# What is the density operator for an isothermal–isobaric ensemble (T,p,N)?

In the microcanonical ensemble $(E,V,N)$, the density operator is $$\hat{\rho}=\frac{\delta(\hat{H}-E\,\hat{I})}{Tr(\delta(\hat{H}-E\,\hat{I}))}$$

Where $\hat{H}$ is the Hamiltonian of the system and $\hat{I}$ is the identity operator.

In the canonical ensemble $(T,V,N)$, the density operator is $$\hat{\rho}=\frac{exp\{\hat{H}/(k_B\,T)\}}{Tr(exp\{\hat{H}/(k_B\,T)\})}$$

Where $k_B$ is Boltzmann constant.

In the macrocanonical ensemble $(T,V,\mu)$, the density operator is $$\hat{\rho}=\frac{exp\{(\hat{H}-\mu\,\hat{N})/(k_B\,T)\}}{Tr(exp\{(\hat{H}-\mu\,\hat{N})/(k_B\,T)\})}$$

Where $\hat{N}$ is the particle number operator.

Now, in the isothermal–isobaric ensemble $(T,p,N)$ the expression for the density operator should be similar to the expression used in the canonical ensemble but with the Boltzmann factor multiplied by $exp\{-p\,V/(k_B\,T)\}$.

My question is: How do you write the density operator for an isothermal–isobaric ensemble $(T,p,N)$ in terms of operators? Shall the volume $V$ of the system be written as an operator?

• I think it will be a Laplace transform w.r.t. $V$ of the canonical ($V$ being continuous where $N$ isn't). I'll see if I can find a reference. (But yes, I think if $N$ is an operator, so will probably $V$ be). Dec 4, 2014 at 15:30
• Thanks for your comment @alarge. Then, what would be the interpretation of a "volume operator"? What would be its eigenvalues and its eigenvectors? How do you create such a volume operator? Would it be simply by writing the product between the three cartesian position operators $\hat{X}$, $\hat{Y}$ and $\hat{Z}$?
– roy
Dec 4, 2014 at 17:44

So for the grand canonical the particle operator $\hat{N}$ is defined through $\hat{N}|i, N\rangle = N|i, N\rangle$, and so $\hat{\rho} = \frac{1}{Z} \sum_i \sum_N e^{-\beta(E_i-\mu N)}|i, N\rangle\langle i, N| = \frac{\exp(-\beta(\hat{\mathcal{H}}-\mu\hat{N}))}{\mathrm{Tr}\exp(-\beta(\hat{\mathcal{H}}-\mu\hat{N}))}$
Now in the isothermal-isobaric we should have $\hat{V}|i, V\rangle = V |i, V\rangle$, i.e. $\hat{\rho} = \frac{1}{Z} \sum_i \int\mathrm{d}V e^{-\beta(E_i+pV)}|i, V\rangle\langle i, V| = \frac{\exp(-\beta(\hat{\mathcal{H}}+p\hat{V}))}{\mathrm{Tr}\exp(-\beta(\hat{\mathcal{H}}+p\hat{V}))}$
I don't think that $\hat{V}$ has to have anything to do with $\hat{x}$, $\hat{y}$, $\hat{z}$, but am not certain of this.
I didn't find any particularly good references, but Statistical Mechanics: Theory and Molecular Simulation by Mark Tuckerman (p. 401) does list the density operator etc. for the isothermal-isobaric ensemble (in fact the author avoids using the operator notation for both $V$ and $N$, preferring to leave these out of the trace as separate sums/integrals).