4.4: Ordinary Canonical Ensemble (OCE)
- Page ID
- 18565
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\( \newcommand\Vlambda{\vec\lambda}\)
\( \newcommand\Vmu
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\( \newcommand\Vnu
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\( \newcommand\Vxi
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\( \newcommand\Vom
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\( \newcommand\Vpi
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\( \newcommand\Vvarpi
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\( \newcommand\Vrho
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\( \newcommand\Vvarrho
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\( \newcommand\Vsigma
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\( \newcommand\Vvarsigma
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\( \newcommand\Vtau
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\( \newcommand\Vupsilon
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\( \newcommand\Vphi
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\( \newcommand\Vvarphi
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\( \newcommand\Vchi
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\( \newcommand\Vpsi
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\( \newcommand\Vomega
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\( \newcommand\VGamma
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\( \newcommand\VDelta
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Canonical Distribution and Partition Function
Consider a system \(S\) in contact with a world \(W\), and let their union \(U=W\cup S\) be called the ‘universe’. The situation is depicted in Figure [universe]. The volume \(V\ns_{\ssr{S}}\) and particle number \(N\ns_{\ssr{S}}\) of the system are held fixed, but the energy is allowed to fluctuate by exchange with the world \(W\). We are interested in the limit \(N\ns_{\ssr{S}}\to\infty\), \(N\ns_{\ssr{W}}\to\infty\), with \(N\ns_{\ssr{S}}\ll N\ns_{\ssr{W}}\), with similar relations holding for the respective volumes and energies. We now ask what is the probability that \(S\) is in a state \(\sket{n}\) with energy \(E\ns_n\). This is given by the ratio
\[\begin{align} P\ns_n&=\lim_{\RDelta E\to 0}\ {D\ns_{\ssr{W}}(E\ns_{\ssr{U}}-E\ns_n)\,\RDelta E\over D\ns_{\ssr{U}}(E\ns_{\ssr{U}})\,\RDelta E}\label{OCErat}\\ &={\hbox{ # of states accessible to $W$ given that $E\ns_{\ssr{S}}=E\ns_n$}\over \hbox{ total # of states in $U$}}\ .\bvph \end{align}\]
Then
\[\begin{align} \ln P\ns_n&=\ln D\ns_{\ssr{W}} (E\ns_{\ssr{U}}-E\ns_n) - \ln D\ns_{\ssr{U}} (E\ns_{\ssr{U}})\\ &=\ln D\ns_{\ssr{W}} (E\ns_{\ssr{U}})- \ln D\ns_{\ssr{U}}(E\ns_{\ssr{U}}) - E\ns_n\>{\pz\ln D\ns_{\ssr{W}}(E)\over \pz E}\bigg|\nd_{E=E\ns_{\ssr{U}}} \!\! + \ldots\\ &\equiv -\alpha-\beta E\ns_n\ . \end{align}\]
The constant \(\beta\) is given by
\[\beta={\pz\ln D\ns_{\ssr{W}}(E)\over\pz E}\bigg|\nd_{E=E\ns_{\ssr{U}}} = {1\over \kT}\ .\]
Thus, we find \(P\ns_n=e^{-\alpha}\,e^{-\beta E\ns_n}\). The constant \(\alpha\) is fixed by the requirement that \(\sum_n P\ns_n=1\):
\[P\ns_n={1\over Z}\, e^{-\beta E\ns_n}\qquad,\qquad Z(T,V,N)=\sum_n e^{-\beta E\ns_n}=\Tra e^{-\beta \HH}\ .\]
We’ve already met \(Z(\beta)\) in Equation \ref{Zlap} – it is the Laplace transform of the density of states. It is also called the partition function of the system \(S\). Quantum mechanically, we can write the ordinary canonical density matrix as
\[\vrhhat={e^{-\beta\HH}\over \Tra e^{-\beta\HH}}\quad,\]
which is known as the Gibbs distribution. Note that \(\big[\vrhhat,\HH\big]=0\), hence the ordinary canonical distribution is a stationary solution to the evolution equation for the density matrix. Note that the OCE is specified by three parameters: \(T\), \(V\), and \(N\).
