2.4.13 · D1Maxwell-Boltzmann distribution — full derivation
Ek gas mein billions molecules random speeds pe ud rahe hain, aur Maxwell-Boltzmann distribution bas ek hill-shaped graph hai jo batata hai ki har speed kitni common hai . Derivation mein sab kuch do facts se banta hai: fast molecules energetically expensive hote hain (wo rare ho jaate hain), lekin fast hone ke zyada tarike hote hain (velocity space mein ek bada "shell") — aur neeche poora page tumhe exactly woh symbols sikhata hai jo in dono facts ko encode karte hain.
Yeh page kuch bhi assume nahi karta . Isse pehle ki tum full derivation ko chhuao, har letter, squiggle, aur picture jo woh use karta hai, yahan se ground up mein banaya gaya hai, ek aisi order mein jahan har idea apne pehle waale idea pe lean karta hai.
Definition Velocity vectorv direction mein move karta hai. Hum uski motion ko ek arrow v x y z Components v x , v y , v z
Socho ek molecule ek room mein shoot kar raha hai. Uska east-west floor pe shadow v x v y v z
v Speed arrow ki length hai — kitni tezi, direction ignore karke. Agar tum teen components jaante ho, toh 3D mein Pythagoras deta hai:
v = v x 2 + v y 2 + v z 2 .
Square root of sum of squares kyun? Kyunki yeh bas ruler-distance formula ka 3D version hai: ek box ka diagonal jiski sides v x , v y , v z
Intuition Topic ko DONO kyun chahiyeDerivation components se start karta hai (woh independent hain — koi force east-motion ko up-motion se nahi jodata) aur baad mein unhe speed mein assemble karta hai (jo cheez hum actually measure aur plot karte hain). Dono ko alag rakhna hi poora trick hai.
Energy-cost factor banane se pehle, humein woh teen physical quantities chahiye jo isse banti hain.
Definition Teen physical inputs
T absolute temperature kelvin (K) mein. T = 0 T k B Boltzmann's constant = 1.38 × 1 0 − 23 J/K unit converter hai: temperature (kelvin) ko energy (joules) mein convert karta hai, kyunki k B T m ek single molecule ki mass kilograms mein. Molar mass nahi! (Molar mass M mole hai; m
m M Log kyun trip karte hain: tables molar mass M Fix: ya toh per-molecule m k B M R m R M k B
Combination k B T natural energy scale hai — roughly "ek degree of freedom kitni kinetic energy carry karta hai." Yahi Equipartition theorem ka message hai.
e e − x e ≈ 2.718 e − x x placeholder input hai — koi bhi real number daalo (x e − x
x = 0 1 jaise x 0
kabhi actually 0 "kitna bacha hai" ki tarah socho — jaise ek hot cup thanda ho raha ho: pehle fast drop hota hai, phir slower, hamesha room temperature approach karta raha.
Exactly yeh curve kyun aur, say, seedha line down kyun nahi? Kyunki e − x yeh exponent mein ek sum ko alag factors ke product mein convert karta hai . Matlab, e − ( a + b ) = e − a ⋅ e − b split kar sakta hai, ek per axis.
Definition Boltzmann factore − E / k B T x E / k B T k B T E e − E / k B T 2 1 m v 2 m e − m v 2 /2 k B T energy-cost aadha hissa hai.
Tum is factor se properly Boltzmann factor and partition function mein miloge — abhi ke liye bas ise "badi energy ⇒ chhoti probability" ki tarah padho.
Intuition Exponent mein sum kyun teen component factors mein split hota haiKinetic energy sum v x 2 + v y 2 + v z 2 e − m ( v x 2 + v y 2 + v z 2 ) /2 k B T e − 2 k B T m ( v x 2 + v y 2 + v z 2 ) = e − 2 k B T m v x 2 ⋅ e − 2 k B T m v y 2 ⋅ e − 2 k B T m v z 2 . sirf ek axis hai. Toh joint density teen independent single-axis pieces mein toot jaati hai — isi liye derivation v x , v y , v z
Intuition Tum nahi pooch sakte "EXACTLYv = 422 band ke baare mein poochte hain, v v + d v
f ( v ) d v
d v f ( v ) probability density : probability per unit speed . Yeh khud probability nahi hai.f ( v ) d v [ v , v + d v ]
Ek speeds ka histogram socho. Jaise bars infinitely thin hoti jaati hain, tops ek smooth curve f ( v ) area (height f ( v ) d v
∫ ∫ 0 ∞ f ( v ) d v 0 sab thin bars ke areas add karo." Kyunki har molecule ki koi na koi speed hoti hai, yeh fractions total 1 100% ∫ 0 ∞ f ( v ) d v = 1. normalization kehlata hai — yeh formula mein bache huye constants fix karta hai.
