I Liouville equation with Dirac delta as probability density

AI Thread Summary
The discussion centers on solving the Liouville equation with a Dirac delta initial condition. The proposed solution suggests that the density function ρ(t) takes the form of a Dirac delta centered on the trajectories defined by Hamilton's equations. Participants highlight the need to verify this solution by substituting it back into the Liouville equation. Confusion arises regarding the treatment of the Dirac delta function and its derivatives, prompting suggestions to apply standard derivative rules for distributions. The conversation emphasizes a methodical approach to proving the equality in the equation.
andresB
Messages
625
Reaction score
374
I would like to know the solution to Liouville equation
∂ρ/∂t=-{ρ,H}

given the initial condition

ρ(t=0)=δ(q,p)

where δ(q,p) is a dirac delta centered in some point (q,p) in phase space.

I have the feeling, but I'm not sure, that the solution is of the form

ρ(t)=δ(q(t),p(t))

where q(t) and p(t) are the trajectories from Hamilton equations.

Any help?
 
Physics news on Phys.org
Did you try inserting your suspicion into the Liouville equation to see if it does solve it?

Note that your suspicion is actually
$$
\rho(q,p,t) = \delta(q-q(t),p-p(t)),
$$
where ##q(t)## and ##p(t)## are just functions of time (that satisfy the Hamilton equations of motion), when expressed properly.
 
Orodruin said:
Did you try inserting your suspicion into the Liouville equation to see if it does solve it?

I tried, but it is confusing to me due to the nature of Dirac delta. Liouville equation read
##∂δ/∂t= -(∂δ/∂q)(∂H/∂p) + (∂δ/∂p)(∂H/∂q)## (1)

Equality should be understood in the distribution sense. Using a test function ##F(q,p)##, I get for the left hand side of (1)

##∫F(q,p) ∂δ/∂t dqdp= ∂/∂t∫F(q,p)δ(q-q(t),p-p(t) dqdp= ∂F(q(t),p(t))/∂t##

But I'm unsure how to prove the equality to the right hand side of (1).
 
You do not need to use test functions. Just apply the standard rules of derivatives that hold also for distributions. Take it step by step and show your work.
 
Thread 'Gauss' law seems to imply instantaneous electric field propagation'
Imagine a charged sphere at the origin connected through an open switch to a vertical grounded wire. We wish to find an expression for the horizontal component of the electric field at a distance ##\mathbf{r}## from the sphere as it discharges. By using the Lorenz gauge condition: $$\nabla \cdot \mathbf{A} + \frac{1}{c^2}\frac{\partial \phi}{\partial t}=0\tag{1}$$ we find the following retarded solutions to the Maxwell equations If we assume that...
Dear all, in an encounter of an infamous claim by Gerlich and Tscheuschner that the Greenhouse effect is inconsistent with the 2nd law of thermodynamics I came to a simple thought experiment which I wanted to share with you to check my understanding and brush up my knowledge. The thought experiment I tried to calculate through is as follows. I have a sphere (1) with radius ##r##, acting like a black body at a temperature of exactly ##T_1 = 500 K##. With Stefan-Boltzmann you can calculate...
Thread 'A scenario of non-uniform circular motion'
(All the needed diagrams are posted below) My friend came up with the following scenario. Imagine a fixed point and a perfectly rigid rod of a certain length extending radially outwards from this fixed point(it is attached to the fixed point). To the free end of the fixed rod, an object is present and it is capable of changing it's speed(by thruster say or any convenient method. And ignore any resistance). It starts with a certain speed but say it's speed continuously increases as it goes...
Back
Top