Boltzmann equation

3070: 2570: 3065:{\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}}} 2329: 1966: 8655:. 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 1136: 8889: 2324:{\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}}} 53: 772: 3309: 4477: 1849: 3524: 1405: 7532: 3110: 1131:{\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}}} 6827: 7280: 4298: 4095:

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.

5423: 267:, 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. 3304:{\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} 5429: 5715: 4100: 4900: 4520: 6063: 4051: 5581: 1629: 4472:{\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}},} 2532: 7072: 7680:. 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 1844:{\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} } 7692:

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

6931: 5074: 6219: 3623: 3519:{\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} }} 1400:{\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}},} 6462: 5330: 6573: 7527:{\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,} 3609:

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

6822:{\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,} 1955: 1899: 554: 7144: 5162: 5980: 7275:{\displaystyle {\hat {\mathbf {L} }}_{\mathrm {NR} }={\frac {\partial }{\partial t}}+{\frac {\mathbf {p} }{m}}\cdot \nabla +\mathbf {F} \cdot {\frac {\partial }{\partial \mathbf {p} }}\,.} 4999: 6138: 5111: 4782: 2451: 7333:

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

7156:(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 5570:{\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 ),} 193: 171: 5831:{\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,} 6092: 5225: 1571: 4215:{\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),} 1409:

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

6936: 5908: 5881: 5279: 5252: 5189: 4744:{\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'} ,} 4269: 4242: 5012: 8850: 7773: 3620:

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

5704:{\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 ,} 8660: 8384:

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.

8345:"Aerodynamic optimisation of a hypersonic reentry vehicle based on solution of the Boltzmann–BGK equation and evolutionary optimisation" 767:

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

8312:"A discontinuous finite element solution of the Boltzmann kinetic equation in collisionless and BGK forms for macroscopic gas flows" 8255:

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

2374: 7648:) are generally used to find approximate solutions to the various forms of the Boltzmann equation. Example applications range from 4956: 9205: 9200: 7697:. The collision term is modified in Enskog equations such that particles have a finite size, for example they can be modelled as 6355:{\displaystyle {\frac {\partial }{\partial t}}(\rho V_{i})+{\frac {\partial }{\partial x_{j}}}(\rho V_{i}V_{j}+P_{ij})-nF_{i}=0,} 3884:{\displaystyle \left({\frac {\partial f}{\partial t}}\right)_{\text{coll}}=\iint gI(g,\Omega )\,d\Omega \,d^{3}\mathbf {p} _{B},} 1150:

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

5418:{\displaystyle \int A{\frac {\partial f}{\partial t}}\,d^{3}\mathbf {p} ={\frac {\partial }{\partial t}}(n\langle A\rangle ),} 8836: 8648: 8433: 8265:

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".

7637: 1906: 1912: 1856: 511: 3581: 39: 7313:. It is not a priori clear that the state of a quantum system can be characterized by a classical phase space density 5116: 8454: 8013: 254: 7778: 9235: 7673: 7297:

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".

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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

8981: 8897: 8656: 7763: 7681: 7290: 6058:{\displaystyle {\frac {\partial }{\partial t}}\rho +{\frac {\partial }{\partial x_{j}}}(\rho V_{j})=0,} 4069: 2552: 1523: 1485: 724: 682: 201: 122: 8579:"Multi-Cue Kinetic Model with Non-Local Sensing for Cell Migration on a Fiber Network with Chemotaxis" 8428:

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

5921: 241:, when a fluid is in transport. One may also derive other properties characteristic to fluids such as 9220: 8818: 8525:

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

176: 154: 9004: 8994: 8920: 8547: 7768: 4895:{\displaystyle g_{ij}=|\mathbf {p} _{i}-\mathbf {p} _{j}|=|\mathbf {p'} _{i}-\mathbf {p'} _{j}|,} 3544: 1902: 1624:{\displaystyle \mathbf {r} +\Delta \mathbf {r} =\mathbf {r} +{\frac {\mathbf {p} }{m}}\,\Delta t} 250: 6579:

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

2527:{\displaystyle {\frac {df}{dt}}=\left({\frac {\partial f}{\partial t}}\right)_{\mathrm {coll} }} 9051: 8393: 7604:

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

7067:{\textstyle J_{qi}={\tfrac {1}{2}}\rho \langle (v_{i}-V_{i})(v_{k}-V_{k})(v_{k}-V_{k})\rangle } 764: 6068: 5977:

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

5194: 8936: 8859: 7083: 6465: 3349: 7974: 9079: 8755: 8718: 8678: 8230: 8195: 8143: 8035: 7927: 7661: 7610: 7322: 7294: 5886: 5859: 5853: 5257: 5230: 5167: 4247: 4046:{\displaystyle g=|\mathbf {p} _{B}-\mathbf {p} _{A}|=|\mathbf {p'} _{B}-\mathbf {p'} _{A}|} 246: 89: 73: 46: 4227: 8: 9145: 9124: 7975: 7705: 7342: 3557: 359: 8759: 8722: 8682: 8234: 8199: 8147: 8039: 7931: 479:{\displaystyle d^{3}\mathbf {r} \,d^{3}\mathbf {p} =dx\,dy\,dz\,dp_{x}\,dp_{y}\,dp_{z}.} 233:

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

9172: 9084: 9061: 9042: 8986: 8800: 8771: 8734: 8694: 8611: 8578: 8526: 8292: 8274: 8159: 8133: 7950: 7917: 7905: 7883: 7783: 7728: 7589: 7577: 7542: 7349: 7298: 7153: 4921:. The integration is over the momentum components in the integrand (which are labelled 3567:

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

668:{\displaystyle dN=f(\mathbf {r} ,\mathbf {p} ,t)\,d^{3}\mathbf {r} \,d^{3}\mathbf {p} } 7337:; in galaxies, physical collisions between the stars are very rare, and the effect of 1909:). However, since collisions do occur, the particle density in the phase-space volume 9177: 9074: 9047: 8958: 8948: 8941: 8915: 8775: 8738: 8698: 8644: 8616: 8598: 8450: 8429: 8411: 8366: 8296: 8163: 8101: 8073: 8009: 7986: 7955: 7834: 7823: 7597: 7585: 4054: 1154: 1141: 8288: 9069: 8792: 8763: 8726: 8686: 8606: 8590: 8559: 8478: 8403: 8356: 8323: 8284: 8238: 8203: 8155: 8151: 8043: 7945: 7935: 7906:"Global classical solutions of the Boltzmann equation with long-range interactions" 7875: 7815: 7649: 3606: 1242: 261: 81: 9119: 1234:

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).

9194: 9150: 8709:(1972). "On the Boltzmann equation part II: The full initial value problem". 8602: 8415: 8370: 7736: 7596:, particles are continuously created and annihilated. In such an environment 7317:. However, for a wide class of applications a well-defined generalization of 6926:{\textstyle u={\tfrac {1}{2}}\rho \langle (v_{i}-V_{i})(v_{i}-V_{i})\rangle } 4929:). The sum of integrals describes the entry and exit of particles of species 1151: 113: 7940: 9167: 8706: 8620: 8563: 8047: 7959: 7622: 7601: 7310: 5069:{\displaystyle \langle A\rangle ={\frac {1}{n}}\int Af\,d^{3}\mathbf {p} .} 3595: 7814:. Lecture Notes in Physics (LNP, vol. 660). Berlin, Heidelberg: Springer. 8976: 8971: 8548:"Modeling cell movement in anisotropic and heterogeneous network tissues" 8531: 7687: 7618: 7306: 4077: 3352: 3104: 293: 289: 109: 8828: 8008:

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

8966: 8804: 8767: 8730: 8690: 7887: 7740: 6457:{\displaystyle P_{ij}=\rho \langle (v_{i}-V_{i})(v_{j}-V_{j})\rangle } 9027: 8746:

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

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

7788: 7592:. In the very dense medium formed by the primordial plasma after the 1414: 1410: 328: 242: 8796: 7879: 7809: 6568:{\displaystyle A={\frac {m(v_{i})^{2}}{2}}={\frac {p_{i}p_{i}}{2m}}} 1853:

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

7732: 7712: 7698: 7660:, solution of the Boltzmann equation can be represented by an 2419:{\displaystyle d^{3}\mathbf {r} \,d^{3}\mathbf {p} \,\Delta t} 556:, is in question, at the heart of the equation is a quantity 85: 7285:

Quantum theory and violation of particle number conservation

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Boltzmann-like equations are also used for the movement of

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

5974: 3552: 1235: 234: 173:, and has momentum nearly equal to a given momentum vector 8220: 8185: 8095: 7162:

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

7613:. This includes the formation of the light elements in 275: 8502:"Enskog Equation - an overview | ScienceDirect Topics" 7811:

