FACTOID # 16: In the 2000 Presidential Election, Texas gave Ralph Nader the 3rd highest popular vote count of any US state.
 
 Home   Encyclopedia   Statistics   States A-Z   Flags   Maps   FAQ   About 
   
 
WHAT'S NEW
 

SEARCH ALL

FACTS & STATISTICS    Advanced view

Search encyclopedia, statistics and forums:

 

 

(* = Graphable)

 

 


Encyclopedia > Maxwell Speed Distribution

In the classical picture of an ideal gas, molecules bounce around at a variety of different velocities, never interacting with each other. Though this qualitative picture is obviously flawed (since molecules always do interact), it is a useful model for situations where the particle density is very low; in a more quantitative sense, this means that the particles themselves are very small when compared to the volume between them. An ideal gas or perfect gas is a hypothetical gas consisting of identical particles of negligible volume, with no intermolecular forces. ... In science, a molecule is the smallest particle of a pure chemical substance that still retains its chemical composition and properties. ...


Accordingly, we will want to know exactly how many of these molecules are moving around at a given speed. The Maxwell Speed Distribution (MSD) is a probability distribution describing the "spread" of these molecular speeds; it is derived, and therefore only valid, assuming that we're dealing with an ideal gas. Again, no gas is truly ideal, but our own atmosphere at STP behaves enough like the ideal situation that the MSD can be used. An ideal gas or perfect gas is a hypothetical gas consisting of identical particles of negligible volume, with no intermolecular forces. ... The TLA STP has several meanings: Standard Temperature and Pressure is a chemistry term for a specific controlled experimental or reaction condition. ...


Note that speed is a scalar quantity, describing how fast the particles are moving, regardless of direction; velocity also describes the direction that the particles are moving. Speed (symbol: v) is the rate of motion, or equivalently the rate of change of position, expressed as distance d moved per unit of time t. ... The velocity of an object is simply its speed in a particular direction. ...


It is elementary using statistical mechanics to find that the MSD must be proportional to the probability that a particle is moving at a given speed. Another important element is the fact that space is three dimensional, which implies that for any given speed, there are many possible velocity vectors. Statistical mechanics is the application of statistics, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ... The word vector means carrier in Latin; it is derived from the Latin verb vehere, which means to carry. ...


The probability of a molecule having a given speed can be found by using Boltzmann factors; considering the energy to be dependent only on the kinetic energy, we find that:



Here, m is the mass of the molecule, k is Boltzmann's constant, and T is the temperature. The Boltzmann constant (k or kB) is the physical constant relating temperature to energy. ...


In 3-dimensional velocity space, the velocity vectors corresponding to a given speed v live on the surface of a sphere with radius v. The larger v is, the bigger the sphere, and the more possible velocity vectors there are. So the number of possible velocity vectors for a given speed goes like the surface area of a sphere of radius v.



Multiplying these two functions together gives us the distribution, and normalizing this gives us the MSD in its entirety.



(Again, m is the mass of the molecule, k is Boltzmann's constant, and T is the temperature.) The Boltzmann constant (k or kB) is the physical constant relating temperature to energy. ...


As this formula is a normalized probability distribution, it gives the probability of a molecule having a speed between v and v + dv. If you want to find the probability of a particle to be between two different velocities v0 and v1, simply integrate this function with those numbers as the bounds.


Averages

There are three general methods for finding the "average" value of the speed of the Maxwell Speed Distribution.


Firstly, by differentiating the MSD and finding its maximum, we can determine the most probable speed. Calling this vmax, we find that:



Second, we can find the root mean square of the speed by finding the average value of v2. (Alternatively, and much simply, we can solve it by using the equipartition theorem.) Calling this vrms: In mathematics, the root mean square or rms is a statistical measure of the magnitude of a varying quantity. ... The Equipartition Theorem is a principle of classical (non-quantum) statistical mechanics which states that the internal energy of a system composed of a large number of particles will distribute itself evenly among each of the degrees of freedom allowed to the particles of the system. ...



Third and finally, we can find the average value of v from the MSD. Calling this bar{v}:


bar{v} = left ( frac{8 k T}{pi m} right )^{1/2}



Notice that:


These three different ways of understanding the average velocity, though not equivalent numerically, do not each describe different physics. They are each a different way of "book-keeping," if you will. It is simply very important to be consistent in which quantity being used, and to be clear which quantity is being used.

