783: 1779: 1775: 1772: 1769: 1765: 1759: 1755: 1749: 1743: 1739: 1733: 1729: 1726: 1723: 1720: 1717: 1714: 1711: 1706: 1702: 1698: 1694: 1691: 1688: 1682: 1678: 1672: 1668: 1663: 1659: 1609: 1606: 1600: 1596: 1590: 1586: 1582: 1578: 1574: 1571: 1567: 1529: 1523: 1519: 1491: 1485: 1481: 1439: 1436: 1385: 1375: 1369: 1366: 1361: 1358: 1352: 1347: 1337: 1331: 1328: 1323: 1320: 1314: 1309: 1299: 1293: 1290: 1285: 1282: 1276: 1271: 1265: 1262: 1257: 1254: 1239: 1236: 1210: 1199: 1190: 1181: 1172: 1163: 1110: 1106: 1102: 1096: 1092: 1088: 1082: 1078: 1074: 1070: 1067: 1063: 1060: 1056: 1053: 1049: 1046: 1043: 1038: 1034: 1030: 1025: 1021: 1017: 1012: 1008: 1004: 1001: 998: 995: 992: 989: 986: 983: 977: 974: 971: 968: 965: 962: 959: 956: 953: 948: 940: 937: 934: 931: 928: 925: 922: 917: 913: 910: 908: 906: 903: 900: 897: 893: 889: 885: 881: 878: 873: 867: 863: 856: 853: 850: 847: 844: 841: 838: 835: 832: 827: 822: 816: 812: 805: 802: 799: 796: 793: 790: 787: 782: 778: 775: 773: 771: 768: 767: 730: 724: 720: 708:momentum space 688: 682: 678: 652: 646: 642: 636: 630: 626: 621: 618: 615: 611: 607: 603: 599: 596: 593: 590: 587: 533: 527: 523: 517: 511: 507: 464: 459: 455: 451: 445: 441: 437: 431: 427: 423: 419: 416: 412: 409: 405: 402: 399: 395: 389: 385: 379: 373: 369: 357:volume element 341: 337: 333: 311: 302: 293: 277:is called the 266: 263: 261: 258: 207: 201: 197: 186:momentum space 172: 150: 128: 122: 118: 107:volume element 15: 9: 6: 4: 3: 2: 9237: 9226: 9223: 9221: 9218: 9216: 9213: 9211: 9208: 9206: 9203: 9201: 9198: 9196: 9193: 9191: 9188: 9187: 9185: 9168: 9165: 9163: 9160: 9158: 9155: 9154: 9153: 9150: 9148: 9145: 9141: 9140:superfluidity 9138: 9136: 9133: 9132: 9131: 9128: 9127: 9125: 9121: 9115: 9112: 9110: 9107: 9105: 9102: 9100: 9097: 9095: 9092: 9091: 9089: 9087: 9083: 9075: 9072: 9070: 9067: 9066: 9065: 9062: 9060: 9057: 9056: 9054: 9052: 9048: 9042: 9038: 9035: 9033: 9030: 9028: 9025: 9023: 9020: 9018: 9015: 9013: 9010: 9009: 9007: 9003: 8995: 8992: 8990: 8987: 8986: 8985: 8981: 8977: 8974: 8972: 8969: 8967: 8964: 8962: 8959: 8958: 8957: 8954: 8953: 8951: 8949: 8945: 8939: 8936: 8932: 8929: 8927: 8924: 8922: 8919: 8917: 8914: 8913: 8911: 8908: 8906: 8903: 8901: 8898: 8896: 8893: 8892: 8890: 8888: 8884: 8869: 8866: 8864: 8861: 8860: 8858: 8854: 8850: 8843: 8838: 8836: 8831: 8829: 8824: 8823: 8820: 8814: 8811: 8809: 8806: 8805: 8795: 8791: 8787: 8783: 8779: 8775: 8770: 8766: 8762: 8758: 8754: 8750: 8746: 8742: 8738: 8733: 8729: 8725: 8721: 8717: 8713: 8709: 8705: 8701: 8697: 8696:Arkeryd, Leif 8693: 8689: 8685: 8681: 8677: 8673: 8669: 8665: 8661: 8656: 8655: 8651: 8647: 8646:Fokker–Planck 8641: 8635: 8631: 8630: 8624: 8623: 8611: 8607: 8602: 8597: 8593: 8589: 8585: 8581: 8577: 8573: 8569: 8562: 8554: 8550: 8546: 8542: 8538: 8531: 8522: 8517: 8510: 8496: 8492: 8486: 8484: 8474: 8469: 8465: 8461: 8457: 8451: 8445: 8444:0-521-40844-X 8441: 8437: 8431: 8424: 8420: 8414: 8406: 8402: 8398: 8394: 8389: 8384: 8380: 8376: 8369: 8361: 8357: 8352: 8347: 8343: 8339: 8335: 8328: 8319: 8314: 8310: 8306: 8302: 8295: 8287: 8283: 8279: 8275: 8270: 8265: 8261: 8257: 8250: 8241: 8233: 8229: 8225: 8221: 8217: 8213: 8206: 8198: 8194: 8190: 8186: 8182: 8178: 8171: 8162: 8154: 8150: 8146: 8142: 8138: 8134: 8129: 