Variables of the Canonical Ensemble

The Canonical Ensemble partition function depends on variables including the composition (N), volume (V) and temperature (T) of a given system, where the above partition function equation is still valid with ${\displaystyle Q_{N,V,T}=Q(N,V,T)}$.

The partition function of the canonical ensemble is defined as the sum over all states of a particular system involving each states respective energies, and represented by the following equation,

${\displaystyle Q=\sum _{j}\exp \left({\frac {-E_{j}}{k_{B}T}}\right)}$

In this equation, ${\displaystyle k_{B}}$ represents the Boltzmann Constant with a value of ${\displaystyle k_{B}=1.3806503\times 10^{-23}JK^{-1}}$, T represents the temperature in kelvin and ${\displaystyle E_{j}}$ is the energy at state j.

Internal Energy

The partition function can also be related to all state functions from classical thermodynamics, such as U, A, G and S. The ensemble average of the internal energy in a given system is the thermodynamic equivalent to internal energy, as stated by the Gibbs postulate, and defined by,

${\displaystyle U=\langle E\rangle ={\frac {\sum _{j}E_{j}\exp \left({\frac {-E_{j}}{k_{B}T}}\right)}{Q}}}$

Where the variable Q here represents the partition function term. For a canonical ensemble, the internal energy can be derived from the above equation by considering the derivative of Q with respect to T,

{\displaystyle {\begin{alignedat}{7}{\frac {\partial {}}{\partial {T}}}Q={\frac {\partial {}}{\partial {T}}}\sum _{j}\exp \left({\frac {-E_{j}}{k_{B}T}}\right)&\\=\sum _{j}{\frac {\partial {}}{\partial {T}}}\exp \left({\frac {-E_{j}}{k_{B}T}}\right)&\\=\sum _{j}\exp \left({\frac {-E_{j}}{k_{B}T}}\right){\frac {\partial {}}{\partial {T}}}\left({\frac {-E_{j}}{k_{B}T}}\right)&\\={\frac {E_{j}}{k_{B}}}{\frac {1}{T^{2}}}\exp \left({\frac {-E_{j}}{k_{B}T}}\right)&\\\end{alignedat}}}
{\displaystyle {\begin{alignedat}{7}{\frac {\partial {}}{\partial {T}}}Q=\sum _{j}{\frac {E_{j}}{k_{B}}}{\frac {1}{T^{2}}}\exp \left({\frac {-E_{j}}{k_{B}T}}\right)={\frac {1}{k_{B}T^{2}}}\sum _{j}E_{j}\exp \left({\frac {-E_{j}}{k_{B}T}}\right)&\\={\frac {Q}{Q}}{\frac {1}{k_{B}T^{2}}}\sum _{j}E_{j}\exp \left({\frac {-E_{j}}{k_{B}T}}\right)&\\=Q{\frac {1}{k_{B}T^{2}}}{\frac {\sum _{j}E_{j}\exp \left({\frac {-E_{j}}{k_{B}T}}\right)}{Q}}&\\={\frac {Q}{k_{B}T^{2}}}{\frac {\sum _{j}E_{j}\exp \left({\frac {-E_{j}}{k_{B}T}}\right)}{Q}}={\frac {Q}{k_{B}T^{2}}}\langle E\rangle \end{alignedat}}}

Rearranging this resulting equation for ${\displaystyle \langle E\rangle }$ yields,

${\displaystyle \langle E\rangle =k_{B}T^{2}{\frac {1}{Q}}\left({\frac {\partial {Q}}{\partial {T}}}\right)_{N,V}}$

and from the Gibbs Postulate,

${\displaystyle U=k_{B}T^{2}\left({\frac {\partial {\textrm {ln}}Q}{\partial T}}\right)_{N,V}}$

Helmholtz Energy

The Helmholtz Energy, or Helmholtz Free Energy, of a Canonical Ensemble represents the amount of work and energy obtainable by a certain closed system under constant concentration, volume and temperature. The expression for A utilizes both the internal energy and entropy of the system, and is derived by,

${\displaystyle A=U-TS}$

Where A is the Helmholtz Energy term, U represents the internal energy and S represents the entropy of the system being studied. Substituting the per-determined values for the canonical ensemble,

${\displaystyle A=\langle E\rangle -T\left({\frac {\langle E\rangle }{T}}+k_{B}\ln {Q}\right)}$

${\displaystyle A=\langle E\rangle -\langle E\rangle -k_{B}T\ln {Q}}$

${\displaystyle A=-k_{B}T\ln \left(Q\right)}$

Q represents the partition function for this particular system, and is proportional to the absolute free energy. Thus, an increase in ${\displaystyle \ln(Q)}$ will occur with an increase in the number of total accessible states, and a more negative free energy.

