1.3: Transport Phenomena

a. Boltzmann Transport Equation

It is one thing to describe the behavior of matter and photons in equilibrium, but stars shine. Therefore energy must flow from the interior to the surface regions of the star and the details of the flow play a dominant role in determining the resultant structure and evolution of the star. We now turn to an extremely simple description of how this flow can be quantified; this treatment is due to Ludwig Boltzmann and should not be confused with the Boltzmann formula of Maxwell-Boltzmann statistics. Although the ideas of Boltzmann are conceptually simple, many of the most fundamental equations of theoretical physics are obtained from them.

Basically the Boltzmann transport equation arises from considering what can happen to a collection of particles as they flow through a volume of phase space. Our prototypical volume of phase space was a six-dimensional "cube", which implies that it has five-dimensional "faces". The Boltzmann transport equation basically expresses the change in the phase density within a differential volume, in terms of the flow through these faces, and the creation or destruction of particles within that volume.

For the moment, let us call the six coordinates of this space xi remembering that the first three refer to the spatial coordinates and the last three refer to the momentum coordinates. If the "area" of one of the five-dimensional "faces" is A, the particle density is N/V, and the flow velocity is v , then the inflow of particles across that face in time dt is \[v \frac{N}{V} A d t=\left[\frac{d x_i}{d t} f\left(x_i, t\right) \frac{d V}{d x_i}\right] d t\label{1.2.1}\]

Similarly, the number of particles flowing out of the opposite face located \(\mathrm{dx}_{\mathrm{i}}\) away is \[\left[\frac{d x_i}{d t} f\left(x_i+d x_i, t\right) \frac{d V}{d x_i}\right] d t=\text { number of outflowing particles }\label{1.2.2}\]

The net change due to flow in and out of the six-dimensional volume is obtained by calculating the difference between the inflow and outflow and summing over all faces of the volume: \[\sum_i \frac{d x_i}{d t}\left[\frac{f\left(x_i+d x_i, t\right)-f\left(x_i, t\right)}{d x_i}\right] d V d t=\sum_i \frac{d x_i}{d t} \frac{\partial f}{\partial x_i} d V d t\label{1.2.3}\]

Note that the sense of equation \ref{1.2.3} is such that if the inflow exceeds the outflow, the net flow is considered negative. Now this flow change must be equal to the negative time rate of change of the phase density (i.e., \(\mathrm{d} f / \mathrm{dt}\)). We can split the total time rate of change of the phase density into that part which represents changes due to the differential flow \(\ll \mathrm{f} / \ll \mathrm{t}\) and that part which we call the creation rate S. Equating the flow divergence with the local temporal change in the phase density, we have \[\sum_i \frac{d x_i}{d t} \frac{\partial f}{\partial x_i} d V d t=-\left(\frac{\partial f}{\partial t}-S\right) d V d t\label{1.2.4}\]

Rewriting our phase space coordinates xi in terms of the spatial and momentum coordinates and using the old notation of Isaac Newton to denote total differentiation with respect to time (i.e., the dot .) we get \[\frac{\partial f\left(x_i, p_i, t\right)}{\partial t}+\sum_{i=1}^3\left(\dot{x}_i \frac{\partial f}{\partial x_i}+\dot{p}_i \frac{\partial f}{\partial p_i}\right)=S\label{1.2.5}\]

This is known as the Boltzmann transport equation and can be written in several different ways. In vector notation we get \[\frac{\partial f}{\partial t}+\vec{v} \cdot \nabla f-\frac{1}{m} \nabla \Phi \cdot \nabla_v f=S\label{1.2.6}\]

Here the potential gradient \(\nabla \Phi\) has replaced the momentum time derivative while \(\nabla\mathrm{v}\) is a gradient with respect to velocity. The quantity m is the mass of a typical particle. It is also not unusual to find the Boltzmann transport equation written in terms of the total Stokes time derivative \[\frac{D}{D t} \equiv \frac{\partial}{\partial t}+\vec{v} \cdot \nabla\label{1.2.7}\]

