I Liouville's Theorem and Poincare Recurrence Theorem

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Liouville's Theorem asserts that the volume of a region in phase space remains constant along Hamiltonian flows, implying that the total time derivative of volume, dV/dt, equals zero. In contrast, the Poincare Recurrence Theorem suggests that the volume swept out in time t, V(t), has a constant rate of change, dV/dt = C, where C is non-negative. This leads to confusion, as Liouville's Theorem would imply that C should be zero. The discussion highlights the need for clarity regarding whether the reference is to the total or partial time derivative in the context of these theorems. Further context from the lecture notes is necessary to resolve the apparent contradiction.
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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
 
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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?
 
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 ?
 
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.
 
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