Necessary Background of Statistical Physics(2)
Necessary Background of Statistical Physics (2)
Temperature A(3M-2)-K(p n=3 when the velocity of the center of mass is fixed to zero Kinetic energy. Kp 2m
Temperature = = N i N p m K p i 1 2 2 1 ( ) Kinetic energy: − = ( ) (3 ) 2 p N c K k N N T Nc = 3 when the velocity of the center of mass is fixed to zero
Pressure Definition aF = oNe( n Working expression 6w
Pressure ( ) 3 1 1 r r N N i i i U N P = = − '( ) ( , ) 6 1 1 2 (2) r 1 r 2r1 2 r1 2 r r d d u N = − ( ) =− = V Z TV N V F T N T N P k T ln ( , , ) , , Definition: Working expression:
Chemical potential μ=(OF/ON)ry Widom method(test particle method): u=FN+I-FN=-kTIn(ZN+ZN=kTInA3-kTIn(QN+/QN) A=(2B 2/m)2-thermal wave length Q、FxF O n dp 'exp()) 长(exp(-0) where is the interaction potential between the N+l th particle with with all the others X=μ-=-kTln(<exp(-βφ)>) Lid=kT In(p a)-chemical potential of the ideal gas Drawback: break down at high densities
Chemical potential m = (F/N)T,V Widom method (test particle method): m = FN+1 - FN = -kTln(ZN+1/ZN) = kTln3 - kTln(QN+1/QN) = (2p 2 /m)1/2 - thermal wave length Drawback: break down at high densities! where is the interaction potential between the N+1 th particle with with all the others, = + = N i r N ri u 1 1 ( , ) exp( ) exp( ( )) exp( ( ( ) )) 1 − = − − + = + N N N N N N N N N N V d d N V r U r r U r Q Q m ex = m - m id = -kT ln(<exp(-)>N) m id = kT ln( 3 ) - chemical potential of the ideal gas
Properties which can be determined from fluctuations Important remark Fluctuations are ensemble dependent! But averages are not e. g <AH(p, q)2>=0 in the microcanonical ensemble but is non zero in the canonical ensemble Heat capacit kT2Cv=<△H(p、q)Nr <(H(PN, q)-<H(p, q)>nvT NVT <H(P q)>NVT -<H(p q) 2 NVT Remark: In general, the numerical precision on fluctuations is poorer than that for the averages <△H(p、qN)N=<△U2N+<△K2Nvr <△K2 NVT 3N(kT)2/2
Properties which can be determined from fluctuations Important remark: Fluctuations are ensemble dependent! But averages are not. e.g., <ΔH(p N ,q N) 2>=0 in the microcanonical ensemble but is non zero in the canonical ensemble. Heat capacity kT2CV = <ΔH(p N ,q N) 2>NVT = <(H(p N ,q N) - <H(p N ,q N) >NVT) 2 >NVT = <H(p N ,q N) 2>NVT - <H(p N ,q N)>NVT 2 Remark: In general, the numerical precision on fluctuations is poorer than that for the averages. <ΔH(p N ,q N) 2>NVT = <ΔU2>NVT + <ΔK2>NVT <ΔK2>NVT = 3N(kT)2 /2