Necessary Background
of Statistical
Physics (2)
Temperature
N
i
N p
mK ip 1
2
2 1)(
Kinetic energy:
)()3( 2 p Nc KNNkT?
Nc = 3 when the velocity of the center of mass is fixed to zero.
Pressure
)(31
1 rr
NN
i ii
UNP
),()('61 21)2(121221 rrrrrr uddN
V NVTZVF
NTNT
kTP ),,(ln
,,
Definition:
Working expression:
Chemical potential
m = (?F/?N)T,V
Widom method (test particle method):
m = FN+1 - FN = -kTln(ZN+1/ZN) = kTln?3 - 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 iN rr
u
1 1
),(
)e x p ())(e xp( )))((e xp(
1
N
N
N
N
N
N
N
N
N NV
d
d
NV rUr rUrQ
Q
mex = m - mid = -kT ln(<exp(-)>N)
mid = 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.,<ΔH(pN,qN)2>=0 in the microcanonical ensemble but
is non zero in the canonical ensemble,
Heat capacity
kT2CV = <ΔH(pN,qN)2>NVT
= <(H(pN,qN) - <H(pN,qN) >NVT)2 >NVT
= <H(pN,qN)2>NVT - <H(pN,qN)>NVT2
Remark,In general,the numerical precision on fluctuations
is poorer than that for the averages,
<ΔH(pN,qN)2>NVT = <ΔU2>NVT + <ΔK2>NVT
<ΔK2>NVT = 3N(kT)2/2
Isothermal compressibility
kTχT = <DN2>mVT/<N>mVT
TT PVV 1?
kT(?<N>mVT /?m)V,T = <DN2>mVT
Transport properties
Self-diffusion coefficient
Einstein formula:
<|r(t) - r(0)|2>? 6Dt t
Expression in terms of velocity auto-correlation function (VACF):
0 )0()( vtvdtD
Diffusion equation:
),(),( 2 trDt tr
Solution with the initial condition,?(r,t=0) =?(r - r0):
)4ex p (1),(
2
0
2/3)4( Dttr
rr
Dt
p
Mean square displacement Velocity auto-correlation function
0 t
1
)0()0(
)0()(
vv
vtv
t
<|r(t) - r(0)|2>
Remark:
The method of mean square displacement provides a better
numerical precision in the calculation of D.
of Statistical
Physics (2)
Temperature
N
i
N p
mK ip 1
2
2 1)(
Kinetic energy:
)()3( 2 p Nc KNNkT?
Nc = 3 when the velocity of the center of mass is fixed to zero.
Pressure
)(31
1 rr
NN
i ii
UNP
),()('61 21)2(121221 rrrrrr uddN
V NVTZVF
NTNT
kTP ),,(ln
,,
Definition:
Working expression:
Chemical potential
m = (?F/?N)T,V
Widom method (test particle method):
m = FN+1 - FN = -kTln(ZN+1/ZN) = kTln?3 - 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 iN rr
u
1 1
),(
)e x p ())(e xp( )))((e xp(
1
N
N
N
N
N
N
N
N
N NV
d
d
NV rUr rUrQ
Q
mex = m - mid = -kT ln(<exp(-)>N)
mid = 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.,<ΔH(pN,qN)2>=0 in the microcanonical ensemble but
is non zero in the canonical ensemble,
Heat capacity
kT2CV = <ΔH(pN,qN)2>NVT
= <(H(pN,qN) - <H(pN,qN) >NVT)2 >NVT
= <H(pN,qN)2>NVT - <H(pN,qN)>NVT2
Remark,In general,the numerical precision on fluctuations
is poorer than that for the averages,
<ΔH(pN,qN)2>NVT = <ΔU2>NVT + <ΔK2>NVT
<ΔK2>NVT = 3N(kT)2/2
Isothermal compressibility
kTχT = <DN2>mVT/<N>mVT
TT PVV 1?
kT(?<N>mVT /?m)V,T = <DN2>mVT
Transport properties
Self-diffusion coefficient
Einstein formula:
<|r(t) - r(0)|2>? 6Dt t
Expression in terms of velocity auto-correlation function (VACF):
0 )0()( vtvdtD
Diffusion equation:
),(),( 2 trDt tr
Solution with the initial condition,?(r,t=0) =?(r - r0):
)4ex p (1),(
2
0
2/3)4( Dttr
rr
Dt
p
Mean square displacement Velocity auto-correlation function
0 t
1
)0()0(
)0()(
vv
vtv
t
<|r(t) - r(0)|2>
Remark:
The method of mean square displacement provides a better
numerical precision in the calculation of D.
