Classical Maxwell-Boltzmann statistics

In the case of classical distinguishable particles, there are g ways to put n, particles in the same level i, and there are 

We include the constraints through the Lagrange multipliers a and j8, and perform the variation of W with respect to Hi to obtain 

Example We consider the case of e classical ideal gas which consists of particles of mass m, with the only interaction between the particles being binary hard-sphere collisions. The particles are contained in a volume 2, and the gas has density n = N/Q. In this case, the energy e, of a particle, with momentum p and position r, and its multiplicity gi are given by 

Then summation over all the values of the index i gives 

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In a special case where g, nu Eq. (5.22) can be reduced to the corrected Maxwell-Boltzmann statistics (note the classical Maxwell-Boltzmann statistics is for distinguishable particles) as... 

In Fermi-Dirac statistics each state can accommodate at most only two particles with opposed spins. In Bose-Einstein statistics, just as in the classical Maxwell-Boltzmann statistics, there is no limitation to the number of particles in a given state. In classical statistics the particles in the same state were assumed to be distinguishable one from the other. As this assumption has been shown in quantum theory to be incorrect the particles in the same state in Bose-Einstein quantum statistics are indistinguishable. Interchanges of two of the par-... 

Turning to electric fields and classical Maxwell-Boltzmann statistics, soluble analytical models now exist which allow calculations of non-degenerate electron densities as a function of thermodynamic state in intense electric fields (low density high temperature). Semiclassical methods are available for switching on atomic potentials to models studied presently, though numerical results are not yet available here. 

Let us investigate the nonrelativistic ideal gas of identical particles governed by the classical Maxwell-Boltzmann statistics in the framework of the Tsallis and Boltzmann-Gibbs statistical mechanics. 

Tet us consider the nonrelativistic ideal gas of N identical particles governed by the classical Maxwell-Boltzmann statistics in the framework of the Tsallis and Boltzmann-Gibbs statistics in the microcanonical ensemble. For this special model, the statistical weight (111) can be written as (see [6] and reference therein)... 

Ludwig Boltzmann (1844-1906), the Austrian physicist, is famous for his outstanding contributions to heat transfer, thermodynamics, statistical mechanics, and kinetic theory of gases. Boltzmann was a student of Josef Stefan and received his doctoral degree in 1866 under his supervision. The Stefan-Boltzmann law (1884) for black body radiation is the result of the associated work of Josef Stefan and Boltzmann in the field of heat transfer. Boltzmann s most significant works were in kinetic theory of gases in the form of Maxwell-Boltzmann distribution and Maxwell-Boltzmann statistics in classical statistical mechanics. 

Classically, the dependence of a number of objects with their energy-per-object is modeled with statistics. When objects are independent of each other (more exactly, are sulgected to independent probabilities of presence) and in suffidaitly large numbers for allowing the approximation of factorials by exponentials, this dependence is modeled by Maxwell-Boltzmann statistics. 

Over the course of history, pure metal has been described by a variety of models. The initial model, attributable to Drude, considered the metal to comprise a gas of electrons enveloping positive ions in a constant potential. Drude applies Maxwell-Boltzmann statistics to that electron gas. In fact, as the electrons are fermions, it is most appropriate to apply Fermi-Dirac statistics to them, as Sommerfeld did in his model, still using a constant potential. Unlike with molecules, though, because of their low mass, the electrons cannot be used for the approximation of the classic limit statistics given by ... 

In classical Newtonian physics the elementar volume of a configurational space cell is infinitesimal (it looks like Planck s constant ti is accepted to be zero) the electron distribution upon their energy is given by Maxwell-Boltzmann statistics there are large amounts of particles, all of which tend to occupy the state with the lowest energy, though chaotic temperature motion, on the other hand, scatters them on different energies. This process is described by the Boltzmann factor. 

In summary, Eq. (86) is a general expression for the number of particles in a given quantum state. If t = 1, this result is appropriate to Fenni-rDirac statistics, or to Bose-Einstein statistics, respectively. However, if i is equated torero, the result corresponds to the Maxwell -Boltzmann distribution. In many cases the last is a good approximation to quantum systems, which is furthermore, a correct description of classical ones - those in which the energy levels fotm a continuum. From these results the partition functions can be calculated, leading to expressions for the various thermodynamic functions for a given system. In many cases these values, as obtained from spectroscopic observations, are more accurate than those obtained by direct thermodynamic measurements. 

In the general approach to classical statistical mechanics, each particle is considered to occupy a point in phase space, i.e., to have a definite position and momentum, at a given instant. The probability that the point corresponding to a particle will fall in any small volume of the phase space is taken proportional to die volume. The probability of a specific arrangement of points is proportional to the number of ways that the total ensemble of molecules could be permuted to achieve the arrangement. When this is done, and it is further required that the number of molecules and their total energy remain constant, one can obtain a description of the most probable distribution of the molecules in phase space. Tlie Maxwell-Boltzmann distribution law results. 