8124: 8120: 8116: 8115:Phys. Lett. B 8109: 8107: 8098: 8092: 8088: 8081: 8079: 8070: 8064: 8060: 8053: 8051: 8049: 8047: 8038: 8034: 8030: 8026: 8022: 8018: 8011: 8004: 8003:0-07-051400-3 8000: 7994: 7992: 7983: 7977: 7973: 7968: 7967: 7958: 7950: 7946: 7941: 7936: 7931: 7926: 7922: 7918: 7913: 7908: 7904: 7900: 7896: 7892: 7886: 7878: 7874: 7870: 7866: 7862: 7858: 7851: 7844: 7838: 7836: 7834: 7832: 7825: 7820: 7814: 7810: 7806: 7802: 7801: 7793: 7789: 7779: 7776: 7774: 7771: 7769: 7766: 7764: 7761: 7759: 7756: 7754: 7751: 7749: 7746: 7744: 7741: 7740: 7733: 7731: 7727: 7726:morphogenesis 7723: 7719: 7715: 7710: 7708: 7703: 7698: 7696: 7691: 7689: 7685: 7674: 7672: 7668: 7664: 7660: 7656: 7653:in powers of 7652: 7648: 7643: 7640: 7636: 7632: 7628: 7623: 7615: 7613: 7609: 7605: 7601: 7596: 7592: 7588: 7584: 7580: 7576: 7572: 7568: 7563: 7562:phase space. 7559: 7555: 7547: 7540: 7533: 7510: 7504: 7498: 7495: 7487: 7483: 7474: 7463: 7459: 7453: 7449: 7443: 7440: 7431: 7423: 7415: 7411: 7402: 7391: 7387: 7383: 7377: 7340: 7335: 7333: 7329: 7325: 7315: 7313: 7309: 7305: 7301: 7297: 7293: 7289: 7285: 7281: 7271: 7258: 7237: 7229: 7223: 7218: 7208: 7202: 7190: 7155: 7149: 7144: 7139: 7122: 7116: 7105: 7099: 7074: 7064: 7042: 7038: 7034: 7029: 7025: 7013: 7009: 7005: 7000: 6996: 6984: 6980: 6976: 6971: 6967: 6957: 6951: 6948: 6942: 6937: 6934: 6930: 6901: 6897: 6893: 6888: 6884: 6872: 6868: 6864: 6859: 6855: 6845: 6839: 6836: 6830: 6827: 6818: 6805: 6802: 6799: 6794: 6790: 6784: 6780: 6776: 6773: 6765: 6761: 6755: 6752: 6748: 6744: 6739: 6736: 6732: 6728: 6723: 6719: 6713: 6709: 6703: 6699: 6695: 6689: 6686: 6680: 6675: 6671: 6667: 6656: 6652: 6639: 6631: 6627: 6621: 6617: 6613: 6607: 6604: 6598: 6595: 6586: 6569: 6567: 6548: 6545: 6538: 6534: 6528: 6524: 6517: 6512: 6506: 6496: 6492: 6485: 6479: 6476: 6465:Second moment 6462: 6460: 6456: 6432: 6428: 6424: 6419: 6415: 6403: 6399: 6395: 6390: 6386: 6376: 6373: 6368: 6365: 6361: 6351: 6338: 6335: 6332: 6327: 6323: 6319: 6316: 6308: 6305: 6301: 6297: 6292: 6288: 6282: 6278: 6274: 6263: 6259: 6246: 6238: 6234: 6230: 6221: 6204: 6202: 6184: 6180: 6176: 6171: 6161: 6157: 6150: 6147: 6144: 6130: 6111: 6107: 6100: 6095: 6091: 6070: 6067: 6064: 6061: 6041: 6038: 6035: 6027: 6023: 6019: 6008: 6004: 5991: 5988: 5982: 5965: 5949: 5946: 5941: 5931: 5927: 5920: 5917: 5914: 5903:Zeroth moment 5900: 5884: 5880: 5857: 5853: 5844: 5839: 5833: 5827: 5814: 5811: 5808: 5799: 5793: 5778: 5765: 5755: 5751: 5739: 5733: 5725: 5716: 5711: 5708: 5700: 5687: 5683: 5675: 5671: 5662: 5653: 5647: 5643: 5639: 5636: 5633: 5623: 5619: 5609: 5605: 5596: 5585: 5581: 5577: 5574: 5566: 5553: 5542: 5538: 5534: 5528: 5517: 5513: 5498: 5495: 5490: 5480: 5476: 5466: 5462: 5453: 5442: 5438: 5433: 5429: 5422: 5414: 5401: 5392: 5386: 5377: 5365: 5355: 5351: 5343: 5335: 5326: 5323: 5315: 5313: 5294: 5286: 5282: 5273: 5255: 5251: 5228: 5224: 5198: 5194: 5187: 5165: 5161: 5138: 5134: 5130: 5127: 5122: 5118: 5087: 5083: 5065: 5052: 5042: 5038: 5033: 5030: 5027: 5022: 5019: 5014: 5008: 4996: 4990: 4977: 4967: 4963: 4958: 4955: 4952: 4949: 4938: 4923: 4921: 4917: 4913: 4909: 4905: 4900: 4891: 4878: 4868: 4862: 4853: 4848: 4842: 4828: 4818: 4808: 4803: 4788: 4783: 4780: 4776: 4767: 4763: 4758: 4754: 4750: 