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

8025: 7328: 7301:, including the formation of the light elements in 7139:{\displaystyle {\hat {\mathbf {L} }}=\mathbf {C} ,} 4274: 84:in 1872. The classic example of such a system is a 7584:often attempt to take into account the effects of 7526: 7274: 7138: 7066: 6925: 6821: 6567: 6456: 6354: 6202: 6132: 6086: 6057: 5965: 5902: 5875: 5830: 5703: 5569: 5417: 5313: 5273: 5246: 5219: 5183: 5156: 5105: 5068: 4993: 4894: 4743: 4471: 4263: 4236: 4214: 4045: 3883: 3518: 3303: 3064: 2526: 2418: 2323: 1949: 1893: 1843: 1623: 1544: 1506: 1399: 1245:, one distribution is needed for each, see below. 1130: 745: 703: 667: 548: 478: 222: 187: 165: 143: 8491: 8471:Philosophical Transactions of the Royal Society A 8309: 7848: 7846: 7844: 7842: 3913:are the momenta of any two particles (labeled as 3244: 3195: 3146: 1420:Note that some authors use the particle velocity 9192: 8819:The Boltzmann Transport Equation by Franz Vesely 8546:Chauviere, A.; Hillen, T.; Preziosi, L. (2007). 5157:{\displaystyle \mathbf {p} \mapsto p_{i}=mv_{i}} 1448: 7910:Proceedings of the National Academy of Sciences 7341:can be neglected for times far longer than the 8524: 7839: 7808:Gorban, Alexander N.; Karlin, Ilya V. (2005). 7561:phase space as opposed to fully contravariant 4936: 4053:is the magnitude of the relative momenta (see 3573:cannot be solved unless the collision term in 1558:instantly acts on each particle, then at time 8844: 8383: 8067: 7774:Derivation of Navier–Stokes equation from LBE 679:have positions lying within a volume element 300:, and three more for each momentum component 292:of the system; in other words a set of three 8782: 8267:Journal of the American Mathematical Society 8119: 8117: 8091: 8089: 8004: 8002: 7981:(Second ed.). New York: Wiley. p. 7865: 7807: 7061: 6971: 6920: 6859: 6451: 6390: 6127: 6114: 5813: 5807: 5558: 5542: 5406: 5400: 5022: 5016: 4994:{\displaystyle n=\int f\,d^{3}\mathbf {p} .} 3611: 3311:is a shorthand for the momentum analogue of 1253:The general equation can then be written as 8552:American Institute of Mathematical Sciences 8342: 6933:is the kinetic thermal energy density, and 6133:{\displaystyle V_{i}=\langle v_{i}\rangle } 5846:is conserved in a collision. The values of 4777:, the magnitude of the relative momenta is 1476:some number of particles all have position 72:) describes the statistical behaviour of a 8851: 8837: 7289:It is possible to write down relativistic 4244:is the molecular collision frequency, and 3600: 3547:acting on the particles in the fluid, and 45:"BTE" redirects here. For other uses, see 8858: 8610: 8577:Conte, Martina; Loy, Nadia (2022-02-12). 8530: 8482: 8397: 8360: 8343:Evans, B.; Walton, S.P. (December 2017). 8327: 8278: 8137: 8114: 8096:Edward Kolb & Michael Turner (1990). 8086: 7999: 7949: 7939: 7921: 7268: 5760: 5628: 5485: 5360: 5281:is a function of position only, and that 5106:{\displaystyle \mathbf {x} \mapsto x_{i}} 5047: 4972: 4941: 4717: 4710: 3855: 3848: 3051: 2841: 2797: 2753: 2706: 2676: 2646: 2611: 2409: 2393: 2301: 2285: 2253: 2237: 2190: 2073: 2057: 1931: 1875: 1825: 1809: 1762: 1746: 1719: 1696: 1614: 1110: 1096: 1082: 1075: 1068: 1061: 886: 649: 633: 530: 459: 445: 431: 424: 417: 392: 8576: 8063: 8061: 8059: 8057: 7852:Encyclopaedia of Physics (2nd Edition), 7077: 5321:. Multiplying the Boltzmann equation by 5191:is the particle velocity vector. Define 51: 8745: 8705: 8668: 8467:"Theme issue 'Hilbert's sixth problem'" 7628: 195:(thus occupying a very small region of 14: 9193: 8636: 8445:Sydney Chapman; Thomas George Cowling 4933:in or out of the phase-space element. 1901:is constant, which can be shown using 1653:. Then, in the absence of collisions, 1248: 8832: 8054: 7972: 4090:Simplifications to the collision term 3943:are the momenta after the collision, 3921:for convenience) before a collision, 8449:, Cambridge University Press, 1970. 8072:. New York: Dover Publications Inc. 7779:Derivation of Jeans equation from BE 6203:{\displaystyle A=m(v_{i})^{1}=p_{i}} 5314:{\displaystyle p_{i}\to \pm \infty } 2563: 2444: 1959: 276:The phase space and density function 116:region of space (mathematically the 8068:de Groot, S. R.; Mazur, P. (1984). 5910:, as they are linearly dependent). 5840:where the last term is zero, since 112:that the particle occupies a given 24: 8824:Boltzmann gaseous behaviors solved 8386:SIAM Journal on Numerical Analysis 7490: 7482: 7438: 7418: 7410: 7379: 7376: 7257: 7253: 7237: 7210: 7206: 7195: 7192: 6659: 6655: 6594: 6590: 6266: 6262: 6229: 6225: 6011: 6007: 5990: 5986: 5786: 5782: 5741: 5733: 5678: 5670: 5612: 5604: 5520: 5516: 5469: 5461: 5385: 5381: 5351: 5343: 5308: 5003:The average value of any function 4714: 4649: 4571: 4568: 4565: 4562: 4547: 4532: 4447: 4432: 4403: 4388: 4361: 4320: 4305: 4170: 4162: 4142: 4115: 4107: 3852: 3683: 3643: 3635: 3510: 3507: 3504: 3501: 3486: 3478: 3456: 3448: 3428: 3401: 3393: 3358: 3287: 3267: 3259: 3218: 3210: 3169: 3161: 3125: 3117: 3032: 3024: 2991: 2973: 2965: 2930: 2922: 2899: 2881: 2873: 2825: 2817: 2781: 2773: 2737: 2729: 2697: 2689: 2667: 2659: 2637: 2629: 2602: 2594: 2518: 2515: 2512: 2509: 2494: 2486: 2410: 2279: 2164: 2149: 2127: 2051: 2045: 2042: 2039: 2036: 2021: 2013: 1992: 1989: 1986: 1983: 1735: 1720: 1697: 1615: 1583: 1375: 1367: 1337: 1329: 1299: 1291: 1241:). For a mixture of more than one 986: 983: 980: 977: 974: 971: 968: 965: 962: 949: 946: 943: 940: 937: 934: 931: 865: 862: 859: 856: 853: 850: 847: 844: 841: 814: 811: 808: 805: 802: 799: 796: 366:. The small volume ("differential 280:The set of all possible positions 25: 9247: 8812: 7600:and the spatial extension of the 1545:{\displaystyle d^{3}\mathbf {p} } 1507:{\displaystyle d^{3}\mathbf {r} } 746:{\displaystyle d^{3}\mathbf {p} } 704:{\displaystyle d^{3}\mathbf {r} } 675:is the number of molecules which 223:{\displaystyle d^{3}\mathbf {p} } 144:{\displaystyle d^{3}\mathbf {r} } 27:Equation of statistical mechanics 8887: 8583:Bulletin of Mathematical Biology 7364: 7329:General relativity and astronomy 7261: 7244: 7225: 7180: 7120: 7097: 6475: 5966:{\displaystyle A=m(v_{i})^{0}=m} 5913: 5772: 5640: 5497: 5372: 5121: 5086: 5059: 4984: 4870: 4850: 4824: 4809: 4730: 4408: 4378: 4336: 4275:General equation (for a mixture) 4174: 4152: 4130: 4024: 4004: 3978: 3963: 3868: 3826: 3817: 3791: 3782: 3749: 3739: 3709: 3699: 3460: 3438: 3416: 3292: 3241: 3192: 3143: 3129: 3047: 3036: 3003: 2948: 2934: 2912: 2405: 2389: 2313: 2297: 2265: 2249: 2224: 2216: 2202: 2186: 2145: 2137: 2118: 2108: 2085: 2069: 1943: 1927: 1887: 1871: 1837: 1821: 1796: 1788: 1774: 1758: 1715: 1707: 1687: 1677: 1605: 1595: 1587: 1576: 1538: 1500: 1453:Consider particles described by 1413:of particles, and "coll" is the 902: 894: 882: 831: 739: 697: 661: 645: 620: 612: 542: 526: 404: 388: 216: 181: 159: 137: 8570: 8539: 8518: 8459: 8439: 8422: 8377: 8336: 8303: 8289:10.1090/S0894-0347-2011-00697-8 8258: 8249: 8214: 8179: 8170: 7904:& Robert M. Strain (2010). 6143: 6140:is the average fluid velocity. 4057:for more on this concept), and 955: 9206:Partial differential equations 9201:Eponymous equations of physics 8349:Applied Mathematical Modelling 8316:Applied Mathematical Modelling 8156:10.1016/j.physletb.2012.11.046 8070:Non-Equilibrium Thermodynamics 8019: 7966: 7894: 7859: 7801: 7518: 7512: 7391: 7385: 7368: 7184: 7130: 7124: 7113: 7107: 7101: 7058: 7032: 7029: 7003: 7000: 6974: 6917: 6891: 6888: 6862: 6781: 6675: 6647: 6603: 6513: 6499: 6448: 6422: 6419: 6393: 6324: 6282: 6254: 6238: 6178: 6164: 6043: 6027: 5948: 5934: 5816: 5801: 5561: 5536: 5409: 5394: 5302: 5214: 5201: 5125: 5090: 4885: 4843: 4835: 4803: 4707: 4655: 4652: 4630: 4206: 4187: 4039: 3997: 3989: 3957: 3845: 3842: 3813: 3807: 3778: 3769: 3735: 3729: 3695: 3689: 3686: 3674: 2234: 2212: 1806: 1784: 1058: 995: 912: 890: 630: 608: 40:Maxwell–Boltzmann distribution 13: 1: 8630: 5227:as some function of momentum 1449:The force and diffusion terms 296:for each position coordinate 265:integro-differential equation 255:convection–diffusion equation 8874:Principle of maximum entropy 8643:. Dover Books. p. 221. 7764:Fokker–Planck equation 6464:is the pressure tensor (the 4515:, and the collision term is 3616:" and is also known as the " 568:probability density function 188:{\displaystyle \mathbf {p} } 166:{\displaystyle \mathbf {r} } 66:Boltzmann transport equation 18:Boltzmann transport equation 7: 8243:10.1016/j.physa.2009.01.009 8208:10.1016/j.physa.2008.12.023 7769:Williams–Boltzmann equation 7759:The Vlasov–Poisson equation 7746: 7291:quantum Boltzmann equations 4937:Applications and extensions 3377: 3365: 3079: 2541: 2367: 2338: 1205:for particle 2, etc. up to 717:and momenta lying within a 331:: a point in this space is 270: 151:) centered at the position 32:Boltzmann's entropy formula 10: 9252: 8898:Statistical thermodynamics 8595:10.1007/s11538-021-00978-1 4070:differential cross section 1905:(see the discussion under 230:), at an instant of time. 44: 29: 9133: 9095: 9060: 9015: 8957: 8896: 8885: 8866: 8748:Arch. Rational Mech. Anal 8711:Arch. Rational Mech. Anal 8671:Arch. Rational Mech. Anal 8408:10.1137/S0036142998343300 8362:10.1016/j.apm.2017.07.024 8329:10.1016/j.apm.2010.07.027 7646:lattice Boltzmann methods 7074:is the heat flux vector. 