統(tǒng)計物理學(xué)習(xí)講義(編輯修改稿)
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【文章內(nèi)容簡介】 al Phenomena amp。 Powerlaw ? 相變 ? SOC, HOT/COLD理論 Entropy Microstate r: a specific configuration of system Macrostate R: an evaluation value ?(R): number of microstates related to a macrostate Microcanonical entropy: S(R)=k log ?(R) More General forms: A macrostate R: {pi} for system be found in a microstate i A distribution of microstates. Gibbs Entropy: S(R) =k ∑pi logpi Maximum ? the most possible distribution of microstates Without constraint on pi, pi=1/N ? S is maximized ?({ni})=M!/n1!n2!...nN!, pi=ni/M With Constraint on pi: Partition Function Z Observable quantity E (Hamiltonian) Ergodic Hypothesis (time average=ensemble average) We know: ? From experiments: E’, ? Ei for all ri, and E’= E= ∑piEi, ∑pi=1. We want to know the most probable distribution of microstates Maximize S=k∑pilogpi and we get: pi=eβEi/Z, Z=∑ieβEi (β=(kT)1) So, {pi} and β is decided by {Ei} and E Knowing βor T and {Ei}, we can define the most possible distribution of microstates {pi} and Z β? ? T? ? E? ? Z? distribution is less symmetrical Toy Example Three microstates: E1=0, E2=2, E3=3 We have p1E1+p2E2+p3E3=E . 2p2+3p3=E, and p1+p2+p3=1 3 temperatures: decreasing order of T E β Z p1 p2 p3 1 2 1 3 Important concepts ? Partition function: Z(T,E)=∑re E(r)/T Knowing this, we can do a lot of things! Variance of E, sol, … ? Free Energy:
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