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Questions tagged [virial-theorem]

In mechanics, the virial theorem relates the average over time of the total kinetic energy, <T>, of a stable system consisting of several particles bound by potential forces, with that of the total potential energy, <V>.

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Why does a drop in gravitational energy result in star contraction

Consider a gas cloud that is radiating away heat. From virial theorem we see that if the cloud radiates away heat and loses energy this will cause its gravitational energy to drop. Why does this mean ...
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How are the virial equations of state (not the co-efficients !) for gases derived?

Virial Equations of the form for real gases : $$Z = 1 + BP + CP^2 ... $$ $$Z=1+B/v+C/v^2...$$ I see these equations on many sites and textbooks, but I can't find from where they come. The closest ...
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Behaviour of a star as it approaches Ultrarelativistc limit

By considering the potential energy of a degenerate star of mass $M$ and radius $R$, I can use dimensional arguments to show that the radius of the star depends on its mass as: $$ R ∝ M^\frac{2-n}{...
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Proof of equipartition theorem for $i$ not equal to $j$

The generalized equipartition theorem (where variables need not be quadratic) states that if $x_i$ is a canonical variable (position or momentum variable), then $$\left\langle x_i \frac{∂H}{∂x_j}\...
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Virial radius of a collection of point masses

I am experimenting with multi body simulations in astronomy ($N$-body algorithms). It is customary to give the results in dimensionless units. One of them is a length unit, and the virial radius is ...
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virial theorem hydrogen atom

I calculated $\langle T \rangle$ and $\langle V \rangle$ as a function of time of a given state of the hydrogen atom $|\psi\rangle=a|1,0,0\rangle+b|2,0,0\rangle$ and I found that $$\langle V \rangle ...
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Derive the virial theorem from hydrostatic equikibrium

Consider the equation of hydrostatic equilibirium as follows as applied to a stellar object: $$ \frac{dP}{dr } = \frac{-GM\rho}{r^2} $$ I know it is intimately related to the virial theorem but I ...
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Violation of Virial theorem as indication to ergodicity breaking

Under which conditions the break of virial theorem implies break of ergodicity? I've seen this question, but it is very limited and not sufficient. To constrain the discussion I'm interested in 1D ...
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Energy of harmonic oscillators

I've calculated the energy of a classical harmonic oscillator (HO) as: \begin{align*} \overline E = \overline{E_K} + \overline{E_P} = \frac{\overline{p^2}}{2m} + \frac{k\overline{x^2}}{2} = \frac{...
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Hyper-Virial Theorem for Central potentials

I saw an exercise on the hyper virial theorem for central potential and, as I was looking for some clues, I found it hasn't been solved here, so after finding the solution I thought I may include it ...
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Why is the kinetic energy of an electron in a hydrogen atom at $n=2$ level, the negative of its total energy?

My textbook [1] says in order to determine the De Broglie wavelength of an $e^-$ in a hydrogen atom at energy level $n=2$, we can determine its kinetic energy first by finding its total energy from $E=...
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Barkana & Loeb's Virial Radius Plot [closed]

Recently I have been trying to teach myself the basics of modern cosmology. To do so I have been using Barkana & Loeb. I've been working my way through and come to virial radius. They give the ...
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How does the virial expansion converging, mean that there is no phase transition?

I have a question regarding how the virial expansion relates to phase transition. I watch the lecture https://www.coursera.org/learn/statistical-mechanics/lecture/1a5Di/tutorial-3-algorithms-exact-...
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How is virial theorem for point particles related to the virial theorem for an ideal gas?

Some presentations of the virial theorem are mechanical (see this page by John Baez for an example). They assume that there is a system of point particles interacting only via Newtonian gravity (along ...
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Intuitive understanding of the virial radius/mass

I'm currently trying with no avail to understand the intricacies that define the virial radius or mass in a way that is different from the actual galactic radius/mass. I understand it is derived in ...
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What does it mean when a system “virializes”? How does this happen?

What does it mean when a gas virializes in a potential? Is it only that it now satisfies the virial theorem? How does it happen?
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Why are there time averages in the virial theorem if one of its hypotheses is that the system is in equilibrium?

