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# 4.4: Ordinary Canonical Ensemble (OCE)

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$$\newcommand\DDelta ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[16], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.04:_Ordinary_Canonical_Ensemble_(OCE)), /content/body/p/span, line 1, column 23 $$
$$\newcommand\DTheta ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[17], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.04:_Ordinary_Canonical_Ensemble_(OCE)), /content/body/p/span, line 1, column 23 $$
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$$\newcommand\Vxi ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[20], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.04:_Ordinary_Canonical_Ensemble_(OCE)), /content/body/p/span, line 1, column 23 $$
$$\newcommand\Vom ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[21], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.04:_Ordinary_Canonical_Ensemble_(OCE)), /content/body/p/span, line 1, column 23 $$
$$\newcommand\Vpi ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[22], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.04:_Ordinary_Canonical_Ensemble_(OCE)), /content/body/p/span, line 1, column 23 $$
$$\newcommand\Vvarpi ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[23], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.04:_Ordinary_Canonical_Ensemble_(OCE)), /content/body/p/span, line 1, column 23 $$
$$\newcommand\Vrho ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[24], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.04:_Ordinary_Canonical_Ensemble_(OCE)), /content/body/p/span, line 1, column 23 $$
$$\newcommand\Vvarrho ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[25], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.04:_Ordinary_Canonical_Ensemble_(OCE)), /content/body/p/span, line 1, column 23 $$
$$\newcommand\Vsigma ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[26], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.04:_Ordinary_Canonical_Ensemble_(OCE)), /content/body/p/span, line 1, column 23 $$
$$\newcommand\Vvarsigma ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[27], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.04:_Ordinary_Canonical_Ensemble_(OCE)), /content/body/p/span, line 1, column 23 $$
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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

$P\ns_n=\lim_{\RDelta E\to 0}{\hbox{ probability that E\ns_{\ssr{S}}\in[E\ns_n,E\ns_n+\RDelta E]}\over \hbox{ \ \# of S states with E\ns_{\ssr{S}}\in [E\ns_n,E\ns_n+\RDelta E]\ }} ={P(E\ns_n)\,\RDelta E\over D\ns_{\ssr{S}}(E\ns_n)\,\RDelta E} = {D\ns_{\ssr{W}}(E\ns_{\ssr{U}}-E\ns_n)\over D\ns_{\ssr{U}}(E\ns_{\ssr{U}})}\ .$

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,

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