4746: 4740: 4727: 4722: 4712: 4708: 4700: 4691: 4687: 4681: 4677: 4673: 4669: 4665: 4661: 4656: 4652: 4648: 4635: 4630: 4627: 4623: 4614: 4611: 4607: 4601: 4598: 4594: 4590: 4585: 4580: 4577: 4574: 4570: 4566: 4545: 4539: 4529: 4525: 4515: 4505: 4501: 4496: 4492: 4488: 4484: 4477: 4468: 4455: 4445: 4439: 4429: 4425: 4415: 4410: 4402: 4385: 4381: 4371: 4363: 4358: 4354: 4347: 4340: 4336: 4330: 4318: 4312: 4302: 4298: 4283: 4276: 4272: 4261: 4245: 4241: 4220: 4211: 4198: 4192: 4189: 4184: 4180: 4173: 4170: 4154: 4145: 4137: 4134: 4128: 4123: 4113: 4107: 4099: 4085: 4076: 4072: 4068: 4060: 4054: 4050: 4045: 4022: 4016: 4007: 4002: 3996: 3982: 3972: 3962: 3957: 3942: 3939: 3929: 3925: 3918: 3914: 3909: 3905: 3899: 3895: 3888: 3884: 3867: 3862: 3850: 3846: 3838: 3828: 3825: 3820: 3810: 3799: 3793: 3790: 3785: 3775: 3764: 3761: 3755: 3752: 3747: 3741: 3732: 3721: 3715: 3712: 3707: 3701: 3692: 3681: 3669: 3666: 3660: 3657: 3654: 3651: 3641: 3635: 3627: 3618: 3608: 3603: 3597: 3582: 3580: 3579:Bose–Einstein 3576: 3572: 3566: 3560: 3554: 3552: 3548: 3543: 3535: 3529: 3525: 3521: 3515: 3484: 3478: 3470: 3461: 3456: 3440: 3431: 3423: 3420: 3414: 3409: 3399: 3393: 3385: 3371: 3369: 3368: 3362: 3357: 3356: 3345: 3343: 3340: 3334: 3330: 3323: 3319: 3312: 3308: 3287: 3272: 3264: 3260: 3251: 3240: 3223: 3215: 3211: 3202: 3191: 3174: 3166: 3162: 3153: 3142: 3125: 3109: 3095: 3090: 3085: 3072: 3065: 3063: 3061: 3044: 3041: 3032: 3016: 3007: 3004: 3001: 2996: 2986: 2983: 2977: 2974: 2971: 2965: 2957: 2948: 2946: 2933: 2930: 2914: 2905: 2897: 2894: 2891: 2885: 2882: 2879: 2873: 2865: 2856: 2854: 2845: 2839: 2835: 2831: 2822: 2818: 2809: 2800: 2795: 2791: 2787: 2778: 2774: 2765: 2756: 2751: 2747: 2743: 2734: 2730: 2721: 2711: 2707: 2703: 2699: 2696: 2689: 2681: 2672: 2669: 2666: 2659: 2651: 2642: 2639: 2636: 2629: 2621: 2611: 2607: 2604: 2601: 2594: 2586: 2577: 2575: 2570: 2567: 2555: 2554: 2551: 2548: 2543: 2534: 2527: 2525: 2523: 2492: 2486: 2478: 2469: 2464: 2458: 2455: 2450: 2447: 2436: 2435: 2432: 2428: 2420: 2402: 2388: 2384: 2372: 2368: 2359: 2358: 2352: 2347: 2342: 2331: 2324: 2322: 2320: 2296: 2292: 2280: 2276: 2271: 2265: 2263: 2248: 2244: 2232: 2228: 2220: 2217: 2209: 2198: 2195: 2185: 2181: 2169: 2165: 2160: 2156: 2150: 2147: 2144: 2141: 2130: 2122: 2119: 2111: 2101: 2092: 2088: 2085: 2083: 2068: 2064: 2052: 2048: 2043: 2019: 2013: 2005: 1996: 1991: 1989: 1967: 1963: 1951: 1950: 1947: 1926: 1922: 1910: 1906: 1897: 1893: 1870: 1866: 1854: 1850: 1840: 1820: 1816: 1804: 1800: 1792: 1789: 1781: 1770: 1767: 1757: 1753: 1741: 1737: 1731: 1727: 1721: 1718: 1715: 1712: 1700: 1692: 1689: 1680: 1670: 1661: 1657: 1649: 1648:must satisfy 1646: 1640: 1636: 1632: 1628: 1624: 1620:and momentum 1607: 1598: 1588: 1580: 1569: 1555: 1551: 1545: 1541:. If a force 1521: 1517: 1507: 1503:and momentum 1483: 1479: 1469: 1459: 1456: 1451: 1446: 1435: 1432: 1429: 1425: 1419: 1413: 1407: 1405: 1401: 1396: 1383: 1373: 1367: 1359: 1350: 1345: 1335: 1329: 1321: 1312: 1307: 1297: 1291: 1283: 1274: 1269: 1263: 1260: 1255: 1252: 1235: 1233: 1226: 1221: 1219: 1216:for particle 1213: 1209: 1202: 1198: 1189: 1180: 1171: 1162: 1156: 1150: 1145: 1142: 1141:deterministic 1137: 1132: 1127: 1108: 1104: 1100: 1094: 1090: 1086: 1080: 1076: 1072: 