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  • Relation in Pathria's book:

(relevant quotes from the text that put this equation and my doubts in context is given below) $$V^{\frac{2}{3}}E=constant$$

  • The system:

$N$ number of classical particles of mass m each are trapped in three dimensions under the influence of harmonic oscillator potential. The Hamiltonian is given by: $$H=\sum_{i=1}^{3N}\bigg(\frac{p_i^2}{2m}+\frac{1}{2}\lambda q_i^2\bigg)$$ Considering the system to be an isolated one with total energy between $E$ and $E+\Delta E$ with $\Delta E<<<E$,

Consider now the following question:

What is the average pressure exerted by this ensemble?

I calculate the pressure two times while using the above relation between energy and volume only for the second time. I get two different answers and I am not sure which one is correct.

  • Ist Attempt:

The total number of microstates for the system can be calculated as follows:

Noting that for small $\Delta E$ the phase space volume can be written as $\Delta E$ $\times$ $Area\ of\ the\ hyper$$surface\ $: $$\Omega_{MC}=\frac{1}{h_o^{3N}}\ \Delta E \int \rho(q,p,t)\ d^{3N}q\ d^{3N}p $$

Taking, $\rho=c\delta(H-E)$ (as for an isolated system each micro-state is equally probable),

$$\Omega_{MC}=\frac{c}{h_o^{3N}} \Delta E \int \delta\bigg(\small\sum_{i=1}^{3N}\big(\frac{p_i^2}{2m}+\frac{1}{2}\lambda q_i^2\big)-E\bigg)\ d^{3N}q\ d^{3N}p$$

The integral simply gives the surface area of a $3N$ dimensional unit sphere and the above expression gives $\Omega_{MC} \propto E^{3N-1}$ while being independent of $V$.

I calculated the average pressure $P_{avg}$ to be zero using :$$P=T\bigg(\frac{\partial S}{\partial V }\bigg)_{E,N}=0$$

as $S$ (entropy) is indepedendent of volume $V$.

This answer is also supported in the following online pdf where the authors, under the section "Nonrelativistic Harmonic Oscillator", say:

entropy S is not a function of volume in the case of the harmonic oscillator. Hence pressure $P=0$

  • IInd Attempt

It comes from using the density of states and the relation between volume and energy.

In Pathria's book on Statistical Mechanics, I found that he had discussed, under section 1.4 titled "The classical ideal gas" that for :

Hence, for the constancy of S and N, which defines a reversible adiabatic process,$\ \ $ $V^{\frac{2}{3}}E=constant$

and then after two lines wrote:

It should be noted here that, since an explicit computation of the number $\Omega$has not yet been done, results (9) and (10) hold for quantum as well as classical statistics; equally general is the result obtained by combining these...

Equation (9) is the one I quoted above that relates volume and energy.

It seems then that this result is quite general since it doesn't depend on $\Omega$ as such and is not restricted to only classical ideal gas system.

If I try to calculate the pressure of the above-given hamiltonian using this relation I get:

$$P=T\frac {\partial S}{\partial E}\frac{\partial E}{\partial V}=\frac{2NKT}{V}$$

which is not zero.

Where is the problem? Is there a fault in the logic used in the former calculation?

OR

Is it incorrect to assume that the relation given in Pathria that relates volume to energy is only applicable for ideal gas systems and if so why?

The relation does not depend on the number of microstates but on the constancy of $S$ and $N$ which I believe the microcanonical ensemble of harmonic oscillator satisfies.

So which of the above is incorrect and why?

Any help is appreciated.

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Is it incorrect to assume that the relation given in Pathria that relates volume to energy is only applicable for ideal gas systems and if so why?

Yes, the relation $V^\frac{2}{3}E = \text{const.}$ is only satisfies for ideal gas, because it is derived from the Hamiltonian of free particles,

$$ \hat{H} = \frac{\hat{p}^2}{2m} \implies E = \frac{h^2}{2m}(n_x^2 + n_y^2 + n_z^2) \tag{free particles} $$

whereas, harmonic oscillators are simply free particles subjected to harmonic potential, which are a completely different from ideal gas system.

The relation does not depend on the number of microstates but on the constancy of S and N which I believe the microcanonical ensemble of harmonic oscillator satisfies.

Yes, microcanonical ensemble of harmonic oscillators satisfies this, but it does not satisfy the specific relation $S(N,V,E) = S(N,V^{2/3}E)$, instead entropy $S$ of harmonic oscillator is independent of the volume $V$.

$$ S(N,V,E) = S(N,V^0E^\alpha) \tag{harmonic oscillator} $$

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