Lecture 41 Statistical Mechanicsand Boltzmann factor. What is statistical mechanics?· Microscopic state and ergodic principle.BoltzmannfactorIdeal gas in gravitational field· Equipartition of energy and heat capacity
Lecture 41 Statistical Mechanics and Boltzmann factor • What is statistical mechanics? • Microscopic state and ergodic principle • Boltzmann factor • Ideal gas in gravitational field • Equipartition of energy and heat capacity
Statistical mechanicsStatistical mechanics is a science that describes theproperties of a thermal system starting from itsmicroscopic structure.? It started from study of gases from molecular view. The microscopic description of a system starts withthe energy of the system in terms of microscopicvariables.N+TE(ri,pi) =Vij订
Statistical mechanics • Statistical mechanics is a science that describes the properties of a thermal system starting from its microscopic structure. • It started from study of gases from molecular view. • The microscopic description of a system starts with the energy of the system in terms of microscopic variables. 𝐸 𝑟𝑖 , 𝑝𝑖 = 𝑖 𝑝𝑖 2 2𝑚𝑖 + 𝑖𝑗 𝑉𝑖𝑗
Ergodic principle· Microscopic states: defined by specifications ofcoordinates and momenta of all particles in thesystem.: Statistical hypothesis: in a thermal equilibrium witha fixed energy, all microscopic states can beaccessed with equal probability. This the fundamental principle (ergodic principle)of statistical mechanics, sometime it is called micro-canonicalensemble
Ergodic principle • Microscopic states: defined by specifications of coordinates and momenta of all particles in the system. • Statistical hypothesis: in a thermal equilibrium with a fixed energy, all microscopic states can be accessed with equal probability. • This the fundamental principle (ergodic principle) of statistical mechanics, sometime it is called microcanonical ensemble
Practical example. If we have a fair coin, every time the probability ofgetting head or tail is the same, Y.If wethrowcoin3 times,we have will 2^3=8possible results (states): HHh, HHt, Hth, Thh, Tth, Tht, Htt, TtT· Ergodic principle states that all results come withequalprobability
Practical example • If we have a fair coin, every time the probability of getting head or tail is the same, ½ • If we throw coin 3 times, we have will 2^3=8 possible results (states): • HHH, HHT, HTH, THH, TTH, THT, HTT, TTT • Ergodic principle states that all results come with equal probability
Microscopic state· For particles with position r and momentum p, themicroscopic state is defined by r and p.The combinedr and p space is called phase space.·Any any time, the stateof a particleis defined by apoint inphasespaceThephasespacevolumeelementisd3rd3p
Microscopic state • For particles with position 𝑟 Ԧ and momentum 𝑝 Ԧ, the microscopic state is defined by 𝑟 Ԧ and 𝑝 Ԧ • The combined 𝑟 Ԧ and 𝑝 Ԧ space is called phase space. • Any any time, the state of a particle is defined by a point in phase space. • The phase space volume element is 𝑑 3 𝑟 Ԧ𝑑 3𝑝 Ԧ