What is Phase space: Definition and 132 Discussions

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. T

    A First Order in Time Derivatives + Phase Space

    Hey all, I am reading David Tong's notes on Chern-Simons: https://www.damtp.cam.ac.uk/user/tong/qhe/five.pdf and he makes the following statement that doesn't make much sense to me: "Because the Chern-Simons theory is first order in time derivatives, these Wilson loops are really parameterising...
  2. Maumas

    Invariance of a volume element in phase space, What does it mean?

    The invariance of this volume element is shown by writing the infinitesimal volume elements $$d\eta$$ and $$d\rho$$ $$d\eta=dq_1.....dq_ndp_1......dp_n$$ $$d\rho=dQ_1.......dQ_ndP_1....dP_n$$ and we know that both of them are related to each other by the absolute value of the determinant of...
  3. George Wu

    A Relativistically invariant 2-body phase space integral

    I encounter a function that I don‘t know in the calculation of Relativistically invariant 2-body phase space integral: in this equation, ##s##is the square of total energy of the system in the center-of-mass frame(I think) I don't know what the function ##\lambda^{\frac{1}{2}}## is. There are...
  4. simonjech

    I Lorentz invariant phase space and cross section

    Can someone please explain to me how can we obtain this integral in eq. 5.27 from eq. 5.26? I quite do not understand how is it possible to make this adjustment and why the (p_(f))^2 appeared there in the numerator and also why a solid angle appeared there suddenly.
  5. L

    I Why do phase trajectories point upwards and downwards in a quadratic potential?

    Hi, I am currently preparing for my exam and have just watched a video about motion in phase space. From minute 4 a quadratic potential is introduced and then from minute 6 minute the phase trajectory. Here are the pictures quadratic potential phase trajectory Regarding phase...
  6. D

    A Energy hypersurface in a phase space (statistical physics)

    what is the reason for that the energy hypersurfaces in a phase space, which belong to systems with constant energy are closed? ( see picture )
  7. K

    A How Does Quantum Theory Provide a Measure for Microstates in Phase Space?

    *Pathria, Statistical mechanics*"The microstate of a given classical system, at any time, may be defined by specifying the instantaneous positions and momenta of all the particles constituting the system. Thus. If ##N## is the number of particles in the system, the definition of a microstate...
  8. Kaguro

    A Meaning of density of microstates in phase space

    Hello all. I am studying stat mech from Pathria's book. It says a system is completely described by all positions and momenta of all the N particles. This maybe represented by a single point in 6N-D gamma space. So, each point is a (micro)state. Now if we restrict the system (N,V,E to E+ΔE)...
  9. raisins

    I Phase space integral in noninteracting dipole system

    Hi all, Consider a system of ##N## noninteracting, identical electric point dipoles (dipole moment ##\vec{\mu}##) subjected to an external field ##\vec{E}=E\hat{z}##. The Lagrangian for this system is...
  10. M

    I Dependency of phase space generator to differential distributions

    I attatched an example plot where I created the histogram for the differential distribution with respect to the energy of the d-quark produced in the scattering process. My conception is that the phase space generator can "decide" how much of the available energy it assigns to the respective...
  11. RicardoMP

    A How to reduce an integral in phase space to a one-dimensional form?

    I've been trying for a very long time to show that the following integral: $$ I_D=2{\displaystyle \int} \, {\displaystyle \prod_{i=1}^3} d \Pi_i \, (2\pi )^4\delta^4(p_H-p_L-p_R) |{\cal M}({e_L}^c e_R \leftrightarrow h^*)|^2 f_{L}^0f_{R}^0(1+f_{H}^0). $$ can be reduced to one dimension: $$ I_D...
  12. J

    I Energy in the Hamiltonian formalism from phase space evolution

    The hamiltonian ´for a free falling body is $$H = \dfrac{p^2}{2m} + mgy$$ and since we are using cartesian coordinates that do not depend on time and the potential only depends on the position, we know that ##H=E##. For this hamiltonian, using the Hamilton's equations and initial conditions...
  13. S

