# Area of Phase Space and Dependence on Energy

The phase curve for a system is made for some configuration, for example - The Harmonic Oscillator. Now as we increase the energy, the phase curve enlarges i.e. area enclosed by the curve increases.

My question is how does the area of phase space curve depends upon changing the energy of the particle.

## 1 Answer

There isn't a general answer to this question because only periodic motions produce closed curves in phase-space. Any aperiodic motion, whether chaotic or not, doesn't produce a closed curve.

That said, if we assume that the motion is periodic and that the phase-space is 2-dimensional, so that the area enclosed by the curve is unambiguous, I'm still not sure that the area enclosed by the curve has any physical meaning.

The two simplest examples I can thing of are: the simple harmonic oscillator, and the particle hitting perfectly reflecting walls. For the simple harmonic oscillator, the area enclosed is given by $$A_{sho} = \pi p_{\mathrm{max}} x_{\mathrm{max}}.$$ This is just the formula for the area of an ellipse with the semi-major and semi-minor axes given by the amplitude in the momentum direction and the real-space direction. You could rewrite that as: $$A_{sho} = \pi m\omega A^2$$.

Similarly, the particle reflecting off of perfect walls traces out a rectangle in phase space, and the area of that rectangle is: $$A_{box} = (2p) W$$ where $$p$$ is the momentum of the particle and $$W$$ is the width of the box.

For the simple harmonic oscillator the area looks like the energy, but as the box shows us, that's a coincidence. I believe that there may be thermodynamic statements related to the entropy of the system, but for a single particle this isn't very well defined. For example, if we say that the entropy is related to the phase-space volume occupied by the system, I believe that you can show that for a wall that moves much slower than the ball in the box ($$v_{\mathrm{wall}}\ll p/m$$) that the volume of the rectangle will be conserved (to first order in $$m v_{\mathrm{wall}} / p$$). The challenge there is that this is a Maxwell's demon type situation, where if the wall moves rapidly in discrete steps then it's possible to change the volume enclosed in ways that would increase or decrease entropy without doing any work, violating the second law of thermodynamics. This isn't, technically, a problem since that only applies to many particle states, but it makes trying to use the model for that purpose questionable.

You might try to think about Liouville's theorem, but that's about an area enclosing the current state of the system and how that area evolves in time as it moves with and around the system, not the area enclosed by the system's path in phase space.