Canonical ensemble — partition function Z: Visual walkthrough

2.4.10 · D2 · Physics › Thermodynamics & Statistical Mechanics (Advanced) › Canonical ensemble — partition function Z

Hum ek-ek karke earn karenge: microstate, reservoir, entropy, temperature, exponential weight, normalizer $Z$ , aur $Z$ master key kyun hai.

Step 1 — "Microstate" kya hota hai? Box draw karo.

KYA. Socho ek tiny system — maano ek single atom — jo kai alag-alag configurations mein se kisi ek mein ho sakta hai. Har ek precise configuration ek microstate hai. Hum unhe $1, 2, 3, \dots$ se label karte hain aur har ek ke paas ek energy hoti hai jise hum $E_{i}$ kehte hain (padho: "microstate $i$ ki energy").

KYUN. Yeh poochne se pehle ki "system ke yahan hone ki kitni probability hai?", humein yeh agree karna hoga ki "yahan" ka matlab kya hai. "Yahan" = ek microstate. Poori theory ek rule hai jo har labeled box ko ek probability $P_{i}$ assign karti hai.

PICTURE. Neeche: teen boxes height = energy ke hisaab se stack kiye hue hain. Sabse neeche wala box lowest energy hai (ground state), upar wale boxes zyada energy lete hain. Red arrow us ek microstate $i$ ki taraf point kar raha hai jise hum track karenge.

Step 2 — System akela NAHI hai: reservoir add karo.

KYA. Hamara chhota system ek bahut bade object se chipka hua hai — ek heat reservoir (the "bath") — itna bada ki thodi si energy dene ya lene se usmein koi khaas fark nahi padta. Saath mein, system + reservoir milkar ek isolated super-object banate hain jis ki total energy $E_{tot}$ kabhi nahi badlti.

KYUN. Akela system fixed energy par boring hota hai (woh Microcanonical ensemble hai). Asli cheezein ek environment mein fixed temperature par rehti hain. System se energy sirf bath mein ja sakti hai — toh system ko samajhne ke liye, humein bath ko dekhna hoga.

PICTURE. Ek chhota red circle (system, energy $E_{i}$ ) ek giant black region (reservoir, energy $E_{tot} - E_{i}$ ) mein embedded hai. Dashed outer line isolating wall hai: iske paar koi energy nahi jaati.

Step 3 — Equal-likelihood: probability = bath ke ways ginne ke barabar.

KYA. Fundamental postulate: fixed total energy par ek isolated super-object ke liye, poore system ka har ek accessible microstate equally likely hota hai. Toh system ke box $i$ mein hone ki probability us number of ways ke proportional hai jis tarah reservoir khud ko arrange kar sakta hai jabki baaki energy $E_{tot} - E_{i}$ hold kar raha ho. Us count ko $Ω_{R}$ kehte hain.

KYUN. System ka apna box ek single fixed choice hai jab hum $i$ choose karte hain. Saari freedom — saare "ways" — giant reservoir mein hai. Zyada ways ⇒ zyada likely.

PICTURE. Do scenarios side by side. Left: system thodi energy leta hai → reservoir zyaadatar rakhta hai → bahut saare black dots (bahut saare arrangements). Right: system bahut zyada leta hai → reservoir starved → kam dots. Red bar system ka slice hai.

Step 4 — Entropy par switch karo, kyunki $Ω$ astronomically large hai.

KYA. $Ω_{R}$ ek aisa number hai jaise $1 0^{1 0^{23}}$ — seedha Taylor-expand karna impossible hai. Iska logarithm smooth aur manageable hai. Entropy define karo $S_{R} = k_{B} ln Ω_{R}$ , taaki $Ω_{R} = e^{S_{R} / k_{B}}$ .

KYUN yeh tool (logarithm)? Humein chhote $E_{i}$ ke liye $Ω_{R}$ ko expand karna hai. Ek raw exponential-of-huge $1 0^{billions}$ ke factors se badlta hai — hopeless. Lekin $ln Ω_{R}$ (yaani $S_{R}$ ) chhote range $E_{i}$ par gently aur linearly badlta hai. Log ek runaway product ko ek tame straight line mein badal deta hai jise hum approximate kar sakte hain. Yahi poori wajah hai ki entropy yahan aati hai.

PICTURE. Left axis: $Ω_{R}$ ek cliff ki tarah shoot karta hai (unusable). Right axis: $S_{R} = k_{B} ln Ω_{R}$ ek smooth gentle curve hai. Red tangent line woh hai jo hum Step 5 mein use karenge.

Step 5 — Taylor-expand karo: ek straight line kaafi hai.

KYA. Kyunki $E_{i}$ , $E_{tot}$ ke muqable mein tiny hai, humein reservoir ki entropy sirf $E_{tot}$ par aur ek chhote linear correction ke saath chahiye — Taylor expansion ka pehla term (ek curve ko ek point ke paas uski tangent line se approximate karo): $S_{R} (E_{tot} - E_{i}) \approx constant, no i S_{R} (E_{tot}) - E_{i} red tangent ki slope \frac{\partial S _{R}}{\partial E}_{E_{tot}}$

KYUN. Higher-order terms mein $E_{i} / E_{tot}$ ki powers hoti hain, jo effectively zero hain kyunki bath bahut bada hai. Tangent line hi physics hai.

PICTURE. Entropy curve ka zoom $E_{tot}$ ke paas. Red tangent line curve se chipki hui hai; $E_{i}$ ki width pe uska downward step hi poora correction hai.

Step 6 — Temperature aata hai; Boltzmann factor paida hota hai.

