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 BogoliubovâBornâGreenâKirkwoodâYvon 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:
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}}}
772:
8889:
3309:
53:
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:
1664:
5575:
5836:
4749:
4220:
3386:
1256:
5709:
4094:
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
7355:
3555:
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.
6584:
7171:
6360:
3889:
96:
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
484:
673:
7545:
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
7652:
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.
7608:
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
3579:
is known. This term cannot be found as easily or generally as the others â it is a statistical term representing the particle collisions, and requires knowledge of the statistics the particles obey, like the
2575:
1971:
8823:
2424:
8176:
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}
7715:
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
777:
7704:
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:
8850:
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
6208:
5319:
7633:
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.
4946:
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
56:
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)
5078:
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
3946:
1550:
1512:
751:
709:
228:
149:
8026:
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".
5971:
373:
592:
7731:
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:
8905:
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 ,}
8910:
8843:
8384:
Pareschi, L.; Russo, G. (2000-01-01). "Numerical Solution of the Boltzmann Equation I: Spectrally Accurate Approximation of the Collision Operator".
4271:
is the local Maxwellian distribution function given the gas temperature at this point in space. This is also called "relaxation time approximation".
108:
of each particle in the fluid but rather by considering a probability distribution for the position and momentum of a typical particleâthat is, the
9205:
9200:
6834:
7580:
the fully covariant approach has been used to study the cosmic microwave background radiation. More generically the study of processes in the
562:
which gives this probability per unit phase-space volume, or probability per unit length cubed per unit momentum cubed, at an instant of time
8836:
8648:
8433:
8105:
8077:
7990:
7827:
6367:
7637:
6483:
1906:
7089:
1417:
term â 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 BoltzmannâBGK 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:
8447:
The mathematical theory of non-uniform gases: an account of the kinetic theory of viscosity, thermal conduction, and diffusion in gases
3581:
39:
254:
8454:
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
8013:
2374:
7648:) are generally used to find approximate solutions to the various forms of the Boltzmann equation. Example applications range from
4956:
9235:
7697:. The collision term is modified in Enskog equations such that particles have a finite size, for example they can be modelled as
7673:
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 ),}
3589:
9210:
8265:
Philip T. Gressman; Robert M. Strain (2011). "Global Classical Solutions of the Boltzmann Equation without Angular Cut-off".
7657:
9104:
8221:
Debbasch, Fabrice; Willem van Leeuwen (2009). "General relativistic Boltzmann equation II: Manifestly covariant treatment".
31:
8783:
DiPerna, R. J.; Lions, P.-L. (1989). "On the Cauchy problem for Boltzmann equations: global existence and weak stability".
7866:
DiPerna, R. J.; Lions, P.-L. (1989). "On the Cauchy problem for Boltzmann equations: global existence and weak stability".
1912:
1856:
511:
264:
3585:
9215:
8873:
567:
7677:
7313:. It is not a priori clear that the state of a quantum system can be characterized by a classical phase space density
5116:
8656:
7763:
8981:
8897:
7681:
7290:
4069:
2552:
7694:
7669:
7297:
quantum systems in which the number of particles is not conserved in collisions. This has several applications in
9220:
7773:
7645:
6097:
77:
8186:
Debbasch, Fabrice; Willem van Leeuwen (2009). "General relativistic Boltzmann equation I: Covariant treatment".
35:
9230:
9225:
9157:
9140:
7614:
7302:
5081:
7856:, G. L. Trigg, VHC publishers, 1991, ISBN (Verlagsgesellschaft) 3-527-26954-1, ISBN (VHC Inc.) 0-89573-752-3.
