Average energy from partition function: Foundations

2.4.11 · D1 · Physics › Thermodynamics & Statistical Mechanics (Advanced) › Average energy from partition function

Hum formula $⟨ E ⟩ = - \partial_{β} ln Z$ ko padhne se pehle, iske har piece ko samajhna zaroori hai. Neeche, har symbol ko teen cheezein milti hain: simple words, ek picture, aur topic ko yeh kyun chahiye. Yeh is order mein hain ki har ek sirf upar waalon par depend karta hai.

1. Ek microstate, aur label $i$

Picture. Ek jar socho jisme raffle tickets bhari hain. Har ticket ek microstate hai. Index $i$ sirf ticket par likha number hai — ek name tag, aur kuch nahi.

Topic ko yeh kyun chahiye. Jo kuch bhi hum sum karte hain woh "saare microstates" hote hain. Unhe list karne ke tarike ke bina ( $i = 1, 2, \dots$ ) hum $\sum_{i}$ sum bilkul likh hi nahi sakte.

Figure — Average energy from partition function

2. Ek state ki energy, $E_{i}$

Picture. Ek vertical "energy axis" banao (upar = zyada energy). Har microstate ek horizontal shelf hai height $E_{i}$ par. Yahi woh energy-level diagram hai jo is chapter mein baar baar dikhta hai.

Topic ko yeh kyun chahiye. Poora sawaal — "average energy kya hai?" — tabhi meaningful hai jab har state ek number $E_{i}$ carry kare. Yahi woh values hain jinhe hum average karenge.

3. Temperature $T$ aur heat bath

Picture. Ek choti cup of water jo ek bade lake mein float kar rahi hai. Lake ki temperature kabhi nahi hilti; cup iske saath energy swap karte rehti hai aur lake ki temperature par settle hoti hai.

Topic ko yeh kyun chahiye. "Temperature $T$ par heat bath ke saath contact mein" yahi poora setting hai. $T$ woh knob hai jo decide karta hai ki low-energy states dominate karein (cold) ya sab states roughly equal hon (hot).

4. Boltzmann's constant $k_{B}$

Picture. Ek currency exchange rate: ek taraf kelvin, doosri taraf joules. Temperature ko $k_{B}$ se multiply karo aur tum "us temperature par jiggling ki natural energy scale" paate ho, yaani $k_{B} T$ .

Topic ko yeh kyun chahiye. Energies $E_{i}$ joules mein hain; temperatures kelvin mein. Unhe compare karne ke liye (jaise hume exponent ke andar karna hai) hume $k_{B}$ chahiye taaki units match ho sakein.

5. Inverse temperature $β$

Picture. Ek see-saw: jaise $T$ badhta hai, $β$ ghatta hai, aur vice versa. Jab $β$ bahut bada ho (cold), exponent $- β E_{i}$ high-energy states ke liye ek bada negative number hota hai, unhe crush kar deta hai.

Topic ko yeh kyun chahiye. Final formula hai $⟨ E ⟩ = - \partial_{β} ln Z$ — yeh $β$ ke saath respect mein derivative hai, $T$ ke nahi. $β$ mein kaam karna algebra ko clean rakhta hai kyunki $β$ exponent mein $E_{i}$ ko directly multiply karta hai (koi messy $1/ T$ andar nahi chupta).

6. Exponential $e^{- β E_{i}}$ — Boltzmann weight

Ab hum pieces ko combine karte hain. Sabse pehle exponential hi kyun, koi aur function kyun nahi?

Picture. Ek curve jo upar se shuru hoti hai aur decay karti hai. Low-energy shelves ko lambi bars milti hain; high-energy shelves ko choti bars. Cold ( $β$ bada) curve ko tezi se neeche laata hai; hot ( $β$ chota) ise flatten kar deta hai.

Topic ko yeh kyun chahiye. Yahi weights sab kuch ka raw material hain. Partition function inका sum hai; probabilities inhe normalize karke milti hain; average energy inke sum ka derivative hai.

7. Partition function $Z$

Picture. Pichli figure ke har bar ko leke end to end ek lambi bar mein stack karo. Woh total length $Z$ hai. Jaise system ko warm karo, zyada bars badhte hain, aur $Z$ increase hota hai.

Topic ko yeh kyun chahiye. Do kaam ek saath:

Normalizer — har weight ko $Z$ se divide karna scores ko genuine probabilities mein badalta hai (jo 1 mein add hote hain).
Generator — kyunki $Z$ mein har $e^{- β E_{i}}$ hai, $Z$ ko differentiate karna andar jaake energies kheeench laata hai. Yahi parent note ka poora trick hai.

8. Probability $p_{i}$ aur sum $\sum_{i}$

Picture. Ek pie chart slice karo. Har slice ek microstate hai; slice ka angle $p_{i}$ hai. Bada Boltzmann weight → moti slice. Poora pie $Z$ hai, toh ek slice ka pie ka fraction $w_{i} / Z = p_{i}$ hai.

Topic ko yeh kyun chahiye. Average energy $⟨ E ⟩ = \sum_{i} p_{i} E_{i}$ hai — literally "value × probability ka sum." Is line ka koi matlab hi nahi jab tak $\sum_{i}$ aur $p_{i}$ dono exist na karein.

