Introduction to & Overview of Statistical & Thermal Physics

Basis of Thermodynamics + Some HistoryA Quote from The Feynman Lectures on PhysicsVolume 1, Chapter 1. Feynman is discussing the fundamental importance of the atomic hypothesis to physics:

“If, in some cataclysm, all of scientificknowledge were to be destroyed, & only one sentence passed on to the nextgenerations 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)

Mgh = W = Q & Q = mcT

Work can raise the water temperature.

Joule’s Experimental

Setup

Carnot: “Engine of Highest Possible Efficiency”

• “Carnot” is the name of a famous French family in politics & science.

Nicolas Léonard Sadi Carnot (1796-1832)

The Carnot Family• Lazare Nicolas Marguerite Carnot

(1753-1823), mathematician & politician. • Nicolas Léonard Sadi Carnot (1796-

1832), mathematician & eldest son of Lazare, a pioneer of thermodynamics.

• Hippolyte Carnot (1801-1888), politician & second son of Lazare.

• Marie François Sadi Carnot (1837-1894), son of Hippolyte,

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.

Nicolas Léonard Sadi Carnot (1796-1832)

Importance of the Atomic Hypothesis• Boltzmann's contribution to

19th Century science was vital, but 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 and the debates became increasingly bitter.

Ludwig Boltzmann (1844-1906)

Boltzmann• 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!

• If despair over rejection, or frustration over being unable to prove his point, were contributing factors the irony would be great indeed. Soon after Boltzmann's death, clinching evidence was found for atoms, & few would ever doubt their existence again.

Boltzmann

Robert Brown & Brownian MotionBrown (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 ContributionProved that non-organic particles also have Brownian motion. Thus, Brownian motion is not a manifestation of life.

Robert Brown(A botanist!)

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

Which topic was his PhD thesis?

Which topic was his Nobel Prize?

Albert Einstein1905

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

11

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

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!

1. Thermo: Derives relations between the macroscopic properties (& parameters) of a system (heat capacity, temperature, volume, pressure, ..).

2. 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

BOTHA. Relations between Macroscopic quantities (like Thermo)

ANDB. NUMERICAL VALUES of them (like

Kinetic Theory).

This course covers all three! 1. Thermodynamics 2. Kinetic Theory 3. Statistical

MechanicsStatistical 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)

___________|__________| |

| || |

Thermodynamics Kinetic Theory (a general, macroscopic theory) (a microscopic theory most

easily applicable to gases)

Remarks on Statistical & Thermal PhysicsA brief look at where we are going.

A general survey. No worry about details yet!

The Key Principle of CLASSICALStatistical Mechanics:

• 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)]/ZZ ≡ “Partition Function”, T ≡ Absolute Temperature, k ≡ Boltzmann’s Constant

The Classsical Partition FunctionZ ≡ ∫∫∫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[-E/(kT)]/Z<A>≡ ∫∫∫(A)d3r1d3r2…d3rN d3p1d3p2…d3pNP(E)

Another 6N Dimensional Integral!

Don’t panic! We’ll derive this later!

The Key Principle of QUANTUMStatistical Mechanics:

• 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)]/ZZ ≡ “Partition Function”

T ≡ Absolute Temperaturek ≡ 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

Calculation of Measurable Quantities

The Statistical/Thermal Physics “Mountain”

Equations ofMotion

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.

P(E), Z

Calculation of Measurable Quantities

Equations ofMotion