7321:
exists which is the solution of an effective Boltzmann equation that can be derived from first principles of
9004:
8994:
8920:
7768:
3544:
1902:
250:
92:
in space causing heat to flow from hotter regions to colder ones, by the random but biased transport of the
8124:
M. Drewes; C. Weniger; S. Mendizabal (8 January 2013). "The Boltzmann equation from quantum field theory".
9051:
8393:
6151:
5288:
764:
8936:
8859:
7083:
6465:
5325:
and integrating over momentum yields four terms, which, using integration by parts, can be expressed as
3349:
6058:{\displaystyle {\frac {\partial }{\partial t}}\rho +{\frac {\partial }{\partial x_{j}}}(\rho V_{j})=0,}
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. MĂŒller-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
9079:
8818:
8755:
8718:
8678:
8660:
8525:
van Noije, T. P. C.; Ernst, M. H. (1997-06-03). "Ring Kinetic Theory for an Idealized Granular Gas".
8230:
8195:
8143:
8035:
7927:
7661:
7610:
7322:
7294:
5853:
246:
89:
73:
46:
9145:
9124:
8501:
8398:
7705:
7342:
3557:
359:
7982:
6214:
of the particle, the integrated Boltzmann equation becomes the conservation of momentum equation:
4072:
of the collision, in which the relative momenta of the colliding particles turns through an angle
176:
154:
9172:
9084:
9061:
9042:
8986:
8800:
8771:
8734:
8694:
8547:
8526:
8292:
8274:
8159:
8133:
7917:
7883:
7728:
7589:
7577:
7542:
7349:
7298:
7153:
4895:{\displaystyle g_{ij}=|\mathbf {p} _{i}-\mathbf {p} _{j}|=|\mathbf {p'} _{i}-\mathbf {p'} _{j}|,}
1624:{\displaystyle \mathbf {r} +\Delta \mathbf {r} =\mathbf {r} +{\frac {\mathbf {p} }{m}}\,\Delta t}
7783:
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} }}
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 }
9177:
9074:
9047:
8958:
8948:
8941:
8915:
8644:
8616:
8598:
8450:
8429:
8411:
8366:
8101:
8073:
8009:
7986:
7955:
7823:
7597:
7585:
6068:
5977:
of the particle, the integrated Boltzmann equation becomes the conservation of mass equation:
5194:
4054:
1154:
1141:
9119:
9069:
8792:
8763:
8726:
8686:
8606:
8590:
8559:
8478:
8403:
8356:
8323:
8284:
8238:
8203:
8151:
8043:
7974:
7945:
7935:
7875:
7815:
7649:
3606:
1242:
261:
81:
5886:
5859:
5257:
5230:
5167:
4247:
4046:{\displaystyle g=|\mathbf {p} _{B}-\mathbf {p} _{A}|=|\mathbf {p'} _{B}-\mathbf {p'} _{A}|}
9114:
9109:
9037:
9032:
8999:
8926:
7901:
7778:
7758:
7753:
7717:
7641:
4227:
3617:
3561:
101:
7975:
17:
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
9162:
8878:
8611:
8578:
7950:
7905:
7853:
7724:
7665:
7581:
6576:
4947:
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
718:
668:{\displaystyle dN=f(\mathbf {r} ,\mathbf {p} ,t)\,d^{3}\mathbf {r} \,d^{3}\mathbf {p} }
367:
196:
117:
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
9194:
9150:
8775:
8738:
8698:
8296:
8163:
7834:
7736:
1151:
113:
8288:
9167:
8706:
8155:
7906:"Global classical solutions of the Boltzmann equation with long-range interactions"
7622:
7601:
7310:
1234:
It is assumed the particles in the system are identical (so each has an identical
8638:
8446:
8976:
8971:
8242:
8207:
7618:
7306:
5254:
only, whose total value is conserved in a collision. Assume also that the force
4077:
3352:
3104:
293:
289:
109:
52:
8594:
8344:
8966:
8483:
8466:
8407:
8361:
8328:
8311:
7740:
1469:
not due to other particles (see the collision term for the latter treatment).
8709:(1972). "On the Boltzmann equation part II: The full initial value problem".
