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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 “the System State”.
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 System
with many particles: (~ outline of Ch. 2!)
1. Specify “the System State”.
The state referred to here is the
“System Macrostate”.
• 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!
• If we could set up & solve the coupled, nonlinear differential equations of motion, we
could (in principle) find EXACTLY the
behavior of all particles for all time!
• In practice we don’t have enough information
to do this. Even if we did, such a problem is
Impractical, if not Impossible to solve!
• Instead, we’ll use
Statistical/Probabilistic
Methods.
Statistical/Probabilistic Methods:
Require choosing an Ensemble
• 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 (micro) States.
Webster’s on-line Dictionary:
Definition of “à-priori”
1a : Deductive
1b: Relating to or derived by reasoning
from self-evident propositions.
a 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 Microstates.
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 microstates.
• The hypothesis is that it is equally probable
(equally likely) that the system is in ANY ONE of
it’s accessible microstates.
• 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 (microstate) 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 S = 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 microstates 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 microstate of this system is then specified by
specifying the values of EACH of the quantum
numbers: m1,m2, .. mN.
 There are (2)N unique microstates
of the system! With N ~ 1024,
this number is HUGE!!!
Example 3
• A system with N = 3 Particles, fixed
in position, each with spin = ½
 Each spin is either “up” (↑, m = ½)
or “down” (↓, m = -½).
• Each particle has a vector magnetic moment μ.
• The projection of μ along a “z-axis” is either:
μz = μ, for spin “up”
or μz = -μ, for spin “down”
Possible States of a 3 Spin System
• Given that we know no other information about
this system, all we can say about it is that
It has Equal Probability of Being Found
in Any One of These 8 States.
• Put this system in an External Magnetic Field H.
• Classical E&M tells us that a particle with
magnetic moment μ in an external field H has energy:
ε = - μH
• Combine this with the Quantum Mechanical result:
• This tells us that each particle has 2 possible energies:
ε+ ≡ - μH for spin “up”
ε- ≡ μH for spin “down”
 So, for 3 particles, the Microstate of the
system is specified by specifying each m =  ½
 There are (2)3 = 8 Possible Microstates!!
Example 4
• 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 5
• 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 6
• 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)
• 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 7
• 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 a Classical Description of the
Microstate 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
Microstate of the 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 describe the particle classically. Given q, p
at time t = 0, they can be determined at any other
time t by integrating the
nd
Newton’s 2 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
“Microstate”
• Specification of the
“Particle Microstate”
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 want to describe the particle “Microstate” 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 “Microstate” 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.
“Microstate”
• 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 Microstate
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 “Microstate”
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 “Microstate”
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
• A complete description of the classical “Microstate”
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 “Microstate” 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 2fdimensional 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 Microstate:
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.