    I Interpretations of phase space in Dynamical Systems Theory

    In Dynamical Systems Theory, a point in phase space is interpreted as the state of some system and the system does not exist in two states simultaneously. Can some phase spaces be given an additional interpretation as describing a field of values at different locations that exist...
  14. O

    I Rosetta orbits and phase space

    I was recently working on the two body problem and what I can say about solutions without solving the differential equation. There I came across a problem: Lets consider the Kepler problem (the two body problem with potential ~1/r^2). If I use lagrangian mechanics, I get two differential...
  15. I

    I Phase space path integrals

    I have heard of phase space path integrals, but couldn't find anything in Wikipedia about it, so I am wondering, what does it compute ? In particular, are the endpoints points of definite position and momentum? If so, how does one convert them to quantum states ? Also, how is it related to...
  16. sophiatev

    I What happened to the spatial degrees of freedom for the second particle?

    In Henley and Garcia's Subatomic Physics, they introduce phase space in chapter 10 by considering all the possible locations a particle can occupy in a plot of ##p_x \ vs. x##, ##p_x## being the momentum of the particle in the x direction. They next consider an area pL on this plot, and state...
  17. D

    Phase space of a harmonic oscillator and a pendulum

    Hello everybody, new here. Sorry in advance if I didn't follow a specific guideline to ask this. Anyways, I've got as a homework assignment two cannonical transformations (q,p)-->(Q,P). I have to obtain the hamiltonian of a harmonic oscillator, and then the new coordinates and the hamiltonian...
  18. Riotto

    I Single-particle phase spaces for a system of interacting particles

    For a system of interacting particles, is it possible to define single-particle phase spaces? If not, why?
  19. C

    Phase space volume with a potential (microcanonical ensemble)

    I don't know how to solve that integral, and to calculate the number of microstates first, then aply convolution and then integrate to find the volume of the phase space seems to be more complicated. Any clue on how to solve this? Thank you very much.
  20. DaniV

    A Mapping the Phase Space of Type II Superconductors

    Due to my lab work I want to try map the phase space that created with critical external magnetic field H_c and the critical current J_c through the superconductor of type II. the critical transition happen from the Abrokosov phase to the non-superconductor phase, occurred by the fact that in...
  21. shahbaznihal

    I Phase space density function and Probability density function

    I am reading a text which talks about the WIMP speed distribution in the galactic halo in the frame of the Sun and Earth. The point where I am stuck it is trying to explain the concept of Gravitational Focusing of WIMPs at the location of the Earth due to the gravitational well of the Sun...
  22. A

    Phase space trajectories can't intersect...

    Phase space trajectories can't intersect each other is it due to the fact that at the intersection point there will be more than one possible path for the system to evolve with time??
  23. MattIverson

    What are phase space coordinates and how do you plot them?

    Homework Statement I have phase space coordinates (x0,y0,z0,vx,vy,vz)=(1,0,0,0,1,0). I need to analytically show that these phase space coordinates correspond to a circular orbit. Homework Equations r=sqrt(x^2+y^2+z^2) maybe? The Attempt at a Solution My core problem here is maybe that I...
  24. QuasarBoy543298

    I Volume in phase space- Louviles theorem

    I was looking at the following proof of Louviles theorem : we define a velocity field as V=(dpi/dt, dqi/dt). using Hamilton equations we find that div(V)=0. using continuity equation we find that the volume doesn't change. I couldn't figure out the following : 1- the whole point was to show...
  25. gasar8

    Canonical invariance vs. Lorentz invariance

    Homework Statement I have an assignment to prove that specific intensity over frequency cubed is Lorentz invariant. One of the main tasks there is to prove the invariance of phase space d^3q \ d^3p and I am trying to prove it with symplectic geometry. I am following Jorge V. Jose and Eugene J...
  26. gasar8

