Bose-Einstein statistics — bosons

2.4.16 · Physics › Thermodynamics & Statistical Mechanics (Advanced)

HUM KYA COUNT kar rahe hain?

HUM distribution KAISE derive karte hain (scratch se)

Hum chahte hain average number of bosons ${\bar{n}}_i$ jo ek single-particle energy level ${\epsilon }_i$ mein hain, ek aisi gas ke liye jo temperature $T$ aur chemical potential $\mu$ par heat-and-particle reservoir ke saath contact mein hai.

Step 1 — Grand canonical ensemble use karo. Kyun? Particle number fixed nahi hota (jaise photons appear/disappear karte hain), toh hum energy aur particles dono ko flow karne dete hain. Har level ko ek independent system ki tarah treat kiya jaata hai jo $n_i=0,1,2,\dots$ particles hold kar sakta hai.

Ek level ke liye grand partition function: ${\mathcal{Z}}_i={\sum }_{n_i=0}^{\infty }{e}^{-\beta ({\epsilon }_i-\mu )n_i},\beta =\frac{1}{k_{B}T}.$ Yeh form kyun? Level mein har particle ki cost energy ${\epsilon }_i$ hai aur reservoir se $\mu$ "remove" hota hai; $n_i$ particles ki cost $({\epsilon }_i-\mu )n_i$ hai. Boltzmann weight $e^{-\beta E}$ hota hai.

Step 2 — Geometric series sum karo. Maan lo $x=e^{-\beta ({\epsilon }_i-\mu )}$. Tab ${\mathcal{Z}}_i={\sum }_{n=0}^{\infty }{x}^n=\frac{1}{1-x},\text{(converges tabhi jab }x<1).$ Yeh step kyun? Yeh ek geometric series hai; yahi woh algebraic jagah hai jahan boson ka "unlimited occupancy" enter hota hai — sum $\infty$ tak jaata hai. Convergence ke liye $x<1$ zaroori hai, yani ${\epsilon }_i-\mu >0$, yani $\mu <{\epsilon }_i$ har level ke liye. (Fermions ke liye sum $n=1$ par ruk jaata hai, isliye aisi koi restriction nahi aati.)

Step 3 — Mean occupancy nikalo. Average ${\bar{n}}_i={\sum }_{n}nP(n)$ jahan $P(n)=x^n/{\mathcal{Z}}_i$. Ek neat trick: ${\bar{n}}_i=x\frac{\partial }{\partial x}\ln {\mathcal{Z}}_i=x\frac{\partial }{\partial x}[-\ln (1-x)]=\frac{x}{1-x}.$ Yeh step kyun? $x{\partial }_x\ln \mathcal{Z}$ ek standard generating-function trick hai jo partition sum se $⟨n⟩$ nikaal leta hai.

Step 4 — $x=e^{-\beta ({\epsilon }_i-\mu )}$ wapas substitute karo. $\boxed{{\bar{n}}_i=\frac{1}{e^{\beta ({\epsilon }_i-\mu )}-1}}$

Formula padh rahe hain

Steel-manned mistakes

Flashcards

Connections

• Fermi-Dirac statistics — fermions (the $+1$ sibling)
• Maxwell-Boltzmann statistics (classical high-$\epsilon$ limit)
• Grand canonical ensemble (the framework used)
• Planck radiation law (photons, $\mu =0$)
• Bose-Einstein condensation ($\mu \to {\epsilon }_{\min }^{-}$)
• Chemical potential (why $\mu <{\epsilon }_{\min }$ strictly)
• Density of states (turns $\bar{n}$ into physical spectra)

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