There is a long and twisted history of justifying the fundamental postulate. The main problem here is that on a microscopic basis, dynamics is governed by Newton’s equa- tions, which are reversible, while macroscopic systems decay irreversibly into a unique equilibrium state. A rigorous derivation of the drive towards equilibrium from micro- scopic principles does still not exist and several mechanisms are discussed that some- how smooth out phase space density, such that a homogeneous distribution over the energy shell is achieved. One has to state clearly that statistical mechanics works very well for most physical systems of interest, but it is notoriously difficult to establish a microscopic derivation. This is why we still have to speak of a fundamentalpostulate.
A very instructive approach to this question is to consider Hamiltonian dynamics. In classical systems with N particles, each microstate is a point in 6N-dimensional‘phase space’(~r1, ...,~rn
| {z }
positions
,~p1, ...,p~n
| {z }
momenta
) = (~q,~p). In statistical mechanics, we consider many particle systems and therefore cannot say in which state the system is exactly located, but rather use a statistical ensemble of states. The probability distribution is continuous and the probability that the system is in state(~q,~p)is
ρ(~q(t),~p(t),t)d~qd~p
where ρ is the phase space probability density. For initial conditions(~q(0),~p(0))the system evolves according to Hamilton’s equations:
p˙i =−∂H
∂qi, q˙i = ∂H
∂pi
For an isolated system at rest, energy is conserved due to the time invariance (Noether theorem):
H =const= E
The solutions to the system of ordinary differential equations are unique and do not intersect. Energy conservation reduces phase space to a (6N−1)-dimensional hyper- surface, the energy surface or energy shell.
We now define a phase space velocity
~v:= (~q, ˙˙ ~p) and the corresponding current
~j=ρ~v
For an arbitrary region of phase space, we have a balance equation:
Z
∂VdA~ ·~j=− ∂
∂t Z
Vd~qd~pρ(~q,~p,t)
⇒
Z
d~qd~p ∂ρ
+∇ ·~j
=0
⇒
|{z}
V arbitrary
∂ρ
∂t +∇ ·(ρ~v) =0 continuity equation
Thus the system evolves like a hydrodynamic system, with a probability fluid stream- ing through state space. We now use Hamilton’s equation to show that
0= ∂ρ
∂t +
∑
3N i=1( ∂
∂qi(ρq˙i) + ∂
∂pi(ρp˙i))
= ∂ρ
∂t +
∑
3N i=1(∂ρ
∂qi q˙i+ ∂ρ
∂pi p˙i) +ρ
3N
∑
i=1
(∂q˙i
∂qi +∂p˙i
∂pi)
| {z }
∂2H
∂qi∂pi− ∂2H
∂pi∂qi=0
= dρ
dt =0 Liouville’s theorem
The total derivative of the probability density vanishes. The probability fluid is ‘incom- pressible’ andρ(~q(t),~p(t),t) =ρ(~q(0),~p(0), 0).
Let V0 be the volume of some initial region R0 of phase space. At some time t after- wards, this region can have evolved to some regionRtwith complicated shape, but its volume is unchanged:Vt =V0(Figure 2.13).
p
q
R
0R
tV
t= V
0V
0Figure 2.13: The phase space volume stays constant, although its shape may change.
At this point, we can draw two important conclusions. First the number of occupied microstates does not change, because phase space volume is conserved. More general, entropy does not change, because the system evolves deterministically and thus infor- mation content is not changed. This seems to speak against the fundamental postulate, which requires some kind of dispersion and increase in entropy. However, you can interpret this results also in another manner. Especially because the Hamiltonian sys- tem does not relax into some subset of phase space, it keeps continuing to explore all of phase space in a similar manner and this contributes to the fact that all states are equally likely to be visited.
Secondly, the fact that the phase space volume is conserved does not mean that its shape does not change. In fact for many systems of interest, one finds that a well- defined region of phase space quickly distorts into a very complex shape, especially for chaotic systems. When viewed from a coarse-grained perspective (like in a real world experiment with limited resolution), we will see a smooth distribution. This has been compared with the stirring of oil and water, which keep distinct domains on the microscale (oil and water do not mix), but appear to be uniform on a macroscopic scale.
