2.4.8 · D1Statistical mechanics — microstate, macrostate
Jab aap har particle ko track nahi kar sakte, tab aap iski jagah count karte hain ki kitne microscopic arrangements bahar se ek jaise dikhte hain — aur jo arrangement sabse zyada tareekon se ho sakta hai, wahi aap almost hamesha measure karte hain. Neeche sab kuch sirf woh vocabulary aur pictures hain jo aapko chahiye isse pehle ki woh sentence poori tarah samajh aaye.
Is page par koi bhi assumption nahi hai. Har letter, har symbol, har notation jo parent note use karta hai, woh yahan ground up se build kiya gaya hai, ek aisi order mein jahan har idea pichle wale par tika hua hai.
Kisi bhi physics se pehle, hume counting ke baare mein ek fact chahiye.
Intuition Choices multiply hote hainAgar aap ek choice karte hain a independent choice b a × b a + b 2 × 2 = 4
Definition The multiplication (product) ruleAgar ek process independent stages ki sequence hai jisme n 1 , n 2 , … product n 1 × n 2 × ⋯
Picture: ek branching tree — har level leaves ki sankhya ko multiply karta hai.Ye topic ko kyun chahiye: parent ka Ω total = 2 N N 2 × 2 × ⋯
Intuition Cheezon ko line up karne ke kitne tarike hain?N N N − 1 N − 2 1
N ! = N × ( N − 1 ) × ( N − 2 ) × ⋯ × 2 × 1 N N orderings (ek row mein arrangements) ki sankhya hai.
Special case jo aapko zaroori pata hona chahiye: 0 ! = 1 ek hi tarika hai — kuch mat karo).Picture: N Ye topic ko kyun chahiye: ( n N )
Worked example Chhote factorials1 ! = 1 , 2 ! = 2 , 3 ! = 6 , 4 ! = 24 4 ! = 4 × 3 ! = 4 × 6 = 24
Ye poore topic ki workhorse hai, isliye hum ise slowly build karte hain.
n N n group pick karne ke kitne tarike hain , jahan group ke andar order matter nahi karta (3 ka committee wahi committee hai chahe aap pehle kisi ko bhi naam lo). Saare N !
KYA karte hain: N ! KYU yeh overcount karta hai: humein sirf parwah hai ki kaunse n n n ! N − n ( N − n )! wahi choice milti hai. Toh hum unhe divide karke hatate hain:
( 2 4 ) ( 2 4 ) = 2 ! 2 ! 4 ! = 2 × 2 24 = 6 Har case covered hai: ( 0 4 ) = 1 ( 1 4 ) = 4 ( 2 4 ) = 6 ( 3 4 ) = 4 ( 4 4 ) = 1
Common mistake Bhool jaana ki order matter NAHI kartaKyun sahi lagta hai: lagta hai jaise saare N ! Fix: ek macrostate "kitne upar hain" ke liye, ups ke beech ke labels invisible hain — toh n ! ( N − n )!
Definition Sigma notation∑ n = 0 N f ( n ) = f ( 0 ) + f ( 1 ) + f ( 2 ) + ⋯ + f ( N ) Σ n = 0 shuru hota hai; upar ka number (N rukta hai; f ( n )
Picture: ek machine jo n f ( n ) Ye topic ko kyun chahiye: sanity check n = 0 ∑ N ( n N ) = 2 N
2 N 2 N 2 N double ho jaata hai. Yahi doubling-per-particle ki wajah se real systems ka Ω N ∼ 1 0 23
Intuition Equal-weight countingAgar har microstate equally likely hai (fundamental postulate ), toh ek macrostate ka chance sirf uska share hai total mein: uske kitne microstates hain, divided by saare microstates.
Intuition Zyada particles ke saath peak kyun sharper hoti haiJaise jaise N P ( n ) apna centre N /2 lekin uski width sirf N relative width, width/ ∼ N / N = 1/ N hamesha apne most-probable macrostate mein paaya jaata hai — yahi equilibrium ka asli matlab hai. (Details Binomial and Gaussian distributions mein hain.)
Ye woh "thode se numbers hain jo lab instrument padhta hai" — ek macrostate ke ingredients.
Definition Bulk variables
N number of particles (ek count; picture: box mein kitne molecules hain).V volume (system kitni jagah occupy karta hai; picture: box ka size litres mein).E total energy (saari kinetic + potential energy jodi hui; picture: particles ke beech share hone wala fixed budget).T temperature (kitna garam hai; picture: thermometer reading — gehraai se, yeh batata hai ki entropy added energy par kaise respond karti hai ).P pressure (walls par force per area; picture: ek gauge needle).Ye topic ko kyun chahiye: ek macrostate in mein se kuch ki ek chhoti si mutthi se name hota hai, jaise Ω ( E , V , N )
Ek classical system ke liye, ek microstate ek single point hai jo har position q p ( q 1 , p 1 , … , q 3 N , p 3 N ) phase space hai — N 6 N
Picture: ek dot jo ek enormous multi-dimensional room mein drift kar rahi hai; room ke regions ka study Phase space and Liouville's theorem karta hai.Ye topic ko kyun chahiye: yeh "ek poori microscopic specification" ko ek precise geometric object banata hai jiska aap size measure kar sakte hain.
Binomial coefficient N choose n
Equilibrium is the most probable macrostate
Khud ko test karo — right side cover karo aur zor se jawab do.
Multiplication rule kya kehta hai? Independent choices multiply hote hain: n 1 × n 2 × ⋯
N ! N × ( N − 1 ) × ⋯ × 1 N
0 ! 1
( n N ) n ! ( N − n )! N !
Hum n ! ( N − n )! Chosen aur unchosen groups ke andar ke orderings cancel karne ke liye, kyunki ek macrostate ke liye order matter nahi karta.
( 2 4 ) 6
∑ n = 0 N f ( n ) f ( 0 ) + f ( 1 ) + ⋯ + f ( N )
Total microstates 2 N N
∑ n = 0 N ( n N ) 2 N
Yahan count ko probability mein kaise badlate hain? P ( n ) = Ω ( n ) / Ω total
Kaunse symbols ek macrostate ko name karte hain? Kuch bulk variables jaise E , V , N T , P , N
Phase-space point kya hota hai? Ek microstate: saari positions aur momenta ( q 1 , p 1 , … , q 3 N , p 3 N )
Ω Ek macrostate ke saath consistent microstates ki sankhya — topic ka central object.
