Thermodynamics Maxwell speed distribution

If a system is not uniform, it is not in thermodynamic equilibrium then can obey law of Maxwell s speeds distribution. However, if "equilibrium absence" is not big, can considered like good approximation, all little volume (microscopic scale) is in equilibrium (considered like subsystem). This is for two reasons. First little portions of gas contain a big number of molecules. Second the necessary time for established the equilibrium in a little volume is brief in comparison with necessary time for that transport processes get equilibrium in little volume with rest of system (it is true when concentration, temperature, etc. gradients are not too much big). In consequence, can supjpose that is local thermodynamic equilibrium so speed distribution in any volume element (macroscopic) of medium is Maxwellian, although density, temperature and macroscopic velocity change the position (Duderstadt Martin, 1979 Schwabl, 2002 Bhatnagar et al., 1954 Pai, 1981 Maxwell, 1997 Sued, 2001 Succi, 2002 Cercignani, 1975 Lebowitz Montroll, 1983). 

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]. 

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