A more rigorous way to deal with the coarse-graining issue in a classical framework is the BBGKY-hierarchy (after Bogoliubov, Born, Green, Kirkwood, Yvon). Mixing in phase space is possible even in completely classical systems as proven by deterministic chaos. On the quantum level, one also could argue that completely isolated systems never exist and that coupling to the environment, even if very weak, will eventually lead to smoothing in state space. This aspect seems to suggest how the fundamental postulate might arise.
In order to learn more about the equilibrium state, we next rewrite Liouville’s theorem:
∂ρ
∂t =−
3N
∑
i=1
q˙i ∂ρ
∂qi +p˙i ∂ρ
∂pi
=−
3N
∑
i=1
∂H
∂pi
∂ρ
∂qi −∂H
∂qi
∂ρ
∂pi
=− {H,ρ} Liouville’s equation
Here we used the notation of‘Poisson brackets’in the last step. Liouville’s equation is also known as the collisionless Boltzmann equation because it describes the streaming part for the probability fluid in phase space.
Let us now assume thatρ(~q,~p,t)only depends on the conserved value of energyE.
ρ(~q,~p,t) =Φ(E), dE dt =0 We now get
∂ρ
∂t =− {H,ρ}
=−
3N
∑
i=1
∂H
∂pi
∂ρ
∂qi −∂H
∂qi
∂ρ
∂pi
=−
3N
∑
i=1
∂H
∂pi
∂E
∂qi −∂H
∂qi
∂E
∂pi
| {z }
{H,H}=dEdt=0
∂Φ
∂E =0
This result is also known asJean’s theorem. Thus in this case the state space probability density is constant and has the same value for a given energy. We conclude that once
We next consider an observable A which depends on time only through phase space:
A(t) = A(q~(t),p~(t))
⇒ dA dt =
∑
3N i=1∂A
∂qi q˙i+ ∂A
∂pi p˙i
= {H,A} In equilibrium the ensemble average of an observable
hAi=
Z
d~qd~pρ(~q,~p)A(~q,~p)
will be time-independent. In particular we expect this to apply to all state variables.
Equivalent results can be derived for quantum mechanical systems. However, in this case we cannot use a scalar probability density, because phase space coordinates do not commute. Instead we need to introduce a density operator.
For a given state|Ψi, the observable has the average (projection):
hAi=hΨ|A|Ψi We define the density operator or density matrix:
ρ=|Ψi hΨ| This then yields
hAi=hΨ|A|Ψi=
∑
n
hΨ|A|ni hn|Ψi
=
∑
n
hn|Ψi hΨ|A|ni=
∑
n
hn|ρA|ni
=tr{ρA} average over quantum mechanical distribution of states If we now turn to statistical mechanics, we superimpose a second layer of probability over the quantum mechanical probabilities. We call the states that follow the Schrödinger equationpurestates and then consider mixedstates by adding up several of the pure states in an incoherent manner (no superposition, so probability is not tranfered from one qm state to the other and the weights stay constant).
Using an extended definition of the density matrix:
ρ=
∑
i
pi |Ψii hΨi|
⇒ hAi=
∑
i
pi hΨi|A|Ψii
=
∑
n
∑
i
pi hΨi|A|ni hn|Ψii
=
∑
n
∑
i
pi hn|Ψii hΨi|A|ni
=
∑
n
hn|ρA|ni=tr(ρA)
Again stationary distributions result if ρ is a function of the stationary ensemble in energy representation:
H |ni= En|ni 0=∂tρ= [H,ρ]
⇒ 0=hm| Hρ−ρH |ni= (Em−En)ρmn
⇒ ρmn =0 f or Em 6=En
without degeneracy: ρ=
∑
n
ρ(En)|ni hn|
We now derive the quantum mechanical analogue of Liouville’s equation:
Schrödinger equation: i¯h∂t|Ψi=H |Ψi adjoint Schrödinger equation: i¯h∂thΨ|=hΨ| H
⇒ i¯h∂tρ= i¯h
∑
i
pi Ψ˙i
hΨi|+|ΨiiΨ˙i
=
∑
i
pi (H |Ψii hΨi| − |Ψii hΨi| H)
= [H,ρ] commutator
∂tρ= −i
¯
h [H,ρ] von Neumann equation
Like Liouville’s equation, von Neumann’s equation (also called the quantum Liouville equation) suggests that in equilibrium, probability distributions and state variables are constant.
