In physics, the Maxwell–Boltzmann distribution is a particular probability distribution named after James Clerk Maxwell and Ludwig Boltzmann. It was first defined and used for describing particle speeds in idealized gases, where the particles move inside a stationary container without interacting with one another, except for brief collisions in which they exchange energy and momentum with each other or with their thermal environment; the term "particle" in this context refers to gaseous particles only, the system of particles is assumed to have reached thermodynamic equilibrium. The energies of such particles follow what is known as Maxwell-Boltzmann statistics, the statistical distribution of speeds is derived by equating particle energies with kinetic energy. Mathematically, the Maxwell–Boltzmann distribution is the chi distribution with three degrees of freedom, with a scale parameter measuring speeds in units proportional to the square root of T / m; the Maxwell–Boltzmann distribution is a result of the kinetic theory of gases, which provides a simplified explanation of many fundamental gaseous properties, including pressure and diffusion.
The Maxwell–Boltzmann distribution applies fundamentally to particle velocities in three dimensions, but turns out to depend only on the speed of the particles. A particle speed probability distribution indicates which speeds are more likely: a particle will have a speed selected randomly from the distribution, is more to be within one range of speeds than another; the kinetic theory of gases applies to the classical ideal gas, an idealization of real gases. In real gases, there are various effects that can make their speed distribution different from the Maxwell–Boltzmann form. However, rarefied gases at ordinary temperatures behave nearly like an ideal gas and the Maxwell speed distribution is an excellent approximation for such gases. Ideal plasmas, which are ionized gases of sufficiently low density also have particle distributions that are or Maxwellian; the distribution was first derived by Maxwell in 1860 on heuristic grounds. Boltzmann in the 1870s, carried out significant investigations into the physical origins of this distribution.
The distribution can be derived on the ground. A list of derivations are: Maximum entropy probability distribution in the phase space, with the constraint of conservation of average energy ⟨ H ⟩ = E. Assuming the system of interest contains a large number of particles, the fraction of the particles within an infinitesimal element of three-dimensional velocity space, d 3 v, centered on a velocity vector of magnitude v, is f d 3 v, in which f d 3 v = 3 / 2 e − m v 2 2 k T d 3 v, where m is the particle mass and k T is the product of Boltzmann's constant and thermodynamic temperature. One can write the element of velocity space as d 3 v = d v x d v y d v z, for velocities in a standard Cartesian coordinate system, or as d 3 v = v 2 d v d Ω in a standard spherical coordinate system, where d Ω is an element of solid angle. Here f is given as a probability distribution function, properly normalized so that ∫ f d 3 v over all velocities equals one. In plasma physics, the probability distribution is multiplied by the particle density, so that the integral of the resulting distribution function equals the density.
The Maxwellian distribution function for particles moving in only one direction, if this direction is x, is f d v x = 1 / 2 e − m v x 2 2 k T d v
The Pemigewasset River, known locally as "The Pemi", is a river in the state of New Hampshire, the United States. It is 65.0 miles in length and drains 1,021 square miles. The name "Pemigewasset" comes from the Abenaki word bemijijoasek, meaning "where side current is"; the Pemigewasset originates in the town of Franconia. It flows south through the White Mountains and merges with the Winnipesaukee River to form the Merrimack River at Franklin; the Merrimack flows through southern New Hampshire, northeastern Massachusetts and into the Atlantic Ocean. The Interstate 93 highway runs parallel with the river between New Hampton; the river passes through the communities of Lincoln, North Woodstock, Thornton, Plymouth, Ashland, Bristol, New Hampton, Hill and Franklin. The river descends over waterfalls in Franconia Notch, including "The Basin", passes cascades in North Woodstock, drops over Livermore Falls north of Plymouth; the remainder of the northern Pemi, from Lincoln to Ashland, passes over copious gravel bars and attracts numerous boaters and fishermen.
Below Ashland, the river is impounded by the Ayers Island Dam, a hydroelectric facility, for over five miles. A short stretch of heavy whitewater is found below the dam, before the river reaches the impoundment zone for the Franklin Falls flood control reservoir; the river crosses one additional hydroelectric dam below Franklin Falls before joining the Winnipesaukee River in the center of Franklin. The Pemigewasset watershed consists of over 1,100 miles of rivers and 17,000 acres of lake and reservoir area; the watershed comprises about 20 percent of the Merrimack's total watershed area. Major tributaries include: East Branch of the Pemigewasset River; the East Branch is longer than the main branch of the river. Lost River Mad River Beebe River Baker River Squam River Newfound River Smith River List of New Hampshire rivers The Columbia Gazetteer of North America Merrimack River Watershed Council EPA report on watershed status U. S. Geological Survey Geographic Names Information System: Pemigewasset River
The 1999 Brit Awards were the 19th edition of the biggest annual pop music awards in the United Kingdom. They are run by the British Phonographic Industry and took place on 16 February 1999 at the London Arena in London. In 1999, indie band Belle & Sebastian were nominated for Best British Newcomers, despite having released three albums before the 1999 Awards; the award voted for online by their listeners. At the time, Steps were arguably Britain's biggest boy/girl pop band and were nominated. Despite this, the award was won by Sebastian. On the Saturday after the awards, a story appeared in the press alleging that the group had rigged the vote in their favour, encouraging students from two universities to vote online. However, fans argued that the band had a predominantly large student following, that band member Isobel Campbell had attended one of the universities in question, in particular, the award ought to be given on artistic merit as opposed to popularity or CD sales. 1999 Brit Awards at Brits.co.uk