6094:is the mass density, and 4289:the equation for species 3605:A key insight applied by 1157:systems), since only one 488:Since the probability of 358:, and each coordinate is 9158:Condensed matter physics 9141:Statistical field theory 8637:Harris, Stewart (1971). 7795: 7615:Big Bang nucleosynthesis 7339:gravitational collisions 7303:Big Bang nucleosynthesis 6087:{\displaystyle \rho =mn} 5220:{\displaystyle A(p_{i})} 4086:, due to the collision. 4076:into the element of the 327:. The entire space is 6- 9236:Thermodynamic equations 9016:Mathematical approaches 9005:Lennard-Jones potential 8921:thermodynamic potential 7941:10.1073/pnas.1001185107 7701:having a fixed radius. 7682:Hilbert's sixth problem 7678:Navier–Stokes equations 7650:hypersonic aerodynamics 3601:Two-body collision term 3375:and substituting into ( 1568:their position will be 1459:, each experiencing an 251:electrical conductivity 9052:conformal field theory 8564:10.3934/nhm.2007.2.333 8048:10.1103/PhysRev.94.511 7973:Huang, Kerson (1987). 7711:Many real fluids like 7528: 7348:Its generalization in 7276: 7140: 7068: 6927: 6823: 6569: 6458: 6356: 6204: 6134: 6088: 6059: 5967: 5904: 5877: 5832: 5705: 5571: 5419: 5315: 5275: 5248: 5221: 5185: 5158: 5107: 5070: 4995: 4942:Conservation equations 4896: 4745: 4600: 4473: 4265: 4238: 4216: 4047: 3885: 3612: 3560:, is often called the 3520: 3305: 3066: 2528: 2426:and taking the limits 2420: 2325: 1951: 1895: 1845: 1625: 1546: 1508: 1401: 1132: 747: 705: 669: 550: 480: 224: 189: 167: 145: 57: 9211:Statistical mechanics 8967:Ferromagnetism models 8860:Statistical mechanics 8506:www.sciencedirect.com 8484:10.1098/rsta/376/2118 7977:Statistical Mechanics 7529: 7277: 7141: 7084:Hamiltonian mechanics 7078:Hamiltonian mechanics 7069: 6928: 6824: 6570: 6468:plus the hydrostatic 6466:viscous stress tensor 6459: 6357: 6205: 6135: 6089: 6060: 5968: 5905: 5903:{\displaystyle p_{i}} 5878: 5876:{\displaystyle v_{i}} 5833: 5706: 5572: 5420: 5316: 5276: 5274:{\displaystyle F_{i}} 5249: 5247:{\displaystyle p_{i}} 5222: 5186: 5184:{\displaystyle v_{i}} 5159: 5108: 5071: 4996: 4897: 4746: 4580: 4474: 4266: 4264:{\displaystyle f_{0}} 4239: 4217: 4048: 3886: 3521: 3306: 3067: 2529: 2421: 2326: 1952: 1896: 1846: 1626: 1547: 1509: 1402: 1133: 748: 706: 670: 551: 481: 225: 190: 168: 146: 90:temperature gradients 55: 7706:inelastic collisions 7662:asymptotic expansion 7629:Solving the equation 7617:, the production of 7611:quantum field theory 7356: 7323:quantum field theory 7305:, the production of 7172: 7090: 6937: 6835: 6585: 6484: 6368: 6220: 6152: 6098: 6069: 5981: 5922: 5887: 5860: 5716: 5582: 5430: 5331: 5289: 5258: 5231: 5195: 5168: 5117: 5082: 5013: 4957: 4783: 4521: 4299: 4248: 4237:{\displaystyle \nu } 4228: 4101: 3947: 3624: 3558:Coulomb interactions 3387: 3111: 2571: 2452: 2375: 1967: 1913: 1903:Hamilton's equations 1857: 1665: 1572: 1524: 1486: 1426:instead of momentum 1257: 773: 725: 683: 593: 512: 374: 247:thermal conductivity 202: 177: 155: 123: 80:; it was devised by 74:thermodynamic system 47:BTE (disambiguation) 36:Stefan–Boltzmann law 30:For other uses, see 9216:Transport phenomena 9146:elementary particle 8911:partition functions 8760:1972ArRMA..45....1A 8723:1972ArRMA..45...17A 8683:1972ArRMA..45....1A 8235:2009PhyA..388.1818D 8200:2009PhyA..388.1079D 8148:2013PhLB..718.1119D 8040:1954PhRv...94..511B 7932:2010PNAS..107.5744G 7729:composite particles 7343:age of the universe 4683: 4670: 1907:Liouville's theorem 1249:Principal statement 589:, defined so that, 9173:information theory 9080:correlation length 9075:Critical exponents 9062:Critical phenomena 9043:stochastic process 9023:Boltzmann equation 8916:equations of state 8768:10.1007/BF00253392 8731:10.1007/BF00253393 8691:10.1007/BF00253392 8100:. Westview Press. 8098:The Early Universe 7902:Philip T. Gressman 7743:-related effects. 7727:. Since cells are 7590:general relativity 7578:physical cosmology 7543:Christoffel symbol 7524: 7350:general relativity 7299:physical cosmology 7272: 7154:Liouville operator 7136: 7064: 6966: 6923: 6854: 6819: 6704: 6622: 6565: 6454: 6352: 6200: 6130: 6084: 6055: 5963: 5900: 5873: 5828: 5701: 5567: 5415: 5311: 5271: 5244: 5217: 5181: 5154: 5103: 5066: 4991: 4892: 4741: 4671: 4658: 4469: 4261: 4234: 4212: 4043: 3881: 3516: 3301: 3062: 3060: 2524: 2416: 2321: 2319: 1947: 1891: 1841: 1621: 1542: 1504: 1397: 1128: 1126: 991: 954: 870: 819: 743: 701: 665: 546: 476: 260:The equation is a 220: 185: 163: 141: 76:not in a state of 62:Boltzmann equation 58: 9188: 9187: 9178:Boltzmann machine 9048:mean-field theory 8949:Maxwell relations 8650:978-0-486-43831-3 8434:978-981-4449-53-3 8107:978-0-201-62674-2 8079:978-0-486-64741-8 7992:978-0-471-81518-1 7916:(13): 5744–5749. 7829:978-3-540-22684-0 7735:cells in tissue, 7658:local equilibrium 7638:numerical methods 7598:quantum coherence 7586:quantum mechanics 7504: 7432: 7371: 7266: 7232: 7217: 7187: 7104: 6965: 6853: 6703: 6673: 6621: 6601: 6563: 6526: 6280: 6236: 6025: 5997: 5798: 5793: 5757: 5748: 5692: 5626: 5534: 5512: 5483: 5456: 5392: 5358: 5036: 4554: 4463: 4454: 4419: 4356: 4327: 4179: 4137: 4122: 4055:relative velocity 3659: 3650: 3582:Maxwell–Boltzmann 3528:In this context, 3493: 3465: 3423: 3408: 3281: 3247: 3232: 3198: 3183: 3149: 3134: 3087: 3086: 3041: 3010: 2980: 2939: 2888: 2839: 2795: 2751: 2704: 2674: 2644: 2609: 2549: 2548: 2501: 2473: 2346: 2345: 2125: 2028: 1694: 1612: 1391: 1382: 1353: 1344: 1315: 1306: 1278: 956: 925: 835: 790: 492:molecules, which 16:(Redirected from 9243: 9221:Ludwig Boltzmann 9070:Phase transition 8891: 8890: 8853: 8846: 8839: 8830: 8829: 8808: 8779: 8742: 8702: 8661:Landau equations 8654: 8625: 8624: 8614: 8574: 8568: 8567: 8543: 8537: 8536: 8534: 8532:cond-mat/9706020 8522: 8516: 8515: 8513: 8512: 8498: 8489: 8488: 8486: 8463: 8457: 8443: 8437: 8426: 8420: 8419: 8401: 8392:(4): 1217–1245. 8381: 8375: 8374: 8364: 8340: 8334: 8333: 8331: 8307: 8301: 8300: 8282: 8262: 8256: 8253: 8247: 8246: 8218: 8212: 8211: 8194:(7): 1079–1104. 8183: 8177: 8174: 8168: 8167: 8141: 8132:(3): 1119–1124. 8121: 8112: 8111: 8093: 8084: 8083: 8065: 8052: 8051: 8023: 8017: 8006: 7997: 7996: 7980: 7970: 7964: 7963: 7953: 7943: 7925: 7898: 7892: 7891: 7863: 7857: 7850: 7837: 7833: 7805: 7572: 7560: 7540: 7533: 7531: 7530: 7525: 7505: 7503: 7502: 7501: 7488: 7480: 7478: 7477: 7468: 7467: 7458: 7457: 7449: 7446: 7445: 7433: 7431: 7430: 7429: 7416: 7408: 7406: 7405: 7384: 7383: 7382: 7373: 7372: 7367: 7362: 7281: 7279: 7278: 7273: 7267: 7265: 7264: 7252: 7247: 7233: 7228: 7223: 7218: 7216: 7205: 7200: 7199: 7198: 7189: 7188: 7183: 7178: 7167: 7161: 7151: 7145: 7143: 7142: 7137: 7123: 7106: 7105: 7100: 7095: 7073: 7071: 7070: 7065: 7057: 7056: 7044: 7043: 7028: 7027: 7015: 7014: 6999: 6998: 6986: 6985: 6967: 6958: 6952: 6951: 6932: 6930: 6929: 6924: 6916: 6915: 6903: 6902: 6887: 6886: 6874: 6873: 6855: 6846: 6828: 6826: 6825: 6820: 6809: 6808: 6799: 6798: 6780: 6779: 6770: 6769: 6754: 6753: 6738: 6737: 6728: 6727: 6718: 6717: 6705: 6696: 6690: 6689: 6674: 6672: 6671: 6670: 6654: 6646: 6645: 6636: 6635: 6623: 6614: 6602: 6600: 6589: 6574: 6572: 6571: 6566: 6564: 6562: 6554: 6553: 6552: 6543: 6542: 6532: 6527: 6522: 6521: 6520: 6511: 6510: 6494: 6463: 6461: 6460: 6455: 6447: 6446: 6434: 6433: 6418: 6417: 6405: 6404: 6383: 6382: 6361: 6359: 6358: 6353: 6342: 6341: 6323: 6322: 6307: 6306: 6297: 6296: 6281: 6279: 6278: 6277: 6261: 6253: 6252: 6237: 6235: 6224: 6209: 6207: 6206: 6201: 6199: 6198: 6186: 6185: 6176: 6175: 6139: 6137: 6136: 6131: 6126: 6125: 6110: 6109: 6093: 6091: 6090: 6085: 6064: 6062: 6061: 6056: 6042: 6041: 6026: 6024: 6023: 6022: 6006: 5998: 5996: 5985: 5972: 5970: 5969: 5964: 5956: 5955: 5946: 5945: 5909: 5907: 5906: 5901: 5899: 5898: 5882: 5880: 5879: 5874: 5872: 5871: 5851: 5845: 5837: 5835: 5834: 5829: 5800: 5799: 5796: 5794: 5792: 5781: 5775: 5770: 5769: 5759: 5758: 5755: 5753: 5749: 5747: 5739: 5731: 5710: 5708: 5707: 5702: 5697: 5693: 5691: 5690: 5689: 5676: 5668: 5662: 5661: 5643: 5638: 5637: 5627: 5625: 5624: 5623: 5610: 5602: 5600: 5599: 5576: 5574: 5573: 5568: 5557: 5556: 5535: 5533: 5532: 5531: 5515: 5513: 5505: 5500: 5495: 5494: 5484: 5482: 5481: 5480: 5467: 5459: 5457: 5452: 5448: 5447: 5437: 5424: 5422: 5421: 5416: 5393: 5391: 5380: 5375: 5370: 5369: 5359: 5357: 5349: 5341: 5320: 5318: 5317: 5312: 5301: 5300: 5280: 5278: 5277: 5272: 5270: 5269: 5253: 5251: 5250: 5245: 5243: 5242: 5226: 5224: 5223: 5218: 5213: 5212: 5190: 5188: 5187: 5182: 5180: 5179: 5163: 5161: 5160: 5155: 5153: 5152: 5137: 5136: 5124: 5112: 5110: 5109: 5104: 5102: 5101: 5089: 5075: 5073: 