Phase space Definition and 132 Threads

In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usually consists of all possible values of position and momentum variables. The concept of phase space was developed in the late 19th century by Ludwig Boltzmann, Henri Poincaré, and Josiah Willard Gibbs.

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  1. K

    Phase space element calculation

    Can anyone see what's not right? In the phase space calculation of a 2 to 2 process I get to I=\int dp_1d\Omega \frac{1}{(2\pi)^2}\frac{p_1^2}{2E_12E_2}\delta(E_1+E_2-E) then I use p_1=\sqrt{E_1^2+m_1^2} \Rightarrow dp_1=\frac{E_1}{\sqrt{E_1^2+m_1^2}}dE_1 thus I = \int dE_1d\Omega...
  2. P

    Understanding Phase Space and Building It

    Could anyone explain me what a phase space is and how we can build it?? Thanks in advance.
  3. M

    Calculating Phase Space Volume for Canonical Ensemble

    In calculating entropy in micro canonical ensemble we use KlnW where W is the no. of accessible micro states to the system , now when we move on to canonical ensemble ,we calculate the partition function and from there derive the thermodynamics of the system , and divide it by factor of h to...
  4. N

    Transformations of phase space

    Homework Statement The phase flow is the one-parameter group of transformations of phase space g^t:({\bf{p}(0),{\bf{q}(0))\longmapsto({\bf{p}(t),{\bf{q}(t)) , where {\bf{p}(t) and {\bf{q}}(t) are solutions of the Hamilton's system of equations corresponding to initial condition...
  5. U

    Oscillation / Phase Space Question

    Homework Statement Thornton and Marion, chapter 3, problem 21: Use a computer to produces a phase space diagram similar to Figure 3-11 for the case of critical damping. Show analytically that the equation of the line that the phase paths approach asymptotically is \dot{x}=-\beta x. Show...
  6. I

    Time average vs. phase space average

    Homework Statement For a given total energy E0 compute and compare a time average and a phase space average of x2 for the harmonic oscillator. The one-dimensional Hamiltonian is H = \frac{p^2}{2m}+\frac{m\omega^2}{2}x^2 Reminder: the time average is defined as \langle x^2\rangle...
  7. C

    Phase Space Cells - Statistical Thermodynamics

    Homework Statement 1.) Explain why it is necessary to divide phase space into quantified cells of a finite size. 2.) Why is it necessary to know the size of these cells to over come the Gibb's paradox? Homework Equations The Attempt at a Solution 1.) I think it's something to...
  8. N

    Particle physics: calculating the phase space factor for pion to muon decay

    Show that the phase space factor \rho \propto p^2 dp/dE for the decay \pi\rightarrow \mu + \upsilon is \rho \propto \frac{({m_\pi}^2 - {m_\mu}^2)^2}{{m_\pi}^3}E_\mu where E is the total energy.I can show that p^2 = ({m_\pi}^2 - {m_\mu}^2)^2/4{m_\pi}^2 but then I get stuck, I don't know how...
  9. F

    3 body relativistic phase space

    Hi, Do you know if there is an explicit formula for the integrated 3 body relativistic phase space of 3 particle with the same mass? I.e. M->3m Or an approximate one? Thank you!
  10. C

    How Can I Determine the Bounding Dimensions of a Nonlinear Dynamical System?

    I'm working on a visualizer of sorts for a system: x_{n+1} = sin(a y_n) - cos(b x_n) y_{n+1} = sin(c x_n) - cos(d y_n) with a,b,c,d \in [-2.5, 2.5] So for whatever initial (x_0,y_0) I give the system, I know the next iteration will have both x and y between -2 and 2, and that will be...
  11. Spinnor

    Complex vector X, X=point in 3D phase space, X*X = 0.

    Let X=(x1, x2, x3) be an element of the vector space C^3. The dot product of X with itself, X·X, is (x1x1+x2x2+x3x3). Note that if x1=a+ib then x1x1=x1^2 = a^2 - b^2 + i(2ab), rather that a^2+b^2, which is x1 times the conjugate of x1. Let the real part of C represent the position of a...
  12. Spinnor

    3D phase space of point particle and spinors.

    Can we make a connection? Consider the phase space of a point particle in R^3. Six numbers are required, three for position and three for velocity. Now consider an isotropic vector, X, in C^3 with X*X = 0. X = (x1,x2,x3), X*X = (x1*x1 + x2*x2 + x3*x3), x1 = c1 + i*c2, x1*x1 = (c1*c1 +...
  13. C

    Geometry of phase space and extended phase space

    I just want to clarify the geometrical interpretation of these objects as encountered in the basic theory of ODEs. For discussion let's use the simple set of differential equations found in classical mechanics for a free falling particle: \dot{x} = v;\ \ \dot{v} = -g; Now in phase space the...
  14. D

    Deriving Bundle Map for Tangent Space on Sphere | QM and Phase Space Topology

    Homework Statement Trying to derive a bundle map for a tangent space on a sphere. This is in line with some online courses in QM and phase space topology. I'm not doing an assignment as such (I'm a postgrad though). This is also in line with keeping up with tensor calculus (and the symmetry of...
  15. S

    Can I Use 0 to 2π Limits for Phase Space Volume of a Simple Pendulum?

    Homework Statement Pathria 2.6 (2nd Edition): Phase space volume of a simple pendulum. The total energy can be expressed in the form of the time derivative of the angle + the Sin^2 of that angle. From this I want to calculate the phase space volume. Mathematica gives the solution in the...
  16. J

    How Is the Volume Element in Phase Space Determined for N Harmonic Oscillators?

