Introduction to &
Overview of Statistical & Thermal Physics
• First, some comments on
the importance of
Statistical & Thermal
Physics from some famous
physicists.
• Also, some brief history &
more trivia.
• Most important are some
Basic Definitions &
Terminology.
Basis of Thermodynamics + Some History
• Quote from The Feynman Lectures on Physics
Volume 1, Chapter 1. Feynman is discussing the
fundamental importance of the atomic hypothesis to physics:
“If, in some cataclysm, all of scientific
knowledge were to be destroyed, & only
one sentence passed on to the next
generations of creatures…I believe it is
the atomic hypothesis…all things are
made of atoms; little particles that
move around in perpetual motion,
attracting each other when they are a
little distance apart, but repelling upon
being squeezed into one another”.
Richard P.
Feynman
(1918-1988)
Physics I: “Heat is a Form of Energy”
James Joule’s Experiment
James Joule
(1818-1889)
Joule’s
Experimental
Setup
Mgh = W = Q
& Q = mcT
Work can raise the
water temperature.
Carnot: “Engine of Highest
Possible Efficiency”
Nicolas Léonard
Sadi Carnot
(1796-1832)
• Some history:
“Carnot” is the name of a famous
French family in politics & science.
The Carnot Family
• Lazare Nicolas Marguerite Carnot
(1753-1823), mathematician & politician.
• Nicolas Léonard Sadi Carnot (17961832), mathematician & eldest son of
Lazare, a pioneer of thermodynamics.
• Hippolyte Carnot (1801-1888), politician
& second son of Lazare.
Nicolas Léonard • Marie François Sadi Carnot (1837-1894),
Sadi Carnot
son of Hippolyte,
(1796-1832)
President of France, 1887–1894
• Marie Adolphe Carnot (1839-1920), son
of Hippolyte, mining engineer & chemist.
• A number of streets etc. are named after
this family throughout France.
Importance of the Atomic Hypothesis
Boltzmann's contribution to
19th Century science was vital,
but it had a tragic outcome!
• At the end of the 19th century, several
puzzling facts (which eventually led
to quantum theory), triggered a
reaction against 'materialist' science.
Some people questioned whether
atoms exist.
• Boltzmann, whose work was based
on the atomic concept, found himself
cast as their chief defender & the
debates became increasingly bitter.
Ludwig
Boltzmann
(1844-1906)
Ludwig Boltzmann (1844-1906)
• Prone to bouts of depression,
Boltzmann came to believe that
his life's work had been rejected
by the scientific community.
This wasn’t true!
In 1906, he committed suicide!
Boltzmann
• Despair over rejection, and frustration over being
unable to prove his point, were contributing factors
to his suicide. But the irony of this is huge! Soon
after his death, clinching evidence was found for
atoms, & few would ever doubt their existence again.
Boltzmann’s Gravestone
Boltzmann’s Entropy Formula
Robert Brown & Brownian Motion
Brown (1827)
Observed irregular movement of
pollens in water under a
microscope. [1st observation of
“Brownian motion”: S. Gray, Phil.
Trans. 19, 280, (1696).]
Brown’s Major Contribution
Robert Brown He proved that non-organic
(A botanist!)
particles also have Brownian
motion. Thus, Brownian motion
is not a manifestation of life.
Einstein, Brownian Motion,
& Atomic Hypothesis
“The Miracle Year”
• Einstein published 4 papers in
the Annalen der Physik in 1905.
The Photoelectric Effect
Brownian Motion
Special Theory of Relativity
Albert Einstein • Which topic was his PhD Thesis?
1905
• Which topic was his Nobel Prize?
Einstein on Thermodynamics
• “A theory is the more impressive the
greater the simplicity of its premises, and
the more extended its area of
applicability”.
• “Classical thermodynamics… is the
only physical theory of universal
content which I am convinced that,
within the applicability of its basic
concepts, will never be overthrown”.
Albert Einstein
Eddington on Thermodynamics
• “If someone points out to you that your
pet theory of the universe is in disagreement
with Maxwell’s equations – then so much
the worse for Maxwell’s equations”.
• “But if your theory is found to be against
the second law of thermodynamics I can
offer you no hope; there is nothing for it
but to collapse in deepest humiliation.”
Sir Arthur Eddington, 1929
12
Introduction to Stat Mech
Basic Definitions & Terminology
Thermodynamics (“Thermo”)
is a macroscopic theory!
• Thermo ≡ The study of the
Macroscopic properties of
systems based on a few laws &
hypotheses. It results in
The Laws of Thermodynamics!
Thermodynamics (“Thermo”)
1. Derives relations between the
macroscopic, measureable
properties (& parameters) of a
system (heat capacity, temperature,
volume, pressure, ..).
2. Makes NO direct reference to the
microscopic structure of matter.
Thermo: Makes NO direct reference to
the microscopic structure of matter.
For example, from thermo, we’ll derive
later that, for an ideal gas, the heat
capacities are related by
Cp– Cv = R.
But, thermo gives no prescription for
calculating numerical values for Cp, Cv.
Calculating these requires a microscopic
model & statistical mechanics.
Kinetic Theory is a microscopic theory!
.
1. It applies the Laws of Mechanics (Classical
or Quantum) to a microscopic model of the
individual molecules of a system.
2. It allows the calculation of various
Macroscopically measurable quantities on
the basis of a Microscopic theory applied to
a model of the system.