Why are there time averages in the virial theorem if one of its hypotheses is that the system is in equilibrium? According to my textbook, the virial theorem is that for a gravitationally bound system ...
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When is Virial theorem applicable for superposition of energy eigenstates?

In a note I recently studied, it states that: "All energy eigenstates of harmonic oscillator obey the Virial Theorem, but this theorem does not apply to superpositions of energy eigenstates except ...
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Evolution of the virial ratio in $N$-body sim

I have run an 2D $N$-body sim to simulate the cold collapse of a galaxy. Initial conditions: $N=500$, all velocities=0, positions generated randomly within a circle of radius $R$. The system is ...
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Does the Virial Theorem apply to degenerate matter?

I am trying to understand how the degenerated He core progressively increases its temperature as the star moves up through the Red Giant Branch. A well-known property of the degenerate He core is ...
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Why the He core gets hotter and hotter? Is it because it shrinks?

I know that after a star in the main sequence runs out of H in the core, it will start burning H in the shells surrounding the (now) He core. (1) Why now the Hydrogen shells are hot enough for ...
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Relation of potential energy and total energy in Bohr Model of the hydrogen atom

I am currently in the 11th grade. In the Bohr Model of the hydrogen atom, potential energy is two times that of total energy. It means that magnitude of potential energy is two times that of magnitude ...
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Frictional forces and the virial theorem

Consider a system in which the total force acting on the system consists of conservative forces $F'$ and frictional forces $f^i$, where $f^i$ depends on velocity. It is to show that for such a system ...
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Inconsistency between two formulas for Gravitational Potential Energy that don't yield the same result

According to this site the general form of the Gravitational Potential Energy of mass $m$ is $$U=-\frac{GMm}{r}\tag{1}$$ where $G$ is the gravitation constant, $M$ is the mass of the attracting body, ...
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Why does the virial theorem of quantum mechanics hold for the quantum oscillator but not the infinite square well?

Why does the virial theorem of quantum mechanics hold for the quantum oscillator but not the infinite square well? The proof uses Ehrenfest's theorem, so I was wondering if it had something to do with ...
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Do compact stars heat the CDM halo?

I understand that a single WIMP interacts weakly with individual baryons, or probably even planets, but black holes should be able to slingshot the individual WIMPs they encounter. According to the ...
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Interpretation of scalar moment of inertia used to derive the virial theorem using the Reynolds transport theorem

I have seen an uncommon derivation of the virial theorem and need some help understanding the physical aspects and interpretation. For an integral form of an equation, $$I = \int_{V(t)} \alpha d^3 x ...
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Proving virial theorem

my class is using Griffiths, and I've been assigned to prove the Virial theorem. Of course, there are innumerable resources devoted to working through solutions to his problems, and I use this ...
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Calculate pressure using Monte Carlo: virial expansion vs virial theorem

In Case Study 1 in Frenkel and Smit (2008, 2nd ed, p.51), we calculate the pressure of a Lennard-Jones fluid by a NVT Monte Carlo simulation. We find the pressure from the virial theorem by ...
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On Virial theorem and its Quantum mechanical analog

The relation established by the virial theorem is well pronounced in the case of potential energies that are power law functions of distance. In quantum mechanics, a similar analog is present, ...
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How do I calculate average potential energy for a given potential 'over a long time'?

A classical particle with total energy $E$ moves under the influence of a potential $$V(x,y)= 3x^3+2x^2y+2y^2x+y^3.$$ The average potential energy $\langle V \rangle$, calculated over a long time,...
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Manfred Schroeder Virial Theorem Derivation

I'm working on a project about fractals in physics and I picked up Manfred Schroeder's fascinating book: Fractals, Chaos and Power Laws. In one of the early parts (titled Similarity in Physics: Some ...
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Why is the Virial Theorem not a Special Case of the Ergodic Theorem? What is their Relationship?

The virial theorem involves the time-averages of the potential and kinetic energies if the motion of the system is bounded to a finite region of space. An ergodic theorem relates the time and space ...
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Why is Kinetic Energy = (-) Total Energy and Potential Energy = 2 $\times$ Total Energy?

