Necessary Background of Statistical Physics (1
Necessary Background of Statistical Physics (1)
Why one needs statistics for describing physical phenomena? Any measurable macroscopic property is an average over a very huge number of microscopic configurations Time scale of thermal fluctuations and basic relaxations fs(femto second, 10-15 second) During the time interval necessary for realizing a measurement,a macroscopic system undergoes the change of its microscopic configuration for many many times
Why one needs statistics for describing physical phenomena? Any measurable macroscopic property is an average over a very huge number of microscopic configurations. Time scale of thermal fluctuations and basic relaxations ~ fs (femto second, 10-15 second). During the time interval necessary for realizing a measurement, a macroscopic system undergoes the change of its microscopic configuration for many many times!
Ensembles Microcanonical ensemble Fixed parameters: E-energy, V-volume, N-number of particles ystems isolated with impermeable adiabatic walls No exchange with the environment of any kind
Ensembles Microcanonical ensemble: Fixed parameters: E - energy, V - volume, N - number of particles Systems isolated with impermeable adiabatic walls. No exchange with the environment of any kind
Microcanonical ensemble Distribution function 1/22E,V,N) for E H(p, q) +AE f(p, q) 0 otherwise H(P, q): Hamiltonian p=(pl, p2,.,pN) momenta g=(q1, q2,..., N positions Q2(,V,N: statistical weight (partition function of microcanonical ensemble) This is the fundamental postulate of statistical mechanics
Microcanonical ensemble: Distribution function: 1/(E, V, N) for E H(p N , q N) E+E f(p N , q N) = 0 otherwise H(p N , q N): Hamiltonian p N = (p1 , p2 , …, pN) momenta q N = (q1 , q2 , …, qN) positions (E, V, N): statistical weight (partition function of microcanonical ensemble) This is the fundamental postulate of statistical mechanics
Connection with thermodynamics: Boltzmann formula: S(E, V,N= Q2(E,V,N) SE,V,N: entropy k: Boltzmann constant Remark Q2(E,V,N increases in a spontaneous process S(2E,2V,N)=S1(E,V,N)+S2(E,V,N) But c2(2E,2V,N)=g1(E,V,N)×Ω2(E,V,N)
Connection with thermodynamics: Boltzmann formula: S (E, V, N) = k ln (E, V, N) S (E, V, N): entropy k: Boltzmann constant Remark: (E, V, N) increases in a spontaneous process. 1 2 S(2E, 2V,N) = S1 (E, V, N) + S2 (E, V, N) But (2E, 2V,N) = 1 (E, V, N)×2 (E, V, N)