Chapter 2
Statistical
Description of
Systems of
Particles
Preliminary Comments
• Chs. 2 & 3 are, in some places,
very abstract.
• Difficulties students have with this material
are often because of
the abstract ideas & concepts.
• On the other hand, the math in these
chapters is relatively straightforward.
• I assure you that the material will become less
abstract as we proceed through the course.
Discussion of the General Problem
• We’ve reviewed elementary probability & statistics.
• Now, we are finally ready to talk about
PHYSICS
• In this chapter (& the rest of the course) we’ll
combine statistical ideas with the
Laws of Classical or Quantum
Mechanics ≡ Statistical Mechanics
• We can use either a classical or a quantum
description of a system. Of course, which is
valid obviously depends on the problem!!
Four Essential Ingredients for
A Statistical Description of a System
with many particles: (~ outline of Ch. 2!)
1. Specify of the State of the System
2. Choose a Statistical Ensemble
3. Formulate a Basic
Postulate about à-priori
Probabilities.
4. Do Probability Calculations
Four Essential Ingredients for
A Statistical Description of a Physical
System with many particles: (~ outline of Ch. 2!)
1. Specification of the State of the System:
The state referred to here is the
“Macrostate of the System”.
• We’ll thoroughly discuss what we mean by
“Macrostate” & we’ll also discuss that this is very
different than the “Microstate of the System”!
• We’ll also need a detailed method for specifying the
Macrostate. This is discussed in this chapter.
Specification of the System State
Microstate  Microscopic
System State
• Quantum Description of the System:
This means specifying a (large!) set of
quantum numbers.
• Classical Description of the System:
This means specifying a point in a
large dimensional phase space.
Macrostate  Macroscopic
System State
• Quantum Description of the System:
• For an isolated system, this means
specifying a subset of the quantum
states of the system.
• The system is described by macroscopic
parameters (that can be measured).
2. Statistical Ensemble:
• We need to decide exactly which ensemble to use.
This is also discussed in this chapter.
In either Classical Mechanics
or Quantum Mechanics:
• If we had a detailed knowledge of all positions & momenta
of all system particles & if we knew all inter-particle forces,
we could (in principle) set up & solve the coupled, non-
linear differential equations of motion, we could find
EXACTLY the behavior of all particles for all time!
• In practice we don’t have this information. Even if we
did, such a problem is
Impractical, if not Impossible to solve!
• Instead, we use
Statistical/Probabilistic Methods.
Ensemble
• Now, lets think of doing MANY (≡ N) similar
experiments on the system of particles we are considering.
• In general, the outcome of each experiment will be different.
• So, we ask for the PROBABILITY of a particular
outcome. This PROBABILITY ≡ the fraction of cases
out of N experiments which have that outcome.
• This is how probability is determined by experiment.
A goal of STATISTICAL MECHANICS
is to predict this probability theoretically.
• Next, we need to start somewhere, so we need to assume
3. A Basic Postulate about
à-priori Probabilities.
• “à-priori” ≡ prior (based on our prior
knowledge of the system).
• Our knowledge of a given physical system
leads is to expect that there is NOTHING in
the laws of mechanics (classical or quantum)
which would result in the system preferring to
be in any particular one of it’s
Accessible States.
From Webster’s on-line Dictionary
Definition of “à-priori”
• 1a : Deductive
• 1b: Relating to or derived by reasoning from
self-evident propositions.
Synonym to “à-posteriori”
• 1c: Presupposed by experience.
• 2a : Being without examination or analysis.
analysis : Presumptive
• 2b : Formed or conceived beforehand
3. Basic Postulate about
à-priori Probabilities.
• There is NOTHING in the laws of
mechanics (classical or quantum) which
would result in the system preferring
to be in any particular one of it’s
Accessible States.
3. Basic Postulate about
à-priori Probabilities.
• So, (if we have no contrary experimental
evidence) we make the hypothesis that:
it is equally probable
(or equally likely)
that the system is in ANY ONE
of it’s accessible states.
• The hypothesis is that it is equally probable
(equally likely) that the system is in ANY ONE of
it’s accessible states.