I came across this relation while reading on the Bohr atomic model. Are there any other forces for which these relations hold good?
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How does the total change in energy being negative indicate the direction of the Force?

To be more precise consider the following situation: Suppose a satellite is orbiting at a distance of $r_1$ from the centre of the earth. After some time, rockets are fired such that the new distance $...
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What happens when we use the Virial Theorem iteratively?

Say I want to model the formation of structure in the Universe as a series of events whereby already virialised systems are brought together to create a larger virialised system. I will take the ...
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Interpretation of Virial theorem in the case n=-2

For potentials $V$ that fulfil $V(\lambda \vec{r}_1, \lambda \vec{r}_2, \ldots, \lambda \vec{r}_N) = \lambda^n V(\vec{r}_1,\vec{r}_2,\ldots, \vec{r}_N)$ one can rewrite the virial theorem as $$2\...
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Proving the Virial theorem

Consider the expectation in the canonical ensemble defined by $$\left\langle x_i\frac{\partial \mathcal{H}}{\partial x_j} \right\rangle=\frac{1}{Z}\int d\Gamma x_i\frac{\partial \mathcal{H}}{\...
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Why is the total energy of an orbiting system negative?

Assume it's an circular orbit. Object A orbits around object B. Take object B as frame of reference. .$E=KE_a + GPE$ .$E=\frac 12m_av_a^2 +(-\frac {GM_bm_A}r)$ .$E=\frac 12m_a(GM_br)+(-\frac {...
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How do I find the average kinetic energy and average potential energy of a hydrogen electron in the ground state?

How do I find the average kinetic energy and average potential energy of a hydrogen electron in the ground state? In my modern physics class, we are wrapping up the 3D Schrödinger equation, and I am ...
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For $N$ particles acting under gravity, how long until they settle into a virial equilibrium?

As the title says, if I have a system of particles interacting only due to gravity, over what timescale do we expect them to fall into a virial equilibrium? By virial equilibrium I mean a system that ...
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279 views

Energy in harmonic oscillator [closed]

The expectation value of the potential energy is exactly half the total according to Griffiths. Is that case always true for quantum harmonic oscillator? Is that the case also for classical harmonic ...
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How does dark matter collapse?: Entropy considerations

Inspired by this question. I believe that the usual explanation that preserves the second law of thermodynamics as an astrophysical gas cloud collapses under gravity is that the gas must heat and ...
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How to show that the gravitational potential energy of a two particle system is -2 times the total kinetic energy, without using the virial theorem?

Consider a two-particle system with identical masses, orbiting in circles about their center of mass. I'm supposed to prove that: $$U_p = -2U_k$$ With $U_p$ potential energy of the system, and $U_k$ ...
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Classical vs. quantum energy of the hydrogen atom

If I have an electron and a proton and calculate the classical energy which I get by bringing the electron from infinity to the distance of a Bohr radius to the proton, I get 27.2 eV, but the electron ...
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Origins of many-particle interactions

The internal potential energy of an $N$ particle system is a general function of the coordinates of the particles: $U(r_1,...,r_N)$. In some approximations and expansions - e.g. virial expansion - it ...
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Virial theorem and variational method: a question

I have an hydrogenic atom, knowing that its ground-state wavefunction has the standard form $$ \psi = A e^{-\beta r} $$ with $A = \frac{\beta^3}{\pi}$, I have to find the best value for $\beta$ (...
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Virial of a system

I had obtained $$\overline{E_{kin}} = -\frac{1}{2}\overline{\sum_j\mathbf{r}_j\cdot\mathbf{F}_j}$$ and was asked to show that if the forces are conservative then $$\overline{E_{kin}} = \frac{1}{2}\...
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The virial theorem and a delta function potential

So the virial theorem tells us that: $2\langle T\rangle = \langle \textbf{r}\cdot\nabla V\rangle$. Now I was wondering what would happen if V has te form: $V(\textbf{r}-\textbf{r}') = V_0\delta^{(D)...
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Gravitational Potential Energy (GPE) to Kinetic Energy (KE) transfer in Satellite Orbits

I am stumped by a mundane A Level Physics question (teacher of physics here obviously a bit short of practice!). My colleagues and I are stumped and were wondering if any one could help us. It ...