If we take \(\vec{\mathrm{v}}\) to be a six dimensional "velocity" and \(\nabla\) to be a six- dimensional gradient, then the Boltzmann transport equation takes the form \[\frac{D f}{D t}=S\label{1.2.8}\]

Although this form of the Boltzmann transport equation is extremely general, much can be learned from the solution of the homogeneous equation. This implies that \(S=0\) and that no particles are created or destroyed in phase space.

b. Homogeneous Boltzmann Transport Equation and Liouville's Theorem

Remember that the right-hand side of the Boltzmann transport equation is a measure of the rate at which particles are created or destroyed in the phase space volume. Note that creation or destruction in phase space includes a good deal more than the conventional spatial creation or destruction of particles. To be sure, that type of change is included, but in addition processes which change a particle's position in momentum space may move a particle in or out of such a volume. The detailed nature of such processes will interest us later, but for the moment let us consider a common and important form of the Boltzmann transport equation, namely that where the right-hand side is zero. This is known as the homogeneous Boltzmann transport equation. It is also better known as Liouville's theorem of statistical mechanics. In the literature of stellar dynamics, it is also occasionally referred to as Jeans' theorem² for Sir James Jeans was the first to explore its implications for stellar systems. By setting the right-hand side of the Boltzmann transport equation to zero, we have removed the effects of collisions from the system, with the result that the density of points in phase space is constant. Liouville's theorem is usually generalized to include sets or ensembles of particles. For this generalization phase space is expanded to 6N dimensions, so that each particle of an ensemble has six position and momentum coordinates which are linearly independent of the coordinates of every other particle. This space is often called configuration space, since the entire ensemble of particles is represented by a point and its temporal history by a curve in this 6N-dimensional space. Liouville's theorem holds here and implies that the density of points (ensembles) in configuration space is constant. This, in turn, can be used to demonstrate the determinism and uniqueness of Newtonian mechanics. If the configuration density is constant, it is impossible for two ensemble paths to cross, for to do so, one path would have to cross a volume element surrounding a point on the other path, thereby changing the density. If no two paths can cross, then it is impossible for any two ensembles to ever have exactly the same values of position and momentum for all their particles. Equivalently, the initial conditions of an ensemble of particles uniquely specify its path in configuration space. This is not offered as a rigorous proof, only as a plausibility argument. More rigorous proofs can be found in most good books on classical mechanics^3,4. Since Liouville’s theorem deals with configuration space, it is sometimes considered more fundamental than the Boltzmann transport equation; but for our purposes the expression containing the creation rate S will be required and therefore will prove more useful.

c. Moments of the Boltzmann Transport Equation and Conservation Laws

By the moment of a function we mean the integral of some property of interest, weighted by its distribution function, over the space for which the distribution function is defined. Common examples of such moments can be found in statistics. The mean of a distribution function is simply the first moment of the distribution function, and the variance can be simply related to the second moment. In general, if the distribution function is analytic, all the information contained in the function is also contained in the moments of that function.

The complete solution to the Boltzmann transport equation is, in general, extremely difficult and usually would contain much more information about the system than we wish to know. The process of integrating the function over its defined space to obtain a specific moment removes or averages out much of the information about the function. However, this process also usually yields equations which are much easier to solve. Thus we trade off information for the ability to solve the resulting equations, and we obtain some explicit properties of the system of interest. This is a standard "trick" of mathematical physics and one which is employed over and over throughout this book. Almost every instance of this type carries with it the name of some distinguished scientist or is identified with some fundamental conservation law, but the process of its formulation and its origin are basically the same.

