2.4.10 · D1Canonical ensemble — partition function Z
Ek chhota system jo ek bade warm reservoir ko touch kar raha hai, apni energy fixed nahi rakh sakta — woh constantly chhoti-chhoti amounts lete-deta rehta hai, isliye uski energy kuch average ke aas-paas kaanpti rehti hai. Partition function Z har us state ka weighted headcount hai jisme system ho sakta hai , aur jab ek baar yeh single number aapke paas ho, toh differentiation se har thermodynamic quantity nikal aati hai.
Yeh page assume karta hai ki aapne kuch nahi dekha. Isse pehle ki aap parent note ko touch karein, hum har symbol build karenge — seedhi baatein, ek picture, aur woh reason jiske bina yeh topic exist nahi kar sakta. Upar se neeche padho; har block mein sirf wahi symbols use honge jo pehle se earn kiye ja chuke hain.
Neeche sab kuch pehli figure ki picture ke baare mein hai: ek chhota system jo ek giant heat bath se juda hua hai, dono milke ek rigid box ke andar sealed hain jo kuch bahar nahi jaane deta.
Definition System vs. reservoir (heat bath)System woh chhoti cheez hai jiske baare mein hum care karte hain (ek single spin, ek oscillator, N Reservoir itna bada hai ki thodi si energy dump karne ya nikalne se uska temperature practically nahi badalta. Dono milke isolated hain — total energy locked hai.
Yeh split kyun chahiye? Kyunki fixed temperature wala system woh system hai jise energy swap karne ki permission hai. T E
Microstate system ki ek complete, fully-detailed configuration hai — har position, har spin, har velocity specified. Hum microstates ko i = 1 , 2 , 3 , …
Ek numbered pigeonholes ki row imagine karo; state i
Letter i index hai — ek counter, jaise bachon ko "child 1, child 2" naam dena. Jab baad mein ∑ i
E i i i E i g n
Pigeonholes ko alag-alag unchaiyyon par stacked imagine karo; hole i E i
Yeh topic kyun zaroori hai: statistical mechanics ka poora sawaal yahi hai ki system kitni baar har unchai par baithta hai? — aur jawaab sirf unhi unchaiyyon E i
T T T T
Definition Boltzmann constantk B k B ≈ 1.38 × 1 0 − 23 J/K temperature (kelvin) aur energy (joules) ke beech ka fixed exchange rate hai. Yeh "degrees" ko "joules per particle" mein convert karta hai. Iske bina, T E
β β ≡ k B T 1 β β 1/ energy β E i pure number hai — jo zaroori hai, kyunki aap sirf pure numbers ko exponentiate kar sakte ho.
β k B T E β β Z β steepness knob samjho aane wale exponential ke liye.
Boltzmann factor se pehle, aapko exponential function feel karna hoga.
e x e − x e ≈ 2.718 e − x 1 x = 0 smoothly zero ki taraf decay karta hai jaise x
Ek slide imagine karo jo height 1
Yahi function kyun aur koi straight line kyun nahi? Kyunki yeh sirf wahi shape hai jo reservoir-counting argument (Section 8) se nikaali jaati hai, aur iska ek magical property hai ki d x d e − x = − e − x − 1 β
Yeh ek weight hai, abhi probability nahi — yeh 1 Boltzmann distribution dekho.
Definition Summation symbol∑ ∑ i f i i f i i 1 , 2 , 3 ∑ i f i = f 1 + f 2 + f 3
Z ≤ 1 Z weights ka sum hai, probability nahi. Yeh typically bahut bada hota hai — roughly kitne states thermally reachable hain uska count hai. Jo cheez ≤ 1 P i = e − β E i / Z
g n E n degeneracy g n states ki jagah levels par sum kar sakte ho:
Z = ∑ n g n e − β E n . height E n g n
Ω Ω ( E ) E Ω R ( E ) E Microcanonical ensemble mein.
ln ln x e x inverse hai: yeh jawaab deta hai "e x ln ( ab ) = ln a + ln b ln Z
S S = k B ln Ω define karta hai (agla section).
Definition Partial derivative∂ ∂ x ∂ f f sirf x ∂ d
Intuition Yeh line poore topic ki keystone kyun haiThermodynamics temperature ko define karta hai T 1 = ∂ E ∂ S S R = k B ln Ω R E i e − β E i woh reason hai ki Boltzmann factor exponential hai — parent note ise poori tarah walkthrough karta hai. Baad mein sab kuch (Z ⟨ E ⟩ F
Definition Angle brackets⟨ ⋅ ⟩ ⟨ E ⟩ probability-weighted average energy: ⟨ E ⟩ = ∑ i E i P i
σ E 2 σ E 2 = ⟨ E 2 ⟩ − ⟨ E ⟩ 2 kaampne ki size measure karta hai. Zero variance = energy pinned hai; large variance = wild fluctuations.
C V C V = ∂ T ∂ ⟨ E ⟩ σ E 2 = k B T 2 C V kaampna aur heating ke response ek hi physics hain. Dekho Heat capacity and fluctuations .
N V N V fixed hain; sirf energy fluctuate karne ke liye free hai. Compare karo Grand canonical ensemble — grand partition function se jahan N
F = − k B T ln Z Helmholtz free energy — woh single quantity jo Z S = − ∂ F / ∂ T P = − ∂ F / ∂ V Helmholtz free energy F .
H h H ( p , q ) p q ∑ i phase space par ek integral ban jaata hai, aur h 1/ N ! Gibbs paradox fix karta hai, aur resulting 2 1 k B T Equipartition theorem hai.
Exponential e to the minus x
Boltzmann weight e to minus beta E
Partition function Z = sum of weights
Average energy and F = minus kB T ln Z
Khud test karo — sirf tab reveal karo jab jawaab de chuko.
Index i System ka ek complete microstate (ek fully specified configuration).
β β = 1/ ( k B T ) β E i
β E i Kyunki sirf pure numbers exponential e − β E i
e − x x ≥ 0 1 0
Kya Z Nahi — yeh weights ka sum hai (ek normalizer); probability hai e − β E i / Z
∑ i Index i
Degeneracy g n Distinct microstates ki number jo ek hi energy level E n
ln Z Yeh e x β
Temperature ko entropy ke zariye define karo. 1/ T = ∂ S / ∂ E
Angle brackets ⟨ E ⟩ Probability-weighted average energy, ∑ i E i P i
Canonical ensemble mein N Sirf energy fluctuate karti hai; particles aur volume fixed hain (yahi ( N , V , T )
F F = − k B T ln Z Z
