Download 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 ,... , p3 N , q3 N , t )dp1... p3 N ... dq1... dq3 N   ( p, q, t )d 3 N p d 3 N 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 
3N
3N
d
p
d
q O ( p, q )  ( p, q, t )

3N
3N
d
p
d
q  ( p, q, t )

In thermal equilibrium
 ( p, q, t )   ( p, q )
O 
3N
3N
d
p
d
q O ( p , q )  ( p, q )

3N
3N
d
p
d
q  ( p, q )

The assumption
O 
3N
3N
d
p
d
q O ( p, q )  ( p, q )

3N
3N
d
p
d
q  ( p, q )

T
1
 lim  O (t )dt
T  T
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
H
, qi 
qi
pi
 ( p, q, t )d 3 N p d 3 N q
=probability that a system’s
phase point (p,q) is in
d 3N p d 3N q
with
3N
3N

(
p
,
q
,
t
)
d
p
d
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   (r1 , r 2 , ..., r N )  E   (r1 , r 2 , ..., r N )
Eigenfunctions
labels set of quantum number
Eigenenergies
classical
 ( p, q, t )d 3 N p d 3 N q
quantum
=probability that a system’s
phase point (p,q) is in

=probability of system
being in state label by 
d 3N p d 3N q
with
3N
3N

(
p
,
q
,
t
)
d
p
d
q 1

X   X ( p, q ) ( p, q, t )d 3 N p d 3 N q
with
  1


X    X 

Note: Later we will discuss in more detail the transition from the classical density
function to the quantum mechanical density matrix