• This postulate is reasonable & doesn’t contradict any
laws of mechanics (classical or quantum). Is it correct?
• That can only be confirmed by checking theoretical
predictions & comparing those to experimental
observation!
Physics is an experimental science!!
Sometimes, this postulate is called
The Fundamental Postulate of
(Equilibrium) Statistical Mechanics!
4. Probability Calculations
• Finally, we can do some calculations!
• Once we have the
Fundamental Postulate,
we can use
Probability Theory
to predict the outcome of experiments.
• Now, we will go through steps 1., 2., 3.,
4. again in detail!
Statistical Formulation of the
Mechanical Problem
Section 2.1: Specification of the
System State ≡ Microstate
• Consider any system of particles. We
know that the particles will obey the
laws of Quantum Mechanics (we’ll
discuss the Classical description shortly).
• We’ll emphasize the Quantum treatment.
• Consider any system of particles.
• Using the Quantum treatment,
consider a system with f degrees of
freedom can be described by a (many
particle!) wavefunction
Ψ(q1,q2,….qf,t),
where q1,q2,….qf ≡ Set of f generalized
coordinates required to characterize the
system (needn’t be position coordinates!)
• For a system with f degrees of freedom, the
many particle wavefunction is formally:
Ψ(q1,q2,….qf,t),
q1,q2,….qf ≡ a set of f generalized coordinates
which are required to characterize the system.
• A particular quantum state (macrostate) of the
system is specified by giving values of some
set of f quantum numbers. If we specify Ψ at
a given time t, we can (in principle) calculate
it at any later time by solving the appropriate
Schrödinger Equation
• Now, lets look at some simple
examples, which might also
review a little elementary
Quantum Mechanics
for you.
Example 1
• The system is a single particle, fixed in position.
• It has intrinsic spin ½
– Intrinsic angular momentum = ½ћ.
• In the Quantum Description of this system, the
state of the particle is specified by specifying the
projection m of this spin along a fixed axis
(which we usually call the z-axis).
• The quantum number m can thus have 2 values:
½ (“spin up”) or -½ (“spin down”)
So, there are
2 possible states of the system.
Example 2
• The system is N particles (non-interacting), fixed
in position. Each has intrinsic spin ½ so EACH
particle’s quantum number mi (i = 1,2,…N) can
have one of the 2 values ½.
Suppose that N is HUGE: N ~ 1024.
• The state of this system is then specified by
specifying the values of EACH of the quantum
numbers: m1,m2, .. mN.
 There are (2)N unique states of the
system! With N ~ 1024,
this number is HUGE!!!
Example 3
• The system is a quantum mechanical, onedimensional, simple harmonic oscillator, with
position coordinate x & classical frequency ω.
So the Quantum Energy of this system is:
En = ћω(n + ½),
(n = 0,1,2,3,….).
• The quantum states of this oscillator are then
specified by specifying the quantum number n.
So, there are essentially an
 NUMBER of such states!
Example 4
• The system is N quantum mechanical, onedimensional, simple harmonic oscillators, at positions
xi, with classical frequencies ωi (i = 1,2,.. N).
• The Quantum Energies of each particle in this
system are:
Ei = ћωi(ni + ½),
(ni = 0,1,2,3,….).
• The system’s quantum states are specified by specifying
the values of each quantum number ni.
Here also, there are essentially an
 NUMBER of such states.
• But, there are also a larger number of these than in
Example 3!
Example 5
• The system is one particle, of mass m,
confined to a rectangular box, but otherwise
free. Taking the origin at a corner:
0 ≤ x ≤ Lx, 0 ≤ y ≤ Ly, 0 ≤ z ≤ Lz
The particle is described by the QM
wavefunction ψ(x,y,z), a solution to the
Schrödinger Equation
[-ћ2/(2m)][(∂2/∂x2) + (∂2/∂y2)
+ (∂2/∂z2)]ψ(x,y,z) = Eψ(x,y,z)
• Particle of mass m in a rectangular box:
0 ≤ x ≤ Lx, 0 ≤ y ≤ Ly, 0 ≤ z ≤ Lz
The QM wavefunction ψ(x,y,z), is a solution to
[-ћ2/(2m)][(∂2/∂x2) + (∂2/∂y2)
+ (∂2/∂z2)]ψ(x,y,z) = Eψ(x,y,z)
• Using the boundary condition that ψ = 0 on
the box faces, it can be shown that:
ψ(x,y,z) =
[8/(LxLyLz)]½sin(nxπ/Lx)sin(nyπ/Ly)sin(nzπ/Lz)
• nx, ny, nz are 3 quantum numbers (positive
or negative integers).
