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)
Canonical ensemble Fixed parameters: T-temperature, V-volume, N-number of particles Thermostat at t System enclosed by impermeable diabatic walls Fluctuating parameter: E
Canonical ensemble: Fixed parameters: T - temperature, V - volume, N - number of particles System enclosed by impermeable diabatic walls Fluctuating parameter: E Thermostat at T . . . . .
Canonical ensemble Distribution function f(p, q)=exp( -h(pn, q)/kt)/z ZT,V,N): partition function(normalization factor) N N ZTVM-iJdp dg exp(Hp, q)kD 3N A=(2rB 2/m)/2-thermal wave length o-NSdr exp(F> 2ur r))-configuration integral Connection with thermodynamics: F(T,V,N=-kTIn(T,V,N) F(E, V,N): Helmholtz free energy F=E-TS
Canonical ensemble: Distribution function: f(p N , q N) = exp( - H(p N , q N)/kT)/Z Z(T, V, N): partition function (normalization factor). ( , , ) exp( ( , )/ ) ! 1 3 d d H k T h Z TV N p q p q N N N N N N = − Connection with thermodynamics: F(T, V, N) = - kT ln Z(T, V, N) F(E, V, N): Helmholtz free energy F = E - TS exp( ( , )) ! 1 = − j i i j i N r r r d u N Q = N Q 3 = (2p 2 /m)1/2 - thermal wave length - configuration integral
Grand canonical ensemble Fixed parameters: T,V-volume, H-chemical potential Thermostat and particle reservoir(t, u) System enclosed by permeable diabatic walls Fluctuating parameters: E and N
Grand canonical ensemble: Fixed parameters: T, V - volume, m - chemical potential Thermostat and particle reservoir (T, m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . System enclosed by permeable diabatic walls Fluctuating parameters: E and N
Grand canonical ensemble Distribution function f(pN,qN,N)=exp(-[H(pq)-N以/kT/三(T,V,μ) 三(T,V,μ): partition function (Ty)∑exp(MN/(TyN) Connection with thermodynamics c2(T,V,μ)=-kTln三(T,V,μ) Q2(T,V,H): Grand potential Q=-PV P: pressure
Grand canonical ensemble: Distribution function: f(p N , q N, N) = exp( - [H(p N , q N) - Nm]/kT)/ (T, V, m) (T, V, m): partition function = 0 ( , , ) exp( / ) ( , , ) N TV m mN k T Z TV N Connection with thermodynamics: (T, V, m) = - kT ln (T, V, m) (T, V, m): Grand potential = - PV P: pressure
Isothermal-isobaric ensemble Fixed parameters: T, P-pressure, N Thermostat(t, P piston System enclosed with diabatic walls and connected to the thermostat with a piston Fluctuating parameters: E,V
Isothermal-isobaric ensemble: Fixed parameters: T, P - pressure, N Thermostat (T, P) . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .piston System enclosed with diabatic walls and connected to the thermostat with a piston. Fluctuating parameters: E, V