Internal Energy

Often also noted as $E$. Is equal to ${\epsilon }_{\text{env}}+{\epsilon }_s$, where ${\epsilon }_{\text{env}}$ is the energy of the environment and ${\epsilon }_s$ is the energy of the system occupying a microstate $s$

$U(\sigma ,V,N)\Rightarrow \mathrm{d}U=\frac{\partial U}{\partial \sigma }\mathrm{d}\sigma +\frac{\partial U}{\partial V}\mathrm{d}V+\frac{\partial U}{\partial N}\mathrm{d}N=\tau \mathrm{d}\sigma -p\mathrm{d}V+\mu \mathrm{d}N$

• Fundamental Thermodynamic Identity
• Associated with 1st Law of Thermodynamics
• $\tau \mathrm{d}\sigma =dQ$
• The reason why $-p\mathrm{d}V$ is negative is because that terms is defined to be the work done on the system

$U(\sigma ,V)=-\frac{\partial }{\partial \beta }\ln Z={\tau }^2\frac{\partial }{\partial \tau }\ln Z$

• $Z$ is the [Partition Function]

$U=F+\tau \sigma$

$U\approx \frac{{\pi }^2}{15}\frac{V{\tau }^4}{(\hslash c{)}^3}$

• Internal energy of a volume of photons

$U(\lambda )=\frac{8\pi hc}{{\lambda }^5(e^{hc\tau /\lambda }-1)}$

• Energy of …

$\Delta (U^2)=-\frac{{\partial }^2}{\partial {\beta }^2}\ln Z$

• Shows that higher order fluctuations of internal energy can be generated by taking higher derivatives of $Z$