BPS Explained II: Calculating the Equilibration Rate in the Extreme Quantum Limit

BPS Explained II: Calculating the Equilibration Rate in the Extreme Quantum Limit

Abstract

This is the second in a series of two lectures on the technique of dimensional continuation, a new method for analytically calculating certain energy transport quantities in a weakly to moderately coupled plasma. Recently, this method was employed by Brown, Preston, and Singleton (BPS) to calculate the electron-ion temperature equilibration rate and the charged particle stopping power to leading and next-to-leading order in the plasma coupling. The basic idea is very simple. Concentrating upon the equilibration rate, the calculation consists of the following two steps: (i) perturbatively expand the rate in the form , with the dimensionless expansion parameter being defined by ; (ii) analytically calculate the coefficients and using the method of dimensional continuation. The factor of should be omitted from in nonrationalized electrostatic units. In the first lecture, I presented a basic overview of the requisite theoretical machinery of dimensional continuation imported from particle physics, but in a self-contained manner that assumed no familiarity with quantum field theory. In this lecture, I develop the framework further, and then explicitly calculate the electron-ion temperature equilibration rate in the high temperature limit. In this extreme quantum limit, the calculation of the coefficients and simplifies considerably, allowing us to concentrate on the physics of the method rather than the added complexity of the more general BPS calculation. This method captures all short and long distance physics to second order in , while three-body and higher correlations are contained in the cubic and higher order terms denoted by . In a weakly to moderately coupled plasma, where is small, the error term in this calculation is also small compared to the - and -terms, in which case the BPS methodology is quite accurate. Should higher order contributions be required, they can be calculated systematically, thereby improving the accuracy of the result in a controlled manner. To get a feel for the numbers, one finds at the center of the sun, where the plasma conditions are and . The coupling constant can be scaled to other plasma regimes through the proportionality relation . Of course the application of interest determines the relevant plasma regime, which may or may not lie within the domain of applicability of the BPS calculation. For example, the technique breaks down for warm dense matter where is not very small; however, this analytic perturbative technique is applicable for ignition in inertial confinement fusion and for other processes in hot a weakly coupled plasma.

1

I Introduction and Review

This is the second lecture on dimensional continuation, a new technique (1) recently used to calculate the charged particle stopping power and the temperature equilibration rate in a weakly to moderately coupled plasma (2). In Lecture I (3) of this series, I discussed the basic theoretical machinery of dimensional continuation, and I performed a model calculation of the equilibration rate. Reference (4) also contains a summary of the method in a very readable form. In this lecture, I will present the complete calculation of the electron-ion temperature equilibration rate in the extreme quantum limit, valid to leading and next-to-leading order in the number density (a more general calculation is performed in Section 12 of Ref. (2) to all orders in quantum mechanics, thereby providing an exact interpolation between the extreme classical and quantum limits). This calculation is near exact for a weakly coupled plasma, and it is quite accurate for a moderately coupled plasma. Before proceeding directly to the calculation, however, it might be useful to quickly review some of the more salient features of dimensional continuation discussed in Lecture I.

Under most circumstances, a plasma is not produced in thermal equilibrium; for example, when a laser ionizes a substance, it preferentially heats the electrons over the ions. However, since the electrons are so light, they rapidly come into thermal equilibrium among themselves with temperature ; some time later, the ions too will equilibrate among themselves to a common temperature . Finally, the electrons and ions will begin to equilibrate, and it is this process upon which we shall focus. Let denote the rate per unit volume at which the electron system at temperature exchanges energy with the ion system at temperature through Coulomb interactions (throughout these notes, I will always measure temperature in energy units). The electron-ion equilibration rate is proportional to the temperature difference between the electrons and ions, and can be expressed by

(1)

