Liouville equation with Dirac delta as probability density

  • I
  • Thread starter andresB
  • Start date
  • #1
280
70

Main Question or Discussion Point

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?
 

Answers and Replies

  • #2
Orodruin
Staff Emeritus
Science Advisor
Homework Helper
Insights Author
Gold Member
16,670
6,453
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.
 
  • #3
280
70
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).
 
  • #4
Orodruin
Staff Emeritus
Science Advisor
Homework Helper
Insights Author
Gold Member
16,670
6,453
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.
 

Related Threads on Liouville equation with Dirac delta as probability density

Replies
1
Views
7K
Replies
6
Views
3K
Replies
2
Views
2K
Replies
4
Views
2K
Replies
1
Views
935
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
17
Views
7K
  • Last Post
Replies
4
Views
956
  • Last Post
Replies
2
Views
1K
Replies
1
Views
1K
Top