2.4.2 · D3Legendre transforms connecting them
Intuition Yeh page kis liye haiParent note ne machine banai thi. Yahan hum use har tarah ki road condition pe drive karte hain : clean curves, sign traps, degenerate (straight-line) inputs, limiting behaviour jahan transform blow up karta hai, ek real lab word-problem, aur ek exam twist.
Agar tum koi aisa case dekho jo is page pe nahi hai, woh in cells ka hi ek combination hoga — kuch bhi naya tumhe surprise nahi kar sakta.
Prerequisites jinhe hum use karte hain: Internal Energy and the First Law , Maxwell Relations , Thermodynamic Stability and Convexity , aur mechanics cousin Lagrangian to Hamiltonian (Legendre in Mechanics) . Chemistry payoff Chemical Potential and Phase Equilibrium mein milta hai; F Partition Functions mein hai.
Har Legendre problem in cells mein se ek (ya blend) hoti hai:
Cell
Case class
Kya galat ho sakta hai / kya dhyan rakhna hai
Kis example mein
A Convex power law f = 2 1 a x 2 a > 0
"textbook" case, g
Ex 1
B Sign trap: slope negative hai (− P
negative subtract karo = add karo
Ex 2
C Non-quadratic convex f = e x
p = f ′ ln p > 0 Ex 3
D Degenerate: linear f = c x
slope constant → transform undefined/collapses
Ex 4
E Limiting / boundary: convexity a → 0 +
transform diverge karta hai, stability edge
Ex 5
F Real-world word problem (lab: fixed T , P
sahi potential G
Ex 6
G Real gas pe full thermo swap
P S Ex 7
H Exam twist: double transform (involution)
prove karo ki tum wapas ghar aate ho
Ex 8
Ab hum har cell ko hit karte hain.
f ( x ) = 2 1 a x 2 a > 0 Forecast: padhne se pehle g ( p ) f
Step 1. Slope: p = f ′ ( x ) = a x Yeh step kyun? Legendre transform curve ko uski slope se re-label karta hai, isliye pehli cheez jo hum compute karte hain woh slope function hai.
Step 2. Invert karo: x = p / a Yeh step kyun? g p ka function hona chahiye; har x
Step 3. g = f − p x g ( p ) = 2 1 a ( a p ) 2 − p ⋅ a p = 2 a p 2 − a p 2 = − 2 a p 2 . Yeh step kyun? Yahi definition hai g = f − p x neeche hota hai.
Verify: d p d g = − a p = − x d g / d p = − x a = 2 , x = 3 f = 9 , p = 6 , g = − 9 f − p x = 9 − 18 = − 9
U V negative hai
Maano equilibrium ke paas U ( V ) = U 0 + 2 1 k ( V − V 0 ) 2 S , N k > 0 V Forecast: kya hum U − P V U + P V
Step 1. Slope ko uske sign ke saath padho: ∂ V ∂ U = k ( V − V 0 ) d U = ⋯ − P d V hai − P − P = k ( V − V 0 ) P = − k ( V − V 0 ) Yeh step kyun? Poora trap yahi hai ki − P d V
Step 2. Legendre subtract = slope × variable = ( − P ) V H = U − ( − P ) V = U + P V . Yeh step kyun? "Slope×variable subtract karo" literal hai; yahan slope − P add P V
Step 3. Differential check: d H = d U + P d V + V d P = ( T d S − P d V ) + P d V + V d P = T d S + V d P Yeh step kyun? Confirm karta hai ki − P d V H d P
Verify (numbers): maano k = 4 , V 0 = 1 , U 0 = 0 , V = 3 U = 2 1 ( 4 ) ( 2 ) 2 = 8 P = − 4 ( 2 ) = − 8 H = U + P V = 8 + ( − 8 ) ( 3 ) = 8 − 24 = − 16 U − P V = 8 − ( − 24 ) = 32
H = U − P V U V − P P ( − P ) V P V Hamesha pehle differential se slope ka sign padho.
f ( x ) = e x Forecast: g ( p ) p
Step 1. Slope: p = f ′ ( x ) = e x Yeh step kyun? Wahi pehla move — slope function nikalo.
Step 2. Invert karo: kyunki e x > 0 p > 0 x = ln p Yeh step kyun? p = f ′ ( x ) g domain janm leta hai. e x g p > 0
Step 3. g ( p ) = f − p x = e l n p − p ln p = p − p ln p Yeh step kyun? g = f − p x x = ln p
Verify: d p d g = 1 − ( ln p + 1 ) = − ln p = − x x = 0 p = 1 g = 1 − 1 ⋅ 0 = 1 = f ( 0 ) = f
f ( x ) = c x Forecast: guess karo ki transform exist bhi karta hai ya nahi.
Step 1. Slope: p = f ′ ( x ) = c constant , x Yeh step kyun? Hum hamesha slope se shuru karte hain; yahan woh kabhi change nahi hota.
Step 2. p = c x p = c har x p = c koi x Yeh step kyun? Transform ek one-to-one slope map maangta hai. Ek straight line har jagah ek hi slope rakhti hai → map collapse ho jaata hai.