The difference between \(P(E_n)\) and \(P_n\)
Let the total energy of the Universe be fixed at \(E\ns_{\ssr{U}}\). The joint probability density \(P(E\ns_{\ssr{S}},E\ns_{\ssr{W}})\) for the system to have energy \(E\ns_\RS\) and the world to have energy \(E\ns_{\ssr{W}}\) is
\[P(E\ns_{\ssr{S}},E\ns_{\ssr{W}})=D\ns_{\ssr{S}}(E\ns_{\ssr{S}}) \, D\ns_{\ssr{W}}(E\ns_{\ssr{W}}) \,\delta(E\ns_{\ssr{U}}-E\ns_{\ssr{S}}-E\ns_{\ssr{W}}) \big/ D\ns_{\ssr{U}}(E\ns_{\ssr{U}})\ ,\]
where
\[D\ns_{\ssr{U}}(E\ns_{\ssr{U}})=\impi dE\ns_{\ssr{S}}\>D\ns_{\ssr{S}}(E\ns_{\ssr{S}})\,D\ns_{\ssr{W}}(E\ns_{\ssr{U}}-E\ns_{\ssr{S}})\ ,\]
which ensures that \(\int\!dE\ns_{\ssr{S}}\int\!dE\ns_{\ssr{W}}\,P(E\ns_{\ssr{S}},E\ns_{\ssr{W}})=1\). The probability density \(P(E\ns_{\ssr{S}})\) is defined such that \(P(E\ns_{\ssr{S}})\,dE\ns_{\ssr{S}}\) is the (differential) probability for the system to have an energy in the range \([E\ns_{\ssr{S}},E\ns_{\ssr{S}}+dE\ns_{\ssr{S}}]\). The units of \(P(E\ns_{\ssr{S}})\) are \(E^{-1}\). To obtain \(P(E\ns_{\ssr{S}})\), we simply integrate the joint probability density \(P(E\ns_{\ssr{S}},E\ns_{\ssr{W}})\) over all possible values of \(E\ns_{\ssr{W}}\), obtaining
\[P(E\ns_{\ssr{S}})={D\ns_{\ssr{S}}(E\ns_{\ssr{S}})\,D\ns_{\ssr{W}}(E\ns_{\ssr{U}}-E\ns_{\ssr{S}})\over D\ns_{\ssr{U}}(E\ns_{\ssr{U}})}\ ,\]
as we have in Equation \ref{OCErat}.
Now suppose we wish to know the probability \(P\ns_n\) that the system is in a particular state \(\sket{n}\) with energy \(E\ns_n\). Clearly
\[P\ns_n=\lim_{\RDelta E\to 0}{\hbox{ probability that $E\ns_{\ssr{S}}\in[E\ns_n,E\ns_n+\RDelta E]$}\over \hbox{ \ \# of S states with $E\ns_{\ssr{S}}\in [E\ns_n,E\ns_n+\RDelta E]$\ }} ={P(E\ns_n)\,\RDelta E\over D\ns_{\ssr{S}}(E\ns_n)\,\RDelta E} = {D\ns_{\ssr{W}}(E\ns_{\ssr{U}}-E\ns_n)\over D\ns_{\ssr{U}}(E\ns_{\ssr{U}})}\ .\]
Additional remarks
The formula of Equation \ref{OCErat} is quite general and holds in the case where \(N\ns_{\ssr{S}}/N\ns_{\ssr{W}}=\CO(1)\), so long as we are in the thermodynamic limit, where the energy associated with the interface between S and W may be neglected. In this case, however, one is not licensed to perform the subsequent Taylor expansion, and the distribution \(P\ns_n\) is no longer of the Gibbs form. It is also valid for quantum systems6, in which case we interpret \(P\ns_n=\texpect{n}{\vrh\ns_{\ssr{S}}}{n}\) as a diagonal element of the density matrix \(\vrh\ns_{\ssr{S}}\). The density of states functions may then be replaced by
\[\begin{split} D\ns_{\ssr{W}}(E\ns_{\ssr{U}}-E\ns_n)\,\RDelta E &\to e^{S\ns_{\ssr{W}}(E\ns_{\ssr{U}}-E\ns_n\,,\,\RDelta E)} \equiv \mathop{\textsf{Tra}}_{\ssr{W}} \hskip-0.7cm\int\limits_{E\ns_{\ssr{U}}-E\ns_n}^{E\ns_{\ssr{U}}-E\ns_n+\RDelta E}\hskip-0.7cm dE\>\delta(E-\HH\ns_{\ssr{W}})\\ D\ns_{\ssr{U}}(E\ns_{\ssr{U}})\,\RDelta E &\to e^{S\ns_{\ssr{U}}(E\ns_{\ssr{U}}\,,\,\RDelta E)} \equiv \mathop{\textsf{Tra}}_{\ssr{U}} \hskip-0.4cm\int\limits_{E\ns_{\ssr{U}}}^{E\ns_{\ssr{U}}+\RDelta E}\hskip-0.4cm dE\>\delta(E-\HH\ns_{\ssr{U}})\quad. \end{split}\]
The off-diagonal matrix elements of \(\vrh_{\ssr{S}}\) are negligible in the thermodynamic limit.