∫ S um ke liye stretched-out "S" ki tarah padho. Yeh infinitely many infinitely thin pieces add karne ka tool hai — exactly wahi jo ek continuous distribution maangti hai.
Definition Single-component densityg ( v x ) g probability density hai: g ( v x ) d v x x [ v x , v x + d v x ] f ( v ) ∫ − ∞ ∞ g ( v x ) d v x = 1 y z g
Definition Ek Gaussian (bell curve)A e − b v x 2 e − m v x 2 /2 k B T 0 v x
b width parameter . §3 se compare karte hue, b = 2 k B T m b A height constant , aise choose kiya ki area 1 g ( v x ) = A e − b v x 2
Definition Spherical-shell factor4 π v 2 geometry aadha hissa hai. Socho sabhi velocity arrows jinki same length v v 4 π v 2 zyada room hai. Isi liye f ( 0 ) = 0 0
Intuition Do heroes ko saath rakhnaTeen identical Gaussians (ek per axis) ko shell factor se multiply karo. Teen height constants combine hokar A 3 = ( 2 π k B T m ) 3/2 f ( v ) = zyada room (grows) 4 π v 2 ( 2 π k B T m ) 3/2 energy cost (shrinks) e − m v 2 /2 k B T . hi Maxwell-Boltzmann hai, aur front mein dikhai dene wala messy ( 2 π k B T m ) 3/2 A
velocity components vx vy vz
Gaussian per component g of vx
exponential e to the minus x
sum in exponent splits into product
temperature T and kB and mass m
Gaussian integral fixes A
spherical shell 4 pi v squared
probability density f of v and integral
Shape notice karo: independent components ko har ek ek Gaussian milta hai, Gaussian integral aur normalization constants nail karte hain, shell factor components ko speed mein convert karta hai, aur Boltzmann factor energy cost supply karta hai. Multiply karo — ho gaya.
Khud test karo: right side cover karo aur reveal karne se pehle answer do.
v Ki v
Components se speed v v = v x 2 + v y 2 + v z 2 — 3D Pythagoras / diagonal length.
k B 1.38 × 1 0 − 23 k B T
Hum exact speed ki probability kyun nahi pooch sakte? Speed continuous hai, toh kisi bhi single value ki probability zero hoti hai; hum bands f ( v ) d v
f ( v ) f ( v ) d v f ( v ) per unit speed hai (ek density); f ( v ) d v [ v , v + d v ]
g ( v x ) Ek velocity component ki probability density; g ( v x ) d v x x [ v x , v x + d v x ]
∫ 0 ∞ f ( v ) d v = 1 Har molecule ki koi na koi speed hoti hai, toh saare fractions 100% add hote hain.
Derivation exponential ki kaunsi key property use karta hai? e − ( a + b ) = e − a ⋅ e − b
g ( v x ) = A e − b v x 2 A b b = m /2 k B T (exponent mein width);
A = b / π = m /2 π k B T (height, normalization se fixed).
A Gaussian integral
∫ e − b v x 2 d v x = π / b aur area = 1 se, jo
A π / b = 1 deta hai.
f ( v ) ( 2 π k B T m ) 3/2 A 3
4 π v 2 Un velocity vectors ki sphere ki surface area jo sab length v
f ( 0 ) = 0 e 0 = 1 Shell factor 4 π v 2 v = 0
Gaussian integral ka result kya hai? m M k B Per-molecule mass m k B M R
Ready? Toh full derivation pe jao. Related tools Kinetic theory of gases , Effusion and Graham's law , aur comparison Maxwell-Boltzmann vs Fermi-Dirac vs Bose-Einstein mein milenge.
