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!
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/?(E,V,N) for E? H(pN,qN)? E+?E
f(pN,qN) =
0 otherwise
H(pN,qN),Hamiltonian
pN = (p1,p2,…,pN) momenta
qN = (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) = 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
System enclosed by impermeable diabatic walls
Fluctuating parameter,E
Thermostat at T
.,....
Canonical ensemble:
Distribution function:
f(pN,qN) = exp( - H(pN,qN)/kT)/Z
Z(T,V,N),partition function (normalization factor).
))/,(e x p (),,( !13 kTHddhNVTZ qpqp NNNNNN
Connection with thermodynamics:
F(T,V,N) = - kT ln Z(T,V,N)
F(E,V,N),Helmholtz free energy
F = E - TS
)),(e xp (!1
ij jii
N rrr udNQ
N
Q
3
= (2p 2/m)1/2 - thermal wave length
- configuration integral
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(pN,qN) - Nm]/kT)/?(T,V,m)
(T,V,m),partition function
0 ),,()/e x p (),,( N NVTZkTNVT mm
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:
Distribution function:
f(pN,qN,V) = exp( - [H(pN,qN) + PV]/kT)/ Z(T,P,N)
Z(T,P,N),partition function
V NVTZkTPVNPTZ ),,()/e x p (),,(
Connection with thermodynamics:
G(T,P,N) = - kT ln Z(T,P,N)
G(T,P,N),Gibbs free energy
G = F + PV
N-body distribution functions
One-body distribution functon:
Probability for finding a particle at a given position,r1,
Homogeneous system:
r(1)(r) = constant = r = N/V
r(1)(r1) = 1/(N-1)!? dr2…d rNdpN f(r1,r2,…,rN ; pN)
Normalization:
dr1 r(1)(r1) = N
How to determine r(1)(r) for inhomogeneous systems?
Example:
Fluid density distribution near a solid surface
z
Divide the simulation box into slit bins.
r(1)(z) = <?N(z,z+?z)>/?V
<?N(z,z+?z)>,average particle number in the slit (z,z+?z).
V= A?Lz volume of the slit
A,surface area
Simulation
box
z z+?z
Lz
What can we learn from r(1)(z)?
Microscopic structure of the fluid-solid interface.
r(1)(z)
z
A fluid has a layer structure near a plane solid surface.
Structure of electrolyte solution near an electrode surface
z
anion
cation
solvent
z
r-(1)(z) rd(1)(z)
z
Two-body distribution function:
Probability for finding a pair of particles at r1 and r2.
r(2)(r1,r2) = 1/(N-2)!? dr3,.,drNdpN f(r1,r2,…,rN ; pN) )
Normalization:
dr1? dr2 r(2)(r1,r2) = N(N-1)
Homogeneous systems:
r(2)(r1,r2) = r(2)(|r1 - r2|) = r2g(r12)
g(r12),radial distribution function.
r12 = |r1 - r2|
How to determine g(r12) for homogeneous systems?
rg(r) = <N(r,r+dr)>/?V
V = 4pr2dr
r dr
g(r)
r
1
s
There is a short-range order
in a liquid.
Exercise
Imagine that NCONF configurations of a system of N particles
are stored in a file conf.data,Write a program for calculating
g(r) from the data of this file.
Internal energy
Configuration integral:
)),(e xp (!1
ij jii
N rrr udNQ
),( qp NNHE
)),(ex p (),(! 1 3 prprprh NNNNNNN HHddZN
= <K> + <U> = 3NkT/2 + <U>
<K>,mean kinetic energy
<U>,mean potential energy
),(),(21 21)2(2121 rrrrrr udd r
)),(e x p (),(!1
ij jiiij jii
N rrrrr uudQN
<U>
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!
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/?(E,V,N) for E? H(pN,qN)? E+?E
f(pN,qN) =
0 otherwise
H(pN,qN),Hamiltonian
pN = (p1,p2,…,pN) momenta
qN = (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) = 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
System enclosed by impermeable diabatic walls
Fluctuating parameter,E
Thermostat at T
.,....
Canonical ensemble:
Distribution function:
f(pN,qN) = exp( - H(pN,qN)/kT)/Z
Z(T,V,N),partition function (normalization factor).
))/,(e x p (),,( !13 kTHddhNVTZ qpqp NNNNNN
Connection with thermodynamics:
F(T,V,N) = - kT ln Z(T,V,N)
F(E,V,N),Helmholtz free energy
F = E - TS
)),(e xp (!1
ij jii
N rrr udNQ
N
Q
3
= (2p 2/m)1/2 - thermal wave length
- configuration integral
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(pN,qN) - Nm]/kT)/?(T,V,m)
(T,V,m),partition function
0 ),,()/e x p (),,( N NVTZkTNVT mm
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:
Distribution function:
f(pN,qN,V) = exp( - [H(pN,qN) + PV]/kT)/ Z(T,P,N)
Z(T,P,N),partition function
V NVTZkTPVNPTZ ),,()/e x p (),,(
Connection with thermodynamics:
G(T,P,N) = - kT ln Z(T,P,N)
G(T,P,N),Gibbs free energy
G = F + PV
N-body distribution functions
One-body distribution functon:
Probability for finding a particle at a given position,r1,
Homogeneous system:
r(1)(r) = constant = r = N/V
r(1)(r1) = 1/(N-1)!? dr2…d rNdpN f(r1,r2,…,rN ; pN)
Normalization:
dr1 r(1)(r1) = N
How to determine r(1)(r) for inhomogeneous systems?
Example:
Fluid density distribution near a solid surface
z
Divide the simulation box into slit bins.
r(1)(z) = <?N(z,z+?z)>/?V
<?N(z,z+?z)>,average particle number in the slit (z,z+?z).
V= A?Lz volume of the slit
A,surface area
Simulation
box
z z+?z
Lz
What can we learn from r(1)(z)?
Microscopic structure of the fluid-solid interface.
r(1)(z)
z
A fluid has a layer structure near a plane solid surface.
Structure of electrolyte solution near an electrode surface
z
anion
cation
solvent
z
r-(1)(z) rd(1)(z)
z
Two-body distribution function:
Probability for finding a pair of particles at r1 and r2.
r(2)(r1,r2) = 1/(N-2)!? dr3,.,drNdpN f(r1,r2,…,rN ; pN) )
Normalization:
dr1? dr2 r(2)(r1,r2) = N(N-1)
Homogeneous systems:
r(2)(r1,r2) = r(2)(|r1 - r2|) = r2g(r12)
g(r12),radial distribution function.
r12 = |r1 - r2|
How to determine g(r12) for homogeneous systems?
rg(r) = <N(r,r+dr)>/?V
V = 4pr2dr
r dr
g(r)
r
1
s
There is a short-range order
in a liquid.
Exercise
Imagine that NCONF configurations of a system of N particles
are stored in a file conf.data,Write a program for calculating
g(r) from the data of this file.
Internal energy
Configuration integral:
)),(e xp (!1
ij jii
N rrr udNQ
),( qp NNHE
)),(ex p (),(! 1 3 prprprh NNNNNNN HHddZN
= <K> + <U> = 3NkT/2 + <U>
<K>,mean kinetic energy
<U>,mean potential energy
),(),(21 21)2(2121 rrrrrr udd r
)),(e x p (),(!1
ij jiiij jii
N rrrrr uudQN
<U>
