Table of Contents
Prerequisites and Historical note
This is the notes is the made using lectures of various premium institutes across India.
- Various mathematical concepts such as Advanced calculus which includes multivariable integrations, complex variables, combinatronics and so on.
- Basic programming in a language.
- Knowledge of Classical Mechanics and, Quantum Mechanics upto undergraduate level. ( you can refer to Classical Mechanics by Herbert Goldstein and Introduction to quantum mechanics by David J. Griffiths)
- Historically, Thermodynamics was the conceptual preceder to Statistical Mechanics.
- Since the antiquity to 1600 A.D. notion of heat and temperature were widely accepted. It was believed at that time that heat was associated with the motion of microscopic constituents of matter
- Later during 1700s and first half of 1800s,the wrong notion that heat was instead a fluid like substance started becoming popular.
- In 1850 experiments of James Joule and others show that heat is a form of energy.
- Sadi Carnot had explained the relation between heat and energy. Which was important in the development of steam engines.
- In 1850 Rudolf Clausius and William Thomson (Lord Kelvin) formulated the first law which is the idea that total energy is conserved. The second law of thermodynamics which is the idea that heat cannot be completely converted to work was also formulated.
- In 1738 Daniel Bernoulli has pointed out that gases consist of molecules in motion. Clausius revived this idea in 1857.
- In 1860 James clerk Maxwell derived the expected distribution of molecular speeds in a gas by taking into account molecular conditions.
- In 1877 Ludwig Boltzmann constructed an equation that he thought could describe the detail time evolution of a gas regardless of whether it was in equilibrium or not.
- In the 1860, Clausius had introduced entropy as a ratio of heat to temperature, and had stated the second law in terms of the increase of this quantity.
- Boltzmann then showed that his equation implied the so called H theorem, which states that a quantity equal to entropy in equilibrium must always increase with time. Since molecular collisions were assumed reversible, his derivation could be run in reverse, and would then imply opposite of second law. Boltzmann made the implicit assumption that the motion was uncorrelated before collision but not after which imposes irreversibility.
- In 1900, Gibbs introduced the notion of an ensemble - a collection of many possible states of a system, each assigned a certain probability.
He argued that if the time evolution of a single state were to visit all other states in the ensemble, the so called ergodic hypothesis - then
averaged over a sufficiency long time a single state would behave in a way that was typical of the ensemble. Gibbs also gave qualitative arguments that entropy would increase if it were measured in a “coarse grained” way in which nearby states were not distributed.