    A Lorentz Invariant Phase Space: Symplectic Geometry

    I have an assignment to show that specific intensity over frequency cubed \frac{I}{\nu^3}, is Lorentz invariant and one of the main topics there is to show that the phase space is Lorentz invariant. I did it by following J. Goodman paper, but my professor wants me to show this in another way...
  27. sams

    A Difference between configuration space and phase space

    Lagrangian Mechanics uses generalized coordinates and generalized velocities in configuration space. Hamiltonian Mechanics uses coordinates and corresponding momenta in phase space. Could anyone please explain the difference between configuration space and phase space. Thank you in advance for...
  28. G

    A Phase Space and Its Use in Monte Carlo Simulations of Radiation Beams

    In the field of medical physics, specifically in monte carlo simulation of radiation beams produced by electron accelerators, people call ‘phase space’ to a file that contains the data of a large number of particles when they traverse a reference surface in the machine (usually a plane), i.e...
  29. J

    Decide if the energy surfaces in phase space are bounded

    Homework Statement From Classical Mechanics, Gregory, in the chapter on Hamilton's equations of motion: 14.13: Decide if the energy surfaces in phase space are bounded for the following cases: i.) The two-body gravitation problem with E<0 ii.) The two-body gravitation problem viewed from the...
  30. T

    For the Boltzman equation, why is df/dt=0 when collisionless?

    From wikipedia, The general equation is $$\frac{df}{dt} = (\frac{∂f}{∂t})_{force}+(\frac{∂f}{∂t})_{diff}+(\frac{∂f}{∂t})_{coll}$$ where the "force" term external force, the "diff" term represents the diffusion of particles, and "coll" is the collision term. So shouldn't be df/dt=0 when it is...
  31. R

    Harmonic Oscillator and Volume of Unit Cell in Phase Space

    Long time no see, PhysicsForums. Nevertheless, I have gotten myself into a statistical mechanics class where the prof is pretty brutal and while I can usually manage, this problem finally has me stumped. I'd like to be nudged in the right direction, not outright given the answer if possible. I...
  32. Urs Schreiber

    Mathematical Quantum Field Theory - Reduced Phase Space - Comments

    Greg Bernhardt submitted a new PF Insights post Mathematical Quantum Field Theory - Reduced Phase Space Continue reading the Original PF Insights Post.
  33. Urs Schreiber

    Mathematical Quantum Field Theory - Phase Space - Comments

    Greg Bernhardt submitted a new PF Insights post Mathematical Quantum Field Theory - Phase Space Continue reading the Original PF Insights Post.
  34. S

    I What is the difference between phase space and state-space?

    In state space, the coordinates are the state variables of a system.So,each point in state space represents a specific value of state variables.Thus,state space representation represents the changes in a dynamical system. The state variables are the minimum number of variables which uniquely...
  35. A

    A How can I plot a 3D phase space for a system of differential equations?

    Hi, i would like to know how can i plot a three dimentionnal phase space (mathematica), for this kind of differential equations: x'= (z^2(x-4y+z)+y^2(x+z)+z(x*y-x^2))/(z^2+x^2-2) y'=y(y-3)+z(4y-z)+3(1-x^2)-2(y-3z)(z^2(x-4y+z)+y^2(x+z)+z(x*y-x^2))/x(z^2+x^2-2)...
  36. C

    A Symplectic geometry of phase space

    What is a symplectic manifold or symplectic geometry? (In intuitive terms please) I have a vague understanding that it involves some metric that assigns an area to a position and conjugate momentum that happens to be preserved. What is 'special' about Hamilton's formulation that makes it more...
  37. J

    A Phase Space Factor: Confused About 3D to 4D Conversion

    Hi, I'm recently reading "Particle Physics in the LHC Era" and there is a part about the phase space factor that confuses me. When giving the Lorentz invariant phase space, they wrote: d3p / 2E = θ(E) δ(p2 - m2) d4p This is very confusing as it equates a three dimensional differential to a...
  38. G

