# 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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### 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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### 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 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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### 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}\...
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)...