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\begin{document} 
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\exnumber{2065}
\heading{Classical gas with a general dispersion relation} 
\auname{Konstantin Yavilberg}


{\bf The problem:}

\Dn

Consider a gas of non-interacting particles in a $d$ dimensional box, with kinetic energy of the form $E_p = c|p|^s$, and temperature T.

\Dn

\begin{itemize}

\item[(a)] Find the partition function for $N$ particles in the box.

\item[(b)] Define $\gamma = 1+\frac{s}{d}$ and using (a) show that the energy is $E = \frac{N T}{\gamma-1}$.

\item [(c)] Show that the entropy is $S = \frac{N}{\gamma -1}\,\text{ln} \,(P V^\gamma)+ f(N)$, and prove that in an adiabatic process
 $PV^\gamma = $ const.
 
\item [(d)] Show that the heat capacity ratio is $C_P/C_V = \gamma$.

\end{itemize}

\Dn\Dn


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{\bf The solution:}

\Dn

\begin{itemize}
\item[(a)] The partition function for a single particle, which is a simple phase space integration over Boltzman's factor:
\begin{equation}
	Z_1 = \frac{1}{(2\pi)^d}\int d^d x \int d^dp \;e^{-\beta c |p|^s}
	=  \frac{V}{(2\pi)^d} \int d\Omega_d \int_0^\infty p^{d-1} \;e^{-\beta c p^s}dp
\end{equation}
where we used generalized spherical coordinates to represent the integral, which is in the form of a Gamma function:
\begin{equation}
	\label{partition}
	Z_1 = \frac{V}{(2\pi)^d} \;\Omega_d \frac{(c\beta)^{-d/s}}{s} \;\Gamma\left(\frac{d}{s} \right)
\end{equation}
$\Omega_d$ is the solid angle in the appropriate dimension (e.g. $\Omega_2 = 2\pi$ or $\Omega_3 = 4\pi$).\\ Because the gas is classical, the full partition function is just: $Z = \frac{(Z_1)^N}{N!}$.
\item[(b)] The average energy:
\begin{equation}
	E = - \frac{\partial\, \text{ln}\, Z}{\partial \beta} = - N\frac{\partial\, \text{ln}\, Z_1}{\partial \beta}
		+ \frac{\partial \, \text{ln} \, N!}{\partial \beta}
	  = \frac{N}{\beta} \frac{d}{s} 
\end{equation}
using the definition of $\gamma$, the above fraction becomes:
\begin{equation}
	\label{energy}
	E = N T \frac{d}{d(\gamma - 1)} = \frac{NT}{\gamma -1}
\end{equation}
\item[(c)] By equating the two definitions of the Helmholtz function, we can find the entropy:
\begin{equation}
	F = E - \frac{S}{\beta} = -\frac{\,\text{ln}\,Z}{\beta}
	\quad \Longrightarrow \quad S = \beta E + \text{ln}\,Z
\end{equation}
using (\ref{energy}), we get the expression:
\begin{equation}
	\label{entropy1}
	S = \frac{N}{\gamma -1} + N \,\text{ln}\, Z_1 - \text{ln} \, N! = N\, \text{ln} \left(V\beta^{-d/s} \right) + f(N)
\end{equation}
all of the constants and $N$-dependence was inserted into $f(N)$.\\ Using the fact that any classical non-interacting gas is obeyed by the ideal gas equation $\beta = N/PV$, we rewrite (\ref{entropy1}):
\begin{equation}
S =  N\, \text{ln} \left(V\ \left(\frac{PV}{N}\right)^\frac{1}{\gamma -1} \right) + f(N)
  = N\, \text{ln} \left( \left(\frac{P V^\gamma}{N}\right)^\frac{1}{\gamma-1} \right) + f(N)\\
\end{equation}
and again by implicitly inserting into $f(N)$, we get the desired result:
\begin{equation}
	\label{entropy}
  S=  \frac{N}{\gamma-1}\,\text{ln}\,\left(PV^\gamma\right) + f(N) 
\end{equation}
In the adiabatic process $\,\mathchar'26\mkern-12mu dQ = 0$, which implies $S = \text{const.}$
Thus, from (\ref{entropy}) we get the condition:
\begin{equation}
	S = \frac{N}{\gamma-1}\,\text{ln}\,\left(PV^\gamma\right) = \text{const.}
	\quad \Longrightarrow \quad PV^\gamma = \text{const.}
\end{equation}
\item[(d)] The general form of heat capacity is:
\begin{equation}
	C = \frac{\mathchar'26\mkern-12mu dQ}{dT} = T\frac{dS}{dT}
\end{equation}
we keep the pressure or volume constant for $C_P$ and $C_V$ accordingly:
\begin{equation}
	C_P = \left. T\frac{dS}{dT} \right|_{P = \text{const}} = 
	\frac{NT}{\gamma-1} \frac{d\,\text{ln}\,(PV^\gamma)}{dT} = \gamma\frac{NT}{\gamma-1} \frac{d\,\text{ln}\, V}{dT} 
	= N\frac{\gamma}{\gamma-1}
\end{equation} 
\begin{equation}
	C_V = \left. T\frac{dS}{dT} \right|_{V = \text{const}} = \frac{NT}{\gamma-1}\frac{d\,\text{ln}\,P}{dT}
	=\frac{N}{\gamma-1}
\end{equation}
in both cases we used the ideal gas equation, in performing the derivative. Finally, we obatin the relation:
\begin{equation}
	\frac{C_P}{C_V} = \gamma	
\end{equation}
\end{itemize}
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\end{document} 