ψ(x,y,z) =
[8/(LxLyLz)]½sin(nxπ/Lx)sin(nyπ/Ly)sin(nzπ/Lz)
• nx, ny, nz are 3 quantum numbers
(positive or negative integers).
• So, the particle Quantum Energy is:
E = [(ћ2π2)/(2m)][(nx2/Lx2) + (ny2/Ly2) + (nz2/Lz2)]
• The quantum states of this system are found
by specifying the values of nx, ny, nz.
Again, there are essentially also an
 NUMBER of such states.
Example 6
• N particles, non-interacting, of mass m, confined
to a rectangular box. Take the origin at a corner
0 ≤ x ≤ Lx, 0 ≤ y ≤ Ly, 0 ≤ z ≤ Lz.
Since they are non-interacting, each particle is
described by the QM wavefunction ψi(x,y,z),
which is a solution to the
Schrödinger Equation
[-ћ2/(2m)][(∂2/∂x2) + (∂2/∂y2)
+ (∂2/∂z2)]ψi(x,y,z) = Eiψi(x,y,z)
• N particles of mass m in a rectangular box.
0 ≤ x ≤ Lx, 0 ≤ y ≤ Ly, 0 ≤ z ≤ Lz.
The QM wavefunction ψi(x,y,z), is a solution to
[-ћ2/(2m)][(∂2/∂x2) + (∂2/∂y2)
+ (∂2/∂z2)]ψi(x,y,z) = Eiψi(x,y,z)
• Using the boundary condition that ψ = 0 on
the box faces:
ψi(x,y,z) =
[8/(LxLyLz)]½sin(nxπ/Lx)sin(nyπ/Ly)sin(nzπ/Lz)
• nx, ny, nz are 3 quantum numbers (positive
or negative integers).
ψ(x,y,z) =
[8/(LxLyLz)]½sin(nxπ/Lx)sin(nyπ/Ly)sin(nzπ/Lz)
• nx, ny, nz are 3 quantum numbers (positive
or negative integers).
• Each particle’s Quantum Energy is:
E = [(ћ2π2)/(2m)][(nx2/Lx2) + (ny2/Ly2) + (nz2/Lz2)
• The quantum states of this system are found
by specifying the values of nx, ny, nz. for
each particle.
• Again, there are essentially also an
 NUMBER of such states.
• What about the Classical Description
of the state of a many particle System?
• Of course, the Quantum Description
is always correct!
• However, it is often useful & convenient to
make the
Classical Approximation.
How do we specify the state of
the Classical system then?
Lets start with a very simple case:
A Single Particle in 1 Dimension:
• In classical mechanics, it can be completely
described in terms of it’s generalized coordinate
q & it’s momentum p.
• The usual case is to consider the
Hamiltonian Formulation
of classical mechanics, where we talk of
generalized coordinates q & generalized
momenta p, rather than the
Lagrangian Formulation,
where we talk of coordinates q & velocities (dq/dt).
• Of course, the particle obeys
Newton’s 2nd Law
under the action of the forces on it.
Equivalently, it obeys
Hamilton’s Equations of Motion.
• q & p completely describe the particle
classically. Given q, p at any initial time
(say, t = 0), they can be determined at any other
time t by integrating the equations of motion.
• q & p completely describe the particle classically.
Given q, p at any initial time (say, t = 0), they can be
determined at any other time t by integrating the
Newton’s 2nd Law
Equations of Motion
forward in time.  Knowing q & p at t = 0 in
principle allows us to know them for all time t.
 q & p completely describe the particle for all
time. This situation can be abstractly represented
in a geometric way discussed on the next page.
