# Clarification required for some equations in the formulation of ideal Bose gas

• I
• Wrichik Basu
In summary, the relationship ##g_{3/2}(z) = \frac{\lambda^3}{v}## is valid for all values of ##z## (or equivalently, all temperatures ##T##) as it is derived in the grand canonical ensemble where the number of particles can fluctuate and is chosen to recover the average number of particles in the system.

#### Wrichik Basu

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I am studying Ideal Bose Systems (Chapter 7) from the book "Statistical Mechanics" by Pathria and Beale (4th Ed. - 2021). ##\require{physics}## ##\require{cases}##

The authors have derived the following relations: \begin{align} \dfrac{P}{k_B T} &= \dfrac{1}{\lambda^3} g_{5/2}(z), \label{eq:pres_general} \\[1em] \dfrac{N - N_0}{V} &= \dfrac{1}{\lambda^3} g_{5/2}(z), \end{align} where ##N_0## is the number of particles in the ground state (##\varepsilon = 0##), ##N## is the total number of particles, ##z## is the fugacity defined by $$z = \exp \left(\dfrac{\mu}{k_B T} \right),$$ ##\lambda## is the de Broglie wavelength: $$\lambda = \dfrac{h}{(2\pi m k_B T)^{1/2}},$$ and ##g_\nu (z)## is the Bose-Einstein function: $$g_\nu (z) = \dfrac{1}{\Gamma(\nu)} \int\limits_0^\infty \dfrac{x^{\nu - 1} ~ \mathrm{d}x}{z^{-1} \mathrm{e}^x - 1}.$$ I have also learned that $$g_\nu(z = 1) = \zeta(\nu); \qquad (\nu > 1)$$ where ##\zeta(\nu)## is the Riemann zeta function, and if ##\nu \le 1## in the above equation, then ##g_\nu(z \rightarrow 1)## diverges.

The critical temperature ##T_c## below which the condensate forms, is defined as: $$T_c = \dfrac{h^2}{2\pi m k_B } \left[ \dfrac{N}{V \zeta(3/2)} \right]^{2/3}.$$

My issue starts in the section where the authors describe how ##z## varies with ##v/\lambda^3## where ##v## is the inverse of the particle density: $$v = \dfrac{1}{n} = \dfrac{V}{N}. \label{eq:def_v}$$ Let me first summarize what the authors write:
1. For ##0 \le (v/\lambda^3) \le \qty[\zeta(3/2)]^{-1}## (which corresponds to ##0 \le T \le T_c##), ##z \approx 1.##

2. For ##(v/\lambda^3) > \qty[\zeta(3/2)]^{-1}##, ##z < 1## and is determined from the relationship $$g_{3/2} (z) = \lambda^3 / v < \zeta(3/2$$ or equivalently from $$\dfrac{g_{3/2}(z)}{g_{3/2}(1)} = \qty( \dfrac{T_c}{T} )^{3/2} < 1.$$

3. For ##(v/\lambda^3) \gg 1,## ##g_{3/2}(z) \ll 1## and hence ##z \ll 1.## Therefore, we can expand ##g## in powers of ##z## and write ##g_{3/2}(z) \approx z \Rightarrow z \approx (v/\lambda^3)^{-1}.##
I know that in the domain of ##z## that we are interested in (##z \in [0, 1]##), ##g_{3/2}(z) \le \zeta(3/2).##

The Question: As far as I understand, ##g_{3/2} (z) = \frac{\lambda^3}{v}.## But is this relation valid ##\forall ~ z## (or, equivalently, ##\forall ~ T##) ?

How did this question arise?

In the book, it is nowhere clearly written whether ##g_{3/2} (z) = \frac{\lambda^3}{v}## is always valid. Mostly, the authors have used this equality when ##T > T_c## or ##z < 1,## (for example, point no. 3 in the above quote). But, the following probably suggests that the equality may be valid for all ##T##:

The authors have written the following expressions for the pressure of the ideal Bose gas: $$\begin{subnumcases}{P (T) = } \dfrac{k_B T}{\lambda^3} \zeta(5/2) & \text{for } T < T_c, \label{eq:pres_T_lt_Tc} \\[1em] \dfrac{\zeta(5/2)}{\zeta(3/2)} \dfrac{N}{V} k_B T_c \approx 0.5134~\dfrac{N}{V} k_B T_c & \text{for } T = T_c, \\[1em] \dfrac{N}{V}k_B T \dfrac{g_{5/2}(z)}{g_{3/2}(z)} & \text{for } T > T_c \label{eq:pres_T_gt_Tc} \end{subnumcases}$$
It is clear that Eq. \eqref{eq:pres_T_lt_Tc} has been derived from Eq. \eqref{eq:pres_general} by setting ##z = 1.## However, Eq. \eqref{eq:pres_T_lt_Tc} and Eq. \eqref{eq:pres_T_gt_Tc} are closely related — in the latter, if we set ##z = 1## and substitute ##g_{3/2}(1) = \lambda^3 / v,## and using Eq. \eqref{eq:def_v}, we precisely get back Eq. \eqref{eq:pres_T_lt_Tc}.

I have seen a similar thing in the expressions for ##C_V##. Hence my question.

Edit: Improved grammar.

Last edited:
berkeman
Wrichik Basu said:
The Question: As far as I understand, ##g_{3/2} (z) = \frac{\lambda^3}{v}.## But is this relation valid ##\forall ~ z## (or, equivalently, ##\forall ~ T##) ?
Yes.

Remember that quantum statistical physics is derived in the grand canonical ensemble, where the number of particles can fluctuate, even though the system at hand has a fixed number of particles. To make this work, we require that ##\mu## be chosen such that we recover ##\langle N \rangle \approx N##, i.e., the average number of particles corresponds to the actual number of particles (for big enough ##N## fluctuations are negligible, so we don't care that it is only valid "on average"). If you set up the equation for ##\langle N \rangle##, you will find that setting ##\langle N \rangle = N## leads to ##g_{3/2} (z) = \frac{\lambda^3}{v}##. (All of this assumes we are considering a gas of free particles.)

Edit: post edited since it appears that the braket package is broken in MathJax.

Last edited:
Wrichik Basu
DrClaude said:
Yes.

Remember that quantum statistical physics is derived in the grand canonical ensemble, where the number of particles can fluctuate, even though the system at hand has a fixed number of particles. To make this work, we require that ##\mu## be chosen such that we recover ##\braket{N} \approx N##, i.e., the average number of particles corresponds to the actual number of particles (for big enough ##N## fluctuations are negligible, so we don't care that it is only valid "on average"). If you set up the equation for ##\braket{N}##, you will find that setting ##\braket{N} = N## leads to ##g_{3/2} (z) = \frac{\lambda^3}{v}##. (All of this assumes we are considering a gas of free particles.)
Thank you for the explanation.

berkeman