1068: 1065: 1061: 1058: 1054: 1051: 1044: 1041: 1036: 1032: 1028: 1023: 1019: 1015: 1010: 1006: 1002: 999: 996: 993: 990: 987: 981: 946: 915: 911: 909: 898: 895: 887: 876: 865: 861: 825: 814: 810: 780: 776: 774: 769: 757: 755: 746: 722: 718: 709: 704: 680: 676: 667: 644: 640: 628: 624: 616: 613: 605: 594: 591: 588: 585: 575: 571: 567: 563: 558: 549: 525: 521: 509: 505: 495: 489: 484: 475: 462: 457: 453: 449: 443: 439: 435: 429: 425: 421: 417: 414: 410: 407: 403: 400: 397: 387: 383: 371: 367: 358: 354: 350: 349:parameterized 344: 329: 325: 319: 314: 305: 296: 288: 284: 280: 276: 272: 257: 255: 252: 247: 245: 241: 237: 233: 229: 225: 220: 199: 195: 187: 120: 116: 108: 104: 100: 96: 92: 87: 84: 80: 76: 72: 68: 64: 60: 56: 52: 43: 37: 30: 26: 22: 9123:Applications 9074:size scaling 9011: 8777: 8774:Ann. of Math 8773: 8740: 8736: 8706:(1): 17–34. 8703: 8699: 8663: 8659: 8628: 8575: 8571: 8561: 8544: 8540: 8530: 8509: 8498:. Retrieved 8494: 8463: 8459: 8450: 8430: 8413: 8378: 8374: 8368: 8341: 8337: 8327: 8308: 8304: 8294: 8259: 8255: 8249: 8240: 8215: 8211: 8205: 8180: 8176: 8170: 8161: 8118: 8114: 8086: 8058: 8020: 8016: 8010: 7965: 7957: 7902: 7898: 7885: 7860: 7857:Ann. of Math 7856: 7850: 7843:R. G. Lerner 7799: 7792: 7711: 7699: 7692: 7680: 7644: 7624: 7621: 7612:baryogenesis 7594: 7591:wavefunction 7564: 7557: 7553: 7542: 7538: 7336: 7327: 7323: 7321: 7307: 7303: 7300:baryogenesis 7284:relativistic 7277: 7153: 7147: 7137: 7070: 6819: 6570: 6468: 6352: 6205: 6136: 6133:First moment 5906: 5845:of velocity 5837: 5831: 5828: 5701: 5567: 5415: 5316: 5311: 5274:is zero for 5271: 5066: 4994: 4991: 4942:is given by 4934: 4919: 4915: 4911: 4907: 4903: 4895: 4892: 4768: 4761: 4756: 4752: 4748: 4744: 4741: 4506: 4499: 4494: 4490: 4486: 4479: 4472: 4469: 4284: 4274: 4270: 4267: 4212: 4086: 4082: 4070: 4052: 4048: 3927: 3923: 3916: 3912: 3907: 3903: 3897: 3893: 3886: 3882: 3593: 3564: 3558: 3555: 3527: 3523: 3519: 3516: 3372: 3365: 3360: 3353: 3351: 3342:unit vectors 3332: 3328: 3321: 3317: 3310: 3306: 3088: 3077: 3066: 2556: 2546: 2542:differential 2539: 2528: 2437: 2426: 2418: 2355: 2354:. Dividing ( 2350: 2345: 2340: 2336: 2325: 1952: 1946:changes, so 1841: 1650: 1644: 1638: 1634: 1630: 1626: 1622: 1553: 1549: 1543: 1505: 1467: 1460: 1454: 1449: 1444: 1441: 1430: 1427: 1423: 1417: 1411: 1408: 1397: 1241: 1222: 1217: 1211: 1207: 1200: 1196: 1187: 1178: 1169: 1160: 1154: 1148: 1135: 1128: 758: 744: 702: 665: 573: 569: 565: 561: 555:. This is a 547: 493: 487: 482: 476: 352: 331: 327: 323: 309: 300: 291: 286: 274: 273:and momenta 270: 268: 248: 221: 88: 58: 54: 50: 48: 9114:von Neumann 8984:force field 8976:percolation 8743:(1): 1–16. 8666:(1): 1–16. 8344:: 215–240. 7629:(including 7608:dark matter 7296:dark matter 4067:solid angle 3575:Fermi–Dirac 3534:force field 3094:dot product 1129:which is a 754:Integrating 318:dimensional 283:coordinates 279:phase space 226:energy and 99:probability 67:equilibrium 9184:Categories 8971:Heisenberg 8620:References 8500:2022-05-10 8262:(3): 771. 7730:chemotaxis 3352:Dividing ( 3086:operator, 2540:The total 2431:, we have 2348:change in 748:, at time 332:x, y, z, p 103:very small 9094:Boltzmann 9017:H-theorem 8895:Ensembles 8765:117877311 8728:119481100 8688:117877311 8592:1522-9602 8578:(3): 42. 