8602:
8415:
8370:
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
9027:
7940:
7788:
1414:
1410:
328:
242:
8620:
8563:
8047:
7959:
5069:{\displaystyle \langle A\rangle ={\frac {1}{n}}\int Af\,d^{3}\mathbf {p} .}
7814:. Lecture Notes in Physics (LNP, vol. 660). Berlin, Heidelberg: Springer.
8931:
8548:"Modeling cell movement in anisotropic and heterogeneous network tissues"
8531:
7593:
6469:
6211:
3094:
238:
105:
93:
8828:
8008:
McGraw Hill Encyclopaedia of Physics (2nd Edition), S. P. Parker, 1993,
9096:
8804:
8767:
8730:
8690:
7887:
6457:{\displaystyle P_{ij}=\rho \langle (v_{i}-V_{i})(v_{j}-V_{j})\rangle }
8746:
Arkeryd, Leif (1972). "On the Boltzmann equation part I: Existence".
8669:
Arkeryd, Leif (1972). "On the Boltzmann equation part I: Existence".
7732:
7712:
7698:
7592:. In the very dense medium formed by the primordial plasma after the
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
7819:
253:(by treating the charge carriers in a material as a gas). See also
8279:
8138:
7922:
85:
51:
5974:
4913:
is the differential cross-section, as before, between particles
3552:
1235:
234:
8832:
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
7285:
Quantum theory and violation of particle number conservation
7086:, the Boltzmann equation is often written more generally as
7723:
Boltzmann-like equations are also used for the movement of
7672:
expansion). The first two terms of this expansion give the
173:, and has momentum nearly equal to a given momentum vector
7162:
is the collision operator. The non-relativistic form of
3596:
The collision term (Stosszahlansatz) and molecular chaos
8640:
An introduction to the theory of the Boltzmann equation
7613:. This includes the formation of the light elements in
8502:"Enskog Equation - an overview | ScienceDirect Topics"
7811:
Invariant Manifolds for Physical and Chemical Kinetics
7688:
Limitations and further uses of the Boltzmann equation
6957:
6939:
6845:
6837:
6695:
6613:
4279:
For a mixture of chemical species labelled by indices
1169:
is in question. It is not part of the analysis to use
8310:
Evans, Ben; Morgan, Ken; Hassan, Oubay (2011-03-01).
7358:
7174:
7092:
6587:
6486:
6370:
6222:
6154:
6100:
6071:
5983:
5924:
5889:
5862:
5718:
5584:
5432:
5333:
5291:
5260:
5233:
5197:
5170:
5119:
5084:
5015:
4959:
4785:
4523:
4301:
4250:
4230:
4103:
3949:
3626:
3389:
3113:
2573:
2454:
2377:
1969:
1915:
1859:
1667:
1574:
1526:
1488:
1259:
775:
727:
685:
595:
514:
376:
204:
179:
157:
125:
8496:
8494:
1950:{\displaystyle d^{3}\mathbf {r} \,d^{3}\mathbf {p} }
1894:{\displaystyle d^{3}\mathbf {r} \,d^{3}\mathbf {p} }
1432:; they are related in the definition of momentum by
549:{\displaystyle d^{3}\mathbf {r} \,d^{3}\mathbf {p} }
100:
The equation arises not by analyzing the individual
9133:
9095:
9060:
9015:
8957:
8896:
8866:
7301:, including the formation of the light elements in
7139:{\displaystyle {\hat {\mathbf {L} }}=\mathbf {C} ,}
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:
8471:Philosophical Transactions of the Royal Society A
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
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}}
7910:Proceedings of the National Academy of Sciences
7341:can be neglected for times far longer than the
7808:Gorban, Alexander N.; Karlin, Ilya V. (2005).
7561:phase space as opposed to fully contravariant
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:
7774:Derivation of NavierâStokes 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
8:
8267:Journal of the American Mathematical Society
8119:
8117:
8091:
8089:
8004:
8002:
7981:(Second ed.). New York: Wiley. p.
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
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:
8829:
7289:It is possible to write down relativistic
4244:is the molecular collision frequency, and
3547:acting on the particles in the fluid, and
45:"BTE" redirects here. For other uses, see
8610:
8577:Conte, Martina; Loy, Nadia (2022-02-12).