In summary, the fundamental postulate cannot be proven rigorously and thus stays a postulate. Generations of physicists and mathematicians have tried to improve on the conceptual basis of statistical physics, e.g. trying to prove that certain model sys- tems are ergodic, but usually this created only more riddles, e.g. in dynamical systems theory. We close by noting again that this is an academic problem. In real life, the fun- damental postulate has proven itself beyond doubt due to its success in explaining the physics of many particle systems.
3 The canonical ensemble
3.1 Boltzmann distribution
We consider a system in contact with a‘thermal reservoir’or ‘heat bath’. Then it is tem- peratureT rather than energyE that is fixed. A simple example would be a bottle of beer in a large lake, which eventually will cool down to the temperature of the lake.
To qualify as a thermal reservoir, the surrounding system has to be much larger such that its temperature does not change as energy is exchanged with the system of interest.
Together the two systems form an isolated system for which energy is fixed atEtot.
1 2
Figure 3.1: Two systems in thermal contact. System 2 is considerably larger than system 1 and serves as a ‘thermal reservoir’. Together the two systems are again a microcanoncial ensemble.
We now consider one specific microstateiin system 1. This microstate comes with an energyEi. Its probability to occur is
pi = # favorable outcomes
# possible outcomes = Ωres(Etot−Ei) Ωtot(Etot) = e
Sres(Etot−Ei)/kB
eStot(Etot)/kB
Here we used that the composite system is microcanonical and that we have fixed the microstate in system 1; then the number of accessible microstates is determined by system 2 (the reservoir) only.
We next introduce the average energy of system 1 as the reference energy:
U=hEi=
∑
i
piEi
We now Taylor-expand the entropy of the heat bath:
Sres(Etot−Ei) =Sres(Etot−U+U−Ei) =Sres(Etot−U) + U−Ei T
Here we have used∂S/∂E=1/T. Note that higher order terms do not appear because a reservoir has a constant temperature (first derivative constant, thus the second and the higher order derivatives vanish). We also can use additivity of the entropy to write
Stot(Etot) =S(U) +Sres(Etot−U) Defining the inverse temperature
β:= 1 kB T we thus get
pi = e
βUe−βEieSres(Etot−U)/kB
eS(U)/kBeSres(Etot−U)/kB =eβFe−βEi
where F = U−TS and where the terms with Sres have canceled out. We note that pi ∼ e−βEi and that the prefactoreβFhas the role of a normalization factor. In order to normalize, we use∑ipi =1 to write
pi = 1
Ze−βEi Boltzmann distribution with
Z=
∑
i
e−βEi partition sum
We conclude that the probability for a microstate decreases exponentially with its en- ergy. The newly defined quantity Z is the central concept of the canonical ensemble and plays a similar role as the phase space volumeΩin the microcanonical ensemble.
Comments:
1 We note that the expansion aroundEtot−Uis not required to get the Boltzmann factore−βEi. We would have obtained this result also by expanding simply around Etot, because the derivative would also have given 1/T. The normalization is en- sured anyway by the new quantityZ. The expansion used here becomes impor- tant later because only in this way we get the prefactoreβF. As we will discuss below in more detail, this leads to the important relationF =−kBTlnZconnect- ing thermodynamics (F) and statistics (Z).
2 For classical Hamiltonian systems we have p(~q,~p) = 1
Z N!h3N e−βH(~q,~p)
with the HamiltonianHand the partition sum (or, better, the partition function) is
Z= 1
Z
d~qd~p e−βH(~q,~p)
3 From Liouville’s theorem it follows that the Boltzmann distribution is a stationary distribution asρ= ρ(H). Like for the microcanonical distribution, it is reasonable to associate it with equilibrium.
4 The Boltzmann distribution can also be motivated by information theory. In chap- ter 1 we showed that it maximizes entropy under the condition that:
U=hEi=const