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

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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.
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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 $$\begin{equation}
z = \exp \left(\dfrac{\mu}{k_B T} \right),
\end{equation}$$ ##\lambda## is the de Broglie wavelength: $$\begin{equation}
\lambda = \dfrac{h}{(2\pi m k_B T)^{1/2}},
\end{equation}$$ and ##g_\nu (z)## is the Bose-Einstein function: $$\begin{equation}
g_\nu (z) = \dfrac{1}{\Gamma(\nu)} \int\limits_0^\infty \dfrac{x^{\nu - 1} ~ \mathrm{d}x}{z^{-1} \mathrm{e}^x - 1}.
\end{equation}$$ 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: $$\begin{equation}
T_c = \dfrac{h^2}{2\pi m k_B } \left[ \dfrac{N}{V \zeta(3/2)} \right]^{2/3}.
\end{equation}$$

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: $$ \begin{equation}
v = \dfrac{1}{n} = \dfrac{V}{N}. \label{eq:def_v}
\end{equation}$$ 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 $$\begin{equation}
g_{3/2} (z) = \lambda^3 / v < \zeta(3/2
\end{equation}$$ or equivalently from $$\begin{equation}
\dfrac{g_{3/2}(z)}{g_{3/2}(1)} = \qty( \dfrac{T_c}{T} )^{3/2} < 1.
\end{equation}$$

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.
 
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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.
 
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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.
 
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FAQ: Clarification required for some equations in the formulation of ideal Bose gas

What is an ideal Bose gas?

An ideal Bose gas is a theoretical model used in statistical mechanics to describe the behavior of a large number of identical bosons (particles with integer spin) at low temperatures. It assumes that the particles do not interact with each other and that they can occupy the same quantum state.

What are the equations used to describe an ideal Bose gas?

The main equations used to describe an ideal Bose gas are the Bose-Einstein distribution function, which gives the probability of finding a particle in a particular quantum state, and the Bose-Einstein condensation equation, which determines the critical temperature at which the gas undergoes a phase transition to a condensed state.

Why is clarification needed for some equations in the formulation of ideal Bose gas?

Clarification may be needed for some equations in the formulation of ideal Bose gas because they can be complex and involve advanced mathematical concepts such as quantum mechanics and statistical mechanics. Additionally, the assumptions made in the model may not always hold true in real-world situations.

How does the behavior of an ideal Bose gas differ from that of an ideal Fermi gas?

An ideal Bose gas and an ideal Fermi gas are two different theoretical models used to describe the behavior of a large number of particles at low temperatures. The main difference between them is that bosons can occupy the same quantum state, while fermions (particles with half-integer spin) cannot. This leads to different statistical distributions and behaviors in the two types of gases.

What are some real-world applications of the ideal Bose gas model?

The ideal Bose gas model has been used to study various phenomena in physics, such as superfluidity, Bose-Einstein condensation, and photon statistics in lasers. It also has applications in other fields, such as cosmology and condensed matter physics, and has been used to develop technologies such as Bose-Einstein condensate sensors and quantum computers.

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