8530:
8482:
8397:
8360:
8343:Evans, B.; Walton, S.P. (December 2017).
8327:
8278:
8137:
8096:Edward Kolb & Michael Turner (1990).
7949:
7939:
7921:
7497:
7479:
7473:
7463:
7450:
7448:
7441:
7425:
7407:
7401:
7375:
7374:
7363:
7361:
7360:
7357:
7268:
7260:
7251:
7243:
7224:
7222:
7204:
7191:
7190:
7179:
7177:
7176:
7173:
7119:
7096:
7094:
7093:
7091:
7052:
7039:
7023:
7010:
6994:
6981:
6956:
6944:
6938:
6911:
6898:
6882:
6869:
6844:
6836:
6804:
6794:
6775:
6762:
6746:
6733:
6723:
6713:
6694:
6685:
6666:
6653:
6641:
6631:
6612:
6588:
6586:
6548:
6538:
6531:
6516:
6506:
6493:
6485:
6442:
6429:
6413:
6400:
6375:
6369:
6337:
6315:
6302:
6292:
6273:
6260:
6248:
6223:
6221:
6194:
6181:
6171:
6153:
6121:
6105:
6099:
6070:
6037:
6018:
6005:
5984:
5982:
5951:
5941:
5923:
5894:
5888:
5867:
5861:
5795:
5780:
5771:
5765:
5760:
5754:
5730:
5717:
5685:
5667:
5657:
5639:
5633:
5628:
5619:
5601:
5595:
5583:
5552:
5527:
5514:
5504:
5496:
5490:
5485:
5476:
5458:
5443:
5436:
5431:
5379:
5371:
5365:
5360:
5340:
5332:
5296:
5290:
5281:is a function of position only, and that
5265:
5259:
5238:
5232:
5208:
5196:
5175:
5169:
5148:
5132:
5120:
5118:
5106:{\displaystyle \mathbf {x} \mapsto x_{i}}
5097:
5085:
5083:
5058:
5052:
5047:
5028:
5014:
4983:
4977:
4972:
4958:
4884:
4878:
4868:
4858:
4848:
4842:
4834:
4828:
4823:
4813:
4808:
4802:
4790:
4784:
4728:
4722:
4717:
4710:
4701:
4691:
4675:
4662:
4637:
4621:
4608:
4595:
4584:
4561:
4560:
4539:
4529:
4522:
4460:
4439:
4429:
4412:
4407:
4395:
4385:
4377:
4368:
4350:
4340:
4335:
4332:
4312:
4302:
4300:
4255:
4249:
4229:
4194:
4173:
4159:
4151:
4129:
4127:
4104:
4102:
4038:
4032:
4022:
4012:
4002:
3996:
3988:
3982:
3977:
3967:
3962:
3956:
3948:
3872:
3867:
3860:
3855:
3848:
3830:
3825:
3816:
3795:
3790:
3781:
3757:
3747:
3738:
3717:
3707:
3698:
3656:
3632:
3625:
3500:
3499:
3475:
3459:
3445:
3437:
3415:
3413:
3390:
3388:
3291:
3290:
3274:
3256:
3250:
3239:
3238:
3225:
3207:
3201:
3190:
3189:
3176:
3158:
3152:
3141:
3140:
3128:
3114:
3112:
3051:
3046:
3035:
3021:
3002:
3000:
2962:
2947:
2933:
2919:
2911:
2870:
2849:
2841:
2832:
2814:
2805:
2797:
2788:
2770:
2761:
2753:
2744:
2726:
2706:
2686:
2676:
2656:
2646:
2626:
2611:
2591:
2574:
2572:
2508:
2507:
2483:
2455:
2453:
2409:
2404:
2398:
2393:
2388:
2382:
2376:
2312:
2306:
2301:
2296:
2290:
2285:
2264:
2258:
2253:
2248:
2242:
2237:
2223:
2215:
2201:
2195:
2190:
2185:
2179:
2144:
2136:
2117:
2115:
2107:
2084:
2078:
2073:
2068:
2062:
2057:
2035:
2034:
2010:
1982:
1981:
1970:
1968:
1942:
1936:
1931:
1926:
1920:
1914:
1886:
1880:
1875:
1870:
1864:
1858:
1836:
1830:
1825:
1820:
1814:
1809:
1795:
1787:
1773:
1767:
1762:
1757:
1751:
1746:
1719:
1714:
1706:
1696:
1686:
1684:
1676:
1666:
1614:
1604:
1602:
1594:
1586:
1575:
1573:
1537:
1531:
1525:
1499:
1493:
1487:
1388:
1364:
1350:
1326:
1312:
1288:
1260:
1258:
1118:
1110:
1104:
1096:
1090:
1082:
1075:
1068:
1061:
1046:
1033:
1020:
961:
960:
930:
929:
901:
893:
886:
881:
875:
840:
839:
830:
824:
795:
794:
776:
774:
738:
732:
726:
696:
690:
684:
660:
654:
649:
644:
638:
633:
619:
611:
594:
541:
535:
530:
525:
519:
513:
467:
459:
453:
445:
439:
431:
424:
417:
403:
397:
392:
387:
381:
375:
215:
209:
203:
180:
178:
158:
156:
136:
130:
124:
8063:
8061:
8059:
8057:
7852:Encyclopaedia of Physics (2nd Edition),
5321:. Multiplying the Boltzmann equation by
5191:is the particle velocity vector. Define
8467:"Theme issue 'Hilbert's sixth problem'"
7800:
195:(thus occupying a very small region of
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,
4090:Simplifications to the collision term
3943:are the momenta after the collision,
3921:for convenience) before a collision,
7:
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
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:
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:
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:
5966:{\displaystyle A=m(v_{i})^{0}=m}
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:
8289:10.1090/S0894-0347-2011-00697-8
7904:& Robert M. Strain (2010).
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
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:MaxwellâBoltzmann distribution
1:
5227:as some function of momentum
1449:The force and diffusion terms
296:for each position coordinate
265:integro-differential equation
255:convectionâdiffusion 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
8243:10.1016/j.physa.2009.01.009
8208:10.1016/j.physa.2008.12.023
7769:WilliamsâBoltzmann equation
7759:The VlasovâPoisson equation
7291:quantum Boltzmann equations
4937:Applications and extensions
3377:
3365:
2367:
1205:for particle 2, etc. up to
717:and momenta lying within a
331:: a point in this space is
151:) centered at the position
32:Boltzmann's entropy formula
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:
8885:
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
18:Boltzmann's equation
9158:Condensed matter physics
9141:Statistical field theory
8637:Harris, Stewart (1971).
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:NavierâStokes 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:StefanâBoltzmann 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): 5744â5749.
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:MaxwellâBoltzmann
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:
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): 1217â1245.
8381:
8375:
8374:
8364:
8340:
8334:
8333:
8331:
8307:
8301:
8300:
8282:
8262:
8256:
8253:
8247:
8246:
8218:
8212:
8211:
8194:(7): 1079â1104.
8183:
8177:
8174:
8168:
8167:
8141:
8132:(3): 1119â1124.
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
8782:
8745:
8705:
8668:
8651:
8636:
8633:
8628:
8576:
8575:
8571:
8545:
8544:
8540:
8524:
8523:
8519:
8510:
8508:
8500:
8499:
8492:
8465:
8464:
8460:
8444:
8440:
8427:
8423:
8383:
8382:
8378:
8342:
8341:
8337:
8322:(3): 996â1015.
