Virial theorem

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In mechanics, the virial [G] of a collection of [N] point particles is defined as:

[G = \sum_^ \mathbf_ \cdot \mathbf_ ]

where [\mathbf_] and [\mathbf_] are the position and momentum vectors, respectively, of the [k^}] particle. The expression "virial" derives from vis,viris, the Latin word for "force" or "energy".

Contents

1 Time derivative and averaging
2 The virial theorem
3 Relationship to potential energy
4 Applications to power-law forces
5 Inclusion of electromagnetic fields
6 notes

Time derivative and averaging

The time derivative of the virial can be written

[\frac = \sum_^ \frac_} \cdot \mathbf_ + \sum_^ \mathbf_ \cdot \frac_}]

:[= \sum_^ \mathbf_ \cdot \mathbf_ + \sum_^ m_ \frac_} \cdot \frac_}]

or, more simply,

[\frac = 2 T + \sum_^ \mathbf_ \cdot \mathbf_]

Here [m_] is the mass of the [k^}] particle, [\mathbf_ = \frac_}] is the net force on that particle and [T] is the total kinetic energy of the system

[T \equiv \frac \sum_^ m_ v_^ = \frac \sum_^ m_ \frac_} \cdot \frac_}]

The average of this derivative over a time [\tau] is defined

[\left\langle \frac \right\rangle_ \equiv \frac \int_^ dt \frac = \frac \int_^ dG = \frac]

from which we obtain the exact equation

[\left\langle \frac \right\rangle_ = 2 \left\langle T \right\rangle_ + \sum_^ \left\langle \mathbf_ \cdot \mathbf_ \right\rangle_]

The virial theorem

The virial theorem states that, if [\left\langle \frac \right\rangle_ = 0], then

[2 \left\langle T \right\rangle_ = -\sum_^ \left\langle \mathbf_ \cdot \mathbf_ \right\rangle_]

There are many reasons why the average of the time derivative might vanish, i.e., [\left\langle \frac \right\rangle_ = 0]. One often-cited reason applies to bound systems, i.e., systems that hang together forever. In that case, the virial [G^}] is usually bounded between two extremes, [G_] and [G_], and the average goes to zero in the limit of very long times [\tau]

[\lim_ \left| \left\langle \frac}} \right\rangle_ \right| = \lim_ \left| \frac \right| = \lim_ \frac - G_} = 0]

Even if the the average of the time derivative [\left\langle \frac \right\rangle_ \approx 0] is only approximately zero, the virial theorem holds to the same degree of approximation.

Relationship to potential energy

The total force [\mathbf_] on particle [k] is the sum of all the forces from the other particles [j] in the system

[\mathbf_ = \sum_^ \mathbf_]

where [\mathbf_] is the force applied by particle [j] on particle [k]. Hence, the force term of the virial time derivative can be written

[\sum_^ \mathbf_ \cdot \mathbf_ = \sum_^ \sum_^ \mathbf_ \cdot \mathbf_]

Since no particle acts on itself (i.e., [\mathbf_ = 0] whenever [j=k]), we have

[\sum_^ \mathbf_ \cdot \mathbf_ = \sum_^ \sum_ \mathbf_ \cdot \mathbf_ + \sum_^ \sum_ \mathbf_ \cdot \mathbf_ = \sum_^ \sum_ \mathbf_ \cdot \left( \mathbf_ - \mathbf_ \right)]

where we have assumed that Newton's third law of motion holds, i.e., [\mathbf_ = -\mathbf_] (equal and opposite reaction).

It often happens that the forces can be derived from a potential energy [V] that is a function only of the distance [r_] between the point particles [j] and [k]. Since the force is the gradient of the potential energy, we have in this case

[\mathbf_ \equiv -\nabla__} V = - \frac \frac_ - \mathbf_}}]

which is clearly equal and opposite to [\mathbf_ \equiv -\nabla__} V], the force applied by particle [k] on particle [j], as may be confirmed by explicit calculation. Hence, the force term of the virial time derivative is

[\sum_^ \mathbf_ \cdot \mathbf_ = \sum_^ \sum_ \mathbf_ \cdot \left( \mathbf_ - \mathbf_ \right) =-\sum_^ \sum_ \frac \frac_ - \mathbf_ \right)^2}} = -\sum_^ \sum_ \frac r_]

Applications to power-law forces

It often happens that the potential energy [V] is a power-law function

[V(r_) = \alpha r_^]

where the coefficient [\alpha] and the exponent [n] are constants. In such cases, the force term of the virial time derivative is given by the equation

[-\sum_^ \mathbf_ \cdot \mathbf_ = \sum_^ \sum_ \frac r_ =\sum_^ \sum_ n V(r_) = n U]

where [U] is the total potential energy of the system

[U \equiv \sum_^ \sum_ V(r_)]

In such cases, when the average of the virial time derivative [\left\langle \frac \right\rangle_ = 0] is zero, the general equation holds

[\langle T \rangle_ = -\frac \sum_^ \langle \mathbf_ \cdot \mathbf_ \rangle_ = \frac \langle U \rangle_]

A commonly cited example is gravitational attraction, for which [n=-1].  In that case, the average kinetic energy is simply half of the average negative potential energy

[\langle T \rangle_ = -\frac \langle U \rangle_]

This result is remarkably useful for complex gravitating systems such as solar systems or galaxies, and also holds for electrostatic systems, for which [n=-1] as well.

Although derived for classical mechanics, the virial theorem also holds for quantum mechanics.

Inclusion of electromagnetic fields

The virial theorem can be extended to include electric and magnetic fields. The result isGeorge Schmidt, Physics of High Temperature Plasmas (Second edition), Academic Press (1979), p.72

[\frac\fracI+ \int_Vx_k\fracd^3r = 2(T+U) + W^E + W^M - \int x_k(p_+T_)dS_i],

where I is the moment of inertia, G is the momentum density of the electromagnetic field, T is the kinetic energy of the "fluid", U is the random "thermal" energy of the particles, W^E and W^M are the electric and magnetic energy content of the volume considered. Finally, p_ik is the fluid-pressure tensor expressed in the local moving coordinate system

[p_= \Sigma n^\sigma m^\sigma \langle v_iv_k\rangle^\sigma- V_iV_k\Sigma m^\sigma n^\sigma],

and T_ik is the electromagnetic stress tensor,

[T_= \left( \frac + \frac \right)- \left( \varepsilon_0E_iE_k + \frac \right)].

A plasmoid is a finite configuration of magnetic fields and plasma. With the virial theorem it is easy to see that any such configuration will expand if not contained by external forces. In a finite configuration without pressure-bearing walls or magnetic coils, the surface integral will vanish. Since all the other terms on the right hand side are positive, the acceleration of the moment of inertia will also be positive. It is also easy to estimate the expansion time τ. If a total mass M is confined within a radius R, then the moment of inertial is roughly MR², and the left hand side of the virial theorem is MR²/τ². The terms on the right hand side add up to about pR³, where p is the larger of the plasma pressure or the magnetic pressure. Equating these two terms and solving for τ, we find

[\tau\,\sim R/c_s],

where c_s is the speed of the ion acoustic wave (or the Alfven wave, if the magnetic pressure is higher than the plasma pressure). Thus the lifetime of a plasmoid is expected to be on the order of the acoustic (or Alfven) transit time.

notes

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