    Expectation values as a phase space average of Wigner functions

    Hi. I'm trying to prove that [\Omega] = \int dq \int dp \, \rho_{w}(q,p)\,\Omega_{w}(q,p) where \rho_{w}(q,p) = \frac{1}{2\pi\hbar} \int dy \, \langle q-\frac{y}{2}|\rho|q+\frac{y}{2}\rangle\,\exp(i\frac{py}{\hbar}) is the Wigner function, being \rho a density matrix. On the other hand...
  39. K

    A Total derivative of momentum in quantum mechanics

    In quantum mechanics, the velocity field which governs phase space, takes the form \begin{equation} \boldsymbol{\mathcal{w}}=\begin{pmatrix}\partial_tx\\\partial_tp\end{pmatrix} =\frac{1}{W}\begin{pmatrix}J_x\\J_p\end{pmatrix}...
  40. Gean Martins

    I Phase Space and two dimensional Hilbert Space

    I always had this doubt,but i guess i never asked someone. What's the main difference between the Classical phase space, and the two dimensional Hilbert Space ?
  41. T

    Why is Bernoulli's Equation Isentropic

    I have trouble understanding why we classify an inviscid adiabatic incompressible flow along a streamline as isentropic I understand this from a Thermodynamic definition/explanation $$dS = dQ/T$$ Adiabatic Invsicid $$dQ =0= dS$$ So no heat added or lost no change in entropy I'm fine with that...
  42. D

    A Change of coordinates in quantum phase space

    Hello! I was reading a paper on formulation of QM in phase space (https://arxiv.org/abs/physics/0405029) and I have some doubts related to chapter 5. It seems to me that there is a transformation to modified polar coordinates (instead of radius there is u which is square of radius multiplied by...
  43. B

    I Can a phase space be subject to change, and if so, what are the implications?

    I'm asking if space space is subject to change, if not, why not, and if so, then would it be subject to a subsequent phase space that describes that? Let us make a phase space for a deterministic time evolved dynamic system of three spatial dimensions. Since the actions of the system and the...
  44. binbagsss

    Deriving continuity equation of phase space in Statistical Mechanics

    Hi, So I am aiming to derive the continuity equation using the fact that phase space points are not created/destroyed. So I am going to use the Leibiniz rule for integration extended to 3-d: ## d/dt \int\limits_{v(t)} F dv = \int\limits_{v(t)} \frac{\partial F}{\partial t} dV +...
  45. ShayanJ

    Phase space of spherical coordinates and momenta

    Homework Statement [/B] (a) Verify explicitly the invariance of the volume element ##d\omega## of the phase space of a single particle under transformation from the Cartesian coordinates ##(x, y, z, p_x , p_y , p_z)## to the spherical polar coordinates ##(r, θ, φ, p_r , p_θ , p_φ )##. (b) The...
  46. H

    I Is an integral over phase space reasonable?

    In statistical physics, the partition function should be calculated in the whole phase space.This is finally an integral over the phase space, like ∫d3Nqd3Np... The problem is that the integral covers some cases that different particles have the same generalized coordinates and momenta. So, is...
  47. H

    Why are position and velocity independent variables in a phase space?

    Why do we take a particle's position ##x## and its velocity ##\dot{x}## as independent variables in a phase space when they are dependent in the sense that given the function ##x(t)##, we can get the function ##\dot{x}(t)##? I'm thinking they are independent variables but not independent...
  48. Muthumanimaran

    A Question about Phase Space

    Can Phase Space be break down into different regions, different regions that are not mixed up with one another, if we do so, the different regions can raise to any conservation of physical quantity?
  49. melli1992

    A Massive three particle phase space

    If you produce three massive particles with m1=/=m2=/=m3 near threshold (beta -> 0), the cross section of the production is supressed by a factor beta^4, where beta = sqrt(1-(M_tot)^2/s) and s is COM energy. I have been trying to prove this statement, but I can't seem to manage. Could anybody...