Questions tagged [phase-space]

A notional even-dimensional space representing all relevant states of a dynamical system; it normally consists of all components of position and momentum/velocity involved in that unique specification. Use for both classical and quantum physics.

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200 views

Is it possible to derive Liouville's Theorem purely from maximum differential entropy?

Typically in physics (at least the way I learned mechanics), this is derived using the multi-dimensional divergence theorem on the $2N$-dimensional phase space i.e. $0=\partial_t \rho + \sum\limits_{...
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Quantum versus classical computation of the density of states

If I consider for instance N non interacting particles in a box, I can compute the energy spectrum quantum mechanically, and thus the number of (quantum) microstates corresponding to a total energy ...
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Generating function of point transformation

I am asked to show that the generating function corresponding to a point transformation in Lagrangian mechanics can be taken as null. The point transformation consists of $$ Q_i=Q_i(q,t), $$ and ...
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156 views

What is the topology of phase space of $n$ free relativistic particles in center of mass frame?

Consider an ensemble of $n$ relativistic particles of fixed masses $m_i \geq 0$, $i=1,\ldots,n$ with four momenta $p_i$ such that $p_i^2=m_i^2$. In center of mass frame they sum up to $$P=p_1+\cdots+...
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Star-shaped phase space

I am asked to classify the following phase spaces. The phase spaces 2 and 3 are fairly simple (harmonic oscillator and a elastically reflected particle). However, I fail to classify the phase space ...
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Fluid mechanics, symplectic structure, the Hamiltonian, and vorticity

Consider an inviscid irrotational fluid in two dimensions. There are some explicit connections with symplectic geometry that I do not understand. I am not well versed in the later topic, so please ...
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113 views

Using action-angle variables in non-periodic system

I'm a little confused by the discussion in the last section $\S 50$ of Landau and Lifshitz's (Classical) Mechanics (1960, first English ed.). Here, they consider finite motion of a system whose ...
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201 views

Density of states for a system of $N$ particles (both interacting and non-interacting case)

The density of states $\rho(\epsilon)$ for a single particle (with $g$ number of internal states such as spin) confined in a volume $V$ can be calculated as $$\rho(\epsilon)=\frac{4\pi gV}{h^3}p^2\...
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284 views

Non-canonical transformation

I would like to know any method to transform a known non-canonical set of variables to a canonical set for a given system. The Lagrangian and Hamiltonian are known in the non-canonical variables. I ...
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1answer
62 views

An “upper ceiling” for thermodynamics?

Roger Penrose said in "A Road to Reality" (p.701): “There is a common view that the entropy increase in the second law is somehow just a necessary consequence of the expansion of the universe. ...
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26 views

Relation between Wigner flows and entropy

After reading this question What's the intuitive reason that phase space flow is incompressible in Classical Mechanics but compressible in Quantum Mechanics? and some papers of Steuernagel’s group,...
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Discrepancy regarding Husimi Probability distribution calculation

I am trying to simulate a system of j qubits and for visualization of the dynamics considering the Husimi distribution of the state. To carry out the projection onto coherent states I have proceeded ...
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56 views

The formula for the average number of fermions $\langle N \rangle$

In the context of Fermi gases (or fluids in general), one would typically in the grand-canonical formalism use the formula $\langle N \rangle = -\frac{\partial \psi}{\partial \mu}$, where $\psi$ is ...
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51 views

Probability for a phase space flow to return to its original state

A Hamiltonian system of $n$ interacting atoms, each of mass $M$, is confined within a cubical box of sides $V$. The average initial speed of each particle is $v$. How do I estimate the timescale for ...
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1answer
192 views

Time-reversal operator in Phase Space Representation

Consider the simplest possible case in which the time reversal operator $\hat{\mathrm{T}}$ is given by the operation of complex conjugation $\hat{\mathrm{K}}$. We can view $\mathrm{T}$ is an anti-...
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2answers
259 views

Symplectic form on covariant phase space

Usually, the phase space of a physical system is defined as the cotangent bundle of the configuration space at some fixed time slice $t = t_0$, conveniently co-ordinatized by $\{q^a, p_a\}$ where $$ ...
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1answer
223 views

Liouville's Theorem in Spherical Coordinates

I'm attempting to verify the invariance of $$d\omega=\prod_{i=1}^{3N}dq_idp_i$$ in the case $N=1,$ and under a canonical transformation into spherical coordinates. I know to let $$q_1=r\sin\theta\cos\...
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N-dimensional tori for integrable systems with bounded motion

