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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Volume preserving transformation in the Circular Restricted Three-Body problem
the Lagrangian of the planar circular restricted three-body problem in the rotating coordinate frame is:
$$\mathcal{L}(x,y,\dot{x},\dot{y})=\frac{1}{2}(\dot{x}-\Omega y)^2 + \frac{1}{2}(\dot{y}+\Omega ...
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When is the derivative of Hamilton flow respect to initial conditions independent of time?
Consider a Hamiltonian system with coordinates $\Gamma^A=(q^i,p_i)$ and let $X^A(s,\Gamma_0)$ be the Hamiltonian flow (i.e. a solution to Hamilton's equations) parametrized by time $s$ and initial ...
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Does time average induce phase space propability distribution?
Lets say we have a trajectory (positions and momenta) $(x(t), p(t))$ that is the solution of the equation of motion for a system with Hamiltonian $H(x,p)$. For some function $A(x,p)$, the time average ...
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How does this canonical transformation on a Schwarzschild black hole work?
In this paper "Holography of the Photon Ring" the authors use a canonical transformation in section 2.4 in eqs. (2.52)-(2.55).
It is basically a transformation from spherical coordinates for ...
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How is Hudson's theorem for the Wigner function proved?
Hudson's theorem tells us that a pure state has non-negative Wigner function iff it's Gaussian. This was originally proven in [Hudson 1974], and then generalised to multidimensional systems in [Soto ...
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The establishment of Out-of-Time-Ordered Correlators (OTOCs) from Lyapunov Exponents (LEs)?
the OTOC in quantum system is
$$F(t) = \langle \hat{W}^\dagger(t) \hat{V}^\dagger(0) \hat{W}(t) \hat{V}(0) \rangle_{\beta}
$$
the Lyapunov exponent is
$$\lambda = \lim_{{t \to \infty}} \lim_{{d(0) \to ...
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Generating function condition not satisfied?
We want to find a generating function $S(q_i,P_i,t)$ such that we get the best possible canonical transformations. So it must satisfy the Hamilton-Jacobi equation:
$$H(q_i,\frac{\partial S}{\partial ...
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Dirac delta of operators multiplying matrix element
In playing around with the Wigner-Weyl correspondence, I found myself needing to perform an integral of exponential operators, which I am confused about. TLDR: help to evaluate $$\int d{x}dy\ \delta(x\...
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Lorentz-invariant phase space integral
Consider the following Lorentz invariant integral associated to a $2\to 2$ scattering:
\begin{equation*}
I = \int \frac{d^3\mathbf{p_3}}{(2\pi)^3 2E_3} \int \frac{d^3\mathbf{p_4}}{(2\pi)^3 2E_4} \...
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Problems with Kerson Huang's derivation of NVE ensemble entropy
So I'm currently studying statistical mechanics from different textbooks, but my professor suggested Kerson-Huang for a general derivation of entropy in microcanonical ensembles. In chapter 6.2 is ...
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Why $q,p,Q,P$ are Independent Variables when Using Generating Functions?
In Hamiltonian formalism, specifically generating functions, why do the variables $q, p, Q, P$ are treated as independent when finding the equations that arise from the generating function?
I ...
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Why do we need a Poisson bracket structure?
Let me start by asking why we need a Poisson bracket like structure on the Hamiltonian phase space? Say we have a constraint, why do we go through the trouble of defining a Dirac bracket structure on ...
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Regarding Poisson and Dirac brackets [duplicate]
The question starts with why Poisson brackets (in constrained systems) gives different relation if we substitute the constraints before or after expanding the bracket, and why this difference in ...
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Regarding Poission structure of Hamiltonian phase space
Why exactly do we need $$ \{q^i,p_j\}=\delta^i_j,$$ where $\delta^i_j$ is Kronecker delta and $\{\cdot,\cdot\}$ is the Poisson bracket? What happens to the phase space structure if these fundamental ...
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Decoherence model of two qubits interacting with correlated multimode fields - open quantum system
I read paper on open quantum system, that talk about non-Markovian process and memory effects.
they described the system as a generic decoherence model of two qubits interacting with correlated ...
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