We define the nth moment of a function f as \[\mathrm{M_n[f(x)]=\int x^n f(x) d x}\label{1.2.9}\]

By multiplying the Boltzmann equation by powers of the position and velocity and integrating over the appropriate dimensions of phase space, we can generate equations relating the various moments of the phase density \(\mathrm{f(\vec{x}, \vec{v})}\). In general, such a process always generates two moments of different order \(\mathrm{n}\), so that a succession of moment taking always generates one more moment than is contained in the resulting equations. Some additional property of the system will have to be invoked to relate the last generated higher moment to one of lower order, in order to close the system of equations and allow for a solution. To demonstrate this process, we show how the equation of continuity, the Euler-Lagrange equations of hydrodynamic flow, and the virial theorem can all be obtained from moments of the Boltzmann transport equation.

Continuity Equation and the Zeroth Velocity Moment Although most moments, particularly in statistics, are normalized by the integral of the distribution function itself, we have chosen not to do so here because the integral of the phase density f over all velocity space has a particularly important physical meaning, namely, the local spatial density. \[\rho=m \int_{-\infty}^{+\infty} f(\vec{x}, \vec{v}) d \vec{v}\label{1.2.10}\]

By \(\mathrm{d\vec{v}}\) we mean that the integration is to be carried out over all three velocity coordinates \(\mathrm{v_1}\), \(\mathrm{v_2}\), and \(\mathrm{v_3}\). A pedant might correctly observe that the velocity integrals should only range from \(\mathrm{-c}\) to \(\mathrm{+c}\), but for our purposes the Newtonian view will suffice. Integration over momentum space will properly preserve the limits. Now let us integrate the component form of equation \ref{1.2.5} over all velocity space to generate an equation for the local density. Thus, \[\int_{-\infty}^{+\infty}\left(\frac{\partial f}{\partial t}+\sum_{i=1}^3 v_i \frac{\partial f}{\partial x_i}+\sum_{i=1}^3 \dot{v}_i \frac{\partial f}{\partial v_i}\right) d \vec{v}=\int_{-\infty}^{+\infty} S d \vec{v}\label{1.2.11}\]

Since the velocity and space coordinates are linearly independent, all time and space operators are independent of the velocity integrals. The integral of the creation rate S over all velocity space becomes simply the creation rate for particles in physical space, which we call ℑ. By noting that the two summations in equation \ref{1.2.11} are essentially scalar products, we can rewrite that moment and get \[\frac{\partial}{\partial t} \int_{-\infty}^{+\infty} f d \vec{v}+\int_{-\infty}^{+\infty} \vec{v} \cdot \nabla f d \vec{v}+\int_{-\infty}^{+\infty} \dot{\vec{v}} \cdot \nabla_v f d \vec{v}=\mathfrak{I}\label{1.2.12}\]

It is clear from the definition of ρ that the first term is the partial derivative of the local particle density. The second term can be modified by use of the vector identity \[\vec{v} \cdot \nabla f=\nabla \cdot(f \vec{v})-f \nabla \cdot \vec{v}\label{1.2.13}\]

and by noting that \(\nabla \cdot \overrightarrow{\mathrm{v}}=0\), since space and velocity coordinates are independent. If the particles move in response, to a central force, then we may relate their accelerations \(\dot{\mathrm{\vec{v}}}\) to the gradient of a potential which depends on only position and not velocity. The last term then takes the form \((\nabla \bullet \Phi / \mathrm{m}) \bullet \int \nabla_{\mathrm{v}} \mathrm{f}(\mathrm{v}) \mathrm{d} \vec{\mathrm{v}}\). If we further require that \(\mathrm{f(v)}\) be bounded (i.e., that there be no particles with infinite velocity), then since the integral and gradient operators basically undo each other, the integral and hence the last term of equation \ref{1.2.12} vanish, leaving \[\frac{\partial n}{\partial t}+\nabla \cdot \int_{-\infty}^{+\infty} \vec{v} f d \vec{v}=\mathfrak{I}\label{1.2.14}\]

