554:"In a phase space, every degree of freedom or parameter of the system is represented as an axis of a multidimensional space" I think this phrase is misleading. There is a difference between phase space as defined here and parameter space. The phase space is all the possible states of a given system (usually over time for different initial conditions). Changing the parameters of a system (for example the stiffness of a spring) can give a different phase space. If we allow the parameters to be changed then we obtain a family of phase spaces. Usually, if the system has fixed points one can study the evolution of this fixed points when changing the parameters of the system, leading to the
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947:. However, the "phase" crosses its own path several times, which it cannot do. Since all information needed to determine the evolution of a system's phase can be determined from the phase itself, the time derivative of the phase must also be completely given completely by the phase itself. Hence, a system's phase cannot cross its own path because that would mean that at the crossing point, the phase has multiple time derivatives, which it cannot have in a single point. Therefore, what is shown in the image cannot be the phase path of any system, for the simple reason that the path crosses itself.
535:. By (1) containing all possible states of a system and (2) being visualized by phase portrait, it is essentially identical to the phase space. You claim that the 'state space' is a configuration space, which represents only particle coordinates. Yet, I see the opposite: Instead of being dedicated to the particle potential coordinates, the abstract dynamic system state space permits any state variables. That is, by representing any system, not only particles, state space is more general than your phase space. --
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I agree; the figure also had me baffled. I think that it should be removed since it confuses more than it helps, and as was pointed out above, the path intersects itself. Hmmm... Thinking about this a bit more leads me to believe that the time-dependent forcing term could cause the paths to cross.
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I agree that there's a problem. The prose is ambiguous -- if the "i.e." comment is taken to be explaining only the preceding term "momentum variables", then maybe it's not incorrect, but that's not the natural reading, and in any case, the comment (and the term "cotangent") is out of place in the
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In the text is written "The phase space of a two-dimensional system is called a phase plane, which occurs in classical mechanics for a single particle moving in one dimension, and where the two variables are position and velocity. In this case, a sketch of the phase portrait may give qualitative
1171:"Once the phase integral is known, it may be related to the classical partition function by multiplication of a normalization constant representing the number of quantum energy states per unit phase space. This normalization constant is simply the inverse of Planck's constant
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Really? 6N-dimensional space? So for a system of 10^20 gas particles, we'd have a 6*(10^20)-dimensional phase space? It seems to me that this is a typo and should read 6-dimensional phase space. I'll change it in the next couple of days if nobody objects.
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in which the parameters of the system are the axes of the plot. To summarize, in the phase space you do not represent the parameters as an axis. If the parameter has dynamics then ti can be considered a degree of freedom of the system.
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for a classical system with N particles, each point in the 3N + 3N dimensional phase space is a possible configuration of the system. in statistical mechanics, a "state" is then a probability distribution (in mathematical terms, a
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I've noticed the use of the phrase "phase space" used in the context of ideas. It describes all the possible outcomes of a given scenario. Since this use is not widespread, does it merit mention in this article?
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Since the oscillator is being damped and/or driven by a non-constant force, there doesn't seem to be any reason to expect that the entire dynamics of the system would be captured by position and momentum alone.
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a point in the 6N-dimensional phase space describes the dynamical state of every particle in that system, as each particle is associated with three position variables and three momentum variables
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they are not labeled in the graphic. I fancy myself smart and interested in heady stuff like this, but I think I need to knwo what those axes are to understand the illustration. right?
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Non-intersection of phase curves is a property of autonomous systems. Since the
Duffing oscillator is not autonomous, it will not have that property. One can, of course, introduce
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I assume all of those "dynamical" and "thermodynamical" should actually be "dynamic" and "thermodynamic"? Didn't change them in case it's some weird mathematical usage.
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This article was automatically assessed because at least one WikiProject had rated the article as start, and the rating on other projects was brought up to start class.
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For mechanical systems, the phase space usually consists of all possible values of position and momentum variables (i.e. the cotangent space of configuration space).
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To make Phase Space the main article on the (p, q) space of dynamical systems, and Phase diagram mainly on the physical chemistry uses of that term
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I think this is misleading, because if we have a N-particle System in 3 dimensions with coordinates (p,q) it has 6N Dimsnesions but the power of
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The statement on quantisation needs a tidy - it's the product p.q which has the dimensions of action surely. (so from memory, dp.dq = h) Bob aka
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The images require labels on their axes. The first image has none. The second image's resolution is too poor to decipher the axis labels.
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Speaking of that, it looks like someone erroneously linked the
English article on Phase Space with the French article on Phase Diagrams...
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It seems to me the expression "cotangent space" should be replaced with "cotangent bundle". As I am not 100% sure, I open a talk page.
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https://web.archive.org/web/20081011041922/http://clesm.mae.ufl.edu/wiki.pub/index.php/Configuration_integral_(statistical_mechanics)
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When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.
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Even if the axes were dimensionless (and in this case, they can't be as momentum is expressed in units), they should be labelled.
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is a state; it corresponds to the distribution that is constant everywhere on some constant-enery surface of the phase space.
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information about the dynamics of the system, such as the limit cycle of the Van der Pol oscillator shown in the diagram."
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The animated image showing the evolution of a system's phase claims that what is shown is the phase path of the
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to delete these "External links modified" talk page sections if they want to de-clutter talk pages, but see the
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Now, the statement on quantisation is better: no one will understand, who not already knows! It's a pity!
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If x is position & y is momentum, (commom) then the diagram is flipped - should be clockwise...
491:) on the phase space. so the state space is the family of probability distributions. for example, the
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Isn't "Phase space" = "State space"? They both are the space of all possible states, aren't they? --
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on
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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Done. Howeever, I was sloppy with the links, some may be disambig pages. I have to run ...
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http://clesm.mae.ufl.edu/wiki.pub/index.php/Configuration_integral_(statistical_mechanics)
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1093:. However, any image of this phase space will appear to have intersecting phase curves.
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If you found an error with any archives or the URLs themselves, you can fix them with
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lede, so the best approach is just to delete it all, which I will now do...
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raised to a power equal to the number of degrees of freedom for the system
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Number of dimensions of the configuration space and the phase space
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Integral is still 3N.
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for additional information. I made the following changes:
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are the x & y axes of that first illustration defined?
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I think it might be helpful to contrast phase space with
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