Theorem of the virial

The theorem of the virial is a Théorème Physique which states that:

This result is a simple consequence of the basic principle of dynamics, applied to a whole of masses in interaction reciprocal Gravitation nelle (Problème with NR body).

Total energy E = E_c + E_p is worth thus

E = \ tfrac12 E_p = - E_c

It can also be generalized with other fields, in the form

2 E_c = has · E_p

where has is the power of R in the expression of E_p ; the above mentioned form well is found because has is worth -1 for the gravitational force.

Demonstration

In dynamics with N-body

; Assumption

a system of NR isolated massive body; each body thus undergoes only the only gravitational forces of its neighbors.

The basic principle of dynamics states that for each body I , the gravitational force is written:

$F_i = - \ sum_j G m_i m_j \ frac {r_i-r_j} = \ frac {1} {2} \ sum_i m_i \ left (\ frac {d^2 (r_i^2)}{dt^2} - 2 \ left (\ frac {dr_i} {dt} \ right) ^2 \ right)$

from where finally:

$- \ frac {1} {2} \ sum_ {I, J} G \ frac {m_im_j} + \ sum_i m_i \ left (\ frac {dr_i} {dt} \ right) ^2 = \ frac {1} {2} \ frac {d^2} {dt^2} \ left (\ sum_i m_ir_i^2 \ right)$

One recognizes in this equation:

potential energy

E_p = - \ frac {1} {2} \ sum_ {I, J} G \ frac {m_im_j}

twice kinetic energy

E_c = \ frac {1} {2} \ sum_i m_i \ left (\ frac {dr_i} {dt} \ right) ^2

the derivative second of inertia

I = \ sum_i m_i r_i^2

With balance, I = 0 thus

E_p > + 2 E_c = 0

what it was necessary to show.

In Quantum physics

; Statement: $2 \ langle T \ rangle = N \ langle V \ rangle$

with $\ langle T \ rangle$ corresponds to the median value of the kinetic energy
and $\ langle V \ rangle$ corresponds to the median value of the potential being expressed $V (X) = \ lambda \ cdot x^ {N}$

; Demonstration: Let us show that $\ langle \ rangle = 0$ :

$\ langle \ rangle = \ langle \ phi|HXP| \ phi \ rangle - \ langle \ phi|XPH| \ phi \ rangle$
However, $H| \ phi \ rangle = E | \ phi \ rangle$ and $\ langle \ phi|H = E \ langle \ phi|$
Thus $\ langle \ rangle = E \ langle \ phi|XP| \ phi \ rangle - E \ langle \ phi|XP| \ phi \ rangle$ (1)
Let us work on :

$= HXP - XPH = HXP - XHP +XHP - XPH$
Then, $= P + X$ (2)
Let us express and :

$= - = \ frac {-} {2m} = \ frac {- ih \ cdot P} {m}$
$= = ih \ frac {\ partial V} {\ partial X}$ (3)

Let us return on $\ langle \ rangle = 0$ :

$\ langle \ rangle = 0$
Then, while using (2), one finds:

$0 = \ langle P \ rangle + \ langle X \ rangle$
In the same way, while using (3), one finds

$\ left \ langle \ frac {P^2} {m} \ right \ rangle = N \ langle V \ rangle$

From where the expected result:

$2 \ langle T \ rangle = N \ langle V \ rangle$

In thermodynamics
August 1st
Applications

In astrophysics

The theorem of the virial is very much used in galactic dynamics. It makes it possible for example to quickly obtain an order of magnitude of the total mass M of a cluster of stars if one knows the mean velocity V of the star S in the cluster and the average distance R between two stars of the cluster, which can be estimated starting from the observations:

E_c ~ ½MV²

E_p ~ - GM ²/2R

It comes then 2E_c = - E_p <=> M = 2RV ² /G
Factor 2 comes owing to the fact that for a system of particles it is necessary to avoid twice counting potential energy associated with a couple.

The enigma of the black matter

As it is possible in addition to determine the mass of visible stars starting from their Luminosité, one can compare the total mass obtained by the theorem of the virial with the visible mass. The observation of a considerable difference (factor 10 on a scale Galaxy S and factor 100 on a cluster scale) between the two sizes led the astrophysicist S to suppose the existence of black Matière, i.e. nondetectable by our instruments. Only the other possible explanation would be that the law of the gravitation is not valid with large scales, but no track in this direction gave result to date.
One can show that this black matter dominates the mass of the galaxies outside the disc, in the halation where it extends up to 100-200 kiloparsecs (kpc) - against 10-20 kpc for the visible mass.

In thermodynamics

See also: kinetic Pressure

External resources

the virial reveals the black matter

Where is the mass of the Universe by Albert Bijaoui

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