5072: 5067: 5062: 5057: 5056: 5037: 5029: 5008: 5000: 4998: 4997: 4992: 4987: 4982: 4981: 4952: 4912: 4901: 4899: 4898: 4893: 4888: 4883: 4882: 4877: 4876: 4863: 4862: 4857: 4856: 4846: 4838: 4833: 4832: 4827: 4818: 4817: 4812: 4806: 4798: 4797: 4776: 4750: 4748: 4747: 4742: 4737: 4736: 4727: 4726: 4706: 4705: 4696: 4695: 4679: 4666: 4645: 4644: 4629: 4628: 4616: 4615: 4599: 4594: 4576: 4575: 4574: 4559: 4555: 4553: 4545: 4544: 4543: 4530: 4514: 4478: 4476: 4475: 4470: 4465: 4464: 4461: 4459: 4455: 4453: 4445: 4444: 4443: 4430: 4420: 4418: 4417: 4416: 4411: 4401: 4400: 4399: 4386: 4381: 4373: 4372: 4357: 4355: 4354: 4345: 4344: 4339: 4333: 4328: 4326: 4318: 4317: 4316: 4303: 4292: 4288: 4284:= 1, 2, 3, ..., 4270: 4268: 4267: 4262: 4260: 4259: 4243: 4241: 4240: 4235: 4221: 4219: 4218: 4213: 4199: 4198: 4180: 4178: 4177: 4168: 4160: 4155: 4138: 4133: 4128: 4123: 4121: 4113: 4105: 4085: 4075: 4067: 4052: 4050: 4049: 4044: 4042: 4037: 4036: 4031: 4030: 4017: 4016: 4011: 4010: 4000: 3992: 3987: 3986: 3981: 3972: 3971: 3966: 3960: 3942: 3931: 3912: 3901: 3890: 3888: 3887: 3882: 3877: 3876: 3871: 3865: 3864: 3835: 3834: 3829: 3820: 3800: 3799: 3794: 3785: 3762: 3761: 3756: 3755: 3742: 3722: 3721: 3716: 3715: 3702: 3661: 3660: 3657: 3655: 3651: 3649: 3641: 3633: 3615: 3578: 3572: 3550: 3542: 3525: 3523: 3522: 3517: 3515: 3514: 3513: 3498: 3494: 3492: 3484: 3476: 3466: 3464: 3463: 3454: 3446: 3441: 3424: 3419: 3414: 3409: 3407: 3399: 3391: 3374: 3347: 3336: 3325: 3314: 3310: 3308: 3307: 3302: 3297: 3296: 3295: 3282: 3280: 3279: 3278: 3265: 3257: 3255: 3254: 3249: 3248: 3240: 3233: 3231: 3230: 3229: 3216: 3208: 3206: 3205: 3200: 3199: 3191: 3184: 3182: 3181: 3180: 3167: 3159: 3157: 3156: 3151: 3150: 3142: 3135: 3133: 3132: 3123: 3115: 3102: 3092: 3081: 3071: 3069: 3068: 3063: 3061: 3050: 3042: 3040: 3039: 3030: 3022: 3011: 3006: 3001: 2981: 2979: 2971: 2963: 2955: 2951: 2940: 2938: 2937: 2928: 2920: 2915: 2889: 2887: 2879: 2871: 2863: 2859: 2855: 2854: 2853: 2840: 2838: 2837: 2836: 2823: 2815: 2810: 2809: 2796: 2794: 2793: 2792: 2779: 2771: 2766: 2765: 2752: 2750: 2749: 2748: 2735: 2727: 2717: 2713: 2705: 2703: 2695: 2687: 2675: 2673: 2665: 2657: 2645: 2643: 2635: 2627: 2610: 2608: 2600: 2592: 2564: 2560: 2543: 2533: 2531: 2530: 2525: 2523: 2522: 2521: 2506: 2502: 2500: 2492: 2484: 2474: 2472: 2464: 2456: 2445: 2441: 2433: 2425: 2423: 2422: 2417: 2408: 2403: 2402: 2392: 2387: 2386: 2364: 2354: 2340: 2330: 2328: 2327: 2322: 2320: 2316: 2311: 2310: 2300: 2295: 2294: 2272: 2268: 2263: 2262: 2252: 2247: 2246: 2227: 2219: 2205: 2200: 2199: 2189: 2184: 2183: 2174: 2170: 2148: 2140: 2126: 2121: 2116: 2111: 2092: 2088: 2083: 2082: 2072: 2067: 2066: 2050: 2049: 2048: 2033: 2029: 2027: 2019: 2011: 1997: 1996: 1995: 1960: 1956: 1954: 1953: 1948: 1946: 1941: 1940: 1930: 1925: 1924: 1900: 1898: 1897: 1892: 1890: 1885: 1884: 1874: 1869: 1868: 1850: 1848: 1847: 1842: 1840: 1835: 1834: 1824: 1819: 1818: 1799: 1791: 1777: 1772: 1771: 1761: 1756: 1755: 1745: 1741: 1718: 1710: 1695: 1690: 1685: 1680: 1658: 1652: 1630: 1628: 1627: 1622: 1613: 1608: 1603: 1598: 1590: 1579: 1567: 1557: 1551: 1549: 1548: 1543: 1541: 1536: 1535: 1519: 1513: 1511: 1510: 1505: 1503: 1498: 1497: 1481: 1475: 1472:Suppose at time 1468: 1458: 1444: 1431: 1425: 1406: 1404: 1403: 1398: 1393: 1392: 1389: 1387: 1383: 1381: 1373: 1365: 1355: 1354: 1351: 1349: 1345: 1343: 1335: 1327: 1317: 1316: 1313: 1311: 1307: 1305: 1297: 1289: 1279: 1277: 1269: 1261: 1243:chemical species 1240: 1226: 1215: 1204: 1195: 1187:for particle 1, 1186: 1177: 1168: 1162: 1149: 1137: 1135: 1134: 1129: 1127: 1123: 1122: 1109: 1108: 1095: 1094: 1051: 1050: 1038: 1037: 1025: 1024: 990: 989: 953: 952: 918: 905: 897: 885: 880: 879: 869: 868: 834: 829: 828: 818: 817: 762: 758: 752: 750: 749: 744: 742: 737: 736: 716: 710: 708: 707: 702: 700: 695: 694: 674: 672: 671: 666: 664: 659: 658: 648: 643: 642: 623: 615: 588: 565: 561: 555: 553: 552: 547: 545: 540: 539: 529: 524: 523: 507: 501: 491: 485: 483: 482: 477: 472: 471: 458: 457: 444: 443: 407: 402: 401: 391: 386: 385: 357: 326: 317: 308: 229: 227: 226: 221: 219: 214: 213: 194: 192: 191: 186: 184: 172: 170: 169: 164: 162: 150: 148: 147: 142: 140: 135: 134: 82:Ludwig Boltzmann 21: 9251: 9250: 9246: 9245: 9244: 9242: 9241: 9240: 9231:1872 in Germany 9226:1872 in science 9191: 9190: 9189: 9184: 9129: 9091: 9056: 9038:BBGKY hierarchy 9033:Vlasov equation 9011: 9000:depletion force 8993:Particles with 8953: 8892: 8888: 8883: 8862: 8857: 8815: 8797:10.2307/1971423 8651: 8633: 8628: 8575: 8571: 8544: 8540: 8523: 8519: 8510: 8508: 8500: 8499: 8492: 8465: 8464: 8460: 8444: 8440: 8427: 8423: 8382: 8378: 8341: 8337: 8322:(3): 996–1015. 8308: 8304: 8263: 8259: 8254: 8250: 8219: 8215: 8184: 8180: 8175: 8171: 8122: 8115: 8108: 8094: 8087: 8080: 8066: 8055: 8028:Physical Review 8024: 8020: 8007: 8000: 7993: 7971: 7967: 7899: 7895: 7880:10.2307/1971423 7864: 7860: 7851: 7840: 7830: 7806: 7802: 7798: 7793: 7784:Jeans's theorem 7754:Vlasov equation 7749: 7718:BBGKY hierarchy 7695:Enskog equation 7690: 7674:Euler equations 7642:finite elements 7631: 7562: 7557: 7547: 7539: 7535: 7497: 7493: 7489: 7481: 7479: 7473: 7469: 7463: 7459: 7450: 7448: 7447: 7441: 7437: 7425: 7421: 7417: 7409: 7407: 7401: 7397: 7375: 7374: 7363: 7361: 7360: 7359: 7357: 7354: 7353: 7331: 7287: 7260: 7256: 7251: 7243: 7224: 7222: 7209: 7204: 7191: 7190: 7179: 7177: 7176: 7175: 7173: 7170: 7169: 7163: 7157: 7147: 7119: 7096: 7094: 7093: 7091: 7088: 7087: 7080: 7052: 7048: 7039: 7035: 7023: 7019: 7010: 7006: 6994: 6990: 6981: 6977: 6956: 6944: 6940: 6938: 6935: 6934: 6911: 6907: 6898: 6894: 6882: 6878: 6869: 6865: 6844: 6836: 6833: 6832: 6804: 6800: 6794: 6790: 6775: 6771: 6762: 6758: 6746: 6742: 6733: 6729: 6723: 6719: 6713: 6709: 6694: 6685: 6681: 6666: 6662: 6658: 6653: 6641: 6637: 6631: 6627: 6612: 6593: 6588: 6586: 6583: 6582: 6555: 6548: 6544: 6538: 6534: 6533: 6531: 6516: 6512: 6506: 6502: 6495: 6493: 6485: 6482: 6481: 6478: 6442: 6438: 6429: 6425: 6413: 6409: 6400: 6396: 6375: 6371: 6369: 6366: 6365: 6337: 6333: 6315: 6311: 6302: 6298: 6292: 6288: 6273: 6269: 6265: 6260: 6248: 6244: 6228: 6223: 6221: 6218: 6217: 6194: 6190: 6181: 6177: 6171: 6167: 6153: 6150: 6149: 6146: 6121: 6117: 6105: 6101: 6099: 6096: 6095: 6070: 6067: 6066: 6037: 6033: 6018: 6014: 6010: 6005: 5989: 5984: 5982: 5979: 5978: 5951: 5947: 5941: 5937: 5923: 5920: 5919: 5916: 5894: 5890: 5888: 5885: 5884: 5867: 5863: 5861: 5858: 5857: 5847: 5841: 5795: 5785: 5780: 5779: 5771: 5765: 5761: 5754: 5740: 5732: 5730: 5726: 5725: 5717: 5714: 5713: 5685: 5681: 5677: 5669: 5667: 5663: 5657: 5653: 5639: 5633: 5629: 5619: 5615: 5611: 5603: 5601: 5595: 5591: 5583: 5580: 5579: 5552: 5548: 5527: 5523: 5519: 5514: 5504: 5496: 5490: 5486: 5476: 5472: 5468: 5460: 5458: 5443: 5439: 5438: 5436: 5431: 5428: 5427: 5384: 5379: 5371: 5365: 5361: 5350: 5342: 5340: 5332: 5329: 5328: 5296: 5292: 5290: 5287: 5286: 5265: 5261: 5259: 5256: 5255: 5238: 5234: 5232: 5229: 5228: 5208: 5204: 5196: 5193: 5192: 5175: 5171: 5169: 5166: 5165: 5148: 5144: 5132: 5128: 5120: 5118: 5115: 5114: 5097: 5093: 5085: 5083: 5080: 5079: 5058: 5052: 5048: 5028: 5014: 5011: 5010: 5004: 4983: 4977: 4973: 4958: 4955: 4954: 4950: 4944: 4939: 4910: 4905: 4884: 4878: 4869: 4868: 4867: 4858: 4849: 4848: 4847: 4842: 4834: 4828: 4823: 4822: 4813: 4808: 4807: 4802: 4790: 4786: 4784: 4781: 4780: 4770: 4754: 4729: 4728: 4722: 4718: 4701: 4697: 4691: 4687: 4675: 4662: 4637: 4633: 4621: 4617: 4608: 4604: 4595: 4584: 4561: 4560: 4546: 4539: 4535: 4531: 4529: 4525: 4524: 4522: 4519: 4518: 4508: 4494: 4487: 4482: 4460: 4446: 4439: 4435: 4431: 4429: 4425: 4424: 4412: 4407: 4406: 4402: 4395: 4391: 4387: 4385: 4377: 4368: 4364: 4350: 4346: 4340: 4335: 4334: 4332: 4319: 4312: 4308: 4304: 4302: 4300: 4297: 4296: 4290: 4280: 4277: 4255: 4251: 4249: 4246: 4245: 4229: 4226: 4225: 4194: 4190: 4173: 4169: 4161: 4159: 4151: 4129: 4127: 4114: 4106: 4104: 4102: 4099: 4098: 4092: 4080: 4073: 4058: 4038: 4032: 4023: 4022: 4021: 4012: 4003: 4002: 4001: 3996: 3988: 3982: 3977: 3976: 3967: 3962: 3961: 3956: 3948: 3945: 3944: 3941: 3933: 3930: 3922: 3911: 3903: 3900: 3892: 3872: 3867: 3866: 3860: 3856: 3830: 3825: 3824: 3816: 3795: 3790: 3789: 3781: 3757: 3748: 3747: 3746: 3738: 3717: 3708: 3707: 3706: 3698: 3656: 3642: 3634: 3632: 3628: 3627: 3625: 3622: 3621: 3618:molecular chaos 3613:Stosszahlansatz 3603: 3598: 3592:distributions. 