9. Average (expectation value) $⟨ E ⟩$

Picture. Phir se energy-level shelves: har shelf par ek weight $p_{i}$ rakho aur poori stack ka balance point dhundho. Woh balance height $⟨ E ⟩$ hai. Cold → balance sabse neechi shelf ke paas hota hai; hot → woh pack ke middle ki taraf badhta hai.

Topic ko yeh kyun chahiye. Yahi woh quantity hai jo poora topic compute karta hai. Magic yeh hai ki yeh $- \partial_{β} ln Z$ ke barabar hai, toh hume actually haath se weighted sum nahi karna padta.

10. Derivative $\partial_{β}$ aur logarithm $ln$

Yeh woh do tools hain jo formula use karta hai. Clearly batao ki kyun yahi tool aur koi doosra nahi.

Yahi tool kyun? Hume har weight ke saath ek $E_{i}$ ka factor appear chahiye. $e^{- β E_{i}}$ ko $β$ mein differentiate karna exactly woh factor manufacture karta hai: $\partial_{β} e^{- β E_{i}} = - E_{i} e^{- β E_{i}} .$ Koi aur simple operation kahin se bhi $E_{i}$ produce nahi karta. Yahi ek fact hai ki ek derivative — aur specifically $β$ mein derivative — sahi instrument kyun hai.

Yahi tool kyun? Differentiate karne ke baad hume $- \frac{1}{Z} \partial_{β} Z$ milta hai — ek " $Z$ se divide karo, phir differentiate karo" ka combo. Logarithm us poore combo ko ek clean symbol $\partial_{β} ln Z$ mein bundle karta hai. Bonus: $ln Z$ exactly wahi object hai jo Helmholtz free energy $F = - k_{B} T ln Z$ deta hai, toh ise use karna energy aur free energy ko bina extra mehnat ke link karta hai.

Pieces kaise formula mein assemble hote hain

Upar wali figure ko left-to-right padhna: microstates → energies → weights → unka sum $Z$ → uska log → $β$ mein uski slope → sign flip karo → average energy. Har arrow upar ki ek definition hai.

$§2 E_{i} \to §6 e^{- β E_{i}} \to §7 Z = i \sum e^{- β E_{i}} \to §10 ln Z \to §10 - \partial_{β} ln Z = ⟨ E ⟩ .$

Prerequisite map

Yeh map ek hi destination tak do roads dikhata hai: honest sum road ( $p_{i} \to \sum p_{i} E_{i}$ ) aur generator road ( $ln Z \to - \partial_{β}$ ). Topic ka poora point yahi hai ki yeh do roads milti hain.

Yeh foundations aage kahan le jaate hain

Weights khud Boltzmann distribution hain.
Unka sum Partition function (canonical ensemble) hai.
$ln Z$ ko ek baar differentiate karo → average energy (yeh topic); do baar → Heat capacity and energy fluctuations.
Ise specific systems par apply karo: Two-level system / Schottky anomaly, Quantum harmonic oscillator — thermal, aur Equipartition theorem.
Log khud free energy hai.

Parent par wapas jao: 2.4.11 Average energy from the partition function.

Equipment checklist

Right side cover karo aur reveal karne se pehle jawab do.

Ek sentence mein microstate kya hota hai?

System ke liye ek completely specified possibility (saari positions, saare spins fixed).

E_{i}

kya represent karta hai?

Woh energy jo system ke paas hoti hai jab woh microstate

i

mein baitha ho.

k_{B}

ke units kya hain aur yeh kya karta hai?

Joules per kelvin; yeh temperature ko ek energy scale

k_{B} T

mein convert karta hai.

β

T

ke terms mein likhkhe batao kaunsi direction mein point karta hai.

β = 1/ (k_{B} T)

; cold mein bada, hot mein chota.

State-weight exponential kyun hona chahiye?

Sirf

e^{- β E}

energies add karne par sahi se multiply karta hai, isliye relative likelihoods energy gaps par depend karti hain, energy ke zero par nahi.

Partition function ko ek sum ke roop mein likho.

Z = \sum_{i} e^{- β E_{i}}

Z

ke do kaam batao.

Yeh weights ko probabilities mein normalize karta hai, aur differentiate karne par averages generate karta hai.

Boltzmann probability

p_{i}

express karo.

p_{i} = e^{- β E_{i}} / Z

⟨ E ⟩

ko ek weighted sum ke roop mein define karo.

⟨ E ⟩ = \sum_{i} p_{i} E_{i}

\partial_{β} e^{- β E_{i}}

kya equal hai, aur yeh matter kyun karta hai?

- E_{i} e^{- β E_{i}}

; yeh woh factor

E_{i}

manufacture karta hai jo hume average karne ke liye chahiye.

Woh chain-rule identity state karo jo

ln

ko involve karti hai.

\partial_{β} ln Z = \frac{1}{Z} \partial_{β} Z

Average energy ka final formula assemble karo.

⟨ E ⟩ = - \partial_{β} ln Z

Recall Sabse fast sanity check

Agar har $E_{i} \geq 0$ ho, toh kya $⟨ E ⟩$ positive hona zaroori hai? ::: Haan — $- \partial_{β} ln Z$ mein minus sign exponent ke minus ko cancel kar deta hai, toh ek physically sensible positive average bachta hai.