The second term in equation \ref{1.2.14} is the first velocity moment of the phase density and illustrates the manner by which higher moments are always generated by the moment analysis of the Boltzmann transport equation. However, the physical interpretation of this moment is clear. Except for a normalization scalar, the second term is a measure of the mean flow rate of the material. Thus, we can define a mean flow velocity \(\mathrm{\vec{u}}\) \[\vec{u}=\frac{\int \vec{v} f(v) d v}{\int f(v) d v}\label{1.2.15}\]

which, upon multiplying by the particle mass, enables us to obtain the familiar form of the equation of continuity: \[\frac{\partial \rho}{\partial t}+\nabla \cdot(\rho \vec{u})=\mathfrak{I} m\label{1.2.16}\]

This equation basically says that the explicit time variation of the density plus density changes resulting from the divergence of the flow is equal to the local creation or destruction of material ℑ.

Euler-Lagrange Equations of Hydrodynamic Flow and the First Velocity Moment of the Boltzmann Transport Equation The zeroth moment of the transport equation provided insight into the way in which matter is conserved in a flowing medium. Multiplying the Boltzmann transport equation by the velocity and integrating over all velocity space will produce momentum-like moments, and so we might expect that such operations will also produce an expression of the conservation of momentum. This is indeed the case. However, keep in mind that the velocity is a vector quantity, and so the moment analysis will produce a vector equation rather than the scalar equation, as was the case with the equation of continuity. Multiplying the Boltzmann transport equation by the local particle velocity \(\vec{\mathrm{v}}\), we get \[\int_{-\infty}^{+\infty} \vec{v} \frac{\partial f}{\partial t} d \vec{v}+\int_{-\infty}^{+\infty} \vec{v}(\vec{v} \cdot \nabla f) d \vec{v}+\int_{-\infty}^{+\infty} \vec{v}\left(\dot{\vec{v}} \cdot \nabla_v f\right) d \vec{v}=\int_{-\infty}^{+\infty} \vec{v} S d \vec{v}\label{1.2.17}\]

We can make use of most of the tricks that were used in the derivation of the continuity equation \ref{1.2.16}. The first term can be expressed in terms of the mean flow velocity [equation \ref{1.2.15}] while the second term can be expressed as \[\int_{-\infty}^{+\infty} \vec{v} \nabla \cdot(\vec{v} f) d \vec{v}\label{1.2.18}\]

by using the vector identity given by equation \ref{1.2.13}. Since the quantity in parentheses of the third term in equation \ref{1.2.17} is a scalar and since the particle accelerations depend on position only, we can move them and the vector scalar product outside the velocity integral and re-express them in terms of a potential, so the third term becomes \[\frac{1}{m} \nabla \Phi \cdot \int_{-\infty}^{+\infty}\left(\nabla_v f\right) \vec{v} d \vec{v}\label{1.2.19}\]

The integrand of equation \ref{1.2.19} is not a simple scalar or vector, but is the vector outer, or tensor, product of the velocity gradient of \(f\) The integrand of equation \ref{1.2.19} is not a simple scalar or vector, but is the vector outer, or tensor, product of the velocity gradient of \(\mathrm{\vec{v}}\) itself. However, the vector identity given by equation \ref{1.2.13} still applies if the scalar product is replaced with the vector outer product, so that the integrand in equation \ref{1.2.19} becomes \[\left(\nabla_v f\right) \vec{v}=\nabla_v(f \vec{v})-f \nabla_v \vec{v}=\nabla_v(f \vec{v})-\mathbf{1} f\label{1.2.20}\]

The quantity 1 is the unit tensor and has elements of the Kronecker delta \(\delta_{\mathrm{ij}}\) whose elements are zero when \(\mathrm{i} \neq \mathrm{j}\) and 1 when \(\mathrm{i} = \mathrm{j}\). Again, as long as \(f\) is bounded, the integral over all velocity space involving the velocity gradient of \(f\) will be zero, and the first velocity moment of the Boltzmann transport equation becomes \[\frac{\partial(n \vec{u})}{\partial t}+\int_{-\infty}^{+\infty} \vec{v} \nabla \cdot(\vec{v} f) d \vec{v}-n \nabla \Phi=\int_{-\infty}^{+\infty} S \vec{v} d \vec{v}\label{1.2.21}\]