    Hi all, I am solving a problem for N classic harmonic oscillators. I have the Hamiltonian H = sum(i=1,3N)(p_i^2/(2m) + m*o^2/2 *q_i^2 where p is momentum and q I presume is scaled coordinates. I am given the following hint that the volume in phase space can be written as V(E,N) =...
  17. F

    Phase Space - does each point have a unique time associated with it?

    Phase Space -- does each point have a unique time associated with it? Hi all, If I have an autonomous system: dx/dt=f(x)The k-dimensional state vector x lives in a k-dimensional phase space. Does each point in the k-dimensional phase space have a UNIQUE time associated with it?I don't think...
  18. M

    Can Non-Commutative Geometry Redefine Our Understanding of Phase Space?

    the question is if we have a classical phase space (p,q) the idea is using Heisenberg's uncertainty could we generalize the usual 'geometry' to a non-commutative phase space ? for example we could impose the conditions [ x_i , x_j ]= iL_p \hbar where L_p means Planck's Energy scale and...
  19. E

    Why Can't Flows in Phase Space Cross?

    Hello ladies and gentlemen Why can't flows in phase space cross? Would it imply that the system may be at the same state at some future time and then follow a different trajectory? That is to say that the identical initial condition gives a different final condition. To my mind, flows in...
  20. J

    Graph 6D Phase Space Volume Element: Homework Solutions

    Homework Statement Show Graphically the six dimensional phase-space volume element Homework Equations The Attempt at a Solution I know that the phase space is supposed to incorporate 3 components of position and 3 components of momentum. Would an attempt at this question with a...
  21. M

    Please help; calculation phase space -> how do I use the Delta functions?

    Can somebody help me out? I'm reading about formulas for cross sections for spin1 particles but I don't understand the delta functions, in calculating the 2particle pahse space psi For example the interaction; A+B -> C+D has the formula; psi= (2pi)^2 delta(Pa+Pb-Pc-Pd) d3Pc d3Pd / 4EcEd...
  22. S

    Phase Space GR 3+1 Form: Compact & Non-Compact

    In the 3+1 formulation of GR we have the following basic variables: g_{ij} = \textrm{metric on a spatial surface} \pi^{ij} = \textrm{momentum conjugate to }g_{ij} N^i = \textrm{shift vector} N = \textrm{lapse function} Both N and N^i are purely gauge variables, so are essentially unimportant...
  23. R

    Volume of N dimensional phase space

    Hi guys, I have a volume integral in 3D phase space that looks like: \int \frac{4\pi p^2 dp}{h^3} Now, I want to generalize to N dimensions. How does this look: \int \frac{\frac{2\pi^{d/2}}{\Gamma(\frac{d}{2})}p^N dp}{N!h^{3N}} Essentially, I've changed the 4 pi (which I...
  24. M

    Finding area of elliptical ring (phase space)

    To make a long story short, the problem has an elliptical ring from width E to E+dE in phase space (p on y axis, x on x axis). This is a harmonic oscillator, so the standard equations apply (E=p^2/2m + kx^2/2)... now for the question I need to find the total area in the ring of the ellipse in a...
  25. quasar987

    Calculating Probability of Simple Harmonic Oscillator in Phase Space

    Consider a simple harmonic oscillation in 1 dimension: x(t)=Acos(wt+k). If the energy of this oscillator is btw E and E+\delta E, show that the probability the the position of the oscillator is btw x and x+dx is given by P(x)dx=\frac{1}{\pi}\frac{dx}{\sqrt{A^2-x^2}} Hint: calculate the volume...
  26. R

    Defining Euclidean Norm in Phase Space: A Differential Geometry Analysis

    Hi to everynoe! I have a bit of trouble in understanding the following thing : Suppose we have a phase space, in which a dynamical system evolves: for example a two dimensional vector space: temperature and time. Now, does it make a sense to define the euclidean norm of a vector in such...
  27. Loren Booda

    Phase space: a one-to-one mapping with all quantum dynamics?

    Does the history of wave packets translate exactly onto infinite phase space, or is phase space incompletely (or redundantly) covered by quantum mechanics?
  28. 2

    Boltzmanns statistics and phase space

    can anyone handle this one? when deriving a distribution function using a purely statistical approach, Boltzmann uses some kind of a phase space, that is one with 6N dimentions, 3 for position, 3 for momentum. i see some really short descripions of it but not enough to understand it. all...
  29. 2

    Statistical mechanics and phase space

    it's just not sinking in.. i know a cell in phase space has 6 dimensions, 3 for momentum and the other 3 for position. but i'd like to understand it(phase space). can someone give me an example maybe or tell me why this constuct is needed?? or a link to a very good description?
  30. C

    Does Phase Space Explain Entropy?

    I just read Penrose's explanation of entropy in his book "The Emperor's New Mind". His explanation is completely saturated in an extended discussion of "phase space" . Is this concept of "phase space" absolutely necessary in order to explain or understand entropy ? Celal
  31. T

    Phase space in particle physics: what is it?

    This should be an easy general question to someone out there. My "quarks and Leptons" book by Halzen and Martin introduces the term "phase space" 50 pages before the index reference, and never seems to define it. What is phase space in this context? Thanks
  32. Loren Booda

    Phase space geometry for a deterministic quantum mechanics

    Construct a phase space where every point is center to a circle of radius h, Planck's constant. Particular to such a given point, outside its radius lies conventional phase space and inside, conventional phase space inverted through h - together potentially doubling the effective...
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