Averages within the OCE
To compute averages within the OCE,
\[\big\langle\HA\big\rangle=\Tra\!\big(\vrhhat\,\HA\big) ={\sum_n\texpect{n}{\HA}{n}\>e^{-\beta E\ns_n}\over\sum_n e^{-\beta E\ns_n}}\ ,\]
where we have conveniently taken the trace in a basis of energy eigenstates. In the classical limit, we have
\[\vrh(\Bvphi)={1\over Z}\,e^{-\beta \HH(\Bvphi)} \quad,\quad Z=\Tra e^{-\beta \HH}=\int\!\! d\mu \> e^{-\beta \HH(\Bvphi)}\ ,\]
with \(d\mu=\frac{1}{N!}\prod_{j=1}^N (d^d q\nd_j\,d^d p\nd_j / h^d)\) for identical particles (‘Maxwell-Boltzmann statistics’). Thus,
\[\langle A \rangle =\Tra(\vrh A) = {\int\!\! d\mu\>A(\Bvphi)\,e^{-\beta \HH(\Bvphi)}\over \int\!\! d\mu\> e^{-\beta \HH(\Bvphi)}}\ .\]
Entropy and Free Energy
The Boltzmann entropy is defined by
\[S=-\kB\Tra\!\big(\vrhhat\ln\vrhhat) = -\kB\sum_n P\ns_n\,\ln P\ns_n\ .\]
The Boltzmann entropy and the statistical entropy \(S=\kB\ln D(E)\) are identical in the thermodynamic limit. We define the Helmholtz free energy \(F(T,V,N)\) as
\[F(T,V,N)=-\kT\ln Z(T,V,N)\ ,\]
hence
\[P\ns_n=e^{\beta F}\, e^{-\beta E\ns_n} \qquad,\qquad \ln P\ns_n=\beta F-\beta E\ns_n\ .\]
Therefore the entropy is
\[S=-\kB\sum_n P\ns_n\, \big(\beta F-\beta E\ns_n\big)\\ =-{F\over T} + {\langle \,\HH\,\rangle\over T}\ ,\]
which is to say \(F=E-TS\), where
\[E=\sum_n P\ns_n \,E\ns_n = {\Tra \HH \,e^{-\beta\HH}\over \Tra e^{-\beta\HH}}\]
is the average energy. We also see that
\[Z=\Tra e^{-\beta\HH}=\sum_n e^{-\beta E\ns_n} \quad\Longrightarrow\quad E={\sum_n E\ns_n\,e^{-\beta E\ns_n}\over\sum_n e^{-\beta E\ns_n}}=-{\pz\over\pz\beta}\,\ln Z={\pz\over\pz\beta}\big(\beta F\big)\ .\]
Thus, \(F(T,V,N)\) is a Legendre transform of \(E(S,V,N)\), with
\[dF=-S\,dT - p\,dV + \mu\,dN\ ,\]
which means
\[S=-\pabc{F}{T}{V,N} \qquad,\qquad p=-\pabc{F}{V}{T,N} \qquad,\qquad \mu=+\pabc{F}{N}{T,V}\ .\]
Fluctuations in the OCE
In the OCE, the energy is not fixed. It therefore fluctuates about its average value \(E=\langle \HH\rangle\). Note that
\[\begin{split} -{\pz E\over\pz\beta}&=\kB T^2\,{\pz E\over\pz T}={\pz^2\ln Z\over\pz\beta^2}\\ &={\Tra \HH^2\,e^{-\beta\HH}\over \Tra e^{-\beta\HH}} - \Bigg({\Tra \HH \,e^{-\beta\HH}\over \Tra e^{-\beta\HH}}\Bigg)^{\!\!2}\\ &=\blangle\HH^2\brangle - \blangle\HH\brangle^2\ . \end{split}\]
Thus, the heat capacity is related to the fluctuations in the energy, just as we saw at the end of §4:
\[C\ns_V=\pabc{E}{T}{V,N}={1\over \kB T^2}\, \Big(\blangle \HH^2\brangle - \blangle\HH\brangle^2\Big)\]
For the nonrelativistic ideal gas, we found \(C\ns_V={d\over 2}\,N\kB\), hence the ratio of RMS fluctuations in the energy to the energy itself is
\[{\sqrt{\blangle\,(\RDelta\HH)^2\,\brangle} \over\langle\HH\rangle}= {\sqrt{\kB T^2\,C\ns_V}\over {d\over 2}N\kT} = \sqrt{2\over Nd}\ ,\]
and the ratio of the RMS fluctuations to the mean value vanishes in the thermodynamic limit.