8405:0036-1429 8383:CiteSeerX 8360:0307-904X 8286:115167686 8269:1011.5441 8212:Physica A 8177:Physica A 8153:119253828 8128:1202.1301 7912:1002.3639 7778:H-theorem 7645:Close to 7625:Instead, 7488:α 7480:∂ 7472:∂ 7464:γ 7454:β 7444:γ 7441:β 7432:α 7428:Γ 7424:− 7416:α 7408:∂ 7400:∂ 7392:α 7358:^ 7247:∂ 7243:∂ 7238:⋅ 7227:∇ 7224:⋅ 7200:∂ 7196:∂ 7174:^ 7091:^ 7051:⟩ 7035:− 7006:− 6977:− 6961:⟨ 6958:ρ 6910:⟩ 6894:− 6865:− 6849:⟨ 6846:ρ 6774:− 6696:ρ 6649:∂ 6645:∂ 6614:ρ 6584:∂ 6580:∂ 6441:⟩ 6425:− 6396:− 6380:⟨ 6377:ρ 6317:− 6275:ρ 6256:∂ 6252:∂ 6231:ρ 6219:∂ 6215:∂ 6117:⟩ 6104:⟨ 6062:ρ 6020:ρ 6001:∂ 5997:∂ 5989:ρ 5980:∂ 5976:∂ 5803:⟩ 5797:⟨ 5776:∂ 5772:∂ 5731:∂ 5723:∂ 5709:∫ 5668:∂ 5660:∂ 5637:− 5602:∂ 5594:∂ 5575:∫ 5548:⟩ 5532:⟨ 5510:∂ 5506:∂ 5459:∂ 5451:∂ 5423:∫ 5396:⟩ 5390:⟨ 5375:∂ 5371:∂ 5341:∂ 5333:∂ 5324:∫ 5298:∞ 5295:± 5292:→ 5115:↦ 5080:↦ 5028:∫ 5012:⟩ 5006:⟨ 4956:∫ 4854:− 4809:− 4704:Ω 4674:− 4639:Ω 4591:∬ 4571:∑ 4537:∂ 4522:∂ 4437:∂ 4422:∂ 4393:∂ 4378:∂ 4372:⋅ 4351:∇ 4348:⋅ 4310:∂ 4295:∂ 4221:ν 4190:− 4174:ν 4160:∂ 4152:∂ 4146:⋅ 4132:∇ 4129:⋅ 4105:∂ 4097:∂ 4008:− 3963:− 3842:Ω 3762:− 3673:Ω 3655:∬ 3633:∂ 3625:∂ 3596:Boltzmann 3476:∂ 3468:∂ 3446:∂ 3438:∂ 3432:⋅ 3418:∇ 3415:⋅ 3391:∂ 3383:∂ 3370:) gives: 3339:Cartesian 3277:∇ 3257:∂ 3249:∂ 3234:^ 3208:∂ 3200:∂ 3185:^ 3159:∂ 3151:∂ 3136:^ 3115:∂ 3107:∂ 3033:⋅ 3022:∂ 3014:∂ 2987:⋅ 2981:∇ 2963:∂ 2955:∂ 2931:⋅ 2920:∂ 2912:∂ 2895:⋅ 2889:∇ 2871:∂ 2863:∂ 2815:∂ 2807:∂ 2771:∂ 2763:∂ 2727:∂ 2719:∂ 2687:∂ 2679:∂ 2657:∂ 2649:∂ 2627:∂ 2619:∂ 2592:∂ 2584:∂ 2484:∂ 2476:∂ 2400:Δ 2269:Δ 2196:− 2154:Δ 2139:Δ 2117:Δ 2041:Δ 2011:∂ 2003:∂ 1725:Δ 1710:Δ 1687:Δ 1605:Δ 1573:Δ 1404:collision 1400:diffusion 1365:∂ 1357:∂ 1327:∂ 1319:∂ 1289:∂ 1281:∂ 1144:many-body 947:∭ 916:∭ 826:∫ 781:∫ 251:nonlinear 232:viscosity 91:positions 83:particles 9104:Tsallis 8610:35150333 7949:20231489 7736:See also 7665:and the 7583:Big Bang 6469:Letting 6459:pressure 6201:momentum 6137:Letting 5907:Letting 5684:⟩ 5654:⟨ 5153:, where 4863:′ 4843:′ 4753:p′ 4749:f′ 4745:f′ 4723:′ 4670:′ 4657:′ 4017:′ 3997:′ 3924:p′ 3913:p′ 3742:′ 3702:′ 3084:gradient 1450:external 1133:. While 710:element 351:by time 260:Overview 228:momentum 9099:Shannon 9086:Entropy 8794:1971423 8745:Bibcode 8708:Bibcode 8668:Bibcode 8601:8840942 8220:Bibcode 8185:Bibcode 8133:Bibcode 8025:Bibcode 7940:2851887 7917:Bibcode 7877:1971423 7824:Alt URL 7702:liquids 7688:spheres 7530:is the 7141:is the 5843:moments 4057:is the 3540:is the 3532:is the 3092:is the 3082:is the 2344:is the 1509:within 497:within 287:x, y, z 95:momenta 8948:Models 8856:Theory 8792: 8763: 8726: 8686: 8636: 8608: 8598: 8590: 8442: 8421: 8403: 8385: 8358: 8284: 8151: 8093: 8065: 8001: 7978: 7947: 7937: 7875: 7815: 7728:, and 7722:cancer 7523:where 7135:where 6820:where 6564:, the 6353:where 6199:, the 6054:where 5962:, the 4742:where 4470:where 4213:where 3880:where 3304:, and 3078:where 2337:where 1452:force 742:about 700:about 238:, and 27:, and 9157:chaos 9109:Rényi 8966:Potts 8961:Ising 8790:JSTOR 8776:. 2. 8761:S2CID 8724:S2CID 8684:S2CID 8516:arXiv 8282:S2CID 8264:arXiv 8149:S2CID 8123:arXiv 7907:arXiv 7873:JSTOR 7859:. 2. 