To restate the goal of this lecture more precisely, we shall calculate in the high temperature limit [where two-body scattering is accurately given by the Born approximation], and we will do so exactly to leading and next-to-leading order in the plasma coupling parameter [defined in Lecture I, or in Eq. (5) of this lecture]. Under these conditions, the result takes a particularly simple form (2):

(2)

where is the Euler constant, and are the electron Debye wave number and plasma frequency, and is sum of the squares of the ion plasma frequencies.2

In the form displayed by equation (2), the rate coefficient and the Coulomb logarithm do not explicitly depend upon one’s choice of electrostatic units, and one may calculate the Debye wave numbers and the plasma frequencies in any desired system. For dimensional continuation, however, it is more convenient to use rationalized electrostatic units, and I shall employ this choice from here out. An arbitrary plasma component will be labeled by an index , and is characterized by mass , charge , number density , and temperature . The index can span the electron and ion plasma components, that is to say, with being an arbitrary ion species. Working in three dimensions for now, the Coulomb potential between two charges and separated by a distance is , and in rationalized units, the Debye wave number and the plasma frequency of species take the form3

(3)
(4)

The square of the total Debye wave number is , and the total Debye wave length is .

Ii Calculating the Rate in Perturbation Theory

Reference (2), hereafter referred to as BPS, used a double pronged strategy to calculate the rate coefficient (2). First, a well chosen (5) dimensionless parameter was constructed from the relevant dimensionfull plasma quantities, thereby providing a parameter in which to perform a controlled perturbative expansion to leading and next-to-leading order in . The systematic error of the calculation was estimated by the cubic order term in the expansion, which is quite small for a weakly to moderately coupled plasma. While perturbative calculations are not very common in plasma physics, primarily because of the complexity of the systems of interest and the computational focus within the field, the validity of perturbation theory should nonetheless be clear for a “simple” system such as a weakly coupled and fully ionized plasma. The second part of the BPS argument deployed a powerful technique from quantum field theory allowing one to analytically calculate the coefficients in the -expansion.

ii.1 Perturbative Expansions in Weakly Coupled Plasmas

Let us first concentrate on the perturbative expansion. As demonstrated in Ref. (5), and discussed at length in Lecture I (3), for the case at hand the dimensionless plasma coupling parameter is defined by4

(5)

Note that is the ratio of the Coulomb potential energy of two point-charges, separated by the screening length , to the thermal energy of the plasma. Therefore, can be used to measure the strength of the plasma. To get a feel for the size of in a hot but not too dense plasma, one finds for a hydrogen plasma under the solar-like conditions and . One can scale to other density and temperature regimes by noting that . It was shown in Ref. (5) that plasma quantities always expand in integer powers of the coupling , and therefore is the appropriate parameter in which to perform a controlled perturbative analysis for weakly coupled plasmas.5 The -expansion allows for possible non-analytic terms, such as , and in particular, the electron-ion equilibration rate can be written

(6)

where I have indicated the leading order (LO) and the next-to-leading order (NLO) terms in the expansion. The minus sign on the leading order term of (6) is a matter of convention, and for small values of it renders the coefficient positive when the energy exchange is positive. Provided we can calculate the coefficients and , then (6) will be quite accurate in a weakly to moderately coupled plasma in which is small. Of course this perturbative approach breaks down for strongly coupled plasmas, those for which the value of is of order one or greater, since every term in the expansion becomes equally important in such cases. However, unlike a model or an uncontrolled calculation, the BPS calculation informs us of its domain of validity, and it provides an estimate of its own error through the size of .

ii.2 Calculating the Coefficients of the Expansion

We have now reduced the problem to finding the coefficients and of the rate (6). The coefficient was first obtained long ago by Spitzer (9) (and it can be estimated by dimensional analysis alone). The coefficient , however, was calculated only recently in Ref. (2), which employed a powerful technique from quantum field theory called dimensional regularization, or dimensional continuation as I will call it here. Since this technique is quite subtle and has proven to be somewhat controversial, I should emphasize that the method by which one chooses to calculate these coefficients is immaterial, except to the extent that it must contain enough physics to extract the next-to-leading order coefficient . Techniques other than dimensional continuation could well furnish one with the correct expressions for and , and perhaps in a simpler manner. However, the only relevant point here is that by hook or by crook we must analytically calculate these coefficients, and dimensional continuation is one method of doing this. 6

