---
tags: stat mech, lecture
---

# L03 Entropy, Temperature, Pressure, ChemPot.

## Readings ([reference material](/fy9b5mX0Rui-Sii3wHTncg))
- Reading Kittel Chapter 2
- Sethna Chapter 3

## Notes
[**Lecture 03 class notes**](https://web.physics.utah.edu/~rogachev/7310/L03%20Entropy%20TPM.pdf)
#### 1st Law of Thermodynamics / The Thermodynamic Identity
$dS = (\frac{\partial S}{\partial E})_{VN} dE + (\frac{\partial S}{\partial V})_{EN} dV + (\frac{\partial S}{\partial N})_{EV} dN$

${\hspace{2.5cm} \frac{1}{T} \equiv (\frac{\partial S}{\partial E})_{VN}}$

${\hspace{2.5cm} P \equiv T(\frac{\partial S}{\partial V})_{EN}}$

${\hspace{2.5cm} \mu \equiv -T(\frac{\partial S}{\partial N})_{EV}}$

$\hspace{0.7cm} = \frac{dE}{T} + P\frac{dV}{T} - \mu \frac{dN}{T}$

From which we obtain

$\fbox{dE = TdS - PdV + μdN}$

#### Liouville's theorem
The density of system points in the vicinity of a given system point traveling through phase-space is constant with time. The Liouville equation describes the phase space distribution function's evolution in time. See **canonical transformation** in the [Dictionary](/JUkPDnqjQPmxuCONGrn4yw).

Kittel (Chapter 14, eq (57)):
$f(t+dt, r+dr, v+dv) = f(t,r,v)$

The Liouville Equation:
$\frac{d\rho}{dt}= \frac{\partial\rho}{\partial t} +\sum_{i=1}^n\left(\frac{\partial\rho}{\partial q_i}\dot{q}_i +\frac{\partial\rho}{\partial p_i}\dot{p}_i\right)=0$

#### Partition Functions
Everything we need to find in thermo can b found from the partition function

\begin{array}{lll} &{\rm Function~of~state} & {\rm Statistical~mechanical~expression}\\ \hline\hline U & & -\frac{ d \ln Z}{ d \beta}\\ F & & -k_{ B } T \ln Z\\ \hline S & =-\left(\frac{\partial F}{\partial T}\right)_{V}=\frac{U-F}{T} & k_{ B } \ln Z+k_{ B } T\left(\frac{\partial \ln Z}{\partial T}\right)_{V} \\ p & =-\left(\frac{\partial F}{\partial V}\right)_{T} & k_{ B } T\left(\frac{\partial \ln Z}{\partial V}\right)_{T} \\ H & =U+p V & k_{ B } T\left[T\left(\frac{\partial \ln Z}{\partial T}\right)_{V}+V\left(\frac{\partial \ln Z}{\partial V}\right)_{T}\right] \\ G & =F+p V=H-T S~~~ & k_{ B } T\left[-\ln Z+V\left(\frac{\partial \ln Z}{\partial V}\right)_{T}\right] \\ C_{V} & =\left(\frac{\partial U}{\partial T}\right)_{V} & k_{ B } T\left[2\left(\frac{\partial \ln Z}{\partial T}\right)_{V}+T\left(\frac{\partial^{2} \ln Z}{\partial T^{2}}\right)_{V}\right] \end{array}