Image:Bvn-small.png Probability distributions
Univariate Multivariate
Discrete: Bernoulli | binomial | Boltzmann | compound Poisson | degenerate | degree | Gauss-Kuzmin | geometric | hypergeometric | logarithmic | negative binomial | parabolic fractal | Rademacher | Poisson | Skellam | uniform | Yule-Simon | zeta | Zipf | Zipf-Mandelbrot Ewens | multinomial
Continuous: Beta | Beta prime | Cauchy | chi-square | exponential | exponential power | F | fading | Fisher's z | Fisher-Tippett | Gamma | generalized extreme value | generalized hyperbolic | generalized inverse Gaussian | Hotelling's T-square | hyperbolic secant | hyper-exponential | hypoexponential | inverse chi-square | inverse gamma | Kumaraswamy | Landau | Laplace | Lévy | Lévy skew alpha-stable | logistic | log-normal | Maxwell-Boltzmann | Maxwell speed | normal (Gaussian) | Pareto | Pearson | polar | raised cosine | Rayleigh | relativistic Breit-Wigner | Rice | Student's t | triangular | type-1 Gumbel | type-2 Gumbel | uniform | Voigt | von Mises | Weibull | Wigner semicircle Dirichlet | matrix normal | multivariate normal | Wigner quasi | Wishart
Miscellaneous: Cantor | conditional | exponential family | infinitely divisible | location-scale family | marginal | maximum entropy | phase-type | posterior | prior | quasi | sampling
Edit