7785:Notes 7714:cells 7657:(the 3358:) by 2360:) by 2346:total 1303:force 485:have 330:) = ( 77:with 75:fluid 9039:and 8634:ISBN 8606:PMID 8588:ISSN 8440:ISBN 8419:ISBN 8401:ISSN 8356:ISSN 8091:ISBN 8063:ISBN 7999:ISBN 7976:ISBN 7945:PMID 7813:ISBN 7633:and 7610:and 7577:and 7298:and 7282:for 5964:mass 5786:coll 5745:coll 5102:and 4914:and 4906:and 4893:and 4451:coll 4055:, Ω) 3921:and 3906:and 3891:and 3647:coll 3542:mass 3337:are 2550:is: 2423:and 1379:coll 1341:diff 1225:mass 1152:and 491:and 224:heat 93:and 49:The 8782:doi 8778:130 8753:doi 8716:doi 8676:doi 8648:or 8596:PMC 8580:doi 8549:doi 8468:doi 8464:376 8393:doi 8346:doi 8313:doi 8274:doi 8228:doi 8216:388 8193:doi 8181:388 8141:doi 8119:718 8033:doi 7935:PMC 7925:doi 7903:107 7865:doi 7861:130 7805:doi 7565:In 7341:is 7157:is 7071:In 6461:). 4998:is 4282:is 3577:or 2544:of 2429:→ 0 2421:→ 0 1625:+ Δ 1552:+ Δ 666:all 483:all 340:, p 336:, p 59:BTE 53:or 9186:: 8912:: 8788:. 8759:. 8751:. 8741:45 8739:. 8722:. 8714:. 8704:45 8702:. 8682:. 8674:. 8664:45 8662:. 8604:. 8594:. 8586:. 8576:84 8574:. 8570:. 8543:. 8539:. 8493:. 8482:^ 8462:. 8458:. 8399:. 8391:. 8379:37 8377:. 8354:. 8342:52 8340:. 8336:. 8309:35 8307:. 8303:. 8280:. 8272:. 8260:24 8258:. 8226:. 8214:. 8191:. 8179:. 8147:. 8139:. 8131:. 8117:. 8105:^ 8077:^ 8045:^ 8031:. 8021:94 8019:. 7990:^ 7974:. 7972:53 7943:. 7933:. 7923:. 7915:. 7901:. 7897:. 7871:. 7830:^ 7811:. 7709:. 7697:. 7673:. 7614:. 7556:, 7541:, 7527:βγ 7334:. 7314:. 4898:ij 4760:, 4747:= 4498:, 4489:, 4478:= 3573:, 3553:. 3526:, 3361:dt 3344:. 3326:, 3315:, 3096:, 1633:+ 1629:= 1434:. 1426:= 1220:. 1205:, 1185:, 1167:, 752:. 572:, 568:, 559:: 326:, 307:, 298:, 246:. 234:, 23:, 8931:G 8926:F 8921:H 8916:U 8841:e 8834:t 8827:v 8796:. 8784:: 8767:. 8755:: 8747:: 8730:. 8718:: 8710:: 8690:. 8678:: 8670:: 8652:. 8642:. 8612:. 8582:: 8555:. 8551:: 8545:2 8524:. 8518:: 8503:. 8476:. 8470:: 8425:. 8407:. 8395:: 8362:. 8348:: 8321:. 8315:: 8288:. 8276:: 8266:: 8234:. 8230:: 8222:: 8199:. 8195:: 8187:: 8155:. 8143:: 8135:: 8125:: 8099:. 8071:. 8039:. 8035:: 8027:: 8005:. 7984:. 7951:. 7927:: 7919:: 7909:: 7879:. 7867:: 7821:. 7807:: 7595:f 7560:) 7558:p 7554:x 7552:( 7548:) 7545:i 7543:p 7539:x 7537:( 7525:Γ 7511:, 7508:] 7505:f 7502:[ 7499:C 7496:= 7484:p 7475:f 7460:p 7450:p 7412:x 7403:f 7388:p 7384:= 7381:] 7378:f 7375:[ 7369:R 7366:G 7354:L 7324:f 7308:f 7304:f 7259:. 7251:p 7234:F 7230:+ 7219:m 7215:p 7209:+ 7203:t 7191:= 7185:R 7182:N 7170:L 7154:L 7148:C 7138:L 7123:, 7120:] 7117:f 7114:[ 7110:C 7106:= 7103:] 7100:f 7097:[ 7087:L 7048:) 7043:k 7039:V 7030:k 7026:v 7022:( 7019:) 7014:k 7010:V 7001:k 6997:v 6993:( 6990:) 6985:i 6981:V 6972:i 6968:v 6964:( 6952:2 6949:1 6943:= 6938:i 6935:q 6931:J 6907:) 6902:i 6898:V 6889:i 6885:v 6881:( 6878:) 6873:i 6869:V 6860:i 6856:v 6852:( 6840:2 6837:1 6831:= 6828:u 6806:, 6803:0 6800:= 6795:i 6791:V 6785:i 6781:F 6777:n 6771:) 6766:i 6762:V 6756:j 6753:i 6749:P 6745:+ 6740:j 6737:q 6733:J 6729:+ 6724:j 6720:V 6714:i 6710:V 6704:i 6700:V 6690:2 6687:1 6681:+ 6676:j 6672:V 6668:u 6665:( 6657:j 6653:x 6640:+ 6637:) 6632:i 6628:V 6622:i 6618:V 6608:2 6605:1 6599:+ 6596:u 6593:( 6587:t 6549:m 6546:2 6539:i 6535:p 6529:i 6525:p 6518:= 6513:2 6507:2 6503:) 6497:i 6493:v 6489:( 6486:m 6480:= 6477:A 6438:) 6433:j 6429:V 6420:j 6416:v 6412:( 6409:) 6404:i 6400:V 6391:i 6387:v 6383:( 