(4.5.1)--2.12Boltzmanndistribution.pdf

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1、Boltzmann distribution)exp()0()(RTgzMnznmFormula of distribution in isothermal atmospheric gravity fieldMaxwell velocity distribution2223/2()(,)ddd()expddd22xyzxyzxyzxyzmmfkTkTvvvv v vvvvvvv)exp(kT The distributions are distributed on a particles energy,and they all have a factor called the Boltzman

2、n factor1122exp()NNkT)/exp(kT12 Rule:There are factors in these distributions,called Boltzmann factors.The distribution with Boltzmann factors is called Bortzmann distribution.If n1 and n2 are respectively the population density in a state with particle energy and another state with particle energy

3、in a system with temperature T，then the Boltzmann distribution can be expressed as)exp(2121kTnn The Boltzmann distribution indicates that the probability of particles in each state with the same energy is the same.The probability of particles being in a different state of energy is different,but the

4、 probability of them being in a higher state of energy is smaller-the principle of minimum energy.exp()NkT1）The Boltzmann distribution gives us another expression for temperature1221ln()Tnkn)exp(2121kTnnFor systems where the particles can only take two levels：12 The system that produces the laser is

5、 in a negative temperature state with a population inversion of particles.12Discussion12nn0T 12nn0T 2）When there is an external force field,the molecules are distributed according to the law of energy When molecules are in a conservative force field,the molecular energy has both kinetic energy and p

6、otential energy Molecular kinetic energy is a function of molecular velocity,and molecular potential energy is generally a function of position.The number of molecules is related to velocity and spatial position according to the energy distribution.(p)3k20d()edddd d d2EEkTxyzmNnxyzkTvvv Where n0 rep

7、resents the population density of gas molecules at Ep=0.Law of energy distribution for Boltzmann molecules,d,d,dzzzyyyxxxvvvvvvvvvp222p2pk)(2121EmEmEEEzyxvvvv,d,d,dzzzyyyxxx),(zyx),(zyxvvv.(p)kdeEEkTNC3）Particles in a gravitational field are distributed by heightAccording to the normalization proper

8、ty of Maxwell velocity distribution functionSo the Boltzmann distribution can be written as：Particle number density according to potential energy distribution3k2-()eddd12EkTxyzmkT vvvp-0ded d dEkTNnxyzzyxNnddddP0eEkTnnThe distribution of molecules by potential energy is another common form of Boltzm

9、ann distribution law./3/20d d d()ddd2pkEkTEkTVmn ex y zekTxyzvvvvNIf the conservative external force field is the gravitational field and the potential energy is Ep=mgz(z is the height),thenDistribution of population density by height in gravitational fieldPut it into the ideal gas equation of state then0emgzkTpn kT-0emgzkTp 0eMRTgzpkTghmkTEenenn00pnkTp(p)3k20d()edddd d d2EEkTxyzmNnxyzkTvvvWhere n0 represents the number density of gas molecules at Ep=0.Boltzmanns law of energy distribution for molecules,d,d,dzzzyyyxxxvvvvvvvvv,d,d,dzzzyyyxxxThanks