Differentiating the first term and using the continuity equation \ref{1.2.14} to eliminate \(\partial \mathrm{n} / \partial \mathrm{t}\), we get \[\begin{aligned}
n \frac{\partial \vec{u}}{\partial t} & -(\vec{u} \cdot \nabla n+n \nabla \cdot \vec{u}) \vec{u}+\int_{-\infty}^{+\infty} \vec{v} \nabla \cdot(\vec{v} f) d \vec{v}+\rho \nabla \Phi \\
& =\int_{-\infty}^{+\infty} \vec{v} S d \vec{v}-\int_{-\infty}^{+\infty} \vec{u} S d \vec{v}
\end{aligned}\label{1.2.22}\]

Since \(\nabla \bullet \vec{\mathrm{v}}\) is zero and the velocity and space coordinates are independent, we may rewrite the third term in terms of a velocity tensor defined as \[\mathbf{u}=\frac{\int_{-\infty}^{+\infty} \vec{v} \vec{v} f(v) d \vec{v}}{\int_{-\infty}^{+\infty} f(v) d \vec{v}}\label{1.2.23}\]

Some rearranging and the use of a few vector identities lead to \[\rho \frac{\partial \vec{u}}{\partial t}+\rho(\vec{u} \cdot \nabla) \vec{u}+\nabla \cdot[\rho(\mathbf{u}-\vec{u} \vec{u})]+\rho \nabla \Phi=\int_{-\infty}^{+\infty} m S(\vec{v}-\vec{v}) d \vec{v}\label{1.2.24}\]

The quantity in brackets of the third term is sometimes called the pressure tensor. A density \(\rho\) times a velocity squared is an energy density, which has the units of pressure. We can rewrite that term so it has the form \[\mathbf{P}=\frac{\rho \int_{-\infty}^{+\infty} f(v)(\vec{v}-\vec{u})(\vec{v}-\vec{u}) d v}{\int_{-\infty}^{+\infty} f(v) d \vec{v}}\label{1.2.25}\]

which shows the form of a moment of f. In this instance the moment is a tensor that more or less describes the difference between the local flow indicated by \(\mathrm{\vec{v}}\) and the mean flow \(\mathrm{\vec{u}}\). The form of the moment is that of a variance, and the tensor, in general, consists of nine components. Each component measures the net momentum transfer (or contribution to the local energy density) across a surface associated with that coordinate which results from the net flow coming from another coordinate. Thus the third term is simply the divergence of the pressure tensor which is a vector quantity, and the first velocity moment of the Boltzmann transport equation becomes \[\frac{\partial \vec{u}}{\partial t}+(\vec{u} \cdot \nabla) \vec{u}=-\nabla \Phi-\frac{1}{\rho} \nabla \cdot \mathbf{P}+\frac{1}{\rho} \int_{-\infty}^{+\infty} m S(\vec{v}-\vec{u}) d \vec{v}\label{1.2.26}\]

This set of vector equations is known as the Euler-Lagrange equations of hydrodynamic flow and they are derived here in their most general form.