– For example, it might be able to calculate the specific
heat Cv using Newton’s 2nd Law along with the
known force laws between the particles that make up
the substance of interest.
Kinetic Theory is a
microscopic theory!
3. It uses the microscopic equations
of motion for individual particles.
4. It uses the methods of Probability
& Statistics & the equations of
motion of the particles to calculate
the (thermal average) Macroscopic
properties of a substance.
Statistical Mechanics
(or Statistical Thermodynamics)
1. Ignores a detailed consideration of
molecules as individuals.
2. Is a Microscopic, statistical approach to
the calculation of Macroscopic quantities.
3. Applies the methods of Probability &
Statistics to Macroscopic systems with
HUGE numbers of particles.
Statistical Mechanics
3. For systems with known energy
(Classical or Quantum) it gives
BOTH
A. Relations between Macroscopic
quantities (like Thermo)
AND
B. NUMERICAL VALUES of
them (like Kinetic Theory).
This course covers all three!
1. Thermodynamics
2. Kinetic Theory
3. Statistical Mechanics
Statistical Mechanics:
Reproduces ALL of Thermodynamics
& ALL of Kinetic Theory.
It is more general than either!
A Hierarchy of Theories
(of Systems with a Huge Number of Particles)
Statistical Mechanics
(the most general theory)
___________|__________
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Thermodynamics
Kinetic Theory
(a general, macroscopic theory)
(a microscopic theory most
easily applicable to gases)
Remarks on Statistical & Thermal Physics
A brief look at where we are going.
General survey. No worry about details yet!
The Key Principle of CLASSICAL
Statistical Mechanics is as follows:
• Consider a system containing N particles with 3d
positions r1,r2,r3,…rN, & momenta p1,p2,p3,…pN. The
system is in Thermal Equilibrium at absolute
temperature T. We’ll show that the probability of the
system having energy E is:
P(E) ≡ e[-E/(kT)]/Z
Z ≡ “Partition Function”, T ≡ Absolute
Temperature, k ≡ Boltzmann’s Constant
The Classsical Partition Function
Z ≡ ∫∫∫d3r1d3r2…d3rN d3p1d3p2…d3pN e(-E/kT)
 A 6N Dimensional Integral!
• This assumes that we have already solved
the classical mechanics problem for each
particle in the system so that we know the
total energy E for the N particles as a
function of all positions ri & momenta pi.
E = E(r1,r2,r3,…rN,p1,p2,p3,…pN)D
Don’t panic! We’ll derive this later!
CLASSICAL Statistical Mechanics:
• Let A ≡ any measurable, macroscopic
quantity. The thermodynamic average of A
≡ <A>. This is what is measured. Use
probability theory to calculate <A> :
P(E) ≡
[-E/(kT)]
e
/Z
<A>≡ ∫∫∫(A)d3r1d3r2…d3rN d3p1d3p2…d3pNP(E)
 Another 6N Dimensional Integral!
Don’t panic! We’ll derive this later!
The Key Principle of QUANTUM
Statistical Mechanics is as follows:
• Consider a system which can be in any one of
N quantum states. The system is in Thermal
Equilibrium at absolute temperature T. We’ll
show that the probability of the system being
in state n with energy En is:
P(En) ≡ exp[-En/(kT)]/Z
Z ≡ “Partition Function”
T ≡ Absolute Temperature
k ≡ Boltzmann’s Constant
The Quantum Mechanical
Partition Function
Z ≡ ∑nexp[-En/(kT)]
Don’t panic! We’ll derive this later!
QUANTUM Statistical Mechanics:
• Let A ≡ any measurable, macroscopic
quantity. The thermodynamic average of A
≡ <A>. This is what is measured. Use
probability theory to calculate <A>.
P(En) ≡ exp[(-En/(kT)]/Z
<A> ≡ ∑n <n|A|n>P(En)
<n|A|n> ≡ Quantum Mechanical expectation
value of A in quantum state n.
Don’t panic! We’ll derive this later!
• Question: What is the point of showing this now?
• Question: What is the point of showing this now?
Answer
• Classical & Quantum Statistical Mechanics
both revolve around the calculation of P(E) or P(En).
• Question: What is the point of showing this now?
Answer
• Classical & Quantum Statistical Mechanics
both revolve around the calculation of P(E) or P(En).
• To calculate the probability distribution, we
need to calculate the Partition Function Z
(similar in the classical & quantum cases).
• Question: What is the point of showing this now?
Answer
• Classical & Quantum Statistical Mechanics
both revolve around the calculation of P(E) or P(En).
• To calculate the probability distribution, we
need to calculate the Partition Function Z
(similar in the classical & quantum cases).
Quoting Richard P. Feynman:
“P(E) & Z are at the summit of both
Classical & Quantum
Statistical Mechanics.”
Statistical Mechanics
(Classical or Quantum)
P(E), Z
Equations of
Motion
Calculation of
Measurable
Quantities
The Statistical/Thermal
Physics “Mountain”
P(E), Z
Equations of
Motion
Calculation of
Measurable
Quantities
Statistical/Thermal Physics “Mountain”
• The entire subject is either the “climb” UP to the
summit (calculation of P(E), Z) or the slide DOWN
(use of P(E), Z to calculate measurable properties). On
the way UP: We’ll rigorously define Thermal
Equilibrium & Temperature. On the way
DOWN, we’ll derive all of Thermodynamics
beginning with microscopic theory.