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Proof of Liouville's theorem: Relation between phase space volume and probability distribution function

I understand the proof of Liouville's theorem to the point where we conclude that Hamiltonian flow in phase-space is volume preserving as we flow in the phase space. Meaning the total derivative of ...
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1answer
35 views

Canonical ensemble, energy, heat bath

I am studying through the book Thermodynamics and Statistical Mechanics by Walter Greiner and I’ve got a couple of doubts when I was reading about the classical ensembles, specially the Canonical ...
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659 views

Is there a physical system whose phase space is the torus?

NOTE. This is not a question about mathematics and in particular it's not a question about whether one can endow the torus with a symplectic structure. In an answer to the question What kind of ...
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31 views

Can statistical mechanics be formulated generally in terms of phase space?

In many statistical mechanics books, notably Landau and Lifschitz' volume in the course on theoretical physics, the quantities central to statistical mechanics such as entropy are defined in terms of ...
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56 views

Examples of Weyl transforms of nontrivial operators

I've been able to find examples of Weyl transforms of operators like $\hat{x}$,$\hat{p}$, and $\hat{1}$, but not anything more complicated. Are there derivations of the Weyl transforms of more ...
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26 views

Section of Phase space

Suppose $\mathfrak{X}$ is the configuration space of some system on a Riemannian manifold $\mathcal{M}$. Then, the phase space is $\mathrm{T}^{*}(\mathfrak{X})$. What is a section of ...
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39 views

Deriving probability distributions from the Wigner distribution

I know that I can calculate the probability distributions of $x$ and $p$ from the Wigner quasiprobability distribution, and I can calculate the probability distributions of other operators by ...
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1answer
55 views

Phase space appellation

Does anyone know why they called the momentum-position space the phase space in the first place? To clarify what I mean a bit more, I'll give you an example: The name configuration space for the ...
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3answers
131 views

Probability distribution in phase space and Liouville's theorem?

We can define a probability distribution over phase space (say 1D) $\rho(x,p)$ such that, for example, $$\langle x\rangle = \int x \rho(x,p) dxdp$$ etc. It can be shown here that such a distribution ...
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1answer
53 views

All angle dependence in $\mathrm{d}LIPS_2$?

Recall that $\mathrm{d}LIPS_2$ (one particle decaying into two particles of the same mass) is given by $$\mathrm{d}LIPS_2 = \frac{\vert{\bf k_1'}\vert}{16\pi^2\sqrt{s}}\mathrm{d}\Omega_{cm}.$$ In a ...
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57 views

phase-space volumes or cells for N particle system

For N non interacting spinless particles in a volume, we have 3N degrees of freedom and we can divide the phase space into 6N dimensional cells of volume h raised to power 3N. And each cell ...
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1answer
94 views

Phase Plot for Harmonic Oscillator

This is probably gonna be a dumb question but I don't know exactly where I am making the mistake. I have been taught in highschool that simple harmonic oscillator phase plot is the $sin(\omega t)$: ...
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24 views

Relationship between phase space volume and dynamical entropy

I would like to know if there is any analytical proof or relationship which can show that for a dynamical system in presence of chaos signal, if the phase space volume (PSV) minimization leads to the ...
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79 views

Group of translations in two dimensions - A weird treatment

Again, as usual Schwinger leaves me startled as he writes, the Hermitian displacement operator in 2D is $$ G = p_1\delta x_1 +p_2 \delta x_2 $$ Now, we know clearly that this group is an Abelian ...
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23 views

Pick marginal circles in phase plot of a non-linear dynamic system

Context: I'm studying non-linear dynamics with Mathematica. Part of the problem: Given the following system: $\ddot{x} = x - x^3 - 0.2 \dot{x} + g(\sin(t) + \cos(2t))$, find two values of $g$ that ...
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147 views

Phase Space Flow

Phase space flow shares characteristics with fluid flow such as incompressibility by Liouville's theorem. Extending the similarities one might be curious, does phase space flow have a characteristic ...
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3answers
201 views

Why are we living in the $q$ part of the phase space?

In Hamilton mechanics and quantum mechanics, $p$ and $q$ are almost symmetric. But in the real world, the $p$ space isn't as intuitive as the $q$ space. For example, We can uniquely identify a person ...
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1answer
93 views

Doubts regarding dimension of a system:Definitions and algorithms

I need to do phase reconstruction from time series data. In doing so, I encountered Takens' embedding theorem and Cao's minimum embedding dimension $d$ by nearest neighbor method. In paper "Optimal ...
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1answer
344 views

Ensemble of harmonic oscillators

I have some problems with problem 2.3 from Reif's Fundamentals of statistical and thermal physics: Consider an ensemble of classical one-dimensional harmonic oscillators. a) If we assume ...
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54 views

Current density in phase space

I have a question which arises from looking at the impact free Boltzmann equation. Let $(\vec{x},\vec{v})$ be a vector in our phase space $\Gamma^N = \mathbb{R}^{6N}$. The dynamics of a state are ...
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1answer
102 views