8309:
8308:
8304:
8264:
8263:
8259:
8254:
8250:
8220:
8219:
8215:
8185:
8184:
8180:
8175:
8171:
8123:
8122:
8115:
8108:
8095:
8094:
8087:
8080:
8067:
8066:
8055:
8028:Physical Review
8025:
8024:
8020:
8007:
8000:
7993:
7972:
7971:
7967:
7900:
7899:
7895:
7880:10.2307/1971423
7865:
7864:
7860:
7851:
7840:
7830:
7807:
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:
7493:
7489:
7481:
7469:
7459:
7447:
7437:
7421:
7417:
7409:
7397:
7359:
7354:
7353:
7331:
7287:
7256:
7209:
7175:
7170:
7169:
7163:
7157:
7147:
7088:
7087:
7080:
7048:
7035:
7019:
7006:
6990:
6977:
6940:
6935:
6934:
6907:
6894:
6878:
6865:
6833:
6832:
6800:
6790:
6771:
6758:
6742:
6729:
6719:
6709:
6681:
6662:
6658:
6637:
6627:
6593:
6583:
6582:
6555:
6544:
6534:
6533:
6512:
6502:
6495:
6482:
6481:
6478:
6438:
6425:
6409:
6396:
6371:
6366:
6365:
6333:
6311:
6298:
6288:
6269:
6265:
6244:
6228:
6218:
6217:
6190:
6177:
6167:
6150:
6149:
6146:
6117:
6101:
6096:
6095:
6067:
6066:
6033:
6014:
6010:
5989:
5979:
5978:
5947:
5937:
5920:
5919:
5916:
5890:
5885:
5884:
5863:
5858:
5857:
5847:
5841:
5785:
5779:
5761:
5740:
5732:
5726:
5725:
5714:
5713:
5681:
5677:
5669:
5663:
5653:
5629:
5615:
5611:
5603:
5591:
5580:
5579:
5548:
5523:
5519:
5486:
5472:
5468:
5460:
5439:
5438:
5428:
5427:
5384:
5361:
5350:
5342:
5329:
5328:
5292:
5287:
5286:
5261:
5256:
5255:
5234:
5229:
5228:
5204:
5193:
5192:
5171:
5166:
5165:
5144:
5128:
5115:
5114:
5093:
5080:
5079:
5048:
5011:
5010:
5004:
4973:
4955:
4954:
4950:
4944:
4939:
4910:
4905:
4869:
4867:
4849:
4847:
4822:
4807:
4786:
4781:
4780:
4770:
4754:
4729:
4718:
4697:
4687:
4633:
4617:
4604:
4546:
4535:
4531:
4525:
4524:
4519:
4518:
4508:
4494:
4487:
4482:
4446:
4435:
4431:
4425:
4424:
4406:
4402:
4391:
4387:
4364:
4346:
4334:
4319:
4308:
4304:
4297:
4296:
4290:
4280:
4277:
4251:
4246:
4245:
4226:
4225:
4190:
4169:
4161:
4114:
4106:
4099:
4098:
4092:
4080:
4073:
4058:
4023:
4021:
4003:
4001:
3976:
3961:
3945:
3944:
3941:
3933:
3930:
3922:
3911:
3903:
3900:
3892:
3866:
3856:
3824:
3789:
3748:
3746:
3708:
3706:
3642:
3634:
3628:
3627:
3622:
3621:
3618:molecular chaos
3613:Stosszahlansatz
3603:
3598:
3592:distributions.
