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4.4: Ordinary Canonical Ensemble (OCE)

  • Page ID
    18565
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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 \(y\ns_i\equiv - F\ns_i\).

    [SMfirst] Microscopic, statistical interpretation of the First Law of Thermodynamics.
    Figure \(\PageIndex{1}\): Microscopic, statistical interpretation of the First Law of Thermodynamics.

    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,

    \[\xhi\ns_{kl}={1\over V}{\pz Q\ns_k\over\pz\lambda\ns_l}=-{1\over V}\,{\pz^2\!F\over\pz\lambda\ns_k\,\pz\lambda\ns_l}\ .\]


    This page titled 4.4: Ordinary Canonical Ensemble (OCE) is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Daniel Arovas.