The full distribution function for the energy is
\[P(\CE)=\blangle\delta(\CE-\HH)\brangle={\Tra \delta(\CE-\HH)\,e^{-\beta\HH}\over \Tra e^{-\beta\HH}}={1\over Z}\,D(\CE)\,e^{-\beta \CE}\ .\]
Thus,
\[P(\CE)={e^{-\beta\left[\CE-TS(\CE)\right]}\over\int\!d\CE'\,e^{-\beta\left[\CE'-TS(\CE')\right]}}\ , \label{PEOCE}\]
where \(S(\CE)=\kB\ln D(\CE)\) is the statistical entropy. Let’s write \(\CE=E+\delta \CE\), where \(E\) extremizes the combination \(\CE-T\,S(\CE)\), the solution to \(T\,S'(E)=1\), where the energy derivative of \(S\) is performed at fixed volume \(V\) and particle number \(N\). We now expand \(S(E+\delta \CE)\) to second order in \(\delta \CE\), obtaining
\[S(E+\delta \CE)=S(E) + {\delta \CE\over T} -{\big(\delta \CE\big)^2\over 2 T^2 \,C\ns_V}\, + \ldots\]
Recall that \(S''(E)={\pz\over\pz E}\left({1\over T}\right) = -{1\over T^2 C\ns_V}\). Thus,
\[\CE-T\,S(\CE)=E - T\,S(E) + {(\delta \CE)^2\over 2 T\,C\ns_V} + \CO\big((\delta \CE)^3\big)\ . \label{EminusTS}\]
Applying this to both numerator and denominator of Equation \ref{PEOCE}, we obtain7
\[P(\CE)=\CN\,\exp\Bigg[\!-{(\delta \CE)^2\over 2\kB T^2\,C\ns_V}\Bigg]\ ,\]
where \(\CN=(2\pi\kB T^2 C\ns_V)^{-1/2}\) is a normalization constant which guarantees \(\int\!d\CE\,P(\CE)=1\). Once again, we see that the distribution is a Gaussian centered at \(\langle\CE\rangle = E\), and of width \((\RDelta \CE)\nd_{\ssr{RMS}}=\sqrt{\kB T^2\,C\ns_V}\). This is a consequence of the Central Limit Theorem.
Thermodynamics revisited
The average energy within the OCE is
\[E=\sum_n E\ns_n P\ns_n\ ,\]
and therefore
\[\begin{split} dE=& \sum_n E\ns_n \,dP\ns_n + \sum_n P\ns_n\,dE\ns_n\\ &=\dbar Q-\dbar W\ , \end{split}\label{smfl}\]
where
\[\begin{aligned} \dbar W&=-\sum_n P\ns_n\,dE\ns_n\\ \dbar Q&=\sum_n E\ns_n\,dP\ns_n\ .\end{aligned}\]
Finally, from \(P\ns_n=Z^{-1}\,e^{-E\ns_n/k\ns_\RB T}\), we can write
\[E\ns_n=-\kT\ln Z - \kT\ln P\ns_n\ ,\]
with which we obtain
\[\begin{split} \dbar Q&=\sum_n E\ns_n\,dP\ns_n\\ &=-\kT\ln Z\sum_n dP\ns_n - \kT\sum_n \ln P\ns_n\>dP\ns_n\\ &=T \,d\Big(\!-\kB\sum_n P\ns_n\ln P\ns_n\Big)=T\,dS\ . \end{split}\]
Note also that
\[\begin{align} \dbar W&=-\sum_n P\ns_n \, dE\ns_n \\ &=-\sum_nP\ns_n \Bigg(\!\sum_i {\pz E\ns_n\over\pz X\ns_i}\>dX\ns_i\Bigg)\\ &=-\sum_{n,i} P\ns_n\,\expect{n}{\pz \HH\over\pz X\ns_i}{n}\>dX\ns_i \equiv\sum_i F\ns_i\,dX\ns_i\ , \end{align} \label{workeqn}\]
so the generalized force \(F\ns_i\) conjugate to the generalized displacement \(dX\ns_i\) is
\[F\ns_i=-\sum_n P\ns_n\,{\pz E\ns_n\over\pz X\ns_i}=-\,\bigg\langle {\pz\HH\over\pz X\ns_i}\bigg\rangle\ . \label{thermforce}\]
This is the force acting on the system8. In the chapter on thermodynamics, we defined the generalized force conjugate to \(X\ns_i\) as \(y\ns_i\equiv - F\ns_i\).
