In the kinetic theory of gases in physics , the molecular chaos hypothesis (also called Stosszahlansatz in the writings of Paul and Tatiana Ehrenfest ) is the assumption that the velocities of colliding particles are uncorrelated, and independent of position. This means the probability that a pair of particles with given velocities will collide can be calculated by considering each particle separately and ignoring any correlation between the probability for finding one particle with velocity v and probability for finding another velocity v ' in a small region δr . James Clerk Maxwell introduced this approximation in 1867 although its origins can be traced back to his first work on the kinetic theory in 1860.
17-514: The assumption of molecular chaos is the key ingredient that allows proceeding from the BBGKY hierarchy to Boltzmann's equation , by reducing the 2-particle distribution function showing up in the collision term to a product of 1-particle distributions. This in turn leads to Boltzmann's H-theorem of 1872, which attempted to use kinetic theory to show that the entropy of a gas prepared in a state of less than complete disorder must inevitably increase, as
34-657: Is given by the Liouville equation for the probability density function f N = f N ( q 1 … q N , p 1 … p N , t ) {\displaystyle f_{N}=f_{N}(\mathbf {q} _{1}\dots \mathbf {q} _{N},\mathbf {p} _{1}\dots \mathbf {p} _{N},t)} in 6 N -dimensional phase space (3 space and 3 momentum coordinates per particle) where q i , p i {\displaystyle \mathbf {q} _{i},\mathbf {p} _{i}} are
51-681: Is known as Boltzmann–Grad limit . Schematically, the Liouville equation gives us the time evolution for the whole N {\displaystyle N} -particle system in the form D f N = 0 {\displaystyle Df_{N}=0} , which expresses an incompressible flow of the probability density in phase space. We then define the reduced distribution functions incrementally by integrating out another particle's degrees of freedom f s ∼ ∫ f s + 1 {\textstyle f_{s}\sim \int f_{s+1}} . An equation in
68-471: Is modeled based on the molecular chaos hypothesis ( Stosszahlansatz ). In fact, in the Boltzmann equation f 2 = f 2 ( p 1 , p 2 , t ) {\displaystyle f_{2}=f_{2}(\mathbf {p} _{1},\mathbf {p_{2}} ,t)} is the collision integral. This limiting process of obtaining Boltzmann equation from Liouville equation
85-438: Is the pair potential for interaction between particles, and Φ ext ( q i ) {\displaystyle \Phi ^{\text{ext}}(\mathbf {q} _{i})} is the external-field potential. By integration over part of the variables, the Liouville equation can be transformed into a chain of equations where the first equation connects the evolution of one-particle probability density function with
102-486: The principle of maximum entropy in order to generalize the ansatz to higher-order distribution functions. This article about statistical mechanics is a stub . You can help Misplaced Pages by expanding it . BBGKY hierarchy In statistical physics , the Bogoliubov–Born–Green–Kirkwood–Yvon ( BBGKY ) hierarchy (sometimes called Bogoliubov hierarchy ) is a set of equations describing
119-463: The BBGKY chain is a common starting point for many applications of kinetic theory that can be used for derivation of classical or quantum kinetic equations. In particular, truncation at the first equation or the first two equations can be used to derive classical and quantum Boltzmann equations and the first order corrections to the Boltzmann equations. Other approximations, such as the assumption that
136-411: The BBGKY hierarchy tells us that the time evolution for such a f s {\displaystyle f_{s}} is consequently given by a Liouville-like equation, but with a correction term that represents force-influence of the N − s {\displaystyle N-s} suppressed particles The problem of solving the BBGKY hierarchy of equations is as hard as solving
153-643: The Liouville equation over the variables q s + 1 … q N , p s + 1 … p N {\displaystyle \mathbf {q} _{s+1}\dots \mathbf {q} _{N},\mathbf {p} _{s+1}\dots \mathbf {p} _{N}} . The problem with the above equation is that it is not closed. To solve f s {\displaystyle f_{s}} , one has to know f s + 1 {\displaystyle f_{s+1}} , which in turn demands to solve f s + 2 {\displaystyle f_{s+2}} and all
170-409: The density probability function depends only on the relative distance between the particles or the assumption of the hydrodynamic regime, can also render the BBGKY chain accessible to solution. s -particle distribution functions were introduced in classical statistical mechanics by J. Yvon in 1935. The BBGKY hierarchy of equations for s -particle distribution functions was written out and applied to
187-529: The dynamics of a system of a large number of interacting particles. The equation for an s -particle distribution function (probability density function) in the BBGKY hierarchy includes the ( s + 1)-particle distribution function, thus forming a coupled chain of equations. This formal theoretic result is named after Nikolay Bogolyubov , Max Born , Herbert S. Green , John Gamble Kirkwood , and Jacques Yvon [ fr ] . The evolution of an N -particle system in absence of quantum fluctuations
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#1732775315514204-414: The gas molecules are allowed to collide. This drew the objection from Loschmidt that it should not be possible to deduce an irreversible process from time-symmetric dynamics and a time-symmetric formalism: something must be wrong ( Loschmidt's paradox ). The resolution (1895) of this paradox is that the velocities of two particles after a collision are no longer truly uncorrelated. By asserting that it
221-587: The original Liouville equation, but approximations for the BBGKY hierarchy (which allow truncation of the chain into a finite system of equations) can readily be made. The merit of these equations is that the higher distribution functions f s + 2 , f s + 3 , … {\displaystyle f_{s+2},f_{s+3},\dots } affect the time evolution of f s {\displaystyle f_{s}} only implicitly via f s + 1 . {\displaystyle f_{s+1}.} Truncation of
238-412: The position and momentum for i {\displaystyle i} -th particle with mass m {\displaystyle m} , and the net force acting on the i {\displaystyle i} -th particle is where Φ i j ( q i , q j ) {\displaystyle \Phi _{ij}(\mathbf {q} _{i},\mathbf {q} _{j})}
255-420: The two-particle probability density function, second equation connects the two-particle probability density function with the three-particle probability density function, and generally the s -th equation connects the s -particle probability density function with the ( s + 1)-particle probability density function: The equation above for s -particle distribution function is obtained by integration of
272-754: The way back to the full Liouville equation. However, one can solve f s {\displaystyle f_{s}} , if f s + 1 {\displaystyle f_{s+1}} could be modeled. One such case is the Boltzmann equation for f 1 ( q 1 , p 1 , t ) {\displaystyle f_{1}(\mathbf {q} _{1},\mathbf {p} _{1},t)} , where f 2 ( q 1 , q 2 , p 1 , p 2 , t ) {\displaystyle f_{2}(\mathbf {q} _{1},\mathbf {q} _{2},\mathbf {p} _{1},\mathbf {p} _{2},t)}
289-466: Was acceptable to ignore these correlations in the population at times after the initial time, Boltzmann had introduced an element of time asymmetry through the formalism of his calculation. Though the Stosszahlansatz is usually understood as a physically grounded hypothesis, it was recently highlighted that it could also be interpreted as a heuristic hypothesis. This interpretation allows using
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