# 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. ### 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. ### Exploring the Fundamentals of Physics for Teachers

At the moment, I do not have any supporting material.
3. ### 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...
4. ### 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...
5. ### 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.
6. ### 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...
7. ### 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 )
8. ### 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...
9. ### 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)...
10. ### 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...
11. ### 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...

43. ### 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...
44. ### 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...
45. ### 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 +...
46. ### 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...
47. ### 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...
48. ### 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...
49. ### 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?
50. ### 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...