3574:
3568:
3562:Vlasov equation
3548:
3529:
3485:
3477:
3471:
3470:
3455:
3447:
3400:
3392:
3385:
3384:
3370:
3361:
3359:Final statement
3346:
3338:
3335:
3327:
3324:
3316:
3312:
3286:
3270:
3266:
3258:
3237:
3221:
3217:
3209:
3188:
3172:
3168:
3160:
3139:
3124:
3116:
3109:
3108:
3098:
3090:
3079:
3059:
3058:
3031:
3023:
2972:
2964:
2953:
2952:
2929:
2921:
2880:
2872:
2861:
2860:
2845:
2828:
2824:
2816:
2801:
2784:
2780:
2772:
2757:
2740:
2736:
2728:
2725:
2721:
2696:
2688:
2666:
2658:
2636:
2628:
2625:
2621:
2601:
2593:
2584:
2569:
2568:
2556:
2541:
2493:
2485:
2479:
2478:
2465:
2457:
2450:
2449:
2435:
2427:
2394:
2378:
2373:
2372:
2360:
2349:
2338:
2318:
2317:
2302:
2286:
2270:
2269:
2254:
2238:
2191:
2175:
2106:
2102:
2090:
2089:
2074:
2058:
2020:
2012:
2006:
2005:
1998:
1977:
1965:
1964:
1932:
1916:
1911:
1910:
1876:
1860:
1855:
1854:
1826:
1810:
1763:
1747:
1675:
1671:
1663:
1662:
1654:
1632:
1570:
1569:
1559:
1553:
1527:
1522:
1521:
1515:
1489:
1484:
1483:
1482:within element
1477:
1473:
1464:
1454:
1451:
1433:
1427:
1421:
1374:
1366:
1360:
1359:
1336:
1328:
1322:
1321:
1298:
1290:
1284:
1283:
1270:
1262:
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:
1114:
1100:
1086:
1042:
1029:
1016:
916:
915:
871:
820:
783:
771:
770:
760:
754:
728:
723:
722:
712:
686:
681:
680:
650:
634:
591:
590:
571:
563:
557:
531:
515:
510:
509:
503:
497:
489:
463:
449:
435:
393:
377:
372:
371:
370:") is written
354:
350:
346:
332:
324:
319:
315:
310:
306:
301:
278:
273:
205:
200:
199:
175:
174:
153:
152:
126:
121:
120:
50:
43:
28:
23:
22:
15:
12:
11:
5:
9249:
9247:
9239:
9238:
9233:
9228:
9223:
9218:
9213:
9208:
9203:
9193:
9192:
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:
8858:
8856:
8855:
8848:
8841:
8833:
8827:
8826:
8821:
8814:
8813:External links
8811:
8810:
8809:
8791:(2): 321â366.
8780:
8743:
8703:
8665:
8664:
8649:
8632:
8629:
8627:
8626:
8569:
8558:(2): 333â357.
8538:
8517:
8490:
8477:(2118). 2018.
8458:
8438:
8421:
8399:10.1.1.46.2853
8376:
8335:
8302:
8257:
8248:
8229:(9): 1818â34.
8213:
8178:
8169:
8113:
8106:
8085:
8078:
8053:
8034:(3): 511â525.
8018:
7998:
7991:
7965:
7893:
7874:(2): 321â366.
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:ChapmanâEnskog
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:
24:
14:
13:
10:
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:
8812:
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:FokkerâPlanck
8652:
8646:
8642:
8641:
8635:
8634:
8630:
8622:
8618:
8613:
8608:
8604:
8600:
8596:
8592:
8588:
8584:
8580:
8573:
8570:
8565:
8561:
8557:
8553:
8549:
8542:
8539:
8533:
8528:
8521:
8518:
8507:
8503:
8497:
8495:
8491:
8485:
8480:
8476:
8472:
8468:
8462:
8459:
8456:
8455:0-521-40844-X
8452:
8448:
8442:
8439:
8435:
8431:
8425:
8422:
8417:
8413:
8409:
8405:
8400:
8395:
8391:
8387:
8380:
8377:
8372:
8368:
8363:
8358:
8354:
8350:
8346:
8339:
8336:
8330:
8325:
8321:
8317:
8313:
8306:
8303:
8298:
8294:
8290:
8286:
8281:
8276:
8272:
8268:
8261:
8258:
8252:
8249:
8244:
8240:
8236:
8232:
8228:
8224:
8217:
8214:
8209:
8205:
8201:
8197:
8193:
8189:
8182:
8179:
8173:
8170:
8165:
8161:
8157:
8153:
8149:
8145:
8140:
8135:
8131:
8127:
8126:Phys. Lett. B
8120:
8118:
8114:
8109:
8103:
8099:
8092:
8090:
8086:
8081:
8075:
8071:
8064:
8062:
8060:
8058:
8054:
8049:
8045:
8041:
8037:
8033:
8029:
8022:
8019:
8015:
8014:0-07-051400-3
8011:
8005:
8003:
7999:
7994:
7988:
7984:
7979:
7978:
7969:
7966:
7961:
7957:
7952:
7947:
7942:
7937:
7933:
7929:
7924:
7919:
7915:
7911:
7907:
7903:
7897:
7894:
7889:
7885:
7881:
7877:
7873:
7869:
7862:
7859:
7855:
7849:
7847:
7845:
7843:
7839:
7836:
7831:
7825:
7821:
7817:
7813:
7812:
7804:
7801:
7795:
7790:
7787:
7785:
7782:
7780:
7777:
7775:
7772:
7770:
7767:
7765:
7762:
7760:
7757:
7755:
7752:
7751:
7746:
7744:
7742:
7738:
7737:morphogenesis
7734:
7730:
7726:
7721:
7719:
7714:
7709:
7707:
7702:
7700:
7696:
7687:
7685:
7683:
7679:
7675:
7671:
7667:
7664:in powers of
7663:
7659:
7654:
7651:
7647:
7643:
7639:
7634:
7628:
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:
7328:
7326:
7324:
7320:
7316:
7312:
7308:
7304:
7300:
7296:
7292:
7284:
7282:
7269:
7248:
7240:
7234:
7229:
7219:
7213:
7201:
7166:
7160:
7155:
7150:
7133:
7127:
7116:
7110:
7085:
7077:
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
6475:
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:
6143:
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
5913:
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:
4941:
4936:
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:
4274:
4272:
4256:
4252:
4231:
4222:
4209:
4203:
4200:
4195:
4191:
4184:
4181:
4165:
4156:
4148:
4145:
4139:
4134:
4124:
4118:
4110:
4096:
4089:
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:
3600:
3595:
3593:
3591:
3590:BoseâEinstein
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:
3358:
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:
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1532:
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1227:for particle
1224:
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1152:deterministic
1148:
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1091:
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360:parameterized
355:
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316:
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291:
287:
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275:
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268:
266:
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9134:Applications
9085:size scaling
9022:
8788:
8785:Ann. of Math
8784:
8751:
8747:
8717:(1): 17â34.
8714:
8710:
8674:
8670:
8639:
8586:
8582:
8572:
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8509:. Retrieved
8505:
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8097:
8069:
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8021:
7976:
7968:
7913:
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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:
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7158:
7148:
7081:
6830:
6581:
6479:
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6216:
6147:
6144:First moment
5917:
5856:of velocity
5848:
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5839:
5712:
5578:
5426:
5327:
5322:
5285:is zero for
5282:
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4953:is given by
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3353:unit vectors
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2553:differential
2550:
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2361:
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558:
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311:
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285:
284:and momenta
281:
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9125:von Neumann
8995:force field
8987:percolation
8754:(1): 1â16.
8677:(1): 1â16.
8355:: 215â240.
7640:(including
7619:dark matter
7307:dark matter
4078:solid angle
3586:FermiâDirac
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
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6000:ρ
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3644:∂
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3607:Boltzmann
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3350:Cartesian
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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
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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
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5854:moments
4068:is the
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3103:is the
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508:within
298:x, y, z
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3891:where
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3089:where
2348:where
1463:force
753:about
711:about
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9168:chaos
9120:RĂ©nyi
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
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1314:force
496:have
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88:with
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9050:and
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1236:mass
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502:and
235:heat
104:and
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8659:or
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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:Ξ
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:ĂȘ
3333:y
3329:ĂȘ
3322:x
3318:ĂȘ
3313:â
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:â
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:Î
2430:t
2428:Î
2414:t
2406:p
2400:3
2396:d
2390:r
2384:3
2380:d
2368:1
2362:f
2352:f
2350:Î
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:Î
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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