Equipartition theorem — ½k_BT per quadratic degree of freedom: Foundations

2.4.14 · D1 · Physics › Thermodynamics & Statistical Mechanics (Advanced) › Equipartition theorem — ½k_BT per quadratic degree of freedo

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1. Sabke peeche ki picture: ek coordinate $q$

Figure 1 dekho. Spring pe ek ball left ya right push ki ja sakti hai; $q$ se labelled arrow measure karta hai ki woh resting spot se kitni door hai. Agar hum track karte ki woh kitni tezi se move karti hai, toh woh speed ek aur coordinate hoti. Ek arrow = ek coordinate.

Figure — Equipartition theorem — ½k_BT per quadratic degree of freedom

Topic ko iske liye kyun zaroorat hai: equipartition ek statement hai "energy store karne ke har independent tarike" ke baare mein. Har aisa tarika ek coordinate $q$ hai. Inhe count karne se pehle, humein ek pe point karne mein capable hona chahiye.

2. Energy $E$ , aur "quadratic" ka matlab

Ab key shape. Jab tum spring ko $q$ se stretch karte ho, toh stored energy proportional nahi hoti $q$ se — woh $q$ squared ki tarah badhti hai.

Figure 2 dekho. Energy-vs- $q$ curve ek valley hai (parabola): neeche flat, sides pe steep. $q$ stretch double karo aur energy quadruple ho jati hai, kyunki $2^{2} = 4$ . Woh U-shape "quadratic" ka visual signature hai.

Topic ko iske liye kyun zaroorat hai: poora $\frac{1}{2} k_{B} T$ result sirf $q^{2}$ shape ke liye kaam karta hai. Agar energy $q^{4}$ ya $∣ q ∣$ hoti, toh answer badal jaata. Toh "quadratic" entry ticket hai.

3. Constant $α$ — stiffness ya mass

Bada $α$ matlab narrow, steep valley (stretch karna mushkil); chhota $α$ matlab wide, gentle valley (stretch karna aasan). Figure 2 dono draw karta hai.

Topic ko iske liye kyun zaroorat hai: equipartition ki punchline yeh hai ki $α$ final energy se cancel out ho jaata hai. Tum us surprise ki appreciation nahi kar sakte jab tak tum nahi jaante ki $α$ woh mass/stiffness hai jo matter karne ki expectation rakhta tha.

4. Temperature $T$ aur Boltzmann's constant $k_{B}$

Topic ko iske liye kyun zaroorat hai: $\frac{1}{2} k_{B} T$ poora answer hai. $T$ input hai, $k_{B}$ ise energy mein convert karta hai.

5. Probability, aur Boltzmann weight $e^{- E / k_{B} T}$

Random jiggling saari configurations ko equally visit nahi karti — low-energy waale bahut zyada common hote hain. Kitna zyada? Yeh single most important formula hai jo equipartition ko feed karti hai.

Figure 3 dekho. Jaise energy $E$ badhti hai, yeh weight ek cliff se girta hai. Cliff ki width $k_{B} T$ set karta hai: ek warmer system (bada $k_{B} T$ ) higher-energy configurations tolerate karta hai, toh uski curve zyada gently girti hai.

Topic ko iske liye kyun zaroorat hai: derivation mein har average har $q$ ki value ko is factor se weight karta hai. Yeh "how likely" ingredient hai.

6. Averaging: $⟨ \dots ⟩$ aur integral $\int$

Hum baar baar "average energy" kehte rehte hain. Brackets ka matlab yahan yeh hai.

Pichle teen sections ko jodke, ek quadratic slot mein average energy hai $⟨ α q^{2} ⟩ = \frac{\int _{- \infty}^{\infty} α q ^{2} e ^{- α q^{2} / k_{B} T} d q}{\int _{- \infty}^{\infty} e ^{- α q^{2} / k_{B} T} d q} .$ Plain words mein padho: top = "sum of (energy × uska Boltzmann weight)"; bottom = "weights ka sum" (yeh normalise karta hai, toh total probability 1 hoti hai). Ratio exactly wahi weighted average hai jo upar define kiya.

Topic ko iske liye kyun zaroorat hai: yeh ratio hi theorem ka left-hand side hai. Is section se pehle sab kuch exist karta tha taaki hum yeh ek line honestly likh sakein.

7. Gaussian (bell curve) $e^{- a q^{2}}$

Ek quadratic energy ka Boltzmann weight, $e^{- α q^{2} / k_{B} T}$ , hai hi ek Gaussian jisme $a = α / k_{B} T$ hai. Toh upar ke do integrals bas bell-curve areas hain — isliye messy-looking average ka ek clean answer hai.

8. $β$ , partition function $Z$ , aur derivative trick

Do aakhri shorthands jo parent use karta hai.

Foundations theorem ko kaise feed karte hain

Equipment checklist

Khud ko test karo — sirf out loud jawab dene ke baad reveal karo.

Symbol

q

kya stand karta hai, ek picture mein?

Ek arrow jo ek single tarika measure karta hai jisme koi cheez positioned ya move ho sakti hai — ek coordinate.

"Quadratic" energy

α q^{2}

graph pe kaunsi shape banata hai?

Ek symmetric U-shaped valley (parabola):

q

double karo,

E

quadruple ho jaata hai.

α

physically kya hai, aur bada

α

kaisa dikhta hai?

Stiffness/mass coefficient; bada

α

= steep, narrow valley (stretch karna mushkil).

Bundle

k_{B} T

kaunsi single energy represent karta hai?

Temperature

T

pe thermal energy ka ek typical "ghunt";

k_{B}

kelvin ko joules mein convert karta hai.

Probability weight kyun exponential

e^{- E / k_{B} T}

hai aur kuch linear kyun nahi?

Sirf exponential energies add karne ko probabilities multiply karne mein turn karta hai, jo independent systems require karte hain.

⟨ X ⟩

ka matlab kya hai aur ise yahan kaise compute karte hain?

X

ka probability-weighted mean: (

X

× weight) ka sum divided by weights ka sum.

Integral

\int_{- \infty}^{\infty} f d q

ko picture mein kya dikhate hain?

Curve

f (q)

ke neeche area, har possible

q

par ek smooth total.

Area

\int_{- \infty}^{\infty} e^{- a q^{2}} d q

kya hai?

π / a

— Gaussian bell-curve area.

β

kya hai, aur

Z

kya hai?

β = 1/ k_{B} T

(inverse temperature);

Z

woh normalising "total weight" integral hai, partition function.

Hum

Z

β

ke respect mein differentiate kyun karte hain?

Kyunki

\frac{d}{d β}

exactly woh

- α q^{2}

factor pull down karta hai, average ko

- \frac{d}{d β} ln Z

mein turn kar deta hai.

Jab upar ki har line ek reflex ban jaaye, toh parent Equipartition derivation padho — woh ek story ki tarah padhegi.