Step 3. Interpret karo. p = c g = f − p x = c x − c x = 0 ( c , 0 ) kaunsa x Yeh step kyun? Failure ko geometrically dikhata hai: ek line ka koi curvature nahi hota, isliye "tangent lines" fan out nahi karti — woh sab ek hi line hain, koi positional info nahi.
Verify: Second derivative f ′′ = 0 f ′′ = 0 f ′′ = 0 ⇒ stability (∂ 2 U / ∂ S 2 = T / C V > 0 guarantee karti hai ki thermo transforms kaam karein.
g ( p ) = − 2 a p 2 a → 0 + Forecast: fixed p g
Step 1. p = 0 a → 0 + g = − 2 a p 2 → − ∞ Yeh step kyun? a f ′′ a → 0
Step 2. Meanwhile woh x p x = p / a → ∞ Yeh step kyun? Ek ever-flatter parabola pe slope p − ∞
Step 3. Thermodynamic reading: agar C V → ∞ S U Yeh step kyun? Pure math limit ko real physics se jodta hai: diverging response = instability ki shuruat.
Verify: a = 0.1 , p = 2 g = − 0.2 4 = − 20 a = 0.01 g = − 0.02 4 = − 200 − ∞
Worked example "Ek reaction open beaker mein bench pe chal rahi hai. Main kaunsa potential minimize karun?"Beaker atmosphere ke liye open hai (pressure P T N Forecast: compute karne se pehle potential guess karo.
Step 1. Controlled variables list karo: T P Yeh step kyun? Sahi potential woh hota hai jiske natural variables exactly wahi hain jo tum hold karte ho — phir equilibrium ek clean minimum ban jaata hai.
Step 2. Fundamental relations se match karo. Sirf G d G = − S d T + V d P + μ d N ( T , P , N ) Yeh step kyun? U ( S , V , N ) H ( S , P , N ) F ( T , V , N ) uncontrolled natural variable (S V G
Step 3. Fixed T , P d G = μ d N ⇔ d G = 0 ⇔ Yeh step kyun? Yahi poori wajah hai ki chemists G Chemical Potential and Phase Equilibrium .
Verify (units): [ μ d N ] = ( J/mol ) ( mol ) = J [ G ] = J d G = − S d T + V d P + μ d N d T = d P = 0 μ d N
Worked example Ideal gas keF ( T , V , N ) T , N F = − N k B T ln V + ( terms not depending on V ) Forecast: guess karo ki P = + ∂ F / ∂ V − ∂ F / ∂ V
Step 1. d F = − S d T − P d V + μ d N V ( ∂ V ∂ F ) T , N = − P Yeh step kyun? Sign directly differential se padho (Cell B ka lesson).
Step 2. Compute karo: ∂ V ∂ ( − N k B T ln V ) = − V N k B T − P − P = − V N k B T ⇒ P = V N k B T . Yeh step kyun? Ideal gas law ek potential ki slope ke roop mein nikal aati hai — koi alag postulate nahi chahiye.
Step 3. Entropy dusre slot se: S = − ( ∂ T ∂ F ) V , N Yeh step kyun? Wahi machine, alag variable — ek Maxwell/slope pairing (dekho Maxwell Relations ).
Verify (numbers): N = 2 , k B = 1 T = 300 , V = 10 P = 10 2 ⋅ 1 ⋅ 300 = 60 ∂ F / ∂ V = − 10 2 ⋅ 300 = − 60 = − P N k B T / V m 3 ( J/K ) ( K ) = J/m 3 = Pa
f ( x ) = 2 1 a x 2 do baar transform karo aur dikhao ki tum wapas ghar aate ho
Forecast: guess karo ki second transform kya deta hai — original, ya kuch scaled?
Step 1. Pehla transform (Ex 1 se): g ( p ) = − 2 a p 2 d p d g = − a p Yeh step kyun? Hume g input ke roop mein chahiye.
Step 2. Naya conjugate variable q = d p d g = − a p p = − a q Yeh step kyun? Involution proof ke liye g f
Step 3. Second transform h ( q ) = g − q p = − 2 a p 2 − q ( − a q ) = − 2 a ( − a q ) 2 + a q 2 = − 2 a q 2 + a q 2 = 2 a q 2 Yeh step kyun? Yeh 2 1 a q 2 f
Verify (numbers): a = 2 , q = 3 h = 2 1 ( 2 ) ( 9 ) = 9 = f ( 3 ) = 2 1 ( 2 ) ( 9 ) = 9 char potentials ke across kabhi bhi koi information nahi gayi .
Recall Kaunsa cell fail karta hai, aur kyun?Sirf Cell D (linear f f ′′ = 0 p = f ′ ( x ) x ∂ 2 U / ∂ S 2 = T / C V > 0
Mnemonic Sign padho, phir slope×variable subtract karoHar case is cheez tak reduce hua: (1) slope compute karo, (2) invert karo (yahan domain reveal hota hai aur yeh pata chalta hai ki degenerate hai ya nahi), (3) g = f − ( slope ) ( x ) sign aur flatness (f ′′ = 0
