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By Mark R. Andrews, Raymond Papp

ISBN-10: 0585432058

ISBN-13: 9780585432052

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Each number density is determined by a common temperature, T , and the individual particle properties, most prominently by a particle rest mass, mi and a chemical potential, μi . The respective ideal densities can be obtained as being ni = e μi /T di d3 pe−Ei (p)/T . 39) Here, Ei (p) = p2 + m2i is the energy of a single, freely and relativistically moving particle with rest mass mi and momentum p. The factor di counts for a possible degeneracy due to internal degrees of freedom, not taking part in the particles motion but distinguishing quantum states.

Retrospectively, it is related to the rest mass energy part in the more general energy conservation; classically it was treated separately and expressed in a conservation law for the number of atoms fixing atomic mass unit. In the view of modern physics, however, the number of particles is not a conserved quantity. The Gibbs free energy is to be minimized in this case, the corresponding entropic potential, −G/T = S − E/T + ∑a μ a Qa /T is to be maximized. This leads to the following variational principle N α 1 − ∑ pi + σ ({pi }) − β ∑ pi εi + β ∑ μ a ∑ pi qai i a i = max.

13) not having a maximum. This result is the Raleigh–Jeans radiation law, established in 1900. Raleigh obtained this result by counting standing waves around the frequency ω in a cavity. 14) with e being Euler’s number. Applying now the definition 1/T = ∂ S/∂ E one obtains Wien’s spectral law. Naturally, Planck had chosen the entropy formula for the oscillators just to arrive at the given spectral law. He was not satisfied with this result. Later, in 1900, he obtained his famous interpolating formula between the Raleigh– Jeans and the Wien spectra.

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Application of It for Competitive Advantage at Keane, INC by Mark R. Andrews, Raymond Papp

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