Maxwell-Boltzmann particles are distinguishable, and a partition function, or distribution, of these particles can be derived from classical considerations. Real systems exist in which individual particles are indistinguishable. For example, individual electrons in a solid metal do not maintain positional proximity to specific atoms. These electrons obey Fermi-Dirac statistics (133). In contrast, the quantum effects observed for most normal gases can be correlated with Bose-Einstein statistics (117). The approach to statistical thermodynamics described thus far is referred to as wave mechanics. An equivalent quantum theory is referred to as matrix mechanics (134—136). 

The previously described theory in its original form assumes that the classical kinetic theory of gases is applicable to the electron gas, that is, electrons are expected to have velocities that are temperature dependent according to the Maxwell-Boltzmann distribution law. But, the Maxwell-Boltzmann energy distribution has no restrictions to the number of species allowed to have exactly the same energy. However, in the case of electrons, there are restrictions to the number of electrons with identical energy, that is, the Pauli exclusion principle consequently, we have to apply a different form of statistics, the Fermi-Dirac statistics. 

Maxwell-Boltzmann temperature distribution - based on classical statistics, gives the number of molecules in a gas whose total velocity lies within a given range. 

In classical statistical mechanics, each particle is regarded as occupying a point in phase space, i.e. to have an exact position and momentum at any particular instant. The probability that this point will occupy any small volume of the phase space is taken to be proportional to the volume. The Maxwell-Boltzmann law gives the most probable distribution of the particles in phase space. 

The treatment of the influence phenomenon has led to the formulation of an exponential law that is identical to the one known as Boltzmann s statistics (also known as Maxwell-Boltzmann s statistics). This classical approach relies on the concept of probability, which predicts the number of times an event may occur when a large number of triggering actions are performed. (It must be noted that the notions of event and of sequences of events require the existence of time). In the derivation of Boltzmann s statistics, two assumptions are required (Landau and Lifchitz 1958) ... 

This relation is known as Boltzmann distribution (or Maxwell-Boltzmann distribution) because it is classically established on statistical arguments. 

The three branches of quantum statistics (Maxwell-Boltzmann, Bose-Einstein and Fermi-Dirac) meld into one, known as the classic limiting case, if the following condition is met ... 

The zeroth moment of a distribution is 1, the first moment is < i>, the second moment is < P>, etc. The higher moments of a distribution hence compute successively higher averages of the distributions of the independent variable for example, in classical statistical thermodynamics the mean square velocity is the second moment of the Maxwell-Boltzmann speed distribution for an ideal gas, and is directly related to average kinetic energy < KE > = m < v >/2, and hence to temperature [= 3k TI2 for a monatomic gas]. 

There is no need to stress the confidence we all feel in calculations that are securely founded on classical thermodynamics, but it is important to emphasize that those based on statistical mechanics are not inherently any less secure. This science is now about 100 years old, if we reckon Maxwell, Boltzmann and Gibbs to be its founders, and that is ample time for any faults in the foundations to have revealed themselves. In practice, the calculations may be more speculative because of approximations that we have introduced, but the existence of these approximations is always evident, even if their consequences are not fully known. 

In the subsequent chapters in which we will be investigating the thermal, electrical, optical, and magnetic properties of materials, it will be necessary to be able to determine the energy distribution of electrons, holes, photons, and phonons. To do this, we need to introduce some quantum statistical mechanical concepts in order to develop the distribution fimc-tions needed for this purpose. We will develop the Bose-Einstein (B-E) distribution function that applies to all particles except electrons and holes (and other fermions) that obey the Pauli exclusion principle and show how this function becomes the Maxwell-Boltzmann (M-B) distribution in the classical limit. Also, we will show how the Planck distribution results by relaxing the requirement that particles be conserved. Next we develop the Fermi-Dirac (F-D) distribution that applies to electrons and holes and becomes the basis for imderstanding semiconductors and photonic systems. 

Let us use our statistical approach to obtain the average energy of a molecule of a classical monatomic gas using Maxwell-Boltzmann (M—B) statistics. The average energy per molecule is given by... 

Hendrik Antoon Lorentz, from Leyden (Holland), presided the conference, whose general theme was the Theory of Radiation and the Quanta. The conference5 was opened with speeches by Lorentz and Jeans, one on Applications of the Energy Equipartition Theorem to Radiation, the other on the Kinetic Theory of Specific Heat according to Maxwell and Boltzmann. In their talks, the authors explored the possibility of reconciling radiation theory with the principles of statistical mechanics within the classical frame. Lord Rayleigh, in a letter read to the... 

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