Image File history File links Bvn-small. ... In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ... A multivariate random variable or random vector is a vector X=(X1,...,Xn) whose components are scalar-valued random variables on the same probability space (Ω, P). ... In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ... In probability theory and statistics, the Bernoulli distribution, named after Swiss scientist James Bernoulli, is a discrete probability distribution, which takes value 1 with success probability and value 0 with failure probability . ... See binomial (disambiguation) for a list of other topics using that name. ... The Maxwell-Boltzmann distribution is a probability distribution with applications in physics and chemistry. ... In probability theory, a compound Poisson distribution is the probability distribution of a Poisson-distibuted number of independent identically-distributed random variables. ... In mathematics, a degenerate distribution is the probability distribution of a random variable which always has the same value. ... In the mathematical field of graph theory the degree distribution of a graph is a function describing the total number of vertices in a graph with a given degree (number of connections to other vertices). ... In mathematics, the Gauss-Kuzmin distribution gives the probability distribution of the occurrence of a given integer in the continued fraction expansion of an arbitrary real number. ... In probability theory and statistics, the geometric distribution is either of two discrete probability distributions: the probability distribution of the number X of Bernoulli trials needed to get one success, supported on the set { 1, 2, 3, ...}, or the probability distribution of the number Y = X âˆ’ 1 of failures before... In probability theory and statistics, the hypergeometric distribution is a discrete probability distribution that describes the number of successes in a sequence of n draws from a finite population without replacement. ... In probability and statistics, the logarithmic distribution (also known as the logarithmic series distribution) is a discrete probability distribution. ... In probability and statistics the negative binomial distribution is a discrete probability distribution. ... In the parabolic fractal distribution, the logarithm of the frequency or size of entities in a population is a quadratic polynomial of the logarithm of the rank. ... In probability theory and statistics, the Rademacher distribution is a discrete probability distribution. ... In probability theory and statistics, the Poisson distribution is a discrete probability distribution. ... The Skellam distribution is the discrete probability distribution of the difference N1 − N2 of two correlated or uncorrelated random variables N1 and N2 having Poisson distributions with different expected values μ1 and μ2. ... In probability theory and statistics, the discrete uniform distribution is a discrete probability distribution that can be characterized by saying that all values of a finite set of possible values are equally probable. ... In probability and statistics, the Yule-Simon distribution is a discrete probability distribution. ... In probability theory and statistics, the zeta distribution is a discrete probability distribution. ... Originally, Zipfs law stated that, in a corpus of natural language utterances, the frequency of any word is roughly inversely proportional to its rank in the frequency table. ... In probability theory and statistics, the Zipf-Mandelbrot law is a discrete probability distribution. ... In population genetics, Ewenss sampling formula, introduced by Warren Ewens, states that under certain conditions (specified below), if a random sample of n gametes is taken from a population and classified according to the gene at a particular locus then the probability that there are a1 alleles represented once... In probability theory, the multinomial distribution is a generalization of the binomial distribution. ... In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ... In probability theory and statistics, the beta distribution is a continuous probability distribution with the probability density function (pdf) defined on the interval [0, 1]: where α and β are parameters that must be greater than zero and B is the beta function. ... A Beta Prime Distribution is a distribution with probability function: where is a Beta function. ... The Cauchy-Lorentz distribution, named after Augustin Cauchy, is a continuous probability distribution with probability density function where x0 is the location parameter, specifying the location of the peak of the distribution, and γ is the scale parameter which specifies the half-width at half-maximum (HWHM). ... In probability theory and statistics, the chi-square distribution (also chi-squared distribution), or χ2  distribution, is one of the theoretical probability distributions most widely used in inferential statistics, i. ... In probability theory and statistics, the exponential distributions are a class of continuous probability distribution. ... The exponential power distribution, also known as the generalized error distribution, takes a scale parameter a and exponent b. ... In statistics and probability, the F-distribution is a continuous probability distribution. ... In telecommunication, a fading distribution is the probability distribution that signal fading will exceed a given value relative to a specified reference level. ... Fishers z-distribution is the distribution of half the logarithm of a F distribution variate: It was first described by Ronald Fisher in a paper delivered at the International Mathematical Congress of 1924 in Toronto, entitled On a distribution yielding the error functions of several well-known statistics. Nowadays... In probability theory and statistics the Gumbel distribution is used to find the minimum (or the maximum) of a number of samples of various distributions. ... In probability theory and statistics, the gamma distribution is a continuous probability distribution. ... In probability theory and statistics, the generalized extreme value distribution (GEV) is a family of continuous probability distributions developed within extreme value theory to combine the Gumbel, Fréchet and Weibull families also known as type I, II and III extreme value distributions. ... The generalised hyperbolic distribution is a continuous probability distribution defined by the probability density function where is the modified Bessel function of the second kind. ... In probability theory, the Generalized inverse Gaussian distribution (GIG) is a probability distribution with probability density function It is used extensively in geostatistics, statistical linguistics, finance, etc. ... In statistics, Hotellings T-square statistic, named for Harold Hotelling, is a generalization of Students t statistic that is used in multivariate hypothesis testing. ... In probability theory and statistics, the hyperbolic secant distribution is a continuous probability distribution whose probability density function and characteristic function are proportional to the hyperbolic secant function. ... In probability theory, a hyper-exponential distribution is a continuous distribution such that the probability density function of the random variable X is given by: Where is an exponentially distributed random variable with rate parameter , and is the probability that X will take on the form of the exponential distribution... The hypoexponential distribution is a generalization of Erlang distribution in the sense that the n exponential distributions may have different rates. ... In probability and statistics, the inverse-chi-square distribution is the probability distribution of a random variable whose inverse has a chi-square distribution. ... The inverse gamma distribution has the probability density function over the support with shape parameter and scale parameter . ... In probability theory and statistics, Kumaraswamys double bounded distribution is as versatile as the Beta distribution, but much simpler to use especially in simulation studies as it has a simple closed form solution for both its pdf and cdf. ... The