Statistical mechanics — microstate, macrostate

2.4.8 · Physics › Thermodynamics & Statistical Mechanics (Advanced)

WHY yeh distinction exist karti hai?

WHY: Ek litre gas mein $\sim 1 0^{22}$ molecules hote hain. Tum kabhi bhi saari positions aur velocities likh nahi sakte, aur zaroorat bhi nahi — thermometers aur pressure gauges sirf averages dekhte hain. Isliye physics do descriptions mein split hoti hai:

Microscopic reality — har particle ke har coordinate (bahut zyada information).
Macroscopic measurement — kuch bulk variables (bahut kam information).

Gehra idea yeh hai: unke beech ka bridge counting hai. Equilibrium koi jaadu nahi hai — yeh bas woh macrostate hai jis par sabse zyada microstates correspond karte hain.

HOW count karein: coin / spin model (scratch se derive karein)

WHAT we model: $N$ distinguishable two-state objects (spins up/down, ya coins H/T). Yeh sabse simple non-trivial counting problem hai aur stat-mech ka poora logic pehle se yahan hai.

Microstate: poori ordered list, jaise $N = 4$ ke liye: $↑↓↑↑$ — ek specific sequence.
Macrostate: bas "kitne up hain", $n$ . Hame parwah nahi kaun se hain.

Total microstates $= 2^{N}$ kyun? Har object independently 2 choices rakhta hai, aur choices multiply hote hain (fundamental counting principle): $Ω_{total} = N 2 \times 2 \times \dots \times 2 = 2^{N}$

Macrostate $n$ ki multiplicity $(n N)$ kyun hai? Hame count karna hai ki kitne distinct sequences mein exactly $n$ ups hain. Choose karo ki $N$ slots mein se kaun se $n$ slots up hain — slots ke apne order se koi fark nahi — isliye yeh ek combination hai: $Ω (n) = (n N) = \frac{N !}{n ! ( N - n )!}$

Yeh step kyun? $N!$ saare objects ko order karta hai; lekin $n$ ups ko aapas mein permute karna ( $n!$ ) aur $N - n$ downs ko ( $(N - n)!$ ) ek hi macrostate deta hai, isliye hum unhe divide kar dete hain.

Sanity check (Forecast-then-Verify): saare macrostates ka sum har microstate ko wapas reconstruct karna chahiye: $\sum_{n = 0}^{N} (n N) = 2^{N} ✓ (binomial theorem with 1 + 1)$

Figure — Statistical mechanics — microstate, macrostate

Worked Example 1 — Char coins haath se

Macrostate = heads ki sankhya $n$ .

$n$	microstates	$Ω = (n 4)$
0	TTTT	1
1	HTTT, THTT, TTHT, TTTH	4
2	HHTT, HTHT, HTTH, THHT, THTH, TTHH	6
3	( $n = 1$ ka mirror)	4
4	HHHH	1

Total $= 1 + 4 + 6 + 4 + 1 = 16 = 2^{4}$ . Yeh step kyun? Confirm karta hai ki humne har microstate exactly ek baar count ki.

Most probable macrostate $n = 2$ hai jisme $P = 6/16 = 0.375$ hai. "All heads" macrostate exist karta hai lekin uski $P = 1/16$ hai.

Worked Example 2 — Ek extreme fluctuation ki probability

$N = 100$ spins ke liye, probability kya hai ki sab upar point karein? $P = \frac{Ω ( 100 )}{2 ^{100}} = \frac{1}{2 ^{100}} \approx 8 \times 1 0^{- 31}$ Yeh step kyun? Is macrostate ko sirf ek microstate ( $Ω = 1$ ) realize karta hai, divided by total $2^{100}$ . Ek chote system ke liye bhi, extreme macrostate essentially impossible hai — yeh irreversibility ki statistical origin hai.

Worked Example 3 — Ideal gas, $Ω$ kyun $V$ aur $E$ ke saath badhta hai

$N$ free particles ke liye volume $V$ mein energy $E$ par, microstates ko (position cells)×(momentum-shell states) ki tarah count karo: $Ω \propto V^{N} \times (area of momentum sphere of radius 2 m E) \propto V^{N} E^{3 N /2}$ Yeh step kyun? Har particle $V$ mein kahin bhi ho sakta hai (isliye $V^{N}$ ), aur energy $E = \sum p_{i}^{2} /2 m$ ek $3 N$ -dimensional momentum sphere ki radius fix karti hai; uski surface "volume" $E^{(3 N - 2) /2} \approx E^{3 N /2}$ ki tarah scale karti hai. WHY it matters: zyada $V$ ya $E$ ⇒ vastly zyada microstates ⇒ zyada entropy ⇒ gas spontaneously expand karti hai aur energy share karti hai. Counting thermodynamics predict karta hai.

Recall Feynman: ek 12-saal ke bache ko explain karo

Socho 10 coins flip kar rahe ho. "Macrostate" = kitne heads aaye (ek number, friend ko batana aasaan). "Microstate" = exact pattern jaise H-T-T-H-... Saare 10 heads ke liye sirf ek pattern hai, lekin 5 heads aur 5 tails ke liye 252 patterns hain. Toh jab tum coins shake karte ho tum almost hamesha lagbhag aadhe heads paate ho — isliye nahi ki universe ko yeh pasand hai, balki isliye ki ise karne ke bahut zyada tarike hain. Woh "bahut zyada tarike" hi heat, temperature, aur ek kamre mein smell failne ke peeche ka raaz hai.

Flashcards

Microstate kya hai?

Ek complete microscopic specification — phase space mein ek point (classical) ya ek quantum eigenstate; har particle ke saare coordinates/momenta.

Macrostate kya hai?

Kuch bulk measurable variables se ek specification, jaise

E, V, N

(ya

T, P, N

Multiplicity

Ω

define karo.

Ek diye gaye macrostate ke saath consistent microstates ki sankhya,

Ω (E, V, N)

N

two-state objects ke liye microstates ki total sankhya?

2^{N}

N

mein se "

n

up" macrostate ki multiplicity?

(n N) = \frac{N !}{n ! ( N - n )!}

\sum_{n} (n N) = 2^{N}

kyun hona chahiye?

Saare macrostates par multiplicities ka sum har microstate ko recover karta hai (binomial theorem,

1 + 1 = 2

Statistical mechanics ka fundamental postulate kya hai?

Equilibrium mein, har accessible microstate equally probable hai.

Most-probable macrostate practice mein kyun observe hota hai?

Uska

Ω

doosron ko dwarf kar deta hai; relative fluctuation width

\sim 1/ N \to 0

macroscopic

N

ke liye.

Kya saare microstates equally probable hain? Kya saare macrostates hain?

Microstates: haan. Macrostates: nahi —

Ω

se weighted hote hain.

Ideal-gas

Ω

V

aur

E

ke saath kaise scale karta hai?

Ω \propto V^{N} E^{3 N /2}

Saare

N

spins up point karne ki probability?

1/ 2^{N}

(sirf ek microstate ise realize karta hai).

Equilibrium par microscopically kya hota hai?

Microstate badhta rehta hai; macrostate fixed rehta hai kyunki almost saare microstates ise share karte hain.

Connections

Entropy and Boltzmann's relation S = k ln Ω
Fundamental postulate of statistical mechanics
Phase space and Liouville's theorem
Microcanonical ensemble
Binomial and Gaussian distributions
Second law and irreversibility
Temperature as ∂S/∂E