3574: 3568: 3562:Vlasov equation 3548: 3529: 3500: 3499: 3485: 3477: 3475: 3471: 3470: 3459: 3455: 3447: 3445: 3437: 3415: 3413: 3400: 3392: 3390: 3388: 3385: 3384: 3370: 3361: 3359:Final statement 3346: 3338: 3335: 3327: 3324: 3316: 3312: 3291: 3290: 3286: 3274: 3270: 3266: 3258: 3256: 3250: 3239: 3238: 3237: 3225: 3221: 3217: 3209: 3207: 3201: 3190: 3189: 3188: 3176: 3172: 3168: 3160: 3158: 3152: 3141: 3140: 3139: 3128: 3124: 3116: 3114: 3112: 3109: 3108: 3098: 3090: 3059: 3058: 3046: 3035: 3031: 3023: 3021: 3002: 3000: 2972: 2964: 2962: 2953: 2952: 2947: 2933: 2929: 2921: 2919: 2911: 2880: 2872: 2870: 2861: 2860: 2849: 2845: 2832: 2828: 2824: 2816: 2814: 2805: 2801: 2788: 2784: 2780: 2772: 2770: 2761: 2757: 2744: 2740: 2736: 2728: 2726: 2725: 2721: 2696: 2688: 2686: 2666: 2658: 2656: 2636: 2628: 2626: 2625: 2621: 2601: 2593: 2591: 2584: 2574: 2572: 2569: 2568: 2556: 2508: 2507: 2493: 2485: 2483: 2479: 2478: 2465: 2457: 2455: 2453: 2450: 2449: 2435: 2427: 2404: 2398: 2394: 2388: 2382: 2378: 2376: 2373: 2372: 2360: 2349: 2318: 2317: 2312: 2306: 2302: 2296: 2290: 2286: 2270: 2269: 2264: 2258: 2254: 2248: 2242: 2238: 2223: 2215: 2201: 2195: 2191: 2185: 2179: 2175: 2144: 2136: 2117: 2115: 2107: 2106: 2102: 2090: 2089: 2084: 2078: 2074: 2068: 2062: 2058: 2035: 2034: 2020: 2012: 2010: 2006: 2005: 1998: 1982: 1981: 1977: 1970: 1968: 1965: 1964: 1942: 1936: 1932: 1926: 1920: 1916: 1914: 1911: 1910: 1886: 1880: 1876: 1870: 1864: 1860: 1858: 1855: 1854: 1836: 1830: 1826: 1820: 1814: 1810: 1795: 1787: 1773: 1767: 1763: 1757: 1751: 1747: 1714: 1706: 1686: 1684: 1676: 1675: 1671: 1666: 1663: 1662: 1654: 1632: 1604: 1602: 1594: 1586: 1575: 1573: 1570: 1569: 1559: 1553: 1537: 1531: 1527: 1525: 1522: 1521: 1515: 1499: 1493: 1489: 1487: 1484: 1483: 1482:within element 1477: 1473: 1464: 1454: 1451: 1433: 1427: 1421: 1388: 1374: 1366: 1364: 1360: 1359: 1350: 1336: 1328: 1326: 1322: 1321: 1312: 1298: 1290: 1288: 1284: 1283: 1270: 1262: 1260: 1258: 1255: 1254: 1251: 1238: 1225: 1217: 1214: 1206: 1203: 1197: 1194: 1188: 1185: 1179: 1176: 1170: 1164: 1158: 1145: 1142:6-fold integral 1125: 1124: 1118: 1114: 1104: 1100: 1090: 1086: 1046: 1042: 1033: 1029: 1020: 1016: 961: 960: 930: 929: 916: 915: 901: 893: 881: 875: 871: 840: 839: 830: 824: 820: 795: 794: 783: 776: 774: 771: 770: 760: 754: 738: 732: 728: 726: 723: 722: 712: 696: 690: 686: 684: 681: 680: 660: 654: 650: 644: 638: 634: 619: 611: 594: 591: 590: 571: 563: 557: 541: 535: 531: 525: 519: 515: 513: 510: 509: 503: 497: 489: 467: 463: 453: 449: 439: 435: 403: 397: 393: 387: 381: 377: 375: 372: 371: 370:") is written 354: 350: 346: 332: 324: 319: 315: 310: 306: 301: 278: 273: 215: 209: 205: 203: 200: 199: 180: 178: 175: 174: 158: 156: 153: 152: 136: 130: 126: 124: 121: 120: 50: 43: 28: 23: 22: 15: 12: 11: 5: 9249: 9239: 9238: 9233: 9228: 9223: 9218: 9213: 9208: 9203: 9186: 9185: 9183: 9182: 9181: 9180: 9175: 9170: 9163:Complex system 9160: 9155: 9154: 9153: 9148: 9137: 9135: 9131: 9130: 9128: 9127: 9122: 9117: 9112: 9107: 9101: 9099: 9093: 9092: 9090: 9089: 9088: 9087: 9082: 9072: 9066: 9064: 9058: 9057: 9055: 9054: 9045: 9040: 9035: 9030: 9025: 9019: 9017: 9013: 9012: 9010: 9009: 9008: 9007: 9002: 8991: 8990: 8989: 8984: 8979: 8974: 8963: 8961: 8955: 8954: 8952: 8951: 8946: 8945: 8944: 8939: 8934: 8929: 8918: 8913: 8908: 8902: 8900: 8894: 8893: 8886: 8884: 8882: 8881: 8879:ergodic theory 8876: 8870: 8868: 8864: 8863: 8856: 8855: 8848: 8841: 8833: 8827: 8826: 8821: 8814: 8813:External links 8811: 8810: 8809: 8791:(2): 321–366. 8780: 8743: 8703: 8665: 8664: 8649: 8632: 8629: 8627: 8626: 8569: 8558:(2): 333–357. 8538: 8517: 8490: 8477:(2118). 2018. 8458: 8438: 8421: 8399:10.1.1.46.2853 8376: 8335: 8302: 8257: 8248: 8229:(9): 1818–34. 8213: 8178: 8169: 8113: 8106: 8085: 8078: 8053: 8034:(3): 511–525. 8018: 7998: 7991: 7965: 7893: 7874:(2): 321–366. 7858: 7838: 7828: 7820:10.1007/b98103 7799: 7797: 7794: 7792: 7791: 7786: 7781: 7776: 7771: 7766: 7761: 7756: 7750: 7748: 7745: 7689: 7686: 7670:Chapman–Enskog 7666:Knudsen number 7630: 7627: 7582:early universe 7555: 7537: 7523: 7520: 7517: 7514: 7511: 7508: 7500: 7496: 7492: 7487: 7484: 7476: 7472: 7466: 7462: 7456: 7453: 7444: 7440: 7436: 7428: 7424: 7420: 7415: 7412: 7404: 7400: 7396: 7393: 7390: 7387: 7381: 7378: 7370: 7366: 7330: 7327: 7286: 7283: 7271: 7263: 7259: 7255: 7250: 7246: 7242: 7239: 7236: 7231: 7227: 7221: 7215: 7212: 7208: 7203: 7197: 7194: 7186: 7182: 7135: 7132: 7129: 7126: 7122: 7118: 7115: 7112: 7109: 7103: 7099: 7079: 7076: 7063: 7060: 7055: 7051: 7047: 7042: 7038: 7034: 7031: 7026: 7022: 7018: 7013: 7009: 7005: 7002: 6997: 6993: 6989: 6984: 6980: 6976: 6973: 6970: 6964: 6961: 6955: 6950: 6947: 6943: 6922: 6919: 6914: 6910: 6906: 6901: 6897: 6893: 6890: 6885: 6881: 6877: 6872: 6868: 6864: 6861: 6858: 6852: 6849: 6843: 6840: 6818: 6815: 6812: 6807: 6803: 6797: 6793: 6789: 6786: 6783: 6778: 6774: 6768: 6765: 6761: 6757: 6752: 6749: 6745: 6741: 6736: 6732: 6726: 6722: 6716: 6712: 6708: 6702: 6699: 6693: 6688: 6684: 6680: 6677: 6669: 6665: 6661: 6657: 6652: 6649: 6644: 6640: 6634: 6630: 6626: 6620: 6617: 6611: 6608: 6605: 6599: 6596: 6592: 6577:kinetic energy 6561: 6558: 6551: 6547: 6541: 6537: 6530: 6525: 6519: 6515: 6509: 6505: 6501: 6498: 6492: 6489: 6477: 6474: 6453: 6450: 6445: 6441: 6437: 6432: 6428: 6424: 6421: 6416: 6412: 6408: 6403: 6399: 6395: 6392: 6389: 6386: 6381: 6378: 6374: 6351: 6348: 6345: 6340: 6336: 6332: 6329: 6326: 6321: 6318: 6314: 6310: 6305: 6301: 6295: 6291: 6287: 6284: 6276: 6272: 6268: 6264: 6259: 6256: 6251: 6247: 6243: 6240: 6234: 6231: 6227: 6197: 6193: 6189: 6184: 6180: 6174: 6170: 6166: 6163: 6160: 6157: 6145: 6142: 6129: 6124: 6120: 6116: 6113: 6108: 6104: 6083: 6080: 6077: 6074: 6054: 6051: 6048: 6045: 6040: 6036: 6032: 6029: 6021: 6017: 6013: 6009: 6004: 6001: 5995: 5992: 5988: 5962: 5959: 5954: 5950: 5944: 5940: 5936: 5933: 5930: 5927: 5915: 5912: 5897: 5893: 5883:(and momentum 5870: 5866: 5852:correspond to 5827: 5824: 5821: 5818: 5815: 5812: 5809: 5806: 5803: 5791: 5788: 5784: 5778: 5774: 5768: 5764: 5752: 5746: 5743: 5738: 5735: 5729: 5724: 5721: 5700: 5696: 5688: 5684: 5680: 5675: 5672: 5666: 5660: 5656: 5652: 5649: 5646: 5642: 5636: 5632: 5622: 5618: 5614: 5609: 5606: 5598: 5594: 5590: 5587: 5566: 5563: 5560: 5555: 5551: 5547: 5544: 5541: 5538: 5530: 5526: 5522: 5518: 5511: 5508: 5503: 5499: 5493: 5489: 5479: 5475: 5471: 5466: 5463: 5455: 5451: 5446: 5442: 5435: 5414: 5411: 5408: 5405: 5402: 5399: 5396: 5390: 5387: 5383: 5378: 5374: 5368: 5364: 5356: 5353: 5348: 5345: 5339: 5336: 5310: 5307: 5304: 5299: 5295: 5268: 5264: 5241: 5237: 5216: 5211: 5207: 5203: 5200: 5178: 5174: 5151: 5147: 5143: 5140: 5135: 5131: 5127: 5123: 5100: 5096: 5092: 5088: 5065: 5061: 5055: 5051: 5046: 5043: 5040: 5035: 5032: 5027: 5024: 5021: 5018: 4990: 4986: 4980: 4976: 4971: 4968: 4965: 4962: 4948:number density 4943: 4940: 4938: 4935: 4908: 4891: 4887: 4881: 4875: 4872: 4866: 4861: 4855: 4852: 4845: 4841: 4837: 4831: 4826: 4821: 4816: 4811: 4805: 4801: 4796: 4793: 4789: 4766: 4740: 4735: 4732: 4725: 4721: 4716: 4713: 4709: 4704: 4700: 4694: 4690: 4686: 4682: 4678: 4674: 4669: 4665: 4661: 4657: 4654: 4651: 4648: 4643: 4640: 4636: 4632: 4627: 4624: 4620: 4614: 4611: 4607: 4603: 4598: 4593: 4590: 4587: 4583: 4579: 4573: 4570: 4567: 4564: 4558: 4552: 4549: 4542: 4538: 4534: 4528: 4504: 4492: 4485: 4468: 4458: 4452: 4449: 4442: 4438: 4434: 4428: 4423: 4415: 4410: 4405: 4398: 4394: 4390: 4384: 4380: 4376: 4371: 4367: 4363: 4360: 4353: 4349: 4343: 4338: 4331: 4325: 4322: 4315: 4311: 4307: 4276: 4273: 4258: 4254: 4233: 4211: 4208: 4205: 4202: 4197: 4193: 4189: 4186: 4183: 4176: 4172: 4167: 4164: 4158: 4154: 4150: 4147: 4144: 4141: 4136: 4132: 4126: 4120: 4117: 4112: 4109: 4091: 4088: 4041: 4035: 4029: 4026: 4020: 4015: 4009: 4006: 3999: 3995: 3991: 3985: 3980: 3975: 3970: 3965: 3959: 3955: 3952: 3937: 3926: 3907: 3896: 3880: 3875: 3870: 3863: 3859: 3854: 3851: 3847: 3844: 3841: 3838: 3833: 3828: 3823: 3819: 3815: 3812: 3809: 3806: 3803: 3798: 3793: 3788: 3784: 3780: 3777: 3774: 3771: 3768: 3765: 3760: 3754: 3751: 3745: 3741: 3737: 3734: 3731: 3728: 3725: 3720: 3714: 3711: 3705: 3701: 3697: 3694: 3691: 3688: 3685: 3682: 3679: 3676: 3673: 3670: 3667: 3664: 3654: 3648: 3645: 3640: 3637: 3631: 3602: 3599: 3597: 3594: 3512: 3509: 3506: 3503: 3497: 3491: 3488: 3483: 3480: 3474: 3469: 3462: 3458: 3453: 3450: 3444: 3440: 3436: 3433: 3430: 3427: 3422: 3418: 3412: 3406: 3403: 3398: 3395: 3360: 3357: 3342: 3331: 3320: 3300: 3294: 3289: 3285: 3277: 3273: 3269: 3264: 3261: 3253: 3246: 3243: 3236: 3228: 3224: 3220: 3215: 3212: 3204: 3197: 3194: 3187: 3179: 3175: 3171: 3166: 3163: 3155: 3148: 3145: 3138: 3131: 3127: 3122: 3119: 3085: 3084: 3075: 3073: 3057: 3054: 3049: 3045: 3038: 3034: 3029: 3026: 3020: 3017: 3014: 3009: 3005: 2999: 2996: 2993: 2990: 2987: 2984: 2978: 2975: 2970: 2967: 2961: 2958: 2956: 2954: 2950: 2946: 2943: 2936: 2932: 2927: 2924: 2918: 2914: 2910: 2907: 2904: 2901: 2898: 2895: 2892: 2886: 2883: 2878: 2875: 2869: 2866: 2864: 2862: 2858: 2852: 2848: 2844: 2835: 2831: 2827: 2822: 2819: 2813: 2808: 2804: 2800: 2791: 2787: 2783: 2778: 2775: 2769: 2764: 2760: 2756: 2747: 2743: 2739: 2734: 2731: 2724: 2720: 2716: 2712: 2709: 2702: 2699: 2694: 2691: 2685: 2682: 2679: 2672: 2669: 2664: 2661: 2655: 2652: 2649: 2642: 2639: 2634: 2631: 2624: 2620: 2617: 2614: 2607: 2604: 2599: 2596: 2590: 2587: 2585: 2583: 2580: 2577: 2576: 2547: 2546: 2537: 2535: 2520: 2517: 2514: 2511: 2505: 2499: 2496: 2491: 2488: 2482: 2477: 