It is common to make some further assumptions concerning the flow to further simplify these flow equations. If we consider the common physical situation where there are many collisions in the gas, then there is a tendency to randomize the local velocity field \(\mathrm{\vec{v}}\) and thus to make \(\mathrm{f(\vec{v})=f(-\vec{v})}\). Under these conditions, the pressure tensor becomes diagonal, all elements are equal, and its divergence can be written as the gradient of some scalar which we call the pressure P. In addition, the creation rate S in equation \ref{1.2.26} which involves the effects of collisions will also become symmetric in velocity, which means that the entire integral over velocity space will vanish. This single assumption leads to the simpler and more familiar expression for hydrodynamic flow, namely \[\frac{\partial \vec{u}}{\partial t}+(\vec{u} \cdot \nabla) \vec{u}=-\nabla \Phi-\frac{\nabla P}{\rho}\label{1.2.27}\]

This assumption is necessary to close the moment analysis in that it provides a relationship between the pressure tensor and the scalar pressure P. From the definition of the pressure tensor, under the assumption of a nearly isotropic velocity field, P will be P(ρ) and an expression known as an equation of state will exist. It is this additional equation that will complete the closure of the hydrodynamic flow equations and will allow for solutions. It is also worth remembering that if the mean flow velocity \(\mathrm{\vec{u}}\) is very large compared to the velocities produced by collisions, then the above assumption is invalid, no scalar equation of state will exist, and the full-blown equations of hydrodynamic flow given by equation \ref{1.2.26} must be solved. In addition, a good deal of additional information about the system must be known so that a tensor equation of state can be found and the creation term can be evaluated.

It is worth making one further assumption regarding the flow equations. Consider the case where the flow is zero and the material is quiescent. The entire left-hand side of equation \ref{1.2.27} is now zero, and the assumption of an isotropic velocity field produced by random collisions holds exactly. The Euler-Lagrange equations of hydrodynamic flow now take the particularly simple form \[\nabla P=-\rho \nabla \Phi\label{1.2.28}\] which is known as the equation of hydrostatic equilibrium. This equation is usually cited as an expression of the conservation of linear momentum.Thus the zeroth moment of the Boltzmann transport equation results in the conservation of matter, whereas the first velocity moment yields equations which represent the conservation of linear momentum. You should not be surprised that the second velocity moment will produce an expression for the conservation of energy. So far we have considered moment analysis involving velocity space alone. Later we shall see how moments taken over some dimensions of physical space can produce the diffusion approximation so important to the transfer of photons. As you might expect, moments taken over all physical space should yield "conservation laws" which apply to an entire system. There is one such example worth considering.

Boltzmann Transport Equation and the Virial Theorem The Virial theorem of classical mechanics has a long and venerable history which begins with the early work of Joseph Lagrange and Karl Jacobi. However, the theorem takes its name from work of Rudolf Clausius in the early phases of what we now call thermodynamics. Its most general expression and its relation to both subjects can be nicely seen by obtaining the virial theorem from the Boltzmann transport equation. Let us start with the Euler-Lagrange equations of hydrodynamic flow, which already represent the first velocity moment of the transport equation. These are vector equations, and so we may obtain a scalar result by taking the scalar product of a position vector with the flow equations and integrating over all space which contains the system. This effectively produces a second moment, albeit with mixed moments, of the transport equation. In the 1960s, S. Chandrasekhar and collaborators developed an entire series of Virial-like equations by taking the vector outer (or tensor) product of a position vector with the Euler-Lagrange flow equations. This operation produced a series of tensor equations which they employed for the study of stellar structure. Expressions which Chandrasekhar termed "higher-order virial equations" were obtained by taking additional moments in the spatial coordinate r. However, the use of higher moments makes the relationship to the Virial theorem somewhat obscure.

The origin of the position vector is important only in the interpretation of some of the terms which will arise in the expression. Remembering that the left-hand side of equation \ref{1.2.27} is the total time derivative of the flow velocity \(\vec{\mathrm{u}}\), we see that this first spatial moment equation becomes \[\int_V \rho \vec{r} \cdot \frac{d \vec{u}}{d t} d V+\int_V \rho \vec{r} \cdot \nabla \Phi d V+\int_V \vec{r} \cdot \nabla P d V=0\label{1.2.29}\]

With some generality \[\int_V \rho \frac{d Q}{d t} d V=\frac{d}{d t} \int_V \rho Q d V\label{1.2.30}\]