Before turning to the calculation of the coefficients, allow me to make a comment on the relation between the next-to-leading order -term and the Coulomb logarithm. Writing the leading order coefficient as , and defining the dimensionless coefficient , we can express the rate (6) in the form

(7)

Since the Coulomb logarithm means different things to different people,7 I would like to be quite specific in this lecture. By the words “Coulomb logarithm” I simply mean the term defined in (7), excluding the cubic and higher order terms. Hence, calculating the next-to-leading order coefficient is equivalent to determining the dimensionless coefficient inside the Coulomb logarithm. Finding dimensionless constants is usually a difficult problem, particularly since one cannot appeal to dimensional analysis for an estimate. It should therefore not be surprising that the coefficient varies over an order of magnitude or so across the various models within the literature.

Iii Calculating in Arbitrary Dimensions

Before proceeding directly to the calculation in Sec. IV, let us further develop the basic physics and mathematical machinery necessary to perform calculations in an arbitrary number of dimensions. The motivation for this section is, of course, a thorough exposition of the BPS methodology for calculating Coulomb energy-loss processes in a plasma. However, the material in this section is well known and applicable to a wide variety of other calculations, such as particle decay rates in high energy physics and analytic work in statistical mechanics. For the sake of completeness, however, and to establish some results that will be useful in Sec. IV, I will present a cursory but self-contained treatment here. If this material is familiar, then one may proceed directly to the calculation of the temperature equilibration rate in Sec. IV (given the background material in this section, the calculation itself is less than eight pages in length).

We shall start by developing the hyperspherical coordinate system in dimensions, which is a straightforward generalization of three dimensional spherical coordinates. To illustrate the utility of hyperspherical coordinates, I will calculate the hyperarea and hypervolume of several multidimensional objects by exploiting their spherical and cylindrical symmetries. These results will be used quite extensively in the next section. As a physical application, I then develop the multidimensional analog of the scattering cross section, which will allow us to consistently include short-distance quantum scattering effects in the -expansion (quantum effects manifest themselves through the -dependence of the coefficients in this expansion). Since we are interested in Coulomb energy exchange, we next examine electrostatics in arbitrary dimensions. From the multidimensional form of Gauss’ Law, we shall derive the -dimensional Coulomb potential , and we will see that it depends only upon in such a way as to emphasizes short distance physics when and long distance physics when . In , the short and long distance physics compete with equal strength, giving an infrared and an ultraviolet divergence, and this is what renders the temperature equilibration problem so difficult. To employ the extreme quantum limit, in which the Born approximation for the two-body scattering dominates, we must calculate the Fourier transform of the Coulomb potential in dimensions. Interestingly, we shall find that the Fourier transform of is given by the quite simple expression , the form of which does not depend upon the dimension of space, but only upon the length of the wavenumber . The fact that is so simple greatly facilitates calculations in the extreme quantum limit. With potential in hand, we shall then construct kinetic equations in dimensions. These equations are explicitly finite in all but dimensions, and I will explain the manner by which the BBGKY hierarchy reduces to the Boltzmann equation and the Lenard-Balescu equation (in and respectively).

iii.1 Kinematics and Hyperspherical Coordinates

Hyperspherical Coordinates

Kinematic quantities such as the -dimensional momentum or position vectors are elements of the same -dimensional Euclidean space . For definiteness, I will specialize to the case of position , with the understanding that this vector could also refer to momentum or wavenumber. We can decompose any vector in terms of a rectilinear orthonormal basis , so that , or in component notation . Each component is given by , and a change in the vector corresponds to a change in the rectilinear coordinate . Letting vary successively along the independent directions , we can trace out a small -dimensional hypercube with sides of length ; therefore, the rectilinear volume element is given by the simple form