2471: 2468: 2463: 2460: 2415: 2412: 2407: 2401: 2397: 2391: 2385: 2381: 2344: 2343: 2334: 2332: 2315: 2309: 2305: 2299: 2293: 2289: 2284: 2281: 2278: 2275: 2273: 2271: 2267: 2261: 2257: 2251: 2245: 2241: 2236: 2233: 2230: 2226: 2222: 2218: 2214: 2211: 2208: 2204: 2198: 2194: 2188: 2182: 2178: 2173: 2169: 2166: 2163: 2160: 2157: 2154: 2151: 2147: 2143: 2139: 2135: 2132: 2129: 2124: 2120: 2114: 2110: 2105: 2101: 2098: 2095: 2093: 2091: 2087: 2081: 2077: 2071: 2065: 2061: 2056: 2053: 2047: 2044: 2041: 2038: 2032: 2026: 2023: 2018: 2015: 2009: 2004: 2001: 1999: 1994: 1991: 1988: 1985: 1980: 1976: 1973: 1972: 1945: 1939: 1935: 1929: 1923: 1919: 1889: 1883: 1879: 1873: 1867: 1863: 1839: 1833: 1829: 1823: 1817: 1813: 1808: 1805: 1802: 1798: 1794: 1790: 1786: 1783: 1780: 1776: 1770: 1766: 1760: 1754: 1750: 1744: 1740: 1737: 1734: 1731: 1728: 1725: 1722: 1717: 1713: 1709: 1705: 1702: 1699: 1693: 1689: 1683: 1679: 1674: 1670: 1620: 1617: 1611: 1607: 1601: 1597: 1593: 1589: 1585: 1582: 1578: 1540: 1534: 1530: 1502: 1496: 1492: 1450: 1447: 1396: 1386: 1380: 1377: 1372: 1369: 1363: 1358: 1348: 1342: 1339: 1334: 1331: 1325: 1320: 1310: 1304: 1301: 1296: 1293: 1287: 1282: 1276: 1273: 1268: 1265: 1250: 1247: 1221: 1210: 1201: 1192: 1183: 1174: 1121: 1117: 1113: 1107: 1103: 1099: 1093: 1089: 1085: 1081: 1078: 1074: 1071: 1067: 1064: 1060: 1057: 1054: 1049: 1045: 1041: 1036: 1032: 1028: 1023: 1019: 1015: 1012: 1009: 1006: 1003: 1000: 997: 994: 988: 985: 982: 979: 976: 973: 970: 967: 964: 959: 951: 948: 945: 942: 939: 936: 933: 928: 924: 921: 919: 917: 914: 911: 908: 904: 900: 896: 892: 889: 884: 878: 874: 867: 864: 861: 858: 855: 852: 849: 846: 843: 838: 833: 827: 823: 816: 813: 810: 807: 804: 801: 798: 793: 789: 786: 784: 782: 779: 778: 741: 735: 731: 719:momentum space 699: 693: 689: 663: 657: 653: 647: 641: 637: 632: 629: 626: 622: 618: 614: 610: 607: 604: 601: 598: 544: 538: 534: 528: 522: 518: 475: 470: 466: 462: 456: 452: 448: 442: 438: 434: 430: 427: 423: 420: 416: 413: 410: 406: 400: 396: 390: 384: 380: 368:volume element 352: 348: 344: 322: 313: 304: 288:is called the 277: 274: 272: 269: 218: 212: 208: 197:momentum space 183: 161: 139: 133: 129: 118:volume element 26: 9: 6: 4: 3: 2: 9248: 9237: 9234: 9232: 9229: 9227: 9224: 9222: 9219: 9217: 9214: 9212: 9209: 9207: 9204: 9202: 9199: 9198: 9196: 9179: 9176: 9174: 9171: 9169: 9166: 9165: 9164: 9161: 9159: 9156: 9152: 9151:superfluidity 9149: 9147: 9144: 9143: 9142: 9139: 9138: 9136: 9132: 9126: 9123: 9121: 9118: 9116: 9113: 9111: 9108: 9106: 9103: 9102: 9100: 9098: 9094: 9086: 9083: 9081: 9078: 9077: 9076: 9073: 9071: 9068: 9067: 9065: 9063: 9059: 9053: 9049: 9046: 9044: 9041: 9039: 9036: 9034: 9031: 9029: 9026: 9024: 9021: 9020: 9018: 9014: 9006: 9003: 9001: 8998: 8997: 8996: 8992: 8988: 8985: 8983: 8980: 8978: 8975: 8973: 8970: 8969: 8968: 8965: 8964: 8962: 8960: 8956: 8950: 8947: 8943: 8940: 8938: 8935: 8933: 8930: 8928: 8925: 8924: 8922: 8919: 8917: 8914: 8912: 8909: 8907: 8904: 8903: 8901: 8899: 8895: 8880: 8877: 8875: 8872: 8871: 8869: 8865: 8861: 8854: 8849: 8847: 8842: 8840: 8835: 8834: 8831: 8825: 8822: 8820: 8817: 8816: 8806: 8802: 8798: 8794: 8790: 8786: 8781: 8777: 8773: 8769: 8765: 8761: 8757: 8753: 8749: 8744: 8740: 8736: 8732: 8728: 8724: 8720: 8716: 8712: 8708: 8707:Arkeryd, Leif 8704: 8700: 8696: 8692: 8688: 8684: 8680: 8676: 8672: 8667: 8666: 8662: 8658: 8657:Fokker–Planck 8652: 8646: 8642: 8641: 8635: 8634: 8622: 8618: 8613: 8608: 8604: 8600: 8596: 8592: 8588: 8584: 8580: 8573: 8565: 8561: 8557: 8553: 8549: 8542: 8533: 8528: 8521: 8507: 8503: 8497: 8495: 8485: 8480: 8476: 8472: 8468: 8462: 8456: 8455:0-521-40844-X 8452: 8448: 8442: 8435: 8431: 8425: 8417: 8413: 8409: 8405: 8400: 8395: 8391: 8387: 8380: 8372: 8368: 8363: 8358: 8354: 8350: 8346: 8339: 8330: 8325: 8321: 8317: 8313: 8306: 8298: 8294: 8290: 8286: 8281: 8276: 8272: 8268: 8261: 8252: 8244: 8240: 8236: 8232: 8228: 8224: 8217: 8209: 8205: 8201: 8197: 8193: 8189: 8182: 8173: 8165: 8161: 8157: 8153: 8149: 8145: 8140: 8135: 8131: 8127: 8126:Phys. Lett. B 8120: 8118: 8109: 8103: 8099: 8092: 8090: 8081: 8075: 8071: 8064: 8062: 8060: 8058: 8049: 8045: 8041: 8037: 8033: 8029: 8022: 8015: 8014:0-07-051400-3 8011: 8005: 8003: 7994: 7988: 7984: 7979: 7978: 7969: 7961: 7957: 7952: 7947: 7942: 7937: 7933: 7929: 7924: 7919: 7915: 7911: 7907: 7903: 7897: 7889: 7885: 7881: 7877: 7873: 7869: 7862: 7855: 7849: 7847: 7845: 7843: 7836: 7831: 7825: 7821: 7817: 7813: 7812: 7804: 7800: 7790: 7787: 7785: 7782: 7780: 7777: 7775: 7772: 7770: 7767: 7765: 7762: 7760: 7757: 7755: 7752: 7751: 7744: 7742: 7738: 7737:morphogenesis 7734: 7730: 7726: 7721: 7719: 7714: 7709: 7707: 7702: 7700: 7696: 7685: 7683: 7679: 7675: 7671: 7667: 7664:in powers of 7663: 7659: 7654: 7651: 7647: 7643: 7639: 7634: 7626: 7624: 7620: 7616: 7612: 7607: 7603: 7599: 7595: 7591: 7587: 7583: 7579: 7574: 7573:phase space. 7570: 7566: 7558: 7551: 7544: 7521: 7515: 7509: 7506: 7498: 7494: 7485: 7474: 7470: 7464: 7460: 7454: 7451: 7442: 7434: 7426: 7422: 7413: 7402: 7398: 7394: 7388: 7351: 7346: 7344: 7340: 7336: 7326: 7324: 7320: 7316: 7312: 7308: 7304: 7300: 7296: 7292: 7282: 7269: 7248: 7240: 7234: 7229: 7219: 7213: 7201: 7166: 7160: 7155: 7150: 7133: 7127: 7116: 7110: 7085: 7075: 7053: 7049: 7045: 7040: 7036: 7024: 7020: 7016: 7011: 7007: 6995: 6991: 6987: 6982: 6978: 6968: 6962: 6959: 6953: 6948: 6945: 6941: 6912: 6908: 6904: 6899: 6895: 6883: 6879: 6875: 6870: 6866: 6856: 6850: 6847: 6841: 6838: 6829: 6816: 6813: 6810: 6805: 6801: 6795: 6791: 6787: 6784: 6776: 6772: 6766: 6763: 6759: 6755: 6750: 6747: 6743: 6739: 6734: 6730: 6724: 6720: 6714: 6710: 6706: 6700: 6697: 6691: 6686: 6682: 6678: 6667: 6663: 6650: 6642: 6638: 6632: 6628: 6624: 6618: 6615: 6609: 6606: 6597: 6580: 6578: 6559: 6556: 6549: 6545: 6539: 6535: 6528: 6523: 6517: 6507: 6503: 6496: 6490: 6487: 6476:Second moment 6473: 6471: 6467: 6443: 6439: 6435: 6430: 6426: 6414: 6410: 6406: 6401: 6397: 6387: 6384: 6379: 6376: 6372: 6362: 6349: 6346: 6343: 6338: 6334: 6330: 6327: 6319: 6316: 6312: 6308: 6303: 6299: 6293: 6289: 6285: 6274: 6270: 6257: 6249: 6245: 6241: 6232: 6215: 6213: 6195: 6191: 6187: 6182: 6172: 6168: 6161: 6158: 6155: 6141: 6122: 6118: 6111: 6106: 6102: 6081: 6078: 6075: 6072: 6052: 6049: 6046: 6038: 6034: 6030: 6019: 6015: 6002: 5999: 5993: 5976: 5960: 5957: 5952: 5942: 5938: 5931: 5928: 5925: 5914:Zeroth moment 5911: 5895: 5891: 5868: 5864: 5855: 5850: 5844: 5838: 5825: 5822: 5819: 5810: 5804: 5789: 5776: 5766: 5762: 5750: 5744: 5736: 5727: 5722: 5719: 5711: 5698: 5694: 5686: 5682: 5673: 5664: 5658: 5654: 5650: 5647: 5644: 5634: 5630: 5620: 5616: 5607: 5596: 5592: 5588: 5585: 5577: 5564: 5553: 5549: 5545: 5539: 5528: 5524: 5509: 5506: 5501: 5491: 5487: 5477: 5473: 5464: 5453: 5449: 5444: 5440: 5433: 5425: 5412: 5403: 5397: 5388: 5376: 5366: 5362: 5354: 5346: 5337: 5334: 5326: 5324: 5305: 5297: 5293: 5284: 5266: 5262: 5239: 5235: 5209: 5205: 5198: 5176: 5172: 5149: 5145: 5141: 5138: 5133: 5129: 5098: 5094: 5076: 5063: 5053: 5049: 5044: 5041: 5038: 5033: 5030: 5025: 5019: 5007: 5001: 4988: 4978: 4974: 4969: 4966: 4963: 4960: 4949: 4934: 4932: 4928: 4924: 4920: 4916: 4911: 4902: 4889: 4879: 4873: 4864: 4859: 4853: 4839: 4829: 4819: 4814: 4799: 4794: 4791: 4787: 4778: 4774: 4769: 4765: 4761: 4757: 4751: 4738: 4733: 4723: 4719: 4711: 4702: 4698: 4692: 4688: 4684: 4680: 4676: 4672: 4667: 4663: 4659: 4646: 4641: 4638: 4634: 4625: 4622: 4618: 4612: 4609: 4605: 4601: 4596: 4591: 4588: 4585: 4581: 4577: 4556: 4550: 4540: 4536: 4526: 4516: 4512: 4507: 4503: 4499: 4495: 4488: 4479: 4466: 4456: 4450: 4440: 4436: 4426: 4421: 4413: 4396: 4392: 4382: 4374: 4369: 4365: 4358: 4351: 4347: 4341: 4329: 4323: 4313: 4309: 4294: 4287: 4283: 4272: 4256: 4252: 4231: 4222: 4209: 4203: 4200: 4195: 4191: 4184: 4181: 4165: 4156: 4148: 4145: 4139: 4134: 4124: 4118: 4110: 4096: 4087: 4083: 4079: 4071: 4065: 4061: 4056: 4033: 4027: 4018: 4013: 4007: 3993: 3983: 3973: 3968: 3953: 3950: 3940: 3936: 3929: 3925: 3920: 3916: 3910: 3906: 3899: 3895: 3878: 3873: 3861: 3857: 3849: 3839: 3836: 3831: 3821: 3810: 3804: 3801: 3796: 3786: 3775: 3772: 3766: 3763: 3758: 3752: 3743: 3732: 3726: 3723: 3718: 3712: 3703: 3692: 3680: 3677: 3671: 3668: 3665: 3662: 3652: 3646: 3638: 3629: 3619: 3614: 3608: 3593: 3591: 3590:Bose–Einstein 3587: 3583: 3577: 3571: 3565: 3563: 3559: 3554: 3546: 3540: 3536: 3532: 3526: 3495: 3489: 3481: 3472: 3467: 3451: 3442: 3434: 3431: 3425: 3420: 3410: 3404: 3396: 3382: 3380: 3379: 3373: 3368: 3367: 3356: 3354: 3351: 3345: 3341: 3334: 3330: 3323: 3319: 3298: 3283: 3275: 3271: 3262: 3251: 3234: 3226: 3222: 3213: 3202: 3185: 3177: 3173: 3164: 3153: 3136: 3120: 3106: 3101: 3096: 3083: 3076: 3074: 3072: 3055: 3052: 3043: 3027: 3018: 3015: 3012: 3007: 2997: 2994: 2988: 2985: 2982: 2976: 2968: 2959: 2957: 2944: 2941: 2925: 2916: 2908: 2905: 2902: 2896: 2893: 2890: 2884: 2876: 2867: 2865: 2856: 2850: 2846: 2842: 2833: 2829: 2820: 2811: 2806: 2802: 2798: 2789: 