# Liouville equation with Dirac delta as probability density

• I
• andresB

#### andresB

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?

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.

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.