Transcript File - Statistical Mechanics- PHYS-0704

Statistical Mechanics
Lecture 3-5
Instructor: Suchetana Chatterjee
Presidency University
Physics -0704
The Diatomic Molecule (An example)
https://www.youtube.com/watch?v=0xhtszEjNN0
H = Htrans + Hrot + Hvib + Helec + Hnucleus
Specific Heat of a Diatomic Molecule
Some Essential Points While Calculating the Specific Heat
The Partition function goes as (KT)a/2 , where a is the total number of
squared terms in n the Hamiltonian. Each of these degree of
freedom gets ½ KT (equi-partition)
Why do we do a momentum and position calculation in the classical
case?
Why do we do an energy calculation in the quantum case?
What would we have done if we treated translational motion
quantum mechanically?
It was an act of desperation. For six years I had struggled with the blackbody
theory. I knew the problem was fundamental and I knew the answer. I
had to find a theoretical explanation at any cost, except for the inviolability
of the two laws of thermodynamics.
Max Planck
Planck’s concept of oscillator modes in each frequency interval can be
generalized to the states of a photon gas at a given temperature.
“I have ventured to send you the accompanying
article for your perusal and opinion. I am anxious
to know what you think of it. You will see that I
have tried to deduce the coefficient 8p v2/c3 in
Plank’s Law independent of classical
electrodynamics, only assuming that the
elementary regions in the phase-space has the
content h3. I do not know sufficient German to
translate the paper. If you think the paper worth
publication I shall be grateful if you arrange for
its publication in Zeitschrift für Physic. Though a
complete stranger to you, I do not feel any
hesitation in making such a request. Because we
are all your pupils though profiting only by your
teachings through your writings. I do not know
whether you still remember that somebody from
Calcutta asked your permission to translate your
papers on Relativity in English. You acceded to
the request. The book has since published. I was
the one who translated your paper on Generalised
Relativity.”
Satyen Bose’s letter to Einstein
Cosmic Microwave Background
(the best measured black body in the history of physics)
Cosmic Background Explorer
(1990) : COBE
Relic Radiation of the Big Bang
Phonon Vibrations
In particle and condensed matter physics, Goldstone bosons or
Nambu–Goldstone bosons (NGBs) are bosons that appear
necessarily in models exhibiting spontaneous breakdown of
continuous symmetries.
PHONONS are massless quantum mechanical particles. Phonons are
one example of many like this in many areas of physics. Such
quantum mechanical particles are often called Quasiparticles
Examples of other Quasiparticles:
Rotons: Quantized Normal Modes of molecular rotational excitations.
Magnons: Quantized Normal Modes of magnetic excitations in
magnetic solids
Excitons: Quantized Normal Modes of electron-hole pairs
Polaritons: Quantized Normal Modes of electric polarization
excitations in solids
Problems with the Deby-model (you need periodic lattices and that
changes the dispersion relation)
Ensemble Theory (wikipedia)
Microcanonical: In statistical mechanics, a microcanonical ensemble is the
statistical ensemble that is used to represent the possible states of a mechanical
system which has an exactly specified total energy. The system is assumed to be
isolated in the sense that the system cannot exchange energy or particles with its
environment, so that (by conservation of energy) the energy of the system remains
exactly known as time goes on. The system's energy, composition, volume, and
shape are kept the same in all possible states of the system.
The macroscopic variables of the microcanonical ensemble are quantities such as
the total number of particles in the system (symbol: N), the system's volume
(symbol: V) each which influence the nature of the system's internal states, as well
as the total energy in the system (symbol: E). This ensemble is therefore
sometimes called the NVE ensemble, as each of these three quantities is a
constant of the ensemble.
Very idealistic. Really difficult to find such an ensemble. Universe is close to
it with other complexities
Canonical: In statistical mechanics, a canonical ensemble is the statistical
ensemble that represents the possible states of a mechanical system in thermal
equilibrium with a heat bath at some fixed temperature.[1] The system can
exchange energy with the heat bath, so that the states of the system will differ
in total energy.
The principal thermodynamic variable of the canonical ensemble, determining
the probability distribution of states, is the absolute temperature (symbol: T).
The ensemble typically also depends on mechanical variables such as the
number of particles in the system (symbol: N) and the system's volume
(symbol: V), each of which influence the nature of the system's internal states.
An ensemble with these three parameters is sometimes called the NVT
ensemble.
Typical calculations are done using canonical ensembles.
Grand Canonical In statistical mechanics, a grand canonical ensemble is the
statistical ensemble that is used to represent the possible states of a
mechanical system of particles that is being maintained in thermodynamic
equilibrium (thermal and chemical) with a reservoir.[1] The system is said to
be open in the sense that the system can exchange energy and particles with a
reservoir, so that various possible states of the system can differ in both their
total energy and total number of particles. The system's volume, shape, and
other external coordinates are kept the same in all possible states of the
system.
The thermodynamic variables of the grand canonical ensemble are chemical
potential (symbol: µ) and absolute temperature (symbol: T). The ensemble is
also dependent on mechanical variables such as volume (symbol: V) which
influence the nature of the system's internal states. This ensemble is
therefore sometimes called the µVT ensemble, as each of these three
quantities are constants of the ensemble.
Come on Gibbs! You could have been a bit more creative in giving the
names
Are there other kind of ensembles? Yes we can think of changing
volume.. may be something else depending on the problem. But the
sole idea is that we need to be smart while making our choice of
ensemble calculations.