2785: 2776: 2767: 2762: 2758: 2754: 2745: 2741: 2732: 2722: 2718: 2714: 2710: 2707: 2700: 2692: 2683: 2680: 2677: 2670: 2662: 2653: 2650: 2647: 2640: 2632: 2622: 2618: 2615: 2612: 2605: 2597: 2588: 2586: 2581: 2578: 2566: 2565: 2562: 2559: 2554: 2545: 2538: 2536: 2534: 2503: 2497: 2489: 2480: 2475: 2469: 2466: 2461: 2458: 2447: 2446: 2443: 2439: 2431: 2413: 2399: 2395: 2383: 2379: 2370: 2369: 2363: 2358: 2353: 2342: 2335: 2333: 2331: 2307: 2303: 2291: 2287: 2282: 2276: 2274: 2259: 2255: 2243: 2239: 2231: 2228: 2220: 2209: 2206: 2196: 2192: 2180: 2176: 2171: 2167: 2161: 2158: 2155: 2152: 2141: 2133: 2130: 2122: 2112: 2103: 2099: 2096: 2094: 2079: 2075: 2063: 2059: 2054: 2030: 2024: 2016: 2007: 2002: 2000: 1978: 1974: 1962: 1961: 1958: 1937: 1933: 1921: 1917: 1908: 1904: 1881: 1877: 1865: 1861: 1851: 1831: 1827: 1815: 1811: 1803: 1800: 1792: 1781: 1778: 1768: 1764: 1752: 1748: 1742: 1738: 1732: 1729: 1726: 1723: 1711: 1703: 1700: 1691: 1681: 1672: 1668: 1660: 1659:must satisfy 1657: 1651: 1647: 1643: 1639: 1635: 1631:and momentum 1618: 1609: 1599: 1591: 1580: 1566: 1562: 1556: 1552:. If a force 1532: 1528: 1518: 1514:and momentum 1494: 1490: 1480: 1470: 1467: 1462: 1457: 1446: 1443: 1440: 1436: 1430: 1424: 1418: 1416: 1412: 1407: 1394: 1384: 1378: 1370: 1361: 1356: 1346: 1340: 1332: 1323: 1318: 1308: 1302: 1294: 1285: 1280: 1274: 1271: 1266: 1263: 1246: 1244: 1237: 1232: 1230: 1227:for particle 1224: 1220: 1213: 1209: 1200: 1191: 1182: 1173: 1167: 1161: 1156: 1153: 1152:deterministic 1148: 1143: 1138: 1119: 1115: 1111: 1105: 1101: 1097: 1091: 1087: 1083: 1079: 1076: 1072: 1069: 1065: 1062: 1055: 1052: 1047: 1043: 1039: 1034: 1030: 1026: 1021: 1017: 1013: 1010: 1007: 1004: 1001: 998: 992: 957: 926: 922: 920: 909: 906: 898: 887: 876: 872: 836: 825: 821: 791: 787: 785: 780: 768: 766: 757: 733: 729: 720: 715: 691: 687: 678: 655: 651: 639: 635: 627: 624: 616: 605: 602: 599: 596: 586: 582: 578: 574: 569: 560: 536: 532: 520: 516: 506: 500: 495: 486: 473: 468: 464: 460: 454: 450: 446: 440: 436: 432: 428: 425: 421: 418: 414: 411: 408: 398: 394: 382: 378: 369: 365: 361: 360:parameterized 355: 340: 336: 330: 325: 316: 307: 299: 295: 291: 287: 283: 268: 266: 263: 258: 256: 252: 248: 244: 240: 236: 231: 210: 206: 198: 131: 127: 119: 115: 111: 107: 103: 98: 95: 91: 87: 83: 79: 75: 71: 67: 63: 54: 48: 41: 37: 33: 19: 9134:Applications 9085:size scaling 9022: 8788: 8785:Ann. of Math 8784: 8751: 8747: 8717:(1): 17–34. 8714: 8710: 8674: 8670: 8639: 8586: 8582: 8572: 8555: 8551: 8541: 8520: 8509:. Retrieved 8505: 8474: 8470: 8461: 8441: 8424: 8389: 8385: 8379: 8352: 8348: 8338: 8319: 8315: 8305: 8270: 8266: 8260: 8251: 8226: 8222: 8216: 8191: 8187: 8181: 8172: 8129: 8125: 8097: 8069: 8031: 8027: 8021: 7976: 7968: 7913: 7909: 7896: 7871: 7868:Ann. of Math 7867: 7861: 7854:R. G. Lerner 7810: 7803: 7722: 7710: 7703: 7691: 7655: 7635: 7632: 7623:baryogenesis 7605: 7602:wavefunction 7575: 7568: 7564: 7553: 7549: 7347: 7338: 7334: 7332: 7318: 7314: 7311:baryogenesis 7295:relativistic 7288: 7164: 7158: 7148: 7081: 6830: 6581: 6479: 6363: 6216: 6147: 6144:First moment 5917: 5856:of velocity 5848: 5842: 5839: 5712: 5578: 5426: 5327: 5322: 5285:is zero for 5282: 5077: 5005: 5002: 4953:is given by 4945: 4930: 4926: 4922: 4918: 4914: 4906: 4903: 4779: 4772: 4767: 4763: 4759: 4755: 4752: 4517: 4510: 4505: 4501: 4497: 4490: 4483: 4480: 4295: 4285: 4281: 4278: 4223: 4097: 4093: 4081: 4063: 4059: 3938: 3934: 3927: 3923: 3918: 3914: 3908: 3904: 3897: 3893: 3604: 3575: 3569: 3566: 3538: 3534: 3530: 3527: 3383: 3376: 3371: 3364: 3362: 3353:unit vectors 3343: 3339: 3332: 3328: 3321: 3317: 3099: 3088: 3077: 2567: 2557: 2553:differential 2550: 2539: 2448: 2437: 2429: 2366: 2365:. Dividing ( 2361: 2356: 2351: 2347: 2336: 1963: 1957:changes, so 1852: 1661: 1655: 1649: 1645: 1641: 1637: 1633: 1564: 1560: 1554: 1516: 1478: 1471: 1465: 1460: 1455: 1452: 1441: 1438: 1434: 1428: 1422: 1419: 1408: 1252: 1233: 1228: 1222: 1218: 1211: 1207: 1198: 1189: 1180: 1171: 1165: 1159: 1146: 1139: 769: 755: 713: 676: 584: 580: 576: 572: 566:. This is a 558: 504: 498: 493: 487: 363: 342: 338: 334: 320: 311: 302: 297: 285: 284:and momenta 281: 279: 259: 232: 99: 69: 65: 61: 59: 9125:von Neumann 8995:force field 8987:percolation 8754:(1): 1–16. 8677:(1): 1–16. 8355:: 215–240. 7640:(including 7619:dark matter 7307:dark matter 4078:solid angle 3586:Fermi–Dirac 3545:force field 3105:dot product 1140:which is a 765:Integrating 329:dimensional 294:coordinates 290:phase space 237:energy and 110:probability 78:equilibrium 9195:Categories 8982:Heisenberg 8631:References 8511:2022-05-10 8273:(3): 771. 7741:chemotaxis 3363:Dividing ( 3097:operator, 2551:The total 2442:, we have 2359:change in 759:, at time 343:x, y, z, p 114:very small 9105:Boltzmann 9028:H-theorem 8906:Ensembles 8776:117877311 8739:119481100 8699:117877311 8603:1522-9602 8589:(3): 42. 8416:0036-1429 8394:CiteSeerX 8371:0307-904X 8297:115167686 8280:1011.5441 8223:Physica A 8188:Physica A 8164:119253828 8139:1202.1301 7923:1002.3639 7789:H-theorem 7656:Close to 7636:Instead, 7499:α 7491:∂ 7483:∂ 7475:γ 7465:β 7455:γ 7452:β 7443:α 7439:Γ 7435:− 7427:α 7419:∂ 7411:∂ 7403:α 7369:^ 7258:∂ 7254:∂ 7249:⋅ 7238:∇ 7235:⋅ 7211:∂ 7207:∂ 7185:^ 7102:^ 7062:⟩ 7046:− 7017:− 6988:− 6972:⟨ 6969:ρ 6921:⟩ 6905:− 6876:− 6860:⟨ 6857:ρ 6785:− 6707:ρ 6660:∂ 6656:∂ 6625:ρ 6595:∂ 6591:∂ 6452:⟩ 6436:− 6407:− 6391:⟨ 6388:ρ 6328:− 6286:ρ 6267:∂ 6263:∂ 6242:ρ 6230:∂ 6226:∂ 6128:⟩ 6115:⟨ 6073:ρ 6031:ρ 6012:∂ 6008:∂ 6000:ρ 5991:∂ 5987:∂ 5814:⟩ 5808:⟨ 5787:∂ 5783:∂ 5742:∂ 5734:∂ 5720:∫ 5679:∂ 5671:∂ 5648:− 5613:∂ 5605:∂ 5586:∫ 5559:⟩ 5543:⟨ 5521:∂ 5517:∂ 5470:∂ 5462:∂ 5434:∫ 5407:⟩ 5401:⟨ 5386:∂ 5382:∂ 5352:∂ 5344:∂ 5335:∫ 5309:∞ 5306:± 5303:→ 5126:↦ 5091:↦ 5039:∫ 5023:⟩ 5017:⟨ 4967:∫ 4865:− 4820:− 4715:Ω 4685:− 4650:Ω 4602:∬ 4582:∑ 4548:∂ 4533:∂ 4448:∂ 4433:∂ 4404:∂ 4389:∂ 4383:⋅ 4362:∇ 4359:⋅ 4321:∂ 4306:∂ 4232:ν 4201:− 4185:ν 4171:∂ 4163:∂ 4157:⋅ 4143:∇ 4140:⋅ 4116:∂ 4108:∂ 4019:− 3974:− 3853:Ω 3773:− 3684:Ω 3666:∬ 3644:∂ 3636:∂ 3607:Boltzmann 3487:∂ 3479:∂ 3457:∂ 3449:∂ 3443:⋅ 3429:∇ 3426:⋅ 3402:∂ 3394:∂ 3381:) gives: 3350:Cartesian 3288:∇ 3268:∂ 3260:∂ 3245:^ 3219:∂ 3211:∂ 3196:^ 3170:∂ 3162:∂ 3147:^ 3126:∂ 3118:∂ 3044:⋅ 3033:∂ 3025:∂ 2998:⋅ 2992:∇ 2974:∂ 2966:∂ 2942:⋅ 2931:∂ 2923:∂ 2906:⋅ 2900:∇ 2882:∂ 2874:∂ 2826:∂ 2818:∂ 2782:∂ 2774:∂ 2738:∂ 2730:∂ 2698:∂ 2690:∂ 2668:∂ 2660:∂ 2638:∂ 2630:∂ 2603:∂ 2595:∂ 2495:∂ 2487:∂ 2411:Δ 2280:Δ 2207:− 2165:Δ 2150:Δ 2128:Δ 2052:Δ 2022:∂ 2014:∂ 1736:Δ 1721:Δ 1698:Δ 1616:Δ 1584:Δ 1415:collision 1411:diffusion 1376:∂ 1368:∂ 1338:∂ 1330:∂ 1300:∂ 1292:∂ 1155:many-body 958:∭ 927:∭ 837:∫ 792:∫ 262:nonlinear 243:viscosity 102:positions 94:particles 9115:Tsallis 8621:35150333 7960:20231489 7747:See also 7676:and the 7594:Big Bang 6480:Letting 6470:pressure 6212:momentum 6148:Letting 5918:Letting 5695:⟩ 5665:⟨ 5164:, where 4874:′ 4854:′ 4764:p′ 4760:f′ 4756:f′ 4734:′ 4681:′ 4668:′ 4028:′ 4008:′ 3935:p′ 3924:p′ 3753:′ 3713:′ 3095:gradient 1461:external 1144:. While 721:element 362:by time 271:Overview 239:momentum 9110:Shannon 9097:Entropy 8805:1971423 8756:Bibcode 8719:Bibcode 8679:Bibcode 8612:8840942 8231:Bibcode 8196:Bibcode 8144:Bibcode 8036:Bibcode 7951:2851887 7928:Bibcode 7888:1971423 7835:Alt URL 7713:liquids 7699:spheres 7541:is the 7152:is the 5854:moments 4068:is the 3551:is the 3543:is the 3103:is the 3093:is the 2355:is the 1520:within 508:within 298:x, y, z 106:momenta 8959:Models 8867:Theory 8803: 8774: 8737: 8697: 8647: 8619: 8609: 8601: 8453: 8432: 8414: 8396: 8369: 8295: 8162: 8104: 8076: 8012: 7989: 7958: 7948: 7886: 7826: 7739:, and 7733:cancer 7534:where 7146:where 6831:where 6575:, the 6364:where 6210:, the 6065:where 5973:, the 4753:where 4481:where 4224:where 3891:where 3315:, and 3089:where 2348:where 1463:force 753:about 711:about 249:, and 38:, and 9168:chaos 9120:Rényi 8977:Potts 8972:Ising 8801:JSTOR 8787:. 2. 8772:S2CID 8735:S2CID 8695:S2CID 8527:arXiv 8293:S2CID 8275:arXiv 8160:S2CID 8134:arXiv 7918:arXiv 7884:JSTOR 7870:. 2. 7796:Notes 7725:cells 7668:(the 3369:) by 2371:) by 2357:total 1314:force 496:have 341:) = ( 88:with 86:fluid 9050:and 8645:ISBN 8617:PMID 8599:ISSN 8451:ISBN 8430:ISBN 8412:ISSN 8367:ISSN 8102:ISBN 8074:ISBN 8010:ISBN 7987:ISBN 7956:PMID 7824:ISBN 7644:and 7621:and 7588:and 7309:and 7293:for 5975:mass 5797:coll 5756:coll 5113:and 4925:and 4917:and 4904:and 4462:coll 4066:, Ω) 3932:and 3917:and 3902:and 3658:coll 3553:mass 3348:are 2561:is: 2434:and 1390:coll 1352:diff 1236:mass 1163:and 502:and 235:heat 104:and 60:The 8793:doi 8789:130 8764:doi 8727:doi 8687:doi 8659:or 8607:PMC 8591:doi 8560:doi 8479:doi 8475:376 8404:doi 8357:doi 8324:doi 8285:doi 8239:doi 8227:388 8204:doi 8192:388 8152:doi 8130:718 8044:doi 7946:PMC 7936:doi 7914:107 7876:doi 7872:130 7816:doi 7576:In 7352:is 7168:is 7082:In 6472:). 5009:is 4293:is 3588:or 2555:of 2440:→ 0 2432:→ 0 1636:+ Δ 1563:+ Δ 677:all 494:all 351:, p 347:, p 70:BTE 64:or 9197:: 8923:: 8799:. 