probability distribution for Landau random variates is defined analytically by the complex integral, For numerical purposes it is more convenient to use the following equivalent form of the integral, From GSL manual, used under GFDL. ... In probability theory and statistics, the Laplace distribution is a continuous probability distribution named after Pierre-Simon Laplace. ... In probability theory and statistics, the Lévy distribution, named after Paul Pierre Lévy, is one of the few distributions that are stable and that have probability density functions that are analytically expressible. ... In probability theory, a Lévy skew alpha-stable distribution or just stable distribution, developed by Paul Lévy, is a probability distribution where sums of independent identically distributed random variables have the same distribution as the original. ... In probability theory and statistics, the logistic distribution is a continuous probability distribution. ... In probability and statistics, the log-normal distribution is the probability distribution of any random variable whose logarithm is normally distributed (the base of the logarithmic function is immaterial in that loga X is normally distributed if and only if logb X is normally distributed). ... The Maxwell-Boltzmann distribution is a probability distribution with applications in physics and chemistry. ... The normal distribution, also called Gaussian distribution, is an extremely important probability distribution in many fields. ... The Pareto distribution, named after the Italian economist Vilfredo Pareto, is a power law probability distribution found in a large number of real-world situations. ... The Pearson distribution is a family of probability distributions that are a generalisation of the normal distribution. ... In probability theory, the polar distribution is the probability distribution of angles occurring in a set of two-dimensional vectors, denoted by It is usually graphically represented as a closed curve , where the radius equals the probability . ... In probability theory and statistics, the raised cosine distribution is a probability distribution supported on the interval []. The probability density function is for and zero otherwise. ... In probability theory and statistics, the Rayleigh distribution is a continuous probability distribution. ... NB: The information in this article should be reviewed. ... In probability theory and statistics, the Rice distribution distribution is a continuous probability distribution. ... In probability and statistics, the t-distribution or Students t-distribution is a probability distribution that arises in the problem of estimating the mean of a normally distributed population when the sample size is small. ... In probability theory and statistics, the triangular distribution is a continuous probability distribution. ... In probability theory, the Type-1 Gumbel distribution function is for . Reference Taken from the gsl-ref_19. ... In probability theory, the Type-2 Gumbel distribution function is for . Based on gsl-ref_19. ... In mathematics, the continuous uniform distributions are probability distributions such that all intervals of the same length are equally probable. ... In spectroscopy, the Voigt profile is a spectral line profile named after Woldemar Voigt and found in all branches of spectroscopy in which a spectral line is broadened by two types of mechanisms, one of which alone would produce a Doppler profile, and the other of which would produce a... In probability theory and statistics, the von Mises distribution is a continuous probability distribution. ... In probability theory and statistics, the Weibull distribution (named after Waloddi Weibull) is a continuous probability distribution with the probability density function where and is the shape parameter and is the scale parameter of the distribution. ... The Wigner semicircle distribution, named after the physicist Eugene Wigner, is the probability distribution supported on the interval [−R, R] the graph of whose probability density function f is a semicircle of radius R centered at (0, 0) and then suitably normalized (so that it is really a semi-ellipse... In probability and statistics, the Dirichlet distribution (after Johann Peter Gustav Lejeune Dirichlet) is a continuous multivariate probability distribution. ... The matrix normal distribution is a probability distribution that is a generalization of the normal distribution. ... In probability theory and statistics, a multivariate normal distribution, also sometimes called a multivariate Gaussian distribution, is a specific probability distribution, which can be thought of as a generalization to higher dimensions of the one-dimensional normal distribution (also called a Gaussian distribution). ... The Wigner quasi-probability distribution was introduced by Eugene Wigner in 1932 to study quantum corrections to classical statistical mechanics. ... In statistics, the Wishart distribution, named in honor of John Wishart, is any of a family of probability distributions for nonnegative-definite matrix-valued random variables (random matrices). These distributions are of great importance in the estimation of covariance matrices in multivariate statistics. ... In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ... The Cantor distribution is the probability distribution whose cumulative distribution function is the Cantor function. ... This article defines some terms which characterize probability distributions of two or more variables. ... In probability and statistics, the exponential family is an important class of probability distributions. ... The concept of infinite divisibility arises in different ways in philosophy, physics, economics, order theory (a branch of mathematics), and probability theory (also a branch of mathematics). ... In probability theory, especially as that field is used in statistics, a location-scale family is a set of probability distributions on the real line parametrized by a location parameter μ and a scale parameter σ â‰¥ 0; if X is any random variable whose probability distribution belongs to such a family, then... In probability theory, given two jointly distributed random variables X and Y, the marginal distribution of X is simply the probability distribution of X ignoring information about Y, typically calculated by summing or integrating the joint probability distribution over Y. For discrete random variables, the marginal probability mass function can... In statistics and information theory, a maximum entropy probability distribution is a probability distribution whose entropy is larger than (or equal to) that of all other members of a specified class of distributions. ... In probability theory, a phase-type distribution is a probability distribution that can be represented as the time to absorption in a continuous-time Markov chain with m transient states i = 1, 2, ..., m and one absorbing state 0. ... The posterior probability can be calculated by Bayes theorem from the prior probability and the likelihood function. ... A prior probability is a marginal probability, interpreted as a description of what is known about a variable in the absence of some evidence. ... // Introduction In the most general form, the dynamics of a quantum mechanical system are determined by a master equation - an equation of motion for the density operator (usually written ) of the system. ... In statistics, a sampling distribution is the probability distribution, under repeated sampling of the population, of a given statistic (a numerical quantity calculated from the data values in a sample). ...

References

Schroeder, Daniel V. Thermal Physics. San Francisco: Addison Wesley Longman, 2000. ISBN 0-201-38027-7


 
 

COMMENTARY     


Share your thoughts, questions and commentary here
Your name
Your comments

Want to know more?
Search encyclopedia, statistics and forums:

 


Press Releases |  Feeds | Contact
The Wikipedia article included on this page is licensed under the GFDL.
Images may be subject to relevant owners' copyright.
All other elements are (c) copyright NationMaster.com 2003-5. All Rights Reserved.
Usage implies agreement with terms, 1022, m