I am reading the book Dynamics by N. Rasband. In chapter 9, it is said that the phase space of an integrable system with bounded motion is described in a compact submanifold $M_{I_0}$ which has the ...
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The Berry phase and anomalous commutator

The problem Recently I've read an article "Adiabatic theorem and anomalous commutator", written by Iida and Kuratsuji. In this article the authors relate the Berry phase with anomalous commutator of ...
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38 views

Potential of a particle across a thin razor disk

An example of dynamics in $\lvert x\rvert$ potential is the motion of a particle across a thin infinite disk of matter. A model disk reduces to potential of the form $U = -k/(a+|z|)$, when we consider ...
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297 views

Liouville's theorem for systems with dissipation described by a single hamiltonian

Following this link, one can treat dissipation in the lagrangian by using a factor $e^{\frac{t \beta}{ m}}$ in addition to the Lagrangian $L_0$ of a system without disspation: $ L_0[q, \dot{q}] = \...
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1answer
168 views

Fixing time in Feynman phase space path integral

The phase space version of Feynman's path integral expression for the free particle propagator involves a (formal) sum over paths in phase space with fixed $q$ endpoints and (as far as I'm aware) ...
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419 views

A question on Lagrangian dynamics an the velocity phase space

I've struggled in the past with understanding why we can treat position and velocity as independent variables in the Lagrangian, but I think I may have finally become a bit more enlightened on the ...
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1answer
188 views

Hamilton-Jacobi Equation: Why does any $F(q,Q,0)=f(q,Q)$ lead to a solution?

I) Given the Hamilton-Jacobi equation,$$\frac{\partial F(q,Q,t)}{\partial t}+H\left(q,\frac{\partial F(q,Q,t)}{\partial q},t\right)=0 $$ it is stated that any function $$F(q,Q,t=0)=f(q,Q)\tag{22}$$ ...
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Density matrix and wigner function from first and second moments

Let's say I know the first and second moments of position and momentum for all times. $\langle x\rangle $, $\langle p\rangle $,$\langle x^2\rangle $, $\langle p^2\rangle $, $\langle xp\rangle $, $\...
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86 views

Liouville theorem and the ergodic assumption

I am following a course on statistical mechanics. My instructor presented us the following Liouville theorem in two (claimed) equivalent ways: Differential statement: The probability distribution $\...
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50 views

Quantum mechanics in phase space - what are coordinate components?

I'm trying to understand the answer provided by Qmechanic to this question: What's the intuitive reason that phase space flow is incompressible in Classical Mechanics but compressible in Quantum ...
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Relation between the canonical partition function and the phase space volume

In Kerson Huang's Statistical Mechanics (2nd ed.), it is claimed that the phase space volume occupied by the canonical ensemble is the partition function: $$ Q_N (V, T) \equiv \int \frac{dp dq}{N! h^{...
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Why are 2 dimensions needed for every 1 dimension of space in order to determine the motion of a physical system?

In classical mechanics, the phase space of a mechanical system has twice the number of dimensions of "actual" space (i.e. position space). That is, in phase space, each particle has both a position ...
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Canonical coordinate

Sorry for my broken English. I'm a physics undergrad and quite poor at math. While reading a mechanics textbook, I've found something I cannot understand. There are coordinates, $(q,p,t)$ $\...
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80 views

3-particle phase space in $d$ dimensions

recently I came across a problem concerning the 3-particle phase space. I am trying to show, that the 3-particle phase space for massless particles with momenta $p_1$, $p_2$ and $k$ is given by $$ d\...
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Mean free path for $3\rightarrow 1$ scattering

I want to calculate the mean free path of an antineutrino in nuclear matter where it can undergo the reaction $p+e^-+\bar{\nu} \rightarrow n$, which I imagine will involve calculating the rate of that ...
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383 views

When can action-angle variables be defined?