• Consider the (abstract) 2-dimensional space defined
by q, p: ≡ “Classical Phase Space” of the particle.
• At any time t, stating the (q, p) of
the particle describes it’s
“State”
• Specification of the
“State of the Particle”
is done by stating which point in
this plane the particle “occupies”.
• Of course, as q & p change in time, according to the
equation of motion, the point representing the particle
“State” moves in the plane.
• q, p are continuous variables, so an  number of points are in this
2-Dimensional Classical “Phase Space”
• We’d like to describe the particle “State” classically
in a way that the number of states is countable.
• To do this, it is convenient to
subdivide the ranges of q & p
into very small rectangles of size:
q  p.
• Think of this 2-d phase space as divided into
small cells of equal area: qp ≡ ho
• ho ≡ a small (arbitrary) constant with units of
angular momentum .
• The 2-d phase space has a large number cells of area:
qp = ho.
• The (classical) particle “State” is specified by stating
which cell in phase space the q, p of the particle is in. Or,
by stating that it’s coordinate lies between q & q + q &
that it’s momentum lies between p & p + p.
“State”
• The phase space cell
labeled by the (q,p) that
the particle “occupies”.
• This involves the “small” parameter ho, which is
arbitrary. As a side note, however, we can use
Quantum Mechanics & the
Heisenberg Uncertainty Principle:
“It is impossible to SIMULTANEOUSLY
specify a particle’s position & momentum
to a greater accuracy than qp ≥ ½ћ”
• So, the minimum value of ho is clearly ½ћ.
As ho
½ћ,
• The classical description of the State approaches the
quantum description & becomes more & more accurate.
• Now!! Lets generalize all of this to a
MANY PARTICLE SYSTEM
• 1 particle in 1 dimension means we have to deal
with a 2-dimensional phase space.
• The generalization to N particles is straightforward,
but requires thinking in terms of a very abstract
Multidimensional phase space.
• Consider a system with f degrees of freedom:
 The system is described classically by
f generalized coordinates: q1,q2,q3, …qf
f generalized momenta: p1,p2,p3, …pf.
• A complete description of the classical “State” of
the system requires the specification of:
f generalized coordinates: q1,q2,q3, …qf.
& f generalized momenta: p1,p2,p3, …pf
(N particles, 3-dimensions  f = 3N !!)
• A complete description of the classical “State” of
the system requires the specification of:
f generalized coordinates: q1,q2,q3, …qf.
& f generalized momenta: p1,p2,p3, …pf
(N particles, 3-dimensions  f = 3N !!)
• So, now lets think VERY abstractly in terms of a
2f-dimensional phase space
• The system’s
f generalized coordinates: q1,q2,q3, …qf.
& f generalized momenta: p1,p2,p3, …pf
are regarded as a point in the 2f-dimensional
phase space of the system.
2f-dimensional Phase Space:
f q’s & f p’s:
• Each q & each p label an axis (analogous to the
2-d phase space for 1 particle in 1 dimension).
• Subdivide this phase space into small “cells” of
2f-dimensional “differential volume”:
q1q2q3…qfp1p2p3…p1f ≡ (ho)f
• The classical “State” of the system is then
≡ the cell in this 2f-dimensional phase
space that the system “occupies”.
• Reif, as all modern texts, takes the viewpoint that the
system’s “State” is described by a 2f-dimensional
phase space
≡ “The Gibbs Viewpoint”
The system “State” ≡ The cell in this phase space that
the system “occupies”.
• Older texts take a different viewpoint
≡ “The Boltzmann Viewpoint”:
In this viewpoint, each particle moves in
it’s own 6-dimensional phase space
• In this view, specifying the system “State” requires
specifying each cell in this phase space that each particle
in the system “occupies”.
Summary
Specification of the System State:
In Quantum Mechanics:
• Enumerate & label all possible system quantum states.
In Classical Mechanics:
• Specify which cell in
2f-dimensional phase space
The system is in.
• Need coordinates & momenta of all particles the
system occupies. As ho → ½ћ, the classical &
quantum descriptions become the same.