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Maxwell–BoltzmannProbability density function

Cumulative distribution function

ParametersSupport

PDF

CDF

where erf is the error functionMean

Maxwell–Boltzmann distributionFrom Wikipedia, the free encyclopedia

In statistics the Maxwell–Boltzmann distribution is aparticular probability distribution named after JamesClerk Maxwell and Ludwig Boltzmann. It was firstdefined and used in physics (in particular in statisticalmechanics) for describing particle speeds in idealizedgases where the particles move freely inside astationary container without interacting with oneanother, except for very brief collisions in which theyexchange energy and momentum with each other orwith their thermal environment. Particle in this contextrefers to gaseous particles (atoms or molecules), andthe system of particles is assumed to have reachedthermodynamic equilibrium.[1] While the distributionwas first derived by Maxwell in 1860 on heuristicgrounds,[2] Boltzmann later carried out significantinvestigations into the physical origins of thisdistribution.

A particle speed probability distribution indicateswhich speeds are more likely: a particle will have aspeed selected randomly from the distribution, and ismore likely to be within one range of speeds thananother. The distribution depends on the temperatureof the system and the mass of the particle.[3] TheMaxwell–Boltzmann distribution applies to theclassical ideal gas, which is an idealization of realgases. In real gases, there are various effects (e.g., vander Waals interactions, vortical flow, relativistic speedlimits, and quantum exchange interactions) that maketheir speed distribution sometimes very different fromthe Maxwell–Boltzmann form. However, rarefiedgases at ordinary temperatures behave very nearly likean ideal gas and the Maxwell speed distribution is anexcellent approximation for such gases. Thus, it formsthe basis of the kinetic theory of gases, which providesa simplified explanation of many fundamental gaseousproperties, including pressure and diffusion.[4]

Contents

1 Distribution function

2 Typical speeds

3 Derivation and related distributions

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Mode

Variance

Skewness

Ex.kurtosis

Entropy

The speed probability density functions of the speeds of a fewnoble gases at a temperature of 298.15 K (25 °C). The y­axisis in s/m so that the area under any section of the curve (whichrepresents the probability of the speed being in that range) isdimensionless.

3.1 Distribution for the momentumvector

3.2 Distribution for the energy

3.3 Distribution for the velocity vector

4 See also

5 References

6 Further reading

7 External links

Distribution function

The Maxwell–Boltzmann distribution is thefunction

where is the particle mass and is theproduct of Boltzmann's constant andthermodynamic temperature.

This probability density function gives theprobability, per unit speed, of finding theparticle with a speed near . This equation issimply the Maxwell distribution (given in theinfobox) with distribution parameter

. In probability theory the

Maxwell–Boltzmann distribution is a chidistribution with three degrees of freedom andscale parameter .

The simplest ordinary differential equation satisfied by the distribution is:

or in unitless presentation:

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Typical speeds

The mean speed, most probable speed (mode), and root­mean­square can be obtained from properties of theMaxwell distribution.

The most probable speed, vp, is the speed most likely to be possessed by any molecule (of the samemass m) in the system and corresponds to the maximum value or mode of f(v). To find it, we calculatethe derivative df/dv, set it to zero and solve for v:

which yields:

where R is the gas constant and M = NA m is the molar mass of the substance.

For diatomic nitrogen (N2, the primary component of air) at room temperature (300 K), this gives m/s

The mean speed is the expected value of the speed distribution

The root mean square speed is the second­order moment of speed:

The typical speeds are related as follows:

Derivation and related distributions

The original derivation in 1860 by James Clerk Maxwell was an argument based on demanding certainsymmetries in the speed distribution function.[2] After Maxwell, Ludwig Boltzmann in 1872 derived thedistribution on more mechanical grounds by using the assumptions of his kinetic theory, and showed thatgases should over time tend toward this distribution, due to collisions (see H­theorem). He later (1877)derived the distribution again under the framework of statistical thermodynamics. The derivations in thissection are along the lines of Boltzmann's 1877 derivation, starting with result known as Maxwell–Boltzmann statistics (from statistical thermodynamics). Maxwell–Boltzmann statistics gives the averagenumber of particles found in a given single­particle microstate, under certain assumptions:[1][5]

(1)

where:

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i and j are indices (or labels) of the single­particle micro states,Ni is the average number of particles in the single­particle microstate i,N is the total number of particles in the system,Ei is the energy of microstate i,T is the equilibrium temperature of the system,k is the Boltzmann constant.