KYA. Step 5 mein slope $= 1/ T$ substitute karo, phir exponentiate back karo ( $Ω_{R} = e^{S_{R} / k_{B}}$ ): $S_{R} (E_{tot} - E_{i}) \approx S_{R} (E_{tot}) - \frac{E _{i}}{T}$ $P_{i} \propto e^{S_{R} / k_{B}} = constant in i e^{S_{R} (E_{tot}) / k_{B}} \cdot depends on i e^{- E_{i} / (k_{B} T)}$ Define karo $β \equiv \frac{1}{k _{B} T}$ ("coldness"). Constant kisi bhi comparison se nikal jaata hai: $P_{i} \propto e^{- β E_{i}} yeh hai == Boltzmann factor ==$

KYUN. Exponent ke andar $E_{i} / T$ ka subtraction ek multiplying factor $e^{- β E_{i}}$ ban gaya bahar. Zyada $E_{i}$ ⇒ chhota factor ⇒ exponentially rarer state. Kam $T$ (bada $β$ ) ⇒ steep decay ⇒ system ground state se chipka rehta hai.

PICTURE. Weight $e^{- β E_{i}}$ ko $E_{i}$ ke against do temperatures ke liye plot kiya: ek steep red curve (cold, bada $β$ ) aur ek gentle black curve (hot, chhota $β$ ).

Step 7 — Normalize karo: states ka sum HI $Z$ hai.

KYA. Probability list ka total $1$ hona chahiye. Hamare paas $P_{i} \propto e^{- β E_{i}}$ hai; ise equality banane ke liye hum saare weights ke sum se divide karte hain. Us sum ka ek naam hai — partition function: $P_{i} = \frac{e ^{- β E_{i}}}{j \sum e ^{- β E_{j}}}, Z = i \sum e^{- β E_{i}}$

KYUN. $Z$ khud koi probability nahi hai — yeh total weight hai, woh denominator jo har raw Boltzmann factor ko rescale karta hai taaki woh 1 mein sum ho jaayein. (Iska German naam Zustandssumme = "sum over states" literally definition hi hai.)

PICTURE. Har microstate ke liye height $e^{- β E_{i}}$ ki bars. Unki total stacked height = $Z$ (red brace). Har bar ko $Z$ se divide karne par probability slice milta hai.

Step 8 — $Z$ master key kyun hai: energy pull down karne ke liye differentiate karo.

KYA. Kyunki $β$ har $E_{i}$ ko $Z$ ke andar multiply karta hai, $Z$ ko $β$ ke respect mein differentiate karne par har term par $- E_{i}$ ka factor aa jaata hai — bilkul wahi ingredient jo energy average ke liye chahiye: $\frac{\partial Z}{\partial β} = \sum_{i} (- E_{i}) e^{- β E_{i}} \Rightarrow ⟨ E ⟩ = \frac{\sum _{i} E _{i} e ^{- β E_{i}}}{Z} = - \frac{1}{Z} \frac{\partial Z}{\partial β} = - \frac{\partial l n Z}{\partial β}$

KYUN yeh tool ( $β$ -derivative)? Hum chahte hain $⟨ E ⟩ = \sum_{i} E_{i} P_{i}$ , energies ka weighted average. Derivative $\partial_{β}$ woh machine hai jo free mein har term mein $- E_{i}$ ka factor daal deta hai. Koi nayi physics nahi — sirf calculus jo $Z$ mein stored information harvest kar raha hai.

PICTURE. $ln Z$ ka curve $β$ ke against; kisi bhi point par uski downhill slope (red tangent) $- ⟨ E ⟩$ ke barabar hai. Steeper slope = zyada average energy.

Edge & degenerate cases (reader ko kabhi stranded mat chhodna)

Ek-picture summary

Upar sab kuch ek causal chain hai: isolated whole ki equal-likelihood → bath ke ways gino → log karke entropy banao → tangent line → temperature aata hai → exponential weight → use $Z$ mein sum karo → thermodynamics ke liye $Z$ differentiate karo.

Recall Feynman retelling (plain words mein bolo)

Socho ek marble (hamaara system) ek giant sandbox (the bath) mein baitha hai. Sandbox aur marble dono ek box mein band hain jismein fixed amount of energy hai. Nature fair hai: poore sealed box ke har allowed arrangement ki equally likely hone ki chance hai. Ab — agar marble greedily bahut zyada energy le, toh sandbox ke paas kam energy bachti hai, aur kam energy wale sandbox ke paas bahut kam ways hote hain apne grains arrange karne ke. Toh marble ka bahut zyada energy lena ek rare arrangement hai. Sandbox ke arrangements ginना unmanageable hai ( $1 0^{1 0^{23}}$ jaise numbers), toh hum logarithm lete hain aur use entropy kehte hain; energy ke saath uski gentle slope hi woh hai jise hum $1/ T$ (temperature) define karte hain. Marble ki energy ko $E_{i}$ se slide karna sandbox ki entropy ko $E_{i} / T$ se neeche slide karta hai, aur use un-log karne par woh "neeche by $E_{i} / T$ " ek multiplying factor $e^{- E_{i} / k_{B} T}$ ban jaata hai — Boltzmann factor. In raw weights ko honest probabilities mein badalne ke liye hum sab ko add karte hain; woh grand total hai $Z$ , partition function. Aur yahan magic hai: kyunki coldness $β$ har energy ko $Z$ ke andar multiply karta hai, $Z$ ko $β$ mein derivative se poke karne par average energy nikal aati hai — aur thode aur poke se heat capacity, entropy, free energy. Ek sum $Z$ ; thermodynamics ki poori cheez nikal aati hai. Isliye yeh master key hai. (Dekhein bhi Equipartition theorem, Heat capacity and fluctuations, aur Gibbs paradox.)