(8)

In performing integrals over the kinematic variables, however, symmetry usually dictates the use of hyperspherical coordinates rather than rectilinear coordinates. I will therefore review the hyperspherical coordinate system in this subsection, deriving the measure for a -dimensional volume element in terms of hyperspherical coordinates. For our purposes, the primary utility of hyperspherical coordinates is that the volume element can be written as a product of certain conveniently chosen dimensionless angles, which I will collectively refer to as , and an overall dimensionfull radial factor , so that .

Starting with the usual 3-dimensional spherical coordinates of Fig. 1, let us recall why the three dimensional volume element takes the form (with and , and of course ; the coordinate singularities of the spherical system are not important here). As depicted in the figure, the three dimensional vector has length , and subtends a polar angle relative to the -axis, while its projection onto the - plane subtends an azimuthal angle relative to the -axis. The two angles and specify completely the direction of the unit vector . As we increase the polar angle by a small amount , the vector sweeps out an arc of length ; similarly, a change in the azimuthal angle will cause to sweep out a perpendicular arc (in the - plane) of length . Note that the factor of in arises from the projection of onto the - plane. We can make one more independent displacement by moving units in the radial direction, which results in a line of length . For small displacements in , , and , the vector sweeps out a small cubic volume element with sides of length , , and . The volume of this element is therefore .

Figure 1: Spherical coordinates of a point in three dimensional space: radial distance , polar angle , and azimuthal angle . The angles range over the values and .
Figure 2: Hyperspherical coordinates of a point in four dimensional space. As before, is the radial distance. The angles are defined as follows. (a) First, let be the angle between and the -axis. Let us now project onto the orthogonal three dimensional space, so that . The length of this projection is , and the projection itself is the same as projecting onto the three dimensional hyperplane . (b) The vector can be viewed as a three dimensional vector , which then defines the usual polar and azimuthal angles of Fig. 1, denoted here by and respectively.

Let us now consider the volume element in four dimensional space. Denote the coordinates of a vector by , that is to say, take . Since we cannot visualize four dimensional space,8 let us examine this problem in two steps, each of which can be visualized in either two or three dimensions. First, consider the plane that contains the -axis and the vector , and let be the angle between the -axis and the vector in this plane, as shown in Fig. 2a. We now project onto the hyperplane (a three dimensional slice of four-space), calling the projected vector . Since the three-plane lies perpendicular to each of the axes , , and , the vector lies in the three dimensional space shown in Fig. 2b, and its length is . Let the angle be the polar angle between the -axis and the vector , while is the usual azimuthal angle , as illustrated in Fig. 2b. As we vary the three angles and the radial coordinate, we sweep out a four-dimensional cube (or an approximate cube) with sides of length , , , and . This gives a four dimensional volume element

(9)

where for and . As a useful exercise, we can find the four dimensional hypervolume enclosed by a three-sphere of radius by integrating the volume element over the appropriate bounds,

(10)

The derivative of with respect to gives the hypersurface area of the three-sphere,

(11)

This is analogous to a three dimensional ball of radius and volume bounded by the two-sphere of area .

We can readily generalize this procedure to an arbitrary number of dimensions. Consider a point given by the rectilinear coordinates . Let be the angle between the vector and the -axis, in a manner similar to that of Figs. 1 and 2a. Note that is the arc length swept out by as the angle is incremented by . Let us now project onto the hyperplane , the -plane normal to the -axis and passing through the origin, calling this projection : that is to say, let . The length of this vector is . Let us proceed to the next step and define the angle as the angle between the -axis and , in which case, as the angle is varied by , the vector sweeps out an arc of length . In a similar fashion, project onto the -plane, that is, the plane described by and . This projection is given by , and the length of the projection is .9 For the general iteration, let be the angle between the -axis and , so that . In summary, we define the quantities

(12)
(13)
(14)
(15)

where we have used the fact that . The last two are lines provide the projection for the step. This gives the -dimensional volume element

(16)

For notational convenience, I will write the angular measure in (16) as , so that

(17)

As we proved in the Lecture I, the integration over all angles gives the total solid angle

(18)

and Table 1 illustrates the numerical values of this solid angle over a wide range of dimensions. Note that reaches a maximum around and then slowly decreases.