The assumptions of this equation are that the particles do not interact, and that they are classical; this meansthat each particle's state can be considered independently from the other particles' states. Additionally, theparticles are assumed to be in thermal equilibrium. The denominator in Equation (1) is simply a normalizingfactor so that the Ni/N add up to 1 — in other words it is a kind of partition function (for the single­particlesystem, not the usual partition function of the entire system).

Because velocity and speed are related to energy, Equation (1) can be used to derive relationships betweentemperature and the speeds of gas particles. All that is needed is to discover the density of microstates inenergy, which is determined by dividing up momentum space into equal sized regions.

Distribution for the momentum vector

The potential energy is taken to be zero, so that all energy is in the form of kinetic energy. The relationshipbetween kinetic energy and momentum for massive non­relativistic particles is

(2)

where p2 is the square of the momentum vector p = [px, py, pz]. We may therefore rewrite Equation (1) as:

(3)

where Z is the partition function, corresponding to the denominator in Equation (1). Here m is the molecularmass of the gas, T is the thermodynamic temperature and k is the Boltzmann constant. This distribution ofNi/N is proportional to the probability density function fp for finding a molecule with these values ofmomentum components, so:

(4)

The normalizing constant c, can be determined by recognizing that the probability of a molecule having somemomentum must be 1. Therefore the integral of equation (4) over all px, py, and pz must be 1.

It can be shown that:

(5)

Substituting Equation (5) into Equation (4) gives:

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(6)

The distribution is seen to be the product of three independent normally distributed variables , , and ,with variance . Additionally, it can be seen that the magnitude of momentum will be distributed as aMaxwell–Boltzmann distribution, with . The Maxwell–Boltzmann distribution for themomentum (or equally for the velocities) can be obtained more fundamentally using the H­theorem atequilibrium within the kinetic theory framework.

Distribution for the energy

The energy distribution is found imposing

(7)

where is the infinitesimal phase­space volume of momenta corresponding to the energy interval .Making use of the spherical symmetry of the energy­momentum dispersion relation , thiscan be expressed in terms of as

(8)

Using then (8) in (7), and expressing everything in terms of the energy , we get

and finally

(9)

Since the energy is proportional to the sum of the squares of the three normally distributed momentumcomponents, this distribution is a gamma distribution; in particular, it is a chi­squared distribution with threedegrees of freedom.

By the equipartition theorem, this energy is evenly distributed among all three degrees of freedom, so that theenergy per degree of freedom is distributed as a chi­squared distribution with one degree of freedom:[6]

where is the energy per degree of freedom. At equilibrium, this distribution will hold true for any numberof degrees of freedom. For example, if the particles are rigid mass dipoles of fixed dipole moment, they willhave three translational degrees of freedom and two additional rotational degrees of freedom. The energy ineach degree of freedom will be described according to the above chi­squared distribution with one degree offreedom, and the total energy will be distributed according to a chi­squared distribution with five degrees offreedom. This has implications in the theory of the specific heat of a gas.

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The Maxwell–Boltzmann distribution can also be obtained by considering the gas to be a type of quantumgas.

Distribution for the velocity vector

Recognizing that the velocity probability density fv is proportional to the momentum probability densityfunction by

and using p = mv we get

which is the Maxwell–Boltzmann velocity distribution. The probability of finding a particle with velocity inthe infinitesimal element [dvx, dvy, dvz] about velocity v = [vx, vy, vz] is

Like the momentum, this distribution is seen to be the product of three independent normally distributed

variables , , and , but with variance . It can also be seen that the Maxwell–Boltzmann velocity

distribution for the vector velocity [vx, vy, vz] is the product of the distributions for each of the threedirections:

where the distribution for a single direction is

Each component of the velocity vector has a normal distribution with mean and

standard deviation , so the vector has a 3­dimensional normal distribution, a

particular kind of multivariate normal distribution, with mean and standard deviation

.

