Average energy from partition function: Visual walkthrough

2.4.11 · D2 · Physics › Thermodynamics & Statistical Mechanics (Advanced) › Average energy from partition function

Step 0 — Pieces kya hain? (koi bhi math se pehle vocabulary banao)

Koi bhi symbol se pehle, cast se milte hain.

Jo knob hum actually ghumaenge woh $T$ nahi balki uska inverse hai:

Figure — Average energy from partition function

Figure dekho: cold (left) marble ko sabse neeche waale box mein stack karta hai; hot (right) use saare boxes mein spread kar deta hai. Same boxes, same energies — sirf dial $β$ badla.

Jo prerequisites hum use karte hain woh yahan hain: Boltzmann distribution aur Partition function (canonical ensemble).

Step 1 — Har box kitna likely hai? (Boltzmann weight)

KYA. Har box ko ek raw "unnormalized likeliness" number assign karo.

EXPONENTIAL KYUN, na ki $1/ E_{i}$ ? Kyunki hum chahte hain ki do independent systems ke weights multiply hon (energies add hoti hain: $E = E_{1} + E_{2} \Rightarrow e^{- β (E_{1} + E_{2})} = e^{- β E_{1}} e^{- β E_{2}}$ ). Sirf exponential "energies add karo" ko "weights multiply karo" mein convert karta hai. Yahi woh unique property hai jo hume chahiye.

PICTURE. Har box ko ek bar milta hai jiska height $w_{i}$ hai. High-energy boxes ke tiny bars hote hain; cold dial far bars ko almost invisible kar deta hai.

Step 2 — Weights ko honest probabilities banao ( $Z$ ka entry)

KYA. Bars $w_{i}$ $1$ tak add nahi hote, isliye abhi tak ye probabilities nahi hain. Inhe inke total se divide karo.

Ab box $i$ mein land karne ki honest probability:

KYUN. Ek probability ka $1$ tak sum hona zaroori hai; $Z$ se divide karna woh ek rescaling hai jo yeh karta hai. $Z$ normalizer hai.

PICTURE. Step 2 ke figure jaise hi bars, lekin ab ek full-height column of "shares" mein stack kiye gaye.

Step 3 — Average energy honestly likho

KYA. Average (mean) energy har box ki energy hai, weighted by kitni baar hum use visit karte hain.

KYUN. Yeh average ki single definition hai — value times probability, summed. Abhi tak kuch clever nahi.

Annoyance: woh sum $\sum_{i} E_{i} e^{- β E_{i}}$ mein ek extra factor $E_{i}$ har weight se glued hai. Ise directly, box by box, compute karna tedious hai. Step 4 us tedium ko hata deta hai.

PICTURE. Phir se bars, lekin ab har bar pe uski energy $E_{i}$ tag hai; average "balance point" (centre of mass) hai energy values ka bar height se weighted.

Step 4 — Derivative trick: missing $E_{i}$ manufacture karo

KYA. Poochho: agar hum dial $β$ ko thoda nudge karen toh ek single weight ka kya hoga?

$\frac{\partial}{\partial β} e^{- β E_{i}} = - E_{i} e^{- β E_{i}}$

$\frac{\partial}{\partial β}$ ka matlab hai "rate of change jaise $β$ badhta hai, energies fixed rakhte hue."
Chain rule: $e^{(stuff)}$ ka derivative hai $e^{(stuff)}$ times stuff ka derivative. Yahan stuff $= - β E_{i}$ hai, jiska slope $β$ mein $- E_{i}$ hai.

YEH TOOL KYUN. Hum chahte the ek extra $E_{i}$ har weight ko multiply kare (Step 3 ki annoyance). $β$ mein differentiation exactly woh factor $E_{i}$ produce karta hai — free mein, ek minus sign ke saath attached. Isliye derivative, aur koi bhi operation nahi, sahi key hai.

PICTURE. Dial ko tiny $δ β$ se ghuma; dekho har bar shrink hoti hai. Ek tall-energy bar (bada $E_{i}$ ) zyada shrink hoti hai — kyunki uski shrink rate $- E_{i} w_{i}$ hai, $E_{i}$ ke proportional. Picture tumhe dikhaa rahi hai factor $E_{i}$ ka janam.

Step 5 — Poore sum ko $Z$ ke ek derivative se replace karo

KYA. Step 4 ko har box pe add karo:

$\frac{\partial Z}{\partial β} = \frac{\partial}{\partial β} \sum_{i} e^{- β E_{i}} = \sum_{i} (- E_{i}) e^{- β E_{i}} = - \sum_{i} E_{i} e^{- β E_{i}}$

Sum ka derivative = derivatives ka sum, isliye hum term by term differentiate karte hain.
Result minus wahi sum hai jo Step 3 mein annoying tha.