   2  3  4  5  6 7 8 20
               
 value  6.28  12.6  19.7  26.3  31.0  33.1  32.5 0.516
Table 1: Solid angle as a function of dimension .

As a matter of completeness, let us prove (18) here. First, consider the one-dimensional Gaussian integral

(19)

If we multiply both sides together times (with ), we find

(20)

where the vector in the exponential of the last expression is the -dimensional vector , and  . As in (17), we can factor the angular integrals out of the right-hand-side of (20), and the remaining one-dimensional integral can be converted to a Gamma function with the change of variables  :

(21)

Solving for in (21) gives (18).

In calculating the temperature equilibration rate and the charged particle stopping power, we encounter integrals of the form

(22)
(23)

respectively, with . The exact forms of and do not concern us here, except that their angular dependence is determined by the following considerations: the integral (22) is spherically symmetric since the energy exchange between plasma species is isotropic, while in the latter integral (23), the motion of the charged particle defines a preferred direction around which one must integrate. The integrals and can be viewed as functions defined on the positive integers, and as discussed at length in Lecture I (3), Carlson’s Theorem (8) ensures that there is a unique analytic continuation onto the complex plane. As our first application in this section, let us see how the expressions (22) and (23) provide a means by which to easily and conveniently perform this analytic continuation to complex values of the spatial dimension , thereby rendering truly arbitrary. First, the solid angles and are composed of a simple exponential factor and a Gamma function, whose analytic continuations have been well studied. As for the integrals, simply treat as a complex parameter, performing the one dimensional integral (22) and the double integral (23) in the usual manner of ordinary calculus. This provides functions and of a complex argument , in fulfillment of Carlson’s Theorem. Double integrals of the form (23) were used extensively in Ref. (2) to calculate the stopping power, where the angle is determined by the direction of motion of the charged particle. Calculating the temperature equilibration rate, on the other hand, requires only the simpler one dimensional integral (22), as the energy exchange in this process is isotropic.

The Hypervolume of Spheres, Disks, and Cylinders

Figure 3: A -dimensional sphere of radius bounds the -dimensional ball of radius . By integrating over successive shells of area, we can find the volume by ; or conversely .

We shall now calculate the hypervolume of several useful geometric objects. Let us first consider a -dimensional ball of radius , defined by the set of points for which . We will denote this object by , and in two and three dimensions this is a disk and a spherical, both volume centered at the origin. We can find the -dimensional hypervolume of the ball by simply integrating (16) over all permissible values of the coordinates. It should cause no confusion to denote the hypervolume of the region by the same symbol, and using (18) we find

(24)

The boundary of is a -dimensional sphere defined by , or . By differentiating (24) with respect to the radius , we can also find the hyperarea of a -dimensional sphere of radius in ,

(25)

For brevity, I have denoted the hyperarea by the same symbol as the sphere itself, which is simply the -dimensional boundary of the region . This is illustrated in Fig. 3. The distinction I am making between “hypervolume” and “hyperarea” is somewhat arbitrary, since these are both terms involving regions in a higher dimensional space. When I wish to talk about a -dimensional subregion of the hyperspace , such as , I will use the term hypervolume. On the other hand, when I wish to emphasize a boundary region of a hypervolume, such as , I will use the term “hyperarea.” Regarding the usage of the term “solid angle,” suppose we keep the radius fixed but vary the angles over ranges . The region swept out by this procedure lies on the -dimensional sphere with a hyperarea . We are therefore justified in calling the solid angle in dimensions.