The Maxwell–Boltzmann distribution for the speed follows immediately from the distribution of the velocityvector, above. Note that the speed is

and the volume element in spherical coordinates

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where and are the "course" (azimuth of the velocity vector) and "path angle" (elevation angle of thevelocity vector). Integration of the normal probability density function of the velocity, above, over the course(from 0 to ) and path angle (from 0 to ), with substitution of the speed for the sum of the squares of thevector components, yields the speed distribution.

See also

Maxwell–Boltzmann statisticsMaxwell­Jüttner distributionBoltzmann distributionBoltzmann factorRayleigh distributionKinetic theory

References1. Statistical Physics (2nd Edition), F. Mandl, Manchester Physics, John Wiley & Sons, 2008, ISBN 97804719153312. See:

Maxwell, J.C. (1860) "Illustrations of the dynamical theory of gases. Part I. On the motions and collisionsof perfectly elastic spheres," (http://books.google.com/books?id=­YU7AQAAMAAJ&pg=PA19#v=onepage&q&f=false) Philosophical Magazine, 4th series, 19 : 19­32.Maxwell, J.C. (1860) "Illustrations of the dynamical theory of gases. Part II. On the process of diffusion oftwo or more kinds of moving particles among one another," (http://books.google.com/books?id=DIc7AQAAMAAJ&pg=PA21#v=onepage&q&f=false) Philosophical Magazine, 4th series, 20 : 21­37.

3. University Physics – With Modern Physics (12th Edition), H.D. Young, R.A. Freedman (Original edition),Addison­Wesley (Pearson International), 1st Edition: 1949, 12th Edition: 2008, ISBN (10­) 0­321­50130­6, ISBN(13­) 978­0­321­50130­1

4. Encyclopaedia of Physics (2nd Edition), R.G. Lerner, G.L. Trigg, VHC publishers, 1991, ISBN(Verlagsgesellschaft) 3­527­26954­1, ISBN (VHC Inc.) 0­89573­752­3

5. McGraw Hill Encyclopaedia of Physics (2nd Edition), C.B. Parker, 1994, ISBN 0­07­051400­36. Laurendeau, Normand M. (2005). Statistical thermodynamics: fundamentals and applications. Cambridge

University Press. p. 434. ISBN 0­521­84635­8., Appendix N, page 434 (http://books.google.com/books?id=QF6iMewh4KMC&pg=PA434)

Further readingPhysics for Scientists and Engineers ­ with Modern Physics (6th Edition), P. A. Tipler, G. Mosca,Freeman, 2008, ISBN 0­7167­8964­7Thermodynamics, From Concepts to Applications (2nd Edition), A. Shavit, C. Gutfinger, CRC Press(Taylor and Francis Group, USA), 2009, ISBN (13­) 978­1­4200­7368­3Chemical Thermodynamics, D.J.G. Ives, University Chemistry, Macdonald Technical and Scientific,1971, ISBN 0­356­03736­3Elements of Statistical Thermodynamics (2nd Edition), L.K. Nash, Principles of Chemistry, Addison­Wesley, 1974, ISBN 0­201­05229­6Ward, CA & Fang, G 1999, 'Expression for predicting liquid evaporation flux: Statistical rate theoryapproach', Physical Review E, vol. 59, no. 1, pp. 429­40.Rahimi, P & Ward, CA 2005, 'Kinetics of Evaporation: Statistical Rate Theory Approach', Int. J. ofThermodynamics, vol. 8, no. 9, pp. 1­14.

External links

"The Maxwell Speed Distribution"

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(http://demonstrations.wolfram.com/TheMaxwellSpeedDistribution/) from The WolframDemonstrations Project at Mathworld

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Categories: Continuous distributions Gases James Clerk Maxwell Normal distributionParticle distributions

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