《Simulations Moléculaires》 Cours04II

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