Figure 4: The hyperarea of a hypercylinder of length and radius is , and the hypervolume bounded by the cylinder is .

Finally, let us discuss the -dimensional cylindrical of radius and length . Again, it is easiest to argue from analogy in three dimensions. To form a two-cylinder in , we let a two dimensional disk sweep out a volume as it moves a distance in the orthogonal direction, which is illustrated in Fig. 4a. Similarly, a corresponding -dimensional cylinder is formed by letting a -dimensional ball sweep out a distance along the orthogonal axis, as illustrated in Fig. 4b. Therefore, the hyperarea of the -dimensional cylinder is

(26)

The -dimensional hypervolume enclosed by this cylinder is

(27)

iii.2 The Cross Section

Figure 5: Definition of the cross section in a general number of dimensions. The incident flux of species is the rate of particles per unit hyperarea normal to the flow. The units of are , where L and T denote the units of space and time. By definition, the differential cross section is related to the rate , each at angular position , by . The cross section per unit solid angle about the direction is denoted by . Except for the specification of , this definition does not depend upon the dimensionality of space, and the units of are .

As a physical application of hyperspherical coordinates, let us calculate the form of the classical “cross section” in -dimensions. For simplicity we will consider a projectile striking a fixed target, although we can perform a similar analysis in the center-of-mass frame of the two particles. Such a scattering experiment is illustrated in Fig. 5, in which a beam of incident particles, denoted by the label , is fired at a target with incident flux . The rate at which the scattered -particles enter a given solid angle about the direction is then measured. The flux is a characterization of the rate at which particles move along the beam axis. In dimensions, the spatial region normal to the axis is a -dimensional hyperplane, and the flux is the number of particles per second per unit hyperarea passing through this plane. For example, if the beam direction is , then the number of particles in a time interval passing through a hyperarea normal to is given by . The engineering units of are therefore . In analogy with the usual cross section in three dimensions, we define through

(28)

and therefore has engineering units of .

Suppose the scattering center is a central force, such as the -dimensional Coulomb potential. The particle is confined to a two-dimensional plane for central potential motion, and this holds true even in dimensions. Let denote the impact parameter of projectile. As the particle traverses its plane of motion, its position is uniquely characterized by a function , where is the angle between the beam direction and the projectile (with the scattering center defining the origin). From Fig. 4, the number of particles per unit time passing through the hyperannulus of width and radius is is , and by particle number conservation, the same number of scattered particles reaches the hyperannulus at . The cross section in a -dimensional central potential is therefore given by

(29)

This is Eq. (8.31) of Ref. (2), the starting point for the classical calculation. The cross section will appear in the Boltzmann equation. To include two-body quantum scattering effects, we replace the classical cross section by the quantum cross section:

(30)

where is the quantum scattering amplitude. In the calculations that follow, we shall use work in the extreme quantum limit where the Born approximation for the amplitude can be employed.

iii.3 The Coulomb Potential in Arbitrary Dimensions

Now that we have discussed the cross section in an arbitrary central potential, let us concentrate on the special case of the Coulomb potential. The physics of dimensional continuation is contained in the -dependence of the Coulomb potential in -dimensional space, which ensures that short distance physics is emphasized in and long distance physics in . Changing the spatial dimension about therefore acts as a “physics sieve.” Let us first construct the electric field of a point charge in dimensions. Maxwell’s equations are easily generalized to an arbitrary number of dimensions, and in particular, we can write

(31)

where is the electric field vector and is the -dimensional spatial gradient. The charge density has engineering units of charge divided length to the power, which I will write as . In integral form, the equation can be written

(32)

where is the total electric charge contained in the hypervolume . Note that the dimensionality of space is now explicitly indicated by the integration measure. We can employ the usual symmetry argument to find the electric field of a point source at the origin. Let be the -dimensional ball of radius centered on the point charge , and denote the -dimensional hyperspherical boundary by . By symmetry, the field points radially outward with a magnitude along the direction normal to . The length depends only upon the radial distance and not upon its angular location along . The divergence theorem holds in an arbitrary number of dimensions, and since the hyperarea of is given by (25), we find:

(33)

The electric field is therefore given by

(34)

where we are using the notation , with being a unit vector pointing in the direction of .

