Average energy from partition function — Hinglish notes

2.4.11 · Physics › Thermodynamics & Statistical Mechanics (Advanced)

WHY do we even want this?

Statistical mechanics mein hum ek system ka description karte hain jo temperature $T$ par ek heat bath ke contact mein hai. Har microstate $i$ ki energy $E_{i}$ hoti hai aur yeh Boltzmann probability ke saath occur karta hai: $p_{i} = \frac{e ^{- β E_{i}}}{Z}, β \equiv \frac{1}{k _{B} T} .$ Jo cheez hum measure kar sakte hain woh hai mean energy $⟨ E ⟩ = \sum_{i} p_{i} E_{i}$ . Is sum ko directly compute karna painful hai. Trick yeh hai: notice karo ki wohi exponentials $e^{- β E_{i}}$ $Z$ mein bhi appear karte hain, toh $Z$ ka derivative hamare liye $E_{i}$ ka factor "manufacture" kar deta hai.

HOW: average energy ko scratch se derive karo

Step 1 — Average energy ko seedha likho. $⟨ E ⟩ = \sum_{i} p_{i} E_{i} = \sum_{i} E_{i} \frac{e ^{- β E_{i}}}{Z} = \frac{1}{Z} \sum_{i} E_{i} e^{- β E_{i}} .$ Yeh step kyun? Yeh sirf expectation value ki definition hai — (value × probability) ka sum.

Step 2 — Derivative trick dhundho. Ek single weight ko $β$ ke saath differentiate karo: $\frac{\partial}{\partial β} e^{- β E_{i}} = - E_{i} e^{- β E_{i}} .$ Yeh step kyun? $E_{i}$ ka factor jo hum chahte hain woh naturally (minus sign ke saath) appear hota hai jab hum differentiate karte hain. Toh woh mushkil $\sum_{i} E_{i} e^{- β E_{i}}$ bas $- \partial_{β} Z$ hai.

Step 3 — Sum ko replace karo. Kyunki $\frac{\partial Z}{\partial β} = \sum_{i} (- E_{i}) e^{- β E_{i}}$ hai, $\sum_{i} E_{i} e^{- β E_{i}} = - \frac{\partial Z}{\partial β} .$ Step 1 mein substitute karo: $⟨ E ⟩ = \frac{1}{Z} (- \frac{\partial Z}{\partial β}) = - \frac{1}{Z} \frac{\partial Z}{\partial β} .$

Step 4 — Log se tidy karo. Yaad karo $\frac{\partial ln Z}{\partial β} = \frac{1}{Z} \frac{\partial Z}{\partial β}$ . Isliye:

$ln$ kyun? Kyunki $\frac{1}{Z} \partial_{β} Z$ chain rule se wohi $\partial_{β} ln Z$ hai. Log lene se "pehle $Z$ se divide karo phir differentiate karo" ek clean operation ban jaata hai — aur $ln Z$ bilkul wohi object hai jo free energy $F = - k_{B} T ln Z$ deta hai.

Bonus: energy fluctuations (second derivative)

Ek baar aur differentiate karo aur tumhe variance milega, isliye heat capacity bhi:

C_V=\frac{\partial\langle E\rangle}{\partial T}=k_B\beta^2\,\sigma_E^2.$$ *Care kyun karo?* Yeh dikhata hai ki $Z$ **har** moment generate karta hai: pehla derivative → mean, doosra → variance. ![[2.4.11-Average-energy-from-partition-function.png]] --- ## Worked examples > [!example] 1. Two-level system (jaise spin in a field) > Energies $0$ aur $\varepsilon$ hain. Toh $Z=e^{0}+e^{-\beta\varepsilon}=1+e^{-\beta\varepsilon}$. > **Kyun:** sirf do microstates hain, toh sum mein do terms hain. > $$\langle E\rangle=-\partial_\beta\ln Z=-\frac{-\varepsilon e^{-\beta\varepsilon}}{1+e^{-\beta\varepsilon}}=\frac{\varepsilon}{e^{\beta\varepsilon}+1}.$$ > **Yeh step kyun?** $\partial_\beta\ln Z=\frac1Z\partial_\beta Z$, aur $\partial_\beta Z=-\varepsilon e^{-\beta\varepsilon}$. > **Checks (Forecast-then-verify):** jab $T\to0$ ($\beta\to\infty$), $\langle E\rangle\to 0$ (ground state mein frozen, ✓). Jab $T\to\infty$ ($\beta\to0$), $\langle E\rangle\to \varepsilon/2$ (dono states equally likely, $0,\varepsilon$ ka average, ✓). > [!example] 2. Quantum harmonic oscillator (1D, discrete energy levels) > $E_n=n\hbar\omega$, $n=0,1,2,\dots$ Toh $Z=\sum_{n}e^{-\beta n\hbar\omega}=\dfrac{1}{1-e^{-\beta\hbar\omega}}$ (geometric series). > **Kyun:** ratio $r=e^{-\beta\hbar\omega}<1$ hai, toh $\sum r^n = 1/(1-r)$. > $\ln Z=-\ln(1-e^{-\beta\hbar\omega})$, isliye > $$\langle E\rangle=-\partial_\beta\ln Z=\frac{\hbar\omega\,e^{-\beta\hbar\omega}}{1-e^{-\beta\hbar\omega}}=\frac{\hbar\omega}{e^{\beta\hbar\omega}-1}.$$ > **Yeh step kyun?** Log ko differentiate karo; chain rule se numerator $\hbar\omega e^{-\beta\hbar\omega}$ milta hai. > **Check:** high $T$ → $\langle E\rangle\to k_BT$ (oscillator ke liye classical equipartition, ✓). > [!example] 3. Continuous quadratic mode se equipartition > Ek degree of freedom jisme $E=\tfrac12 ax^2$ hai, $x\in(-\infty,\infty)$: $Z=\int e^{-\beta a x^2/2}dx=\sqrt{2\pi/(\beta a)}\propto\beta^{-1/2}$. > $\ln Z=-\tfrac12\ln\beta+\text{const}$, isliye > $$\langle E\rangle=-\partial_\beta\ln Z=\frac{1}{2\beta}=\tfrac12 k_BT.$$ > **Yeh kyun matter karta hai:** energy mein har quadratic term $\tfrac12 k_BT$ contribute karta hai — yeh **equipartition theorem** hai, $Z$ se sirf do lines mein derive kiya. --- ## Common mistakes (steel-manned) > [!mistake] "$T$ ke saath differentiate karo $\langle E\rangle$ paane ke liye." > **Kyun sahi lagta hai:** energy temperature par depend karti hai, toh $\partial_T$ naturally aata hai. > **Fix:** clean formula hai $\langle E\rangle=-\partial_\beta\ln Z$. Agar aap $T$ par insist karo, > toh Jacobian lana padega: $\langle E\rangle=k_BT^2\,\partial_T\ln Z$ (dhyan do $k_BT^2$ pe, yeh easily drop ho jaata hai). > [!mistake] Minus sign bhool jaana. > **Kyun sahi lagta hai:** "$\ln Z$ ka derivative" positive lagta hai. > **Fix:** $\partial_\beta e^{-\beta E}=-E e^{-\beta E}$ — minus exponent se aata hai. > Sanity check: $\langle E\rangle$ $E_i\ge0$ ke liye **positive** honi chahiye; minus sign yahi deliver karta hai. > [!mistake] $\langle E\rangle = -\partial_\beta Z$ likhna ($1/Z$ bhool jaana). > **Kyun sahi lagta hai:** sum $\sum E_i e^{-\beta E_i}=-\partial_\beta Z$ answer jaisa lagta hai. > **Fix:** woh sum **unnormalized** hai. $Z$ se divide karna zaroori hai, aur isliye $\ln$ form ($\frac1Z\partial_\beta Z$) itna convenient hai. --- > [!recall]- Feynman: 12-saal ke