According to Goldstein, "We can define action-angle variables for [a separable Hamiltonian] system when the orbit equations for all of the $(q_i, p_i)$ pairs describe either closed orbits (...
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1answer
64 views

Relationship between the Galilei Group and the Phase Space

This question is kind of a follow up question to my last question on the need for canonical commutation relations and conjugate observables. A comment from Valter Moretti suggested that, given a ...
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153 views

Showing time-dependent transformations are canonical

In Goldstein's mechanics book he defines a canonical transformation as a transformation from $q, p$ to $Q, P$ with $$ Q = Q(q,p,t) \tag{9.4a} $$ and $$ P = P(q,p,t) \tag{9.4b} $$ such that if $H$ ...
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50 views

Four body decay rate

I would like to calculate a 4-body decay rate and I'm stuck. I have a massive scalar particle (with mass $M$) decaying to four massless particles (2 fermions and 2 scalars). I am not sure how to ...
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1answer
423 views

Invariant Phase space volume under canonical transfromation

I have a volume element in phase space: $$ d\omega = \prod _{i=1}^{N}(dq_{i},dp_{i})$$ Now I should show the invariance of this product under canonical transformations. I think first I would have to ...
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Conceptual Misunderstanding of Hamilton-Jacobi Theory

I will use the example of one dimensional harmonic oscillator to explain my doubt. $$H=\frac{p^2}{2}+\frac{q^2}{2}.$$ We allow, $S=W-\alpha t.$ We get, $$\frac{1}{2}\left(\frac{\partial W}{\partial ...
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Any model integrable but not separable?

In textbooks on classical mechanics, the exactly solvable models are all separable. Is there any model integrable but not separable?
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Is the reversal of the movement of all particles in phase-space, the expansion of space and the development of quantumfields equal to time reversal?

In an earlier question, which was called a duplicate, I already asked a similar question. But I state my question differently now and add something to it. Suppose "we" would reverse the direction of ...
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1answer
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How to find the right constraints on a canonical transformation?

I have a 1-dof Hamiltonian in the variables $p,q$. I am not sure on how to find the constraints on a canonical tansformation $\psi(P,Q)=(p,q)$ such that the Hamiltonian in the new variables $P$ and $Q$...
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Does an stereographical projection of a physical plane have any physical meaning?

Mathematically, an arbitrary 2D plane can be mapped onto a sphere by stereographical projection. Each line on the plane is equivalent to each line on the sphere. If the sphere rotates under the $SO(3)$...
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1answer
245 views

Can the Wigner function be described using coherent states?

The Wigner function for a wave function $\Psi(\vec{r})$ is $$ W(\vec{r},\vec{k}) = \frac{1}{2\pi} \int dy e^{-i \vec{k} \cdot \vec{y}} \Psi^{*}(\vec{r}-\vec{y}/2) \Psi(\vec{r}+\vec{y}/2) . \tag{1} $$ ...
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Commutator implies division of phase space in cells of area $h$?

There is a detail from a very well-written answer here that interested me. Unfortunately Springer is charging 41 Euros to access the paper cited by the author. I wonder if someone could elaborate on ...
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Canonical transformations

I was searching for some more information about canonical transformations. Thus consider canonical coordinates $(p_i,q_i)$ and an other set of canonical coordinates $(P_i, Q_i)$. Both sets will ...
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426 views

Phase plane of simple pendulum

I'm trying to create a phase plane of simple pendulum motion by plotting $\dot\theta$ against $\theta$ in Matlab. I have the equation $\ddot\theta + \sin\theta = 0$, then by integrating I come to the ...
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What physical system does the Hamiltonian $ H = \frac{q^4p^2}{2\mu} + \frac{\lambda}{q^2}$ describe?

My question is basically in the title: I'm training myself on exams of Hamiltonian mechanics and I had this Hamiltonian $$ H = \frac{q^4p^2}{2\mu} + \frac{\lambda}{q^2},$$ without any interpretation. ...
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222 views

Probability conservation versus Liouville's theorem

This question arises to my mind while studying notes on Kinetic theory written by Prof. David Tong. There he derived the Liouville’s equation. The outline of his derivation goes as follows: Our ...
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354 views

What is the phase space volume in terms of angular momentum?

Given a rigid rotor Hamiltonian, defined along the principle axes as $$ H = \sum_{i=1}^3 \frac{L_i^2}{2I_i} $$ say we would like to compute the classical partition function of this system. Is the ...
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90 views

Is time-1 map of a Hamiltonian vector field on a cylinder always twist?

I have a one degree of freedom analytic Hamiltonian $H(q,p)$ defined on a semi-infinite cylinder, i.e. $(q,p) \in \mathbb{T} \times \mathbb{R}^{+}$, such that all level sets $H(q,p)=c$ are closed ...