Figure 6: Short-distance ultraviolet (UV) physics dominates in dimensions . Long-distance or infrared (IR) physics dominates when . UV and IR physics are equally important in .

I find it more convenient to work with the electric potential, a scalar quantity defined by . In fact, I will work with the potential energy , so that

(35)

where I have appended a subscript to the potential energy to remind us that we are working in dimensions. For two charges and separated by a distance , one only need replace by the product . For , the geometric factor in (35) becomes , which is the origin of the of rationalized units. Figure 6 shows the Coulomb potential for , along with two representative dimensions on either side of . As the figure illustrates, the short distance behavior of the Coulomb potential becomes more pronounced in higher dimensions, while long-distances are emphasized in lower dimensions. For aesthetic reasons, the arbitrary integration constant for the potential energy has been adjusted in each case so that all three graphs intersect at a single point. This figure illustrates quite dramatically that by simply dialing the dimension , we can dial a potential that filters either long-distance or short-distance physics.

In the Born approximation to quantum Coulomb scattering, which we shall employ shortly, we need the Fourier transform of the Coulomb potential (35). Unlike the spatial representation , the Fourier representation of the -dimensional Coulomb potential takes the same form in any dimension, namely,

(36)

where is just the square of the norm of the -dimensional wave number  , and I am using the conventions

(37)
(38)

With these conventions, the amplitude in the Born approximation in any dimension is given by

(39)

where is the momentum transfer during the collision. This is a function only of the square of its argument . In particular, the Born approximation does not introduce dependence upon the center-of-momentum energy , and this is what renders its use so convenient.

Expression (36) for the Fourier transform of the potential (35) can be established in a number of ways, perhaps the easiest being an straightforward application of Laplace’s equation,

(40)

Upon inserting (37) for into (40) and using the integral representation of the delta-function, we can write Laplace’s equation in the form

(41)

and solving for provides (36). It might also be informative to prove (36) using the more direct approach of performing the Fourier transform directly. Substituting the Coulomb potential (35) into (38), and then using (23) to rewrite the -dimensional integral, we find

(42)
(43)

where we have made the change of variables . It is convenient to keep the exponential terms in square brackets rather than converting their sum into a cosine term. We will perform the -integration by deforming the contour slightly off the real axis,

(44)
(45)

Upon substituting this back into (43) and changing variables to we can write

(46)

where the second term in the integrand introduces the pole into physical quantities, and the -integral takes the form of the Euler Beta function

(47)

with and . Using gives (36). In Section IV.2 we will need yet another representation of the Beta function, which I record here for convenience:

(48)

iii.4 Kinetic Equations in Arbitrary Dimensions

Distribution Functions

A particle in a space of arbitrary dimension is fully characterized by its position and momentum and , which have rectilinear coordinates and for . I will often denote the square and the magnitude of the momentum by and , respectively. For example, is shorthand for . A swarm of particles distributed over position and momentum values is characterized by a distribution function defined by

number of particles in a hypervolume about (49)

The factor of in the denominator is a conventional normalization factor, and for a spatially uniform distribution this gives the normalization

(50)

where is the number density of -type particles. That is to say, is the number of particles of species in a hypervolume , and the engineering units of are therefore . From (50), we see that a normalized Maxwell-Boltzmann distribution at temperature and number density of is given by

(51)

where is the kinetic energy and is the inverse temperature in energy units. The thermal wave length for species is defined by