# Liouville's Theorem and Poincare Recurrence Theorem

• I
dyn
Hi.
I am working through some notes on the above 2 theorems.
Liouville's Theorem states that the volume of a region of phase space is constant along Hamiltonian flows so i assume this means dV/dt = 0
In the notes on the Poincare Recurrence Theorem it states that if V(t) is the volume of phase space swept out in time t then since volume is preserved dV/dt = C where C≥ 0 is constant. Surely by Liouville's Theorem C should be zero ?
Thanks

## Answers and Replies

Gold Member
2022 Award
Liouville's theorem refers to the phase-space volume of a Hamiltonian system. Let ##q## and ##p## the generalized coordinates and their canonical momenta. Then the trajectory of the system in phase space is determined by Hamilton's canonical equations,
$$\dot{q}=\partial_p H, \quad \dot{p}=-\partial_q H.$$
Let's denote the solution of the corresponding initial-value problem, the "Hamiltonian flow of phase space" with
$$q(q_0,p_0,t),p(q_0,p_0,t).$$
Then the phase-space volume element is
$$\mathrm{d}^{2n} (q,p)=\mathrm{d}^{2n} (q_0,p_0) \mathrm{det} \left (\frac{\partial(q,p)}{\partial(q_0,p_0)} \right)=\mathrm{d}^{2n} (q_0,p_0) D(t,t_0).$$
The time derivative is
$$\mathrm{d}_t \mathrm{d}^{2n} (q,p)|_{t=t_0}=\nabla_{q_0} \cdot \dot{q} + \nabla_{p_0} \cdot \dot{p} = \nabla_{q_0} \cdot \nabla_{p_0} H - \nabla_{p_0} \cdot \nabla_{q_0} H.$$
Now the Jacobi determined fulfills the composition rule
$$D(t,t_0)=D(t,t_1) D(t_1,t_0).$$
From this
$$\partial_t D(t,t_0) = \partial_t D(t,t_1) D(t_1,t_0).$$
For ##t_1 \rightarrow t## we get from the calculation above 0 on the right-hand side and thus
$$\partial_t D(t,t_0)=0.$$
This implies that the phase-space volume does not change under the Hamiltonian flow of phase space.

I don't know, what you mean concerning Poincare's recurrence theorem. Which book/paper are you studying?

dyn
Thanks for your reply. Regarding the volume element in phase space ; is it the total time derivative that is zero ; ie. dV/dt = 0 or the partial derivative wrt time ?
As regards the Poincare Recurrence Theorem i am studying some lecture notes and at the start of the proof it states that if V(t) is the volume of phase space swept out in time t then since volume is preserved dV/dt = C where C≥ 0 is constant.
Surely by Liouville's Theorem C should be zero ?

Gold Member
2022 Award
It's the total time derivative.

I still don't understand the argument you quote from your lecture notes since indeed Liouville's theorem states ##C=0##, but to try to understand what the author means, I'd need more context.

dyn