Entropy

The Gibbs definition of Entropy is described by,

${\displaystyle S=-k_{B}\sum _{j}P_{j}\ln {P_{j}}}$

where ${\displaystyle P_{j}}$ represents the Probability of being in state i and described by the weight of the state divided by the sum of all possible weights, such as,

${\displaystyle P_{i}={\frac {\exp \left({\frac {-E_{i}}{k_{B}T}}\right)}{\sum _{j}\exp \left({\frac {-E_{j}}{k_{B}T}}\right)}}}$

and thus large probabilities of multiple states correlate with large values of S. By using the above Gibbs definition, the entropy of a Canonical Ensemble (NVT) can be derived, in terms of Q, to yield the following expression,

${\displaystyle S={\frac {\langle E\rangle }{T}}+k_{B}\ln {Q}}$ or ${\displaystyle S=k_{B}T\left({\frac {\partial {\ln {Q}}}{\partial {T}}}\right)_{N,V}+k_{B}\ln {Q}}$

The value of Q in this equation can also be represented as ${\displaystyle Q={\frac {q^{n}}{N!}}}$, where q is equivalent to the partition function of the molecules, with N representing the number of molecules, in the system.

The partition function is also represented by the denominator of the probability term for a certain state, given by the following,

${\displaystyle P_{i}={\frac {\exp \left({\frac {-E_{j}}{k_{B}T}}\right)}{Q}}}$

Chemical potential

The Chemical Potential, μ, represents the change in the Helmholtz free energy when an additional particle is added to the canonical ensemble system. A heterogeneous system represents a system containing more than one type of gas and the expression for the chemical potential here is μi, where i represents each differing species. This term is derived from Helmholtz free energy, A, with respect to number of molecules for a certain species i. Thus,

${\displaystyle \mu _{i}=\left({\frac {\partial {A}}{\partial {N_{i}}}}\right)_{T,V,N}}$

And utilizing the above Helmholtz expression, the following can be derived,

${\displaystyle =\left({\frac {\partial {}-k_{B}T\ln {Q}}{\partial {N_{i}}}}\right)_{T,V,N}}$

${\displaystyle =-k_{B}T\left({\frac {\partial {\ln {\frac {q_{i}\left(V,T\right)^{N_{i}}}{N_{i}!}}}}{\partial {N_{i}}}}\right)_{T,V,N}}$

${\displaystyle =-k_{B}T\left({\frac {\partial {\ln {q_{i}\left(V,T\right)^{N_{i}}}}-\ln {N_{i}!}}{\partial {N_{i}}}}\right)_{T,V,N}}$

${\displaystyle =-k_{B}T\left({\frac {\partial {N_{i}\ln {q_{i}\left(V,T\right)}}-N_{i}\ln {N_{i}}+N_{i}}{\partial {N_{i}}}}\right)_{T,V,N}}$

${\displaystyle =-k_{B}T\left(\ln {q_{i}}\left(V,T\right)-\ln {N_{i}}-1+1\right)_{T,V,N}}$

${\displaystyle \mu =-k_{B}T\ln {}\left({\frac {q_{i}\left(V,T\right)}{N_{i}}}\right)}$

This definition helps describe the change in the Helmholtz free energy with a changing system composition, such as a reaction forming and/or consuming certain molecules in a system.

Heat Capacity

In this section, we will complete the deviation of heat capacity in terms of the partition function Q.

${\displaystyle Q={\displaystyle \sum _{j}\exp(-E_{j}/k_{B}T)}}$

According to the Gibbs postulate, the ensemble average is equal to the average internal energy, U

${\displaystyle U=\langle E\rangle ={\frac {\displaystyle \sum _{j}E_{j}\exp(-E_{j}/k_{B}T)}{Q}}}$

Substitute equation for partition function into ensemble average.

${\displaystyle \langle E\rangle ={\frac {\displaystyle \sum _{j}E_{j}\exp(-E_{j}/k_{B}T)}{\displaystyle \sum _{j}\exp(-E_{j}/k_{B}T)}}}$

${\displaystyle \beta ={\frac {1}{k_{B}T}}}$ , which is the Boltzmann's Constant.

Now substituting ${\displaystyle \beta }$ you get,

${\displaystyle \langle E\rangle ={\frac {\displaystyle \sum _{j}E_{j}\exp(-\beta E_{j})}{\displaystyle \sum _{j}\exp(-\beta E_{j})}}}$

${\displaystyle C_{v}=\left({\frac {d\langle E\rangle }{dT}}\right)}$, which is the heat capacity.

To find the equation of heat capacity using Statistical Mechanics we first have to differentiate the ensemble average with respect to ${\displaystyle \beta }$.

${\displaystyle \left({\frac {d\langle E\rangle }{dT}}\right)_{N,V}={\frac {\displaystyle \sum _{j}E_{j}\exp(-\beta E_{j})}{\displaystyle \sum _{j}\exp(-\beta E_{j})}}}$ ${\displaystyle ={\frac {-1}{Q}}\left({\frac {dQ}{d\beta }}\right)_{N,V}}$

Therefore, the equation of heat capacity with respect to the partition function Q is,

${\displaystyle C_{v}={\frac {-1}{Q}}\left({\frac {dQ}{d\beta }}\right)_{N,V}}$