Thus we see from Equation \ref{smfl} that there are two ways that the average energy can change; these are depicted in the sketch of Figure \(\PageIndex{1}\). Starting from a set of energy levels \(\{E\ns_n\}\) and probabilities \(\{P\ns_n\}\), we can shift the energies to \(\{E'_n\}\). The resulting change in energy \((\RDelta E)\ns_{\ssr{I}}=-W\) is identified with the work done on the system. We could also modify the probabilities to \(\{P'_n\}\) without changing the energies. The energy change in this case is the heat absorbed by the system: \((\RDelta E)\ns_{\ssr{II}} = Q\). This provides us with a statistical and microscopic interpretation of the First Law of Thermodynamics.
Generalized Susceptibilities
Suppose our Hamiltonian is of the form
\[\HH=\HH(\lambda)=\HH\ns_0-\lambda\,{\hat Q}\ ,\]
where \(\lambda\) is an intensive parameter, such as magnetic field. Then
\[Z(\lambda)=\Tra e^{-\beta(\HH\ns_0-\lambda{\hat Q})}\]
and
\[{1\over Z}\,{\pz Z\over\pz \lambda}=\beta\cdot{1\over Z}\Tra\Big( {\hat Q}\,e^{-\beta\HH(\lambda)}\Big)=\beta\>\langle{\hat Q}\rangle\ .\]
But then from \(Z=e^{-\beta F}\) we have
\[Q(\lambda,T)=\langle\,{\hat Q}\,\rangle=-\pabc{F}{\lambda}{T}\ .\]
Typically we will take \(Q\) to be an extensive quantity. We can now define the susceptibility \(\xhi\) as
\[\xhi={1\over V}{\pz Q\over\pz\lambda}=-{1\over V}\,{\pz^2\!F\over\pz\lambda^2}\ .\]
The volume factor in the denominator ensures that \(\xhi\) is intensive.
It is important to realize that we have assumed here that \(\big[\HH\ns_0\,,\,{\hat Q}\big]=0\), the ‘bare’ Hamiltonian \(\HH\ns_0\) and the operator \({\hat Q}\) commute. If they do not commute, then the response functions must be computed within a proper quantum mechanical formalism, which we shall not discuss here.
Note also that we can imagine an entire family of observables \(\big\{{\hat Q}\ns_k\big\}\) satisfying \(\big[{\hat Q}\ns_{k\ns}\,,\,{\hat Q}\ns_{k'}\big]=0\) and \(\big[\HH\ns_0\,,\,{\hat Q}\ns_k\big]=0\), for all \(k\) and \(k'\). Then for the Hamiltonian
\[\HH\ns(\Vlambda)=\HH\ns_0-\sum_k\lambda\ns_k\,{\hat Q}\ns_k\ ,\]
we have that
\[Q\ns_k({\Vlambda},T)=\langle\,{\hat Q}\ns_k\,\rangle=-\pabc{F}{\lambda\ns_k}{T,\,N\ns_a,\,\lambda\ns_{k'\ne k}}\]
and we may define an entire matrix of susceptibilities,
\[\xhi\ns_{kl}={1\over V}{\pz Q\ns_k\over\pz\lambda\ns_l}=-{1\over V}\,{\pz^2\!F\over\pz\lambda\ns_k\,\pz\lambda\ns_l}\ .\]