Free energy from partition function — Hinglish notes

2.4.12 · Physics › Thermodynamics & Statistical Mechanics (Advanced)

1. Hum kya bana rahe hain

Goal: yeh deceptively simple result dikhana $F = - k_{B} T ln Z$ aur isse scratch se derive karna, sirf quote nahi karna.

2. First principles se derivation

Step 2a — Ek microstate ki probability

Microstate $i$ ki Boltzmann probability hai $p_{i} = \frac{e ^{- β E_{i}}}{Z} .$ Yeh step kyun? Bath $T$ fix karta hai; fixed average energy par entropy maximize karne se exactly yahi exponential weight milta hai, aur $Z$ se divide karne par $\sum_{i} p_{i} = 1$ normalize ho jaata hai.

Step 2b — $Z$ se internal energy

$U = ⟨ E ⟩ = \sum_{i} p_{i} E_{i} = \frac{1}{Z} \sum_{i} E_{i} e^{- β E_{i}} .$ Dhyan do $\frac{\partial}{\partial β} e^{- β E_{i}} = - E_{i} e^{- β E_{i}}$ , toh $\sum_{i} E_{i} e^{- β E_{i}} = - \frac{\partial Z}{\partial β} .$ Isliye $U = - \frac{1}{Z} \frac{\partial Z}{\partial β} = - \frac{\partial l n Z}{\partial β} .$ Yeh step kyun? Humne weighted sum ko $Z$ ke derivative mein convert kar diya — yahi poora point hai: microstates sum karne ki jagah $ln Z$ differentiate karo.

Step 2c — $Z$ se entropy (Gibbs entropy)

Gibbs/Shannon entropy se shuru karte hain $S = - k_{B} \sum_{i} p_{i} ln p_{i} .$ Yeh step kyun? Yeh entropy ki statistical definition hai; equal-probability microstates ke liye yeh $k_{B} ln Ω$ mein reduce ho jaati hai. $ln p_{i} = - β E_{i} - ln Z$ substitute karo: $S = - k_{B} \sum_{i} p_{i} (- β E_{i} - ln Z) = k_{B} β U i \sum p_{i} E_{i} + k_{B} ln Z = 1 i \sum p_{i} .$ Toh $S = k_{B} β U + k_{B} ln Z = \frac{U}{T} + k_{B} ln Z .$

Step 2d — $F$ assemble karo

$F = U - T S = U - T (\frac{U}{T} + k_{B} ln Z) = U - U - k_{B} T ln Z .$ $F = - k_{B} T ln Z$ Yeh kyun beautiful hai: $U$ exactly cancel ho jaata hai. Free energy pure $ln Z$ hai — yeh intuition confirm karta hai ki $F$ multiplicative $Z$ ka additive (extensive) partner hai.

Figure — Free energy from partition function

3. Baaki sab $F$ se nikal aata hai

Kyunki $d F = - S d T - p d V + μ d N$ , hum padte hain:

Self-consistency check: $U = F + T S$ use karte hue $F = - k_{B} T ln Z$ aur upar wali $S$ ke saath $U = - \partial_{β} ln Z$ wapas milta hai. ✓

4. Worked examples

5. Common mistakes (steel-manned)

Recall Feynman: ek 12-saal ke bache ko explain karo

Socho ek badi bag mein dice hain jo kisi bhi number par gir sakte hain, lekin thande dice "prefer" karte hain chhote numbers aur garam dice wild hote hain. Partition function $Z$ ek aisa number hai jo secretly jaanta hai ki pure bag mein kitne likely outcomes hain. "Free energy" $F$ sirf $Z$ ka ek stretched-out version hai — hum $Z$ ko logarithm se squish karte hain taaki jab aap do bags ko saath rakh do, unke free energies simply add ho sakein multiply karne ki jagah. Us ek stretched number se aap bag ka temperature behavior, uski energy, aur uski messiness (entropy) predict kar sakte ho — bina har ek dice ko ek-ek karke count kiye.

6. Active recall

#flashcards/physics

Canonical partition function define karo. ::: $Z = \sum_{i} e^{- β E_{i}}$ , sabhi microstates par sum, $β = 1/ k_{B} T$ .
$Z$ ke terms mein Helmholtz free energy? ::: $F = - k_{B} T ln Z$ .
$F$ mein logarithm kyun hai (sirf $Z$ nahi)? ::: Independent systems apne $Z$ multiply karte hain; $ln$ product ko sum mein convert karta hai toh $F$ extensive (additive) hota hai.
$Z$ se internal energy? ::: $U = - \partial (ln Z) / \partial β = \frac{1}{Z} (- \partial_{β} Z)$ .
$Z$ aur $U$ se entropy? ::: $S = U / T + k_{B} ln Z = - \partial F / \partial T$ .
$Z$ se pressure? ::: $p = k_{B} T \partial (ln Z) / \partial V = - \partial F / \partial V$ .
$F = U - T S$ derive karte waqt kya cancel hota hai? ::: $U$ exactly cancel ho jaata hai, $- k_{B} T ln Z$ bachta hai.
$N$ identical independent two-level systems ke liye, $F = ?$ ::: $F = - N k_{B} T ln (1 + e^{- β ε})$ .
Classical 1D oscillator $Z$ ? ::: $Z = k_{B} T /ℏ ω$ , giving $U = k_{B} T$ (equipartition).
Entropy formula mein common sign error? ::: $S = - \partial F / \partial T$ (minus, from $d F = - S d T - p d V + μ d N$ ).

7. Connections

Boltzmann distribution and microstates
Helmholtz vs Gibbs free energy
Entropy — Gibbs and Boltzmann definitions
Equipartition theorem
Grand canonical ensemble and grand potential
Legendre transforms in thermodynamics
Ideal gas from the partition function