The Liouville's Theorem & It's Proof
[ 1 ] The Liouville's Theorem :
- The Rate Of Change Of Density Of Phase Point In The
Neighborhood Of A Moving Phase Point In Phase Space Is Zero.
Mathematicaly dp/dt
= 0
- This Part Is Known As Principal Of Conservation Of Density In
Phase Space.
- Any Arbitary Elemants Of Volume In Phase Space Bounded By
Moving Surface Is Constent.
Mathematicaly d(δt)/dt
= 0
- This Part Is Known As Principal Of Conservation Of Extension In
Phase Space.
[ 2 ] Proof :
- dp/dt = 0
- Consider An Element Of Volume δt In Phase Space.
Then The Volume
Element
δt = δq1
δq2....δqf,δp1,δp2....δpf......(1)
- If P Is The Density Of Phase Point Then The No Of Phase Point
In The Given Volume Element
δN = Pδt......(2)
- Differentiation Eq (2) W.R.T To Time t
- The Change In No Of Phase Point Corresponding To Change In
q1.
- Similarly The Change In No Of Phase Point Corresponding To
Change In p1.
- The Net Change In The Numer Of Phase Point In The Given
Volume Elemnet Is
- From Hamilton Equation Of Motion
∂H/∂q1 = - Pi & ∂H/∂Pi = qi
Part - 2
- δN = Pδt
- Differentiation Eq (2) W.R.T To Time t
- Since δN Is Constant Therfor d/dt (δN) = 0
dp/dt = 0
#LiouvillesTheorem
#StatisticalPhysics
#LiouvilleEquation
#PhaseSpace
#MicroscopicDynamics
#HamiltonianDynamics
#Entropy
#Ergodicity
#EnsembleTheory
#BoltzmannEquation
#LiouvilleProof
#CanonicalEnsemble
#MicrocanonicalEnsemble
#GibbsEnsemble
#StatisticalMechanics
#ConservationOfPhaseSpace
#LiouvilleConservation
#PhaseSpaceVolume
#ErgodicHypothesis
#LiouvilleIntegration
#ConservativeDynamics
#LiouvilleDeterminism