bacche ko explain karo > Socho ek bade jar mein dice hain jahan chhote numbers zyada dikhaai dete hain jab thanda hota hai aur sab numbers > equally dikhaai dete hain jab garam hota hai. $Z$ ek magic counter hai jo "har number kitna likely hai" add karta hai. > Yahan cool part yeh hai: agar tum $Z$ ko thoda sa nudge karo (ek derivative lo) aur dekho yeh kaise badalta hai, toh math > *automatically* har outcome ko uske apne number se multiply kar deta hai — bina manually average kiye tumhe **average roll** de deta hai. Toh counter ka ek clever nudge tumhe system ki typical energy bata deta hai. > [!mnemonic] Formula yaad karo > **"Log it, slope it, flip it."** $\ln Z$ lo, $\beta$ mein uska slope lo ($\partial_\beta$), sign flip karo: > $\langle E\rangle=-\partial_\beta\ln Z$. (Flip = Boltzmann exponent se aaya minus.) --- ## Active recall > [!recall] Quick self-test > 1. $Z$ ko differentiate karne se $E_i$ ka factor kyun milta hai? > 2. $\langle E\rangle=-\partial_\beta\ln Z$ mein minus sign kahan se aata hai? > 3. $\ln Z$ ka *doosra* derivative kya deta hai? #flashcards/physics What is the partition function $Z$? ::: $Z=\sum_i e^{-\beta E_i}$, yeh sab microstates par Boltzmann weights ka normalizing sum hai. Formula for average energy from $Z$? ::: $\langle E\rangle=-\dfrac{\partial \ln Z}{\partial\beta}=k_BT^2\dfrac{\partial\ln Z}{\partial T}$. Why does $\partial_\beta$ pull out the energy? ::: Kyunki $\partial_\beta e^{-\beta E_i}=-E_i e^{-\beta E_i}$, toh derivative sum ke andar $E_i$ ka factor generate karta hai. Where does the minus sign come from? ::: Boltzmann exponent se: $e^{-\beta E}$ ko differentiate karne par $-E e^{-\beta E}$ milta hai. What does $\partial^2\ln Z/\partial\beta^2$ equal? ::: Energy variance $\langle E^2\rangle-\langle E\rangle^2=\sigma_E^2$. Average energy of a two-level system ($0,\varepsilon$)? ::: $\langle E\rangle=\dfrac{\varepsilon}{e^{\beta\varepsilon}+1}$. Average energy of a quantum oscillator ($E_n=n\hbar\omega$)? ::: $\langle E\rangle=\dfrac{\hbar\omega}{e^{\beta\hbar\omega}-1}$. Equipartition result for one quadratic mode? ::: $\langle E\rangle=\tfrac12 k_BT$, kyunki $Z\propto\beta^{-1/2}$ hai toh $\langle E\rangle=1/(2\beta)$. Relation between heat capacity and fluctuations? ::: $C_V=k_B\beta^2\sigma_E^2$, matlab heat capacity energy fluctuations measure karta hai. Why use $\ln Z$ instead of $Z$ directly? ::: Kyunki $\frac1Z\partial_\beta Z=\partial_\beta\ln Z$ hai, toh log zaroori $1/Z$ normalization ko ek clean derivative mein fold kar deta hai. --- ## Connections - [[Partition function (canonical ensemble)]] - [[Boltzmann distribution]] - [[Helmholtz free energy F = -kT ln Z]] - [[Equipartition theorem]] - [[Heat capacity and energy fluctuations]] - [[Quantum harmonic oscillator — thermal]] - [[Two-level system / Schottky anomaly]] ## 🖼️ Concept Map ```mermaid flowchart TD Z[Partition function Z] P[Boltzmann probability p_i] AE[Mean energy langle E rangle] DEF[Definition sum e^-beta E_i] DERIV[Differentiate wrt beta] LNZ[ln Z] FORM[langle E rangle = -d ln Z / d beta] VAR[Energy variance] HC[Heat capacity] F[Free energy F = -kT ln Z] Z -->|normalizes| P P -->|weighted sum| AE Z -->|defined by| DEF DERIV -->|pulls out E_i| Z Z -->|log of| LNZ LNZ -->|first derivative| FORM FORM -->|equals| AE LNZ -->|second derivative| VAR VAR -->|yields| HC LNZ -->|gives| F ```