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The Partition Function

[ "article:topic", "Author tag:Tuckerman", "showtoc:no" ]
  • Page ID
    5169
  • 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

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