Phase space
Phase space is the product space of of impulse and position space.
Some basic facts:
- there is a minimal quantum of action as as fundamental constant of nature the Planck constant (h-bar)
this is kind of more a result of math (the Fourier transform) rather than a result of physics,
well actually a result of the wave function describing both position and impulse and thus linking them together - phase space of an isolated system can be locally incompressible sort of – see Liouville's theorem
- phase space is useful in the analysis of chaotic systems
- even for a merely spacially 2D systems the phase space is already 4D do illustration is nontrivial
- in quantum mechanics using the bra-ket notation one can do math while not specifying whether position space or impulse space is to be used.
That is decided at the end by appropriare projection into the space that one wants to look at.
Systems can be described space agnostically by using quantum numbers.
Contents
Components
Position space (sometime real space)
3D spacial Fourier transform of impulse space space.
Juts good old normal space. What one would expect.
Impulse space
3D spacial Fourier transform of position space.
Speeds multiplied by the masses.
Units
It's always pairs complementary values that together give the unit of action Js (but complementary not as in thermodynamics).
- linear position times linear impulse: m * (kg * m/s)
- rotary position times rotary impulse (aka angular momentum) : rad * (m * (kg * m/s))
- energy times time: J s
- other combinations possible
Unit of rad is 1.
Angular momentum having same unit as action is kind of confusing but
it also provides another avenue of intuition for why it's quantized.
Related: Fun with spins
Fourier transform conversions
- a single square profile in 1D (like a wall) corresponds to a sin(x)/x in the Fourier transform
- a sine corresponds to a sine but inverse wavelengths
- a single sharp impulse (dirac delta) corresponds to a constant value
- an infinite set of impulses …
Relation to the arrow of time
Here's an interesting perhaps unusual insight that's usually not talked about:
In a closed system for events to go preferentially forward rather than backward (having an arrow of time)
the number of microstates must monotonically increase.
This can both happen in the position space and in the impulse space.
There may be a decrease in number of microsctates on one of the two side.
But overall there always needs to be an increase.
There is a critical difference though.
Position space has very finite capacity for increase in the number of microstates.
Once one ends up with a maximally chaotic blend of ingredients no more microstates can be achieved.
Impulse space though in real non-closed systems is a boundless sink though.
Heat can easily escape to the the outside as phonons and eventually as infrared photons.
Both of these being bosons may also help. To investigate.
Endoergic reactions – impulse space ordering reactions
I the case of (local) endoergic reactions (that is: chemical reactions that suck up thermal energy from their surroundings)
there is a decrease of microstates in impulse space but
this is overcompensated by an increase of microstates in position space.
The finite capacity of the position space for microstates may be one reason for why
endoergic reactions are rather considered to be a super weird exception rather than the norm.
Anti endoergic reactions – position space ordering reactions
The exact opposite of (local) endoergic reactions is when
there is a decrease of microstates in position space but
that is overcompensated by an increase in microstates in the impulse space.
Basically devaluating some free energy to get more order into position space.
Like in bringing stuff into the machine phase.
Useless reactions
The last of the three possible combinations that runs forward by itself
is when both position and impulse space increase in number of microstates.
That's just a sad waste of free energy and something one should strive to avoid.
On-site disordering Off-site ordering systems
Nanomechanical systems allow for the mechanical transfer of useful free energy to other places in position space
before some hopefully efficient conversion into cooling and/or ordering is done there.
Ordering & cooling like in mechanosyntehsizing solid highly ordered things from raw gaseous or liquid feed stock molecules at low temperatures.
There can obviously be temporary energy storage in elastic forces like springs and eventually in (not nanoscale) flywheels.
But also electrical storage and other exotic things like even gravity.
It should be possible to look at all of this a transfers of phase space volume.
(wiki-TODO: investigatre this deeper)
Related
External links
- https://en.wikipedia.org/wiki/Phase_space
- https://en.wikipedia.org/wiki/Liouville%27s_theorem_(Hamiltonian)
- https://en.wikipedia.org/wiki/Ergodic_hypothesis
- https://en.wikipedia.org/wiki/Action_(physics)
- https://en.wikipedia.org/wiki/Planck_constant
- https://en.wikipedia.org/wiki/Uncertainty_principle