Sign flip karo us sum ko isolate karne ke liye:

$\sum_{i} E_{i} e^{- β E_{i}} = - \frac{\partial Z}{\partial β}$

KYUN. Step 3 ka tedious sum ab bas " $Z$ ka slope" hai. Ek quantity, $Z (β)$ , aur uska slope sab kuch carry karte hain.

PICTURE. $Z$ vs $β$ ke do curves (ek decreasing curve). Ek point par slope ki steepness hai woh weighted energy sum. Jahan $Z$ steeply girta hai, visit ki jaane waali energies badi hain.

Step 6 — Substitute karo, phir log mein fold karo

KYA. Step 5 ko Step 3 mein daalo:

$⟨ E ⟩ = \frac{1}{Z} (- \frac{\partial Z}{\partial β}) = - \frac{1}{Z} \frac{\partial Z}{\partial β}$

Stray $\frac{1}{Z}$ average se abhi bhi wahan hai — woh normalization hai jise hum bhool nahi sakte.

Ab tidy-up. Logarithms aur chain rule ke baare mein ek fact yaad karo:

$\frac{\partial}{\partial β} ln Z = \frac{1}{Z} \frac{\partial Z}{\partial β}$

$ln$ natural logarithm hai; uska derivative rule " $\frac{d}{d x} ln f = \frac{f ^{'}}{f}$ " exactly woh " $\frac{1}{Z} \times$ slope of $Z$ " shape hai jo humare paas hai.

Toh do operations — $Z$ se divide karo aur slope lo — ek operation mein collapse ho jaate hain $ln Z$ par:

$ln$ KYUN. Yeh decoration nahi hai — log exactly woh function hai jiska derivative hai woh messy combination " $\frac{1}{Z} \partial_{β} Z$ ." $ln Z$ use karne ka matlab hai ki hum $1/ Z$ kabhi haath se nahi likhte.

PICTURE. $ln Z$ ko $β$ ke against plot karo (ek downward-bending curve). Uska downhill slope, negated, directly $⟨ E ⟩$ read kar deta hai har temperature par.

Step 7 — Edge & limiting cases (koi gap mat chhodna)

Har derivation ko apne extremes survive karne chahiye. Two-level system par test karo: boxes energies $0$ aur $ε$ par, giving $Z = 1 + e^{- β ε}$ aur $⟨ E ⟩ = \frac{ε}{e ^{β ε} + 1}$ .

PICTURE. Two-level system ke liye $⟨ E ⟩$ -vs- $T$ curve: cold hone par $0$ par flat, rising, phir hot hone par $ε /2$ par saturate. Dono plateaus upar ke limits hain.

Wahi machine, twice differentiate karne par, energy fluctuations deta hai aur Heat capacity and energy fluctuations se link karta hai; continuous quadratic case Equipartition theorem reproduce karta hai; aur discrete ladder Quantum harmonic oscillator — thermal result deta hai. Ek formula, sab ke sab.

The one-picture summary

Boxes → weights → normalize into $Z$ → $ln Z$ ka ek derivative (minus ke saath) → average energy. Poora page ek single arrow chain mein.

Recall Feynman retelling — poora walkthrough plain words mein

Ek row of numbered boxes imagine karo; har box pe ek energy tag hai. Ek temperature dial decide karta hai ki ek jumpy marble har box ko kitni baar visit karta hai — cold matlab woh low boxes se chipka rehta hai, hot matlab woh kahin bhi ghoomta hai. "Kitni baar" ko real chances mein convert karne ke liye, hum har box ki raw likeliness ko ek grand total mein add karte hain jise $Z$ kehte hain, phir har box ka chance uska $Z$ ka slice hota hai. Ab average energy bas har box ki energy times uski chance hai, summed — lekin woh sum pechida hai kyunki har term par ek extra energy tag glued hai. Magic yeh hai: agar tum gently dial ghumaao aur $Z$ ko change hote dekho, math automatically har box ko uski apni energy se multiply karta hai (exactly woh factor jo miss tha), saath mein ek minus sign bhi. $Z$ se divide karo aur tumne ek move mein divide-and-slope kar liya — jo $Z$ ke logarithm ko slope karne jaisa hi hai. Sign flip karo aur woh raha: system ki typical energy, ek counter ko nudge karne se. Cold test karo (energy bottom par freeze ho jaati hai), hot test karo (energy average out ho jaati hai) — dono pass ho jaate hain. Bas yahi poora idea hai.

Visual walkthrough — Average energy from partition function