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The Partition Function

Consider two systems (1 and 2) in thermal contact such that

\(N_2\) \(\gg\) \(N_1\)  
\(E_2\) \(\gg E_1\)     
\(N\) $\textstyle =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ \(N_1 + N_2;              E = E_1 + E_2 \)  
\(\text {dim} (x_1)\) \(\gg\) \(\text {dim} (x_2) \)  


and the total Hamiltonian is just \(H (x) = H_1 (x_1) + H_2 (x_2) \)

Since system 2 is infinitely large compared to system 1, it acts as an infinite heat reservoir that keeps system 1 at a constant temperature \(T\) without gaining or losing an appreciable amount of heat, itself. Thus, system 1 is maintained at canonical conditions, \(N, V, T\).

The full partition function \(\Omega (N, V, E )\) for the combined system is the microcanonical partition function

\[\Omega(N,V,E) = \int dx \delta(H(x)-E) = \int dx_1 dx_2  \delta (H_1(x_1) + H_2(x_2)-E)\]

Now, we define the distribution function, \(f (x_1)\) of the phase space variables of system 1 as

\[ f(x_1) = \int dx_2 \delta (H_1(x_1)+ H_2(x_2)-E)\]

Taking the natural log of both sides, we have

\[ \ln f(x_1) = \ln \int dx_2 \delta (H_1(x_1) + H_2(x_2) - E)\]

Since \(E_2 \gg E_1 \), it follows that \(H_2 (x_2) \gg H_1 (x_1)\), and we may expand the above expression about \(H_1 = 0 \). To linear order, the expression becomes
\(\ln f (x_1)\) $\textstyle =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$

\( \ln \int dx_2 \delta (H_2(x_2)-E) + H_1(x_1) \frac {\partial }{ \partial H_1 (x_1)} \ln \int  dx_2 \delta (H_1(x_1) + H_2(x_2) - E) \vert _{H_1(x_1)=0}\)

 
       
  $\textstyle =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$

\( \ln \int dx_2 \delta (H_2(x_2)-E) -H_1(x_1) \frac {\partial}{\partial E} \ln \int dx_2 \delta (H_2(x_2)-E)\)

 

where, in the last line, the differentiation with respect to \(H_1\) is replaced by differentiation with respect to \(E\). Note that

\( \ln \int dx_2 \delta (H_2( _2)-E)\)

$\textstyle =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ \(\frac {S_2 (E)}{k}\)  
       

\( \frac {\partial}{\partial E} \ln \int dx_2  \delta (H_2(x_2)-E)\)

$\textstyle =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$

\( \frac {\partial}{\partial E} \frac {S_2(E)}{k} = \frac {1}{kT}\)

 


where \(T\) is the common temperature of the two systems. Using these two facts, we obtain

\(\ln f (x_1)\) $\textstyle =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ \(\frac {S_2 (E)}{k} - \frac {H_1 (x_1)}{kT} \)  
       
\(f (x_1)\) $\textstyle =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ \(e^{\frac {S_2(E)}{k}}e^{\frac {-H_1(x_1)}{kT}} \)  

Thus, the distribution function of the canonical ensemble is

\[f(x) \propto e^{\frac {-H(x)}{kT}} \]

The prefactor \(exp (\frac {S_2 (E) }{k} ) \) is an irrelevant constant that can be disregarded as it will not affect any physical properties.

The normalization of the distribution function is the integral:

\[\int dxe^{\frac {-H(x)}{kT}} \equiv Q(N,V,T)\]

where \(Q (N, V, T ) \) is the canonical partition function. It is convenient to define an inverse temperature \(\beta = \frac {1}{kT} \).  \(Q (N, V, T )\) is the canonical partition function. As in the microcanonical case, we add in the ad hoc quantum corrections to the classical result to give

\[ Q(N,V,T) = \frac {1}{N!h^{3N}} \int dx e^{-\beta H(x)}\]

The thermodynamic relations are thus,

Hemlholtz free energy

\[ A (N, V, T ) = - \frac {1}{\beta} \ln Q (N, V, T ) \]

\[ P = -\left ( \frac {\partial A}{\partial V} \right )_{N,T} = kT \left( \frac {\partial \ln Q(N,V,T)}{\partial V} \right )_{N,T}\]

To see that this must be the definition of \(A (N, V, T ) \) , recall the definition of \(A\):

\[ A = E - TS = \langle H (x) \rangle - TS \]

But we saw that

\[ S = - \left ( \frac {\partial A}{\partial T } \right ) _{N,V} \]

 Substituting this in gives

 

\[\frac {\partial A}{\partial T} = \frac {\partial A}{\partial \beta} \frac {\partial \beta }{\partial T} = - \frac {1}{kT^2} \frac {\partial A}{\partial \beta} \]
it follows that

\[ A = \langle H(x) \rangle + \beta \frac {\partial A}{\partial \beta}\]

This is a simple differential equation that can be solved for \(A\). We will show that the solution is

\[ A = - \frac {1}{\beta} \ln Q (\beta)\]

Note that

\[ \beta \frac {\partial A}{\partial \beta} = \frac {1}{\beta} \ln Q (\beta) - \frac {1}{Q} \frac {\partial Q}{\partial \beta} = A - \langle H(x)\rangle\]

Substituting in gives, therefore

\[ A = \langle H(x)\rangle + A - \langle H(x)\rangle = A\]
so this form of \(A\) satisfies the differential equation.Other thermodynamics follow:

\[ A = \langle H(x) \rangle - T \frac {\partial A}{\partial T}\]

or, noting that

Average energy:

\(E\) $\textstyle =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$

\(\langle H(x)\rangle = \frac {1}{Q} C_N \int dx H(x) e^{-\beta H(x)}\)

 
  $\textstyle =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$

\(- \frac {\partial}{\partial \beta} \ln Q(N,V,T)\)

 

Pressure

\[ P = -\left ( \frac {\partial A}{\partial V} \right )_{N,T} = kT \left ( \frac {\partial \ln Q (N,V,T)}{\partial V} \right )_{N,T}\]

Entropy

\(S\) $\textstyle =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$

\(- \frac {\partial A}{\partial T} = - \frac {\partial A}{\partial \beta} \frac {\partial \beta}{\partial T} = \frac {1}{kT^2}  \frac {\partial A}{\partial \beta} \)

 
  $\textstyle =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$

\(k \beta^2 \frac {\partial}{\partial \beta} \left( -\frac {1}{\beta} \ln Q(N,V,T)\right ) = -k \beta \frac {\partial \ln Q}{\partial \beta} + k \ln Q\)

 
  $\textstyle =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$

\( k \beta E + k \ln Q = k \ln Q + \frac {E}{T}\)

 

Heat capacity at constant volume

\[ C_V = \left ( \frac {\partial E}{\partial T} \right )_{N,V} = \frac {\partial E}{\partial \beta} \frac {\partial \beta}{\partial T} = k \beta^2  \frac {\partial}{\partial \beta^2} \ln Q (N,V,T)\]