Study Notes

General
1

Module 1

The Lagrangian L = T-V, where T is the kinetic energy and V the potential energy. Can be re-written in terms of a variable q, which might as well be x. Action is denoted S.

The hamiltonian is the total energy of the system; {\cal H}=T+V

U is internal energy; defined as the sum of all kinetic and potential energies present in the system.
Q is heat
W is work

dU = \delta Q + \delta W; this is called the “total differential of” U

Both Q (heat) and W (work) are path variables.

A subscript variable like \left(\frac{dS}{dT}\right)_{V} means “held constant”.

A “harmonic” potential has V \approx a x^2; that is, F \approx bx.

Conjugate variable

The thermodynamic identity is

dU = T dS - p dV + \sum \mu \ dN

2

Extensive: internal energy depends linearly on the size of the system:

Likewise for the density of a homogeneous system; if the system is divided in half, the extensive properties, such as the mass and the volume, are each divided in half, and the intensive property, the density, remains the same in each subsystem.

http://snst-hu.lzu.edu.cn/zhangyi/ndata/Conjugate_variables_(thermodynamics).html

3

Quantum mechanics

Probability of finding a particle at a point is the squared amplitude.

\Psi is a wave-packet; P = |\Psi(q,t)|^2.

A system satisifying the “markovian property” is memoryless: the future state depends only upon the present state, not past states.

4

Conjugate variables

The thermodynamic identity is

http://snst-hu.lzu.edu.cn/zhangyi/ndata/Conjugate_variables_(thermodynamics).html

5

explains the seebeck effect etc

6

Module 3.1

The n-th “moment” of a probability distribution is
The poisson distribution has a mean value equal to its first moment.

“If the function is a probability distribution, then the first moment is the expected value, the second central moment is the variance, the third standardized moment is the skewness, and the fourth standardized moment is the kurtosis.”