6374:= 6369:j 6366:i 6362:P 6339:, 6336:0 6333:= 6328:i 6324:F 6320:n 6314:) 6309:j 6306:i 6302:P 6298:+ 6293:j 6289:V 6283:i 6279:V 6272:( 6264:j 6260:x 6247:+ 6244:) 6239:i 6235:V 6228:( 6222:t 6185:i 6181:p 6177:= 6172:1 6168:) 6162:i 6158:v 6154:( 6151:m 6148:= 6145:A 6112:i 6108:v 6101:= 6096:i 6092:V 6071:n 6068:m 6065:= 6042:, 6039:0 6036:= 6033:) 6028:j 6024:V 6017:( 6009:j 6005:x 5992:+ 5983:t 5950:m 5947:= 5942:0 5938:) 5932:i 5928:v 5924:( 5921:m 5918:= 5915:A 5885:i 5881:p 5858:i 5854:v 5838:A 5832:A 5815:, 5812:0 5809:= 5806:) 5800:A 5794:n 5791:( 5779:t 5766:= 5762:p 5756:3 5752:d 5740:) 5734:t 5726:f 5717:( 5712:A 5688:, 5676:j 5672:p 5663:A 5648:j 5644:F 5640:n 5634:= 5630:p 5624:3 5620:d 5610:j 5606:p 5597:f 5586:j 5582:F 5578:A 5554:, 5551:) 5543:j 5539:p 5535:A 5529:n 5526:( 5518:j 5514:x 5499:m 5496:1 5491:= 5487:p 5481:3 5477:d 5467:j 5463:x 5454:f 5443:m 5439:A 5434:j 5430:p 5402:, 5399:) 5393:A 5387:n 5384:( 5378:t 5366:= 5362:p 5356:3 5352:d 5344:t 5336:f 5327:A 5312:A 5287:i 5283:p 5272:f 5256:i 5252:F 5229:i 5225:p 5204:) 5199:i 5195:p 5191:( 5188:A 5166:i 5162:v 5139:i 5135:v 5131:m 5128:= 5123:i 5119:p 5111:p 5088:i 5084:x 5076:x 5053:. 5049:p 5043:3 5039:d 5034:f 5031:A 5023:n 5020:1 5015:= 5009:A 4995:A 4978:. 4974:p 4968:3 4964:d 4959:f 4953:= 4950:n 4940:n 4920:i 4916:j 4912:i 4908:j 4904:i 4896:I 4879:, 4875:| 4869:j 4860:p 4849:i 4840:p 4833:| 4829:= 4825:| 4819:j 4814:p 4804:i 4799:p 4793:| 4789:= 4784:j 4781:i 4777:g 4764:) 4762:t 4757:i 4751:( 4728:, 4720:p 4713:3 4709:d 4701:d 4697:] 4692:j 4688:f 4682:i 4678:f 4666:j 4662:f 4653:i 4649:f 4645:[ 4642:) 4636:, 4631:j 4628:i 4624:g 4620:( 4615:j 4612:i 4608:I 4602:j 4599:i 4595:g 4586:n 4581:1 4578:= 4575:j 4567:= 4561:l 4558:l 4555:o 4552:c 4546:) 4540:t 4530:i 4526:f 4516:( 4502:) 4500:t 4495:i 4491:p 4487:r 4485:( 4482:i 4480:f 4475:i 4473:f 4456:, 4446:) 4440:t 4430:i 4426:f 4416:( 4411:= 4403:i 4398:p 4386:i 4382:f 4368:F 4364:+ 4359:i 4355:f 4341:i 4337:m 4331:i 4326:p 4319:+ 4313:t 4303:i 4299:f 4280:i 4275:n 4271:i 4246:0 4242:f 4199:, 4196:) 4193:f 4185:0 4181:f 4177:( 4171:= 4164:p 4155:f 4142:F 4138:+ 4135:f 4124:m 4120:p 4114:+ 4108:t 4100:f 4073:Ω 4071:d 4063:θ 4053:g 4051:( 4049:I 4029:| 4023:A 4014:p 4003:B 3994:p 3987:| 3983:= 3979:| 3973:A 3968:p 3958:B 3953:p 3947:| 3943:= 3940:g 3928:B 3917:A 3908:B 3904:A 3898:B 3894:p 3887:A 3883:p 3868:, 3863:B 3858:p 3851:3 3847:d 3839:d 3835:] 3832:) 3829:t 3826:, 3821:B 3816:p 3811:, 3807:r 3803:( 3800:f 3797:) 3794:t 3791:, 3786:A 3781:p 3776:, 3772:r 3768:( 3765:f 3759:) 3756:t 3753:, 3748:B 3739:p 3733:, 3729:r 3725:( 3722:f 3719:) 3716:t 3713:, 3708:A 3699:p 3693:, 3689:r 3685:( 3682:f 3679:[ 3676:) 3670:, 3667:g 3664:( 3661:I 3658:g 3652:= 3642:) 3636:t 3628:f 3619:( 3599:" 3565:f 3559:f 3538:m 3530:) 3528:t 3524:r 3522:( 3520:F 3500:l 3497:l 3494:o 3491:c 3485:) 3479:t 3471:f 3462:( 3457:= 3450:p 3441:f 3428:F 3424:+ 3421:f 3410:m 3406:p 3400:+ 3394:t 3386:f 3367:2 3355:3 3333:z 3329:ê 3322:y 3318:ê 3311:x 3307:ê 3302:∇ 3288:f 3282:p 3273:= 3265:z 3261:p 3252:f 3241:z 3231:e 3224:+ 3216:y 3212:p 3203:f 3192:y 3182:e 3175:+ 3167:x 3163:p 3154:f 3143:x 3133:e 3126:= 3119:p 3110:f 3089:· 3080:∇ 3071:) 3069:3 3067:( 3045:t 3042:d 3037:F 3026:p 3017:f 