8770:. 8762:. 8752:45 8750:. 8733:. 8725:. 8715:45 8713:. 8693:. 8685:. 8675:45 8673:. 8615:. 8605:. 8597:. 8587:84 8585:. 8581:. 8554:. 8550:. 8504:. 8493:^ 8473:. 8469:. 8410:. 8402:. 8390:37 8388:. 8365:. 8353:52 8351:. 8347:. 8320:35 8318:. 8314:. 8291:. 8283:. 8271:24 8269:. 8237:. 8225:. 8202:. 8190:. 8158:. 8150:. 8142:. 8128:. 8116:^ 8088:^ 8056:^ 8042:. 8032:94 8030:. 8001:^ 7985:. 7983:53 7954:. 7944:. 7934:. 7926:. 7912:. 7908:. 7882:. 7841:^ 7822:. 7720:. 7708:. 7684:. 7625:. 7567:, 7552:, 7538:βγ 7345:. 7325:. 4909:ij 4771:, 4758:= 4509:, 4500:, 4489:= 3584:, 3564:. 3537:, 3372:dt 3355:. 3337:, 3326:, 3107:, 1644:+ 1640:= 1445:. 1437:= 1231:. 1216:, 1196:, 1178:, 763:. 583:, 579:, 570:: 337:, 318:, 309:, 257:. 245:, 34:, 8942:G 8937:F 8932:H 8927:U 8852:e 8845:t 8838:v 8807:. 8795:: 8778:. 8766:: 8758:: 8741:. 8729:: 8721:: 8701:. 8689:: 8681:: 8663:. 8653:. 8623:. 8593:: 8566:. 8562:: 8556:2 8535:. 8529:: 8514:. 8487:. 8481:: 8436:. 8418:. 8406:: 8373:. 8359:: 8332:. 8326:: 8299:. 8287:: 8277:: 8245:. 8241:: 8233:: 8210:. 8206:: 8198:: 8166:. 8154:: 8146:: 8136:: 8110:. 8082:. 8050:. 8046:: 8038:: 8016:. 7995:. 7962:. 7938:: 7930:: 7920:: 7890:. 7878:: 7832:. 7818:: 7606:f 7571:) 7569:p 7565:x 7563:( 7559:) 7556:i 7554:p 7550:x 7548:( 7536:Γ 7522:, 7519:] 7516:f 7513:[ 7510:C 7507:= 7495:p 7486:f 7471:p 7461:p 7423:x 7414:f 7399:p 7395:= 7392:] 7389:f 7386:[ 7380:R 7377:G 7365:L 7335:f 7319:f 7315:f 7270:. 7262:p 7245:F 7241:+ 7230:m 7226:p 7220:+ 7214:t 7202:= 7196:R 7193:N 7181:L 7165:L 7159:C 7149:L 7134:, 7131:] 7128:f 7125:[ 7121:C 7117:= 7114:] 7111:f 7108:[ 7098:L 7059:) 7054:k 7050:V 7041:k 7037:v 7033:( 7030:) 7025:k 7021:V 7012:k 7008:v 7004:( 7001:) 6996:i 6992:V 6983:i 6979:v 6975:( 6963:2 6960:1 6954:= 6949:i 6946:q 6942:J 6918:) 6913:i 6909:V 6900:i 6896:v 6892:( 6889:) 6884:i 6880:V 6871:i 6867:v 6863:( 6851:2 6848:1 6842:= 6839:u 6817:, 6814:0 6811:= 6806:i 6802:V 6796:i 6792:F 6788:n 6782:) 6777:i 6773:V 6767:j 6764:i 6760:P 6756:+ 6751:j 6748:q 6744:J 6740:+ 6735:j 6731:V 6725:i 6721:V 6715:i 6711:V 6701:2 6698:1 6692:+ 6687:j 6683:V 6679:u 6676:( 6668:j 6664:x 6651:+ 6648:) 6643:i 6639:V 6633:i 6629:V 6619:2 6616:1 6610:+ 6607:u 6604:( 6598:t 6560:m 6557:2 6550:i 6546:p 6540:i 6536:p 6529:= 6524:2 6518:2 6514:) 6508:i 6504:v 6500:( 6497:m 6491:= 6488:A 6449:) 6444:j 6440:V 6431:j 6427:v 6423:( 6420:) 6415:i 6411:V 6402:i 6398:v 6394:( 6385:= 6380:j 6377:i 6373:P 6350:, 6347:0 6344:= 6339:i 6335:F 6331:n 6325:) 6320:j 6317:i 6313:P 6309:+ 6304:j 6300:V 6294:i 6290:V 6283:( 6275:j 6271:x 6258:+ 6255:) 6250:i 6246:V 6239:( 6233:t 6196:i 6192:p 6188:= 6183:1 6179:) 6173:i 6169:v 6165:( 6162:m 6159:= 6156:A 6123:i 6119:v 6112:= 6107:i 6103:V 6082:n 6079:m 6076:= 6053:, 6050:0 6047:= 6044:) 6039:j 6035:V 6028:( 6020:j 6016:x 6003:+ 5994:t 5961:m 5958:= 5953:0 5949:) 5943:i 5939:v 5935:( 5932:m 5929:= 5926:A 5896:i 5892:p 5869:i 5865:v 5849:A 5843:A 5826:, 5823:0 5820:= 5817:) 5811:A 5805:n 5802:( 5790:t 5777:= 5773:p 5767:3 5763:d 5751:) 5745:t 5737:f 5728:( 5723:A 5699:, 5687:j 5683:p 5674:A 5659:j 5655:F 5651:n 5645:= 5641:p 5635:3 5631:d 5621:j 5617:p 5608:f 5597:j 5593:F 5589:A 5565:, 5562:) 5554:j 5550:p 5546:A 5540:n 5537:( 5529:j 5525:x 5510:m 5507:1 5502:= 5498:p 5492:3 5488:d 5478:j 5474:x 5465:f 5454:m 5450:A 5445:j 5441:p 5413:, 5410:) 5404:A 5398:n 5395:( 5389:t 5377:= 5373:p 5367:3 5363:d 5355:t 5347:f 5338:A 5323:A 5298:i 5294:p 5283:f 5267:i 5263:F 5240:i 5236:p 5215:) 5210:i 5206:p 5202:( 5199:A 5177:i 5173:v 5150:i 5146:v 5142:m 5139:= 5134:i 5130:p 5122:p 5099:i 5095:x 5087:x 5064:. 5060:p 5054:3 5050:d 5045:f 5042:A 5034:n 5031:1 5026:= 5020:A 5006:A 4989:. 4985:p 4979:3 4975:d 4970:f 4964:= 4961:n 4951:n 4931:i 4927:j 4923:i 4919:j 4915:i 4907:I 4890:, 4886:| 4880:j 4871:p 4860:i 4851:p 4844:| 4840:= 4836:| 4830:j 4825:p 4815:i 4810:p 4804:| 4800:= 4795:j 4792:i 4788:g 4775:) 4773:t 4768:i 4762:( 4739:, 4731:p 4724:3 4720:d 4712:d 4708:] 4703:j 4699:f 4693:i 4689:f 4677:j 4673:f 4664:i 4660:f 4656:[ 4653:) 4647:, 4642:j 4639:i 4635:g 4631:( 4626:j 4623:i 4619:I 4613:j 4610:i 4606:g 4597:n 4592:1 4589:= 4586:j 4578:= 4572:l 4569:l 4566:o 4563:c 4557:) 4551:t 4541:i 4537:f 4527:( 4513:) 4511:t 4506:i 4502:p 4498:r 4496:( 4493:i 4491:f 4486:i 4484:f 4467:, 4457:) 4451:t 4441:i 4437:f 4427:( 4422:= 4414:i 4409:p 4397:i 4393:f 4379:F 4375:+ 4370:i 4366:f 4352:i 4348:m 4342:i 4337:p 4330:+ 4324:t 4314:i 4310:f 4291:i 4286:n 4282:i 4257:0 4253:f 4210:, 4207:) 4204:f 4196:0 4192:f 4188:( 4182:= 4175:p 4166:f 4153:F 4149:+ 4146:f 4135:m 4131:p 4125:+ 4119:t 4111:f 4084:Ω 4082:d 4074:θ 4064:g 4062:( 4060:I 4040:| 4034:A 4025:p 4014:B 4005:p 3998:| 3994:= 3990:| 3984:A 3979:p 3969:B 3964:p 3958:| 3954:= 3951:g 3939:B 3928:A 3919:B 3915:A 3909:B 3905:p 3898:A 3894:p 3879:, 3874:B 3869:p 3862:3 3858:d 3850:d 3846:] 3843:) 3840:t 3837:, 3832:B 3827:p 3822:, 3818:r 3814:( 3811:f 3808:) 3805:t 3802:, 3797:A 3792:p 3787:, 3783:r 3779:( 3776:f 3770:) 3767:t 3764:, 3759:B 3750:p 3744:, 3740:r 3736:( 3733:f 3730:) 3727:t 3724:, 3719:A 3710:p 3704:, 3700:r 3696:( 3693:f 3690:[ 3687:) 3681:, 3678:g 3675:( 3672:I 3669:g 3663:= 3653:) 3647:t 3639:f 3630:( 3610:" 3576:f 3570:f 3549:m 3541:) 3539:t 3535:r 3533:( 3531:F 3511:l 3508:l 3505:o 3502:c 3496:) 3490:t 3482:f 3473:( 3468:= 3461:p 3452:f 3439:F 3435:+ 3432:f 3421:m 3417:p 3411:+ 3405:t 3397:f 3378:2 3366:3 3344:z 3340:ê 3333:y 3329:ê 3322:x 3318:ê 3313:∇ 3299:f 3293:p 3284:= 3276:z 3272:p 3263:f 3252:z 3242:e 3235:+ 3227:y 3223:p 3214:f 3203:y 3193:e 3186:+ 3178:x 3174:p 3165:f 3154:x 3144:e 3137:= 3130:p 3121:f 3100:· 3091:∇ 3082:) 3080:3 3078:( 3056:t 3053:d 3048:F 3037:p 3028:f 3019:+ 3016:t 3013:d 3008:m 3004:p 2995:f 2989:+ 2986:t 2983:d 2977:t 2969:f 2960:= 2949:p 2945:d 2935:p 2926:f 2917:+ 2913:r 2909:d 2903:f 2897:+ 2894:t 2891:d 2885:t 2877:f 2868:= 2857:) 2851:z 2847:p 2843:d 2834:z 2830:p 2821:f 2812:+ 2807:y 2803:p 2799:d 2790:y 2786:p 2777:f 2768:+ 2763:x 2759:p 2755:d 2746:x 2742:p 2733:f 2723:( 2719:+ 2715:) 2711:z 2708:d 2701:z 2693:f 2684:+ 2681:y 2678:d 2671:y 2663:f 2654:+ 2651:x 2648:d 2641:x 2633:f 2623:( 2619:+ 2616:t 2613:d 2606:t 2598:f 2589:= 2582:f 2579:d 2558:f 2544:) 2542:2 2540:( 2519:l 2516:l 2513:o 2510:c 2504:) 2498:t 2490:f 2481:( 2476:= 2470:t 2467:d 2462:f 2459:d 2438:f 2436:Δ 2430:t 2428:Δ 2414:t 2406:p 2400:3 2396:d 2390:r 2384:3 2380:d 2368:1 2362:f 2352:f 2350:Δ 2341:) 2339:1 2337:( 2314:p 2308:3 2304:d 2298:r 2292:3 2288:d 2283:f 2277:= 2266:p 2260:3 2256:d 2250:r 2244:3 2240:d 2235:) 2232:t 2229:, 2225:p 2221:, 2217:r 2213:( 2210:f 2203:p 2197:3 2193:d 2187:r 2181:3 2177:d 2172:) 2168:t 2162:+ 2159:t 2156:, 2153:t 2146:F 2142:+ 2138:p 2134:, 2131:t 2123:m 2119:p 2113:+ 2109:r 2104:( 2100:f 2097:= 2086:p 2080:3 2076:d 2070:r 2064:3 2060:d 2055:t 2046:l 2043:l 2040:o 2037:c 2031:) 2025:t 2017:f 2008:( 2003:= 1993:l 1990:l 1987:o 1984:c 1979:N 1975:d 1944:p 1938:3 1934:d 1928:r 1922:3 1918:d 1888:p 1882:3 1878:d 1872:r 1866:3 1862:d 1838:p 1832:3 1828:d 1822:r 1816:3 1812:d 1807:) 1804:t 1801:, 1797:p 1793:, 1789:r 1785:( 1782:f 1779:= 1775:p 1769:3 1765:d 1759:r 1753:3 1749:d 1743:) 1739:t 1733:+ 1730:t 1727:, 1724:t 1716:F 1712:+ 1708:p 1704:, 1701:t 1692:m 1688:p 1682:+ 1678:r 1673:( 1669:f 1656:f 1650:t 1648:Δ 1646:F 1642:p 1638:p 1634:p 1619:t 1610:m 1606:p 1600:+ 1596:r 1592:= 1588:r 1581:+ 1577:r 1565:t 1561:t 1555:F 1539:p 1533:3 1529:d 1517:p 1501:r 1495:3 1491:d 1479:r 1474:t 1466:F 1456:f 1442:v 1439:m 1435:p 1429:p 1423:v 1395:, 1385:) 1379:t 1371:f 1362:( 1357:+ 1347:) 1341:t 1333:f 1324:( 1319:+ 1309:) 1303:t 1295:f 1286:( 1281:= 1275:t 1272:d 1267:f 1264:d 1239:m 1229:N 1223:N 1219:p 1212:N 1208:r 1202:2 1199:p 1193:2 1190:r 1184:1 1181:p 1175:1 1172:r 1166:p 1160:r 1147:f 1120:z 1116:p 1112:d 1106:y 1102:p 1098:d 1092:x 1088:p 1084:d 1080:z 1077:d 1073:y 1070:d 1066:x 1063:d 1059:) 1056:t 1053:, 1048:z 1044:p 1040:, 1035:y 1031:p 1027:, 1022:x 1018:p 1014:, 1011:z 1008:, 1005:y 1002:, 999:x 996:( 993:f 987:s 984:n 981:o 978:i 975:t 972:i 969:s 966:o 963:p 950:a 947:t 944:n 941:e 938:m 935:o 932:m 923:= 913:) 910:t 907:, 903:p 899:, 895:r 891:( 888:f 883:r 877:3 873:d 866:s 863:n 860:o 857:i 854:t 851:i 848:s 845:o 842:p 832:p 826:3 822:d 815:a 812:t 809:n 806:e 803:m 800:o 797:m 788:= 781:N 761:t 756:p 740:p 734:3 730:d 714:r 698:r 692:3 688:d 662:p 656:3 652:d 646:r 640:3 636:d 631:) 628:t 625:, 621:p 617:, 613:r 609:( 606:f 603:= 600:N 597:d 587:) 585:t 581:p 577:r 575:( 573:f 564:t 559:f 543:p 537:3 533:d 527:r 521:3 517:d 505:p 499:r 490:N 474:. 469:z 465:p 461:d 455:y 451:p 447:d 441:x 437:p 433:d 429:z 426:d 422:y 419:d 415:x 412:d 409:= 405:p 399:3 395:d 389:r 383:3 379:d 364:t 356:) 353:z 349:y 345:x 339:p 335:r 333:( 323:z 321:p 314:y 312:p 305:x 303:p 286:p 282:r 217:p 211:3 207:d 182:p 160:r 138:r 132:3 128:d 68:( 49:. 42:. 20:)

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Knowledge

Boltzmann equation

Index