Since \(\vec{\mathrm{u}}\) is just the time rate of change of position, we can rewrite \(\overrightarrow{\mathrm{r}} \bullet(\mathrm{~d} \overrightarrow{\mathrm{u}} / \mathrm{dt})\) so that the first integral of equation \ref{1.2.29} becomes \[\int_V \rho \vec{r} \cdot \frac{d \vec{u}}{d t} d V=\frac{1}{2} \frac{d^2}{d t^2} \int_V \rho r^2 d V-2 \mathbf{T}\label{1.2.31}\]

Here T is just the total kinetic energy due to the mass motions, as described by \(\mathrm{\vec{u}}\), of the system, and the integral can be interpreted as the moment of inertia about the center, or origin, of the coordinate frame which defines \(\vec{\mathrm{r}}\). The third integral in equation \ref{1.2.29} can be rewritten by using the product law of differentiation and the divergence theorem: \[\int_V \vec{r} \cdot \nabla P d V=\int_V \nabla \cdot(\vec{r} P) d V-\int_V p \nabla \cdot \vec{r} d V=\oint_s P_s \vec{r} \cdot \hat{n} d A-3 \int_V P d V\label{1.2.32}\]

It is also worth noting that \(\nabla \bullet \overrightarrow{\mathrm{r}}=3\). We usually take the volume enclosing the object to be sufficiently large that \(\boldsymbol{P}_{\mathbf{s}}=0\). If we now make use of the ideal gas law [which we derive in the next section along with the fact that the internal kinetic energy density of an ideal gas is \(\left.\varepsilon=\frac{3}{2} \rho \mathrm{kT} /\left(\mu \mathrm{m}_{\mathrm{h}}\right)\right]\), we can replace the pressure P in the last integral of equation \ref{1.2.32} with \((2 / 3) \varepsilon\). The integral then yields twice the total internal kinetic energy of the system, and our moment equation becomes \[\frac{1}{2} \frac{d^2 I}{d t^2}=2(\mathbf{T}+\mathbf{U})-\int_V \rho \vec{r} \cdot \nabla \Phi d V\label{1.2.33}\]

Here I is the moment of inertia about the origin of the coordinate system, and U is the total internal kinetic energy resulting from the random motion of the molecules of the gas. The last term on the right is known as the Virial of Clausius whence the theorem takes its name. The units of that term are force times distance, so it is also an energy-like term and can be expressed in terms of the total potential energy of the system. Indeed, if the force law governing the particles of the system behaves as \(1 / \mathrm{r}^2\), the Virial of Clausius is just the total potential energy⁵. This leads to an expression sometimes called Lagrange's identity which was first developed in full generality by Karl Jacobi and is also called the non-averaged form of the Virial theorem \[\frac{1}{2} \frac{d^2 I}{d t^2}=2(\mathbf{T}+\mathbf{U})+\boldsymbol{\Omega}\label{1.2.34}\]

If we consider a system in equilibrium or at least a long-term steady state, so that the time average of equation \ref{1.2.34} removes the accelerative changes of the moment of inertia (i.e., \(\left\langle\mathrm{d}^2 \mathrm{I} / \mathrm{dt}^2\right\rangle=0\)), then we get the more familiar form of the Virial theorem, namely, \[2\langle\mathbf{T}\rangle+2\langle\mathbf{U}\rangle+\langle\boldsymbol{\Omega}\rangle=0\label{1.2.35}\]

It is worth mentioning that the use of the Virial theorem in astronomy often replaces the time averages with ensemble averages over all available phase space. The theorem which permits this is known as the Ergodic theorem, and all of thermodynamics rests on it. Although such a replacement is legitimate for large systems consisting of many particles, such as a star, considerable care must be exercised in applying it to stellar or extra-galactic systems having only a few members. However, the Virial theorem itself has basically the form and origin of a conservation law, and when the conditions of the theorem's derivation hold, it must apply.