3008:+ 3005:t 3002:d 2997:m 2993:p 2984:f 2978:+ 2975:t 2972:d 2966:t 2958:f 2949:= 2938:p 2934:d 2924:p 2915:f 2906:+ 2902:r 2898:d 2892:f 2886:+ 2883:t 2880:d 2874:t 2866:f 2857:= 2846:) 2840:z 2836:p 2832:d 2823:z 2819:p 2810:f 2801:+ 2796:y 2792:p 2788:d 2779:y 2775:p 2766:f 2757:+ 2752:x 2748:p 2744:d 2735:x 2731:p 2722:f 2712:( 2708:+ 2704:) 2700:z 2697:d 2690:z 2682:f 2673:+ 2670:y 2667:d 2660:y 2652:f 2643:+ 2640:x 2637:d 2630:x 2622:f 2612:( 2608:+ 2605:t 2602:d 2595:t 2587:f 2578:= 2571:f 2568:d 2547:f 2533:) 2531:2 2529:( 2508:l 2505:l 2502:o 2499:c 2493:) 2487:t 2479:f 2470:( 2465:= 2459:t 2456:d 2451:f 2448:d 2427:f 2425:Δ 2419:t 2417:Δ 2403:t 2395:p 2389:3 2385:d 2379:r 2373:3 2369:d 2357:1 2351:f 2341:f 2339:Δ 2330:) 2328:1 2326:( 2303:p 2297:3 2293:d 2287:r 2281:3 2277:d 2272:f 2266:= 2255:p 2249:3 2245:d 2239:r 2233:3 2229:d 2224:) 2221:t 2218:, 2214:p 2210:, 2206:r 2202:( 2199:f 2192:p 2186:3 2182:d 2176:r 2170:3 2166:d 2161:) 2157:t 2151:+ 2148:t 2145:, 2142:t 2135:F 2131:+ 2127:p 2123:, 2120:t 2112:m 2108:p 2102:+ 2098:r 2093:( 2089:f 2086:= 2075:p 2069:3 2065:d 2059:r 2053:3 2049:d 2044:t 2035:l 2032:l 2029:o 2026:c 2020:) 2014:t 2006:f 1997:( 1992:= 1982:l 1979:l 1976:o 1973:c 1968:N 1964:d 1933:p 1927:3 1923:d 1917:r 1911:3 1907:d 1877:p 1871:3 1867:d 1861:r 1855:3 1851:d 1827:p 1821:3 1817:d 1811:r 1805:3 1801:d 1796:) 1793:t 1790:, 1786:p 1782:, 1778:r 1774:( 1771:f 1768:= 1764:p 1758:3 1754:d 1748:r 1742:3 1738:d 1732:) 1728:t 1722:+ 1719:t 1716:, 1713:t 1705:F 1701:+ 1697:p 1693:, 1690:t 1681:m 1677:p 1671:+ 1667:r 1662:( 1658:f 1645:f 1639:t 1637:Δ 1635:F 1631:p 1627:p 1623:p 1608:t 1599:m 1595:p 1589:+ 1585:r 1581:= 1577:r 1570:+ 1566:r 1554:t 1550:t 1544:F 1528:p 1522:3 1518:d 1506:p 1490:r 1484:3 1480:d 1468:r 1463:t 1455:F 1445:f 1431:v 1428:m 1424:p 1418:p 1412:v 1384:, 1374:) 1368:t 1360:f 1351:( 1346:+ 1336:) 1330:t 1322:f 1313:( 1308:+ 1298:) 1292:t 1284:f 1275:( 1270:= 1264:t 1261:d 1256:f 1253:d 1228:m 1218:N 1212:N 1208:p 1201:N 1197:r 1191:2 1188:p 1182:2 1179:r 1173:1 1170:p 1164:1 1161:r 1155:p 1149:r 1136:f 1109:z 1105:p 1101:d 1095:y 1091:p 1087:d 1081:x 1077:p 1073:d 1069:z 1066:d 1062:y 1059:d 1055:x 1052:d 1048:) 1045:t 1042:, 1037:z 1033:p 1029:, 1024:y 1020:p 1016:, 1011:x 1007:p 1003:, 1000:z 997:, 994:y 991:, 988:x 985:( 982:f 976:s 973:n 970:o 967:i 964:t 961:i 958:s 955:o 952:p 939:a 936:t 933:n 930:e 927:m 924:o 921:m 912:= 902:) 899:t 896:, 892:p 888:, 884:r 880:( 877:f 872:r 866:3 862:d 855:s 852:n 849:o 846:i 843:t 840:i 837:s 834:o 831:p 821:p 815:3 811:d 804:a 801:t 798:n 795:e 792:m 789:o 786:m 777:= 770:N 750:t 745:p 729:p 723:3 719:d 703:r 687:r 681:3 677:d 651:p 645:3 641:d 635:r 629:3 625:d 620:) 617:t 614:, 610:p 606:, 602:r 598:( 595:f 592:= 589:N 586:d 576:) 574:t 570:p 566:r 564:( 562:f 553:t 548:f 532:p 526:3 522:d 516:r 510:3 506:d 494:p 488:r 479:N 463:. 458:z 454:p 450:d 444:y 440:p 436:d 430:x 426:p 422:d 418:z 415:d 411:y 408:d 404:x 401:d 398:= 394:p 388:3 384:d 378:r 372:3 368:d 353:t 345:) 342:z 338:y 334:x 328:p 324:r 322:( 312:z 310:p 303:y 301:p 294:x 292:p 275:p 271:r 206:p 200:3 196:d 171:p 149:r 127:r 121:3 117:d 57:( 38:. 31:.

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Knowledge

Boltzmann equation

Index