Internal Energy

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

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

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

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

• $\displaystyle Z$ is the [Partition Function]

$\displaystyle U=F+\tau \sigma$

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

• Internal energy of a volume of photons

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

• Energy of ...

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

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