### Concept of the Gibbsian ensemble

```Statistical Mechanics
Concept of the Gibbsian Ensemble
In classical mechanics a state of a system is determined by knowledge
of position, q, and momentum, p.
p1
A microstate of a gas of N particles is specified by:
3N canonical coordinates q1, q2, …, q3N
6N-dimensional -space
3N conjugate momenta p1, p2, …, p3N
or phase space
A huge number of microstates correspond to the same macrostate
dq1
dp1
q1
Collection of systems (mental copies) macroscopically
identical but in different microstates
 ( p1 , q1 , ... , p 3 N , q 3 N , t ) dp1 ... p 3 N ... dq1 ... dq 3 N   ( p , q , t ) d
3N
pd
3N
q
= #of representative points at t in d3Npd3Nq  probability of finding system in state
with (p,q) in -space element d3Npd3Nq
Another way of looking at the ensemble concept:
dq1
p1
dp1
t3
t2
t1
t2
t3
t4
t5
time
t4
t5
t1
q1
time trajectory spends in d3Npd3Nq  probability of finding
system in d3Npd3Nq
Alternatively to following temporal evolution of trajectory in -space study
copies 1,2,3,4,5 … at a given moment
Density in -space  probability density
Observed value of a dynamical quantity O(p,q)
dq1
p1
dp1
Ensemble average
q1
Only needed when  not normalized
according to  d 3 N p d 3 N q  ( p , q )  1
O 
d
3N
pd
d
3N
3N
q O ( p, q) ( p, q, t)
pd
3N
q  ( p, q, t)
In thermal equilibrium
 ( p, q, t)   ( p, q)
O 
d
3N
pd
d
3N
3N
q O ( p, q) ( p, q)
pd
3N
q  ( p, q)
The assumption
O 
d
3N
pd
d
3N
3N
q O ( p, q) ( p, q)
pd
3N
q  ( p, q)
 lim
T
1
T
T
 O ( t ) dt
0
ergodic hypothesis
Transition from classical to quantum statistics
In classical mechanics a state of a system is determined by knowledge
of position, q, and momentum, p.
Dynamic evolution given by :
trajectory in -space
pi  
H
qi
, qi 
H
pi
 ( p, q, t )d
3N
pd
3N
q =probability that a system’s
phase point (p,q) is in
d
with
  ( p, q, t )d
3N
pd
3N
pd
3N
3N
q
q 1
In quantum mechanics a state of a system is determined by knowledge
of the wave function q    ( q ) .
Thermodynamic description is given in terms of microstates that are the
system’s energy eigenstates determined from
H   ( r 1 , r 2 , ..., r N )  E    ( r 1 , r 2 , ..., r N )
Eigenfunctions
labels set of quantum number
Eigenenergies
classical
 ( p, q, t )d
3N
pd
3N
quantum
q =probability that a system’s
phase point (p,q) is in
d
with
X
  ( p, q, t )d


3N
pd
3N
pd
3N
3N
3N
=probability of system
being in state label by 
q
with
q 1
X ( p , q ) ( p , q , t ) d

pd
3N
q
 
1

X 
  X 

Note: Later we will discuss in more detail the transition from the classical density
function to the quantum mechanical density matrix
```