Probablity density function $f(\mathbf{x},\mathbf{v},t)$

The phase density function is usually denoted as $f(\mathbf{x},\mathbf{v},t)$ which gives probable number of particles moving with velocity $ \mathbf{v}$ at position $\mathbf{x}$ at time t. Also we ...
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1answer
145 views

Peculiar Hamiltonian Phase space

I was solving an exercise of classical mechanics : Consider the following hamiltonian $H(p,q,t) = \frac{p^2}{2m} + \lambda pq + \frac{1}{2}m\lambda^2\frac{q^6}{q^4+\alpha^4}$ Where ...
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253 views

What would happen if energy was conserved but phase space volume wasn't? (and vice-versa)

I'm trying to understand the relationship between the two conservation laws. As I understand, Liouville's result is a weaker condition: it relies merely on the particular form assumed by Hamilton's ...
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1answer
572 views

Phase space in quantum mechanics and Heisenberg uncertainty principle

In my book about quantum mechanics they give a derivation that for one particle an area of $h$ in $2D$ phase space contains exactly one quantum mechanical state. In my book about statistical physics ...
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4answers
227 views

Other application of Liouville's theorem besides thermodynamics

Are there any other important practical and theoretical consequences of Liouville's theorem on the conservation of phase space volume besides the calculation of the microcanonical potential in ...
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37 views

Where can I find the Bohr Sommerfield condition?

I need to solve the Hydrogen Atom using the phase integral [Bohr Sommerfield Condition] but I don't know where can I find it. Help me please!
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1answer
457 views

Postulate of a-priori probabilities

In Statistical Mechanics, we often postulate that for an isolated system, the phase-space density of all accessible microstates (i.e all microstates consistent with the energy) is the same. This is ...
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1answer
406 views

Is there any uncertainty between mass and proper length or time?

I was trying to naively draw a parallel between special relativity and the Heisenberg uncertainty principle. I try to understand uncertainty principle as a consequence of 4-position and 4-momentum ...
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2answers
221 views

Phase volume contraction in dissipative systems

I am confused about phase-volume contraction in dissipative systems. Please help me catch the flaw in my understanding. From a macroscopic point of view I understand that a dynamic system tends to go ...
6
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4answers
576 views

Non-Integrable systems

Integrable systems are systems which have $2n-1$ time-independent, functionally independent conserved quantities (n being the number of degrees of freedom), or n whose Poisson brackets with each other ...
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3answers
472 views

number of microstates associated with two-level quantum systems

this is a very simple question, but apparently one that has no simple answer, at least from standard quantum mechanics theory I'm trying to figure the number of simple quantum states (microstates) of ...
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3answers
290 views

Is particle number a problem for formulating statistical physics in a mathematically rigorous manner?

Quantities like the chemical potential can be expressed as something like $$\mu=-T\left(\tfrac{\partial S}{\partial N}\right)_{E,V}.$$ Now the entropy is the log some volume, which depends on the ...
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1answer
123 views

Where is the critical moment where the microcanonical ensemble enters the justification for the equilibium state?

As explained in many books, for the microscopic justification of the second law of thermodynamics (lets formulate it as the total entropy takes maximum among all possible exchanges of two systems), ...
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3answers
256 views

What are some mechanics examples with a globally non-generic symplecic structure?

In the framework of statistical mechanics, in books and lectures when the fundamentals are stated, i.e. phase space, Hamiltons equation, the density etc., phase space seems usually be assumed to be ...
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4answers
316 views

Books on Hilbert space and phase space?

Can you recommend books or papers that highlight or discuss extensively, or at least more than average, the similarites/differences between phase space and Hilbert space? I am primarily interested in ...
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2answers
254 views

Why doesn't phase space contain acceleration/forces?

I'm watching some Physics lectures on the internet by Leonard Susskind: http://www.youtube.com/watch?v=pyX8kQ-JzHI&feature=BFa&list=PL189C0DCE90CB6D81&lf=plpp_video In this lecture, and ...
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2answers
403 views

Phase space of a discrete dynamical system

Suppose a dynamical system of one variable $x$ with discrete time-steps. I've seen in some papers a type of graph in which $x(n+1)$ is plotted versus $x(n)$. My questions are : 1/ Can this be ...
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402 views

Sympletic structure of General Relativity

Inspired by physics.SE: http://physics.stackexchange.com/questions/15571/does-the-dimensionality-of-phase-space-go-up-as-the-universe-expands/15613 It made me wonder about symplectic structures in ...
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0answers
187 views

What does it mean for a phase space trajectory to be “long” and “stable”?

What does it mean for a phase space trajectory to be "long" and "stable"? I understand the concept of a trajectory in phase space but not how these adjectives can be applied to one. Thanks.
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388 views

Universality in Weak Interactions

I'm currently preparing for an examination of course in introductory (experimental) particle physics. One topic that we covered and that I'm currently revising is the universality in weak ...