Classical relativistic statistical mechanics : The case of a hot dilute plasma

Starting from predictive relativistic mechanics we develop a classical relativistic statistical mechanics. For a system of N particles, the basic distribution function depends, in addition to the 6 N coordinates and velocities, on N times, instead of a single one as in the usual statistical mechanics. This generalized distribution function obeys N (instead of 1) continuity equations, which give rise to N Liouville equations in the case of a dilute plasma (i.e., to lowest, nonzero order in the charges). Hence, the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy for the reduced generalized distribution functions is derived. A relativistic Vlasov equation is obtained in this way. Thermal equilibrium is then considered for a dilute plasma. The calculation is explicitly worked out for a weakly relativistic plasma, up to order 1/c', and known results are recovered.

As an alternative to the manifestly covariant formalism of PRM, we can obtain from the $' the three-accelerations p, , of the system through the relations p'=r '((*.'-, ('.vJ,   ( t)   where v, is the three-velocity of the particle g and y, =(1 -v, ') '~'.I.et c, the speed of light, be equal to 1.In this way we obtain p, , ' as a function of the four-positions x, =(t"x,) and four-veloci- ties u, .Then we can restrict ourselves to simul- taneous configurations, t, =t, Vb, in order to ob- tain the physical three-accel.erations "seen" by a given inertial observer: p, ', ' ~, "Vt~.The canon- ical formulation of this three-dimensional formal- ism of PHM was given first by Hill.' In Ref. 7 the Throughout this paper the Einstein summation con- vention for Greek labels is assumed.
Choosing the signature -2 we have for the four- vector u, where q~i s the Minkowski metric tensor. [In the definition (4) the derivatives 8/su, must be under stood as if the identity (5) did not exist, that is, as if the four components of u', were independent.] Then 2181 three-accelerations, p, ', ~, "are supposed to ex- ist as power series in the coupling constants and then a canonical formalism is proved to exist within this perturbative framework satisfying "good" asymptotic conditions (good meaning that for infinite particle separation in the pastalter- natively in the futurethe canonical coordinates recover their standard free-particle expressions).
The role of good asymptotic conditions was originally brought out by Kerner and Hill.' Accord- ing to the "noninteraction theorems" ' the canon- ical coordinates q, cannot be the three-positions x .
PRM is consistent with the classical theory of fields at least in the perturbative framework."" For example, in the case of electromagnetism, given the retarded Lienard-Wiechert potentials (alternatively advanced or time symmetric) we have a unique four-acceleration ], which reduces to that obtained from these potentials for isotpopic configurations: q z(x, -x~)(x~-x") =0, Vb, a with b4a, Now consider a relativistic macroscopic system of N classical interacting particles, perhaps with an external field in the framework of the PRM.Since, under the conditions stated before, we have a canonical formalism corresponding to the three- accelerations p', ~, " that is, systems of canoni- cal coordinates (q,, pJ and the corresponding Hamiltonian H, we can construct the statistical mechanics of the macroscopic system along the same lines as that in the nonrelativistic case.In the important case of equilibrium we can write for the partition function Z ~P~I Z=--t. )t. -"...dq. d p. , (8) g= 1 where as usual P =1jkT with T the temperature and k tbe Boltzmann constant.Formula (8) sup- poses that the intrinsic angular momentum of the macroscopic system is zero and that we work in a frame relative to which the system is macro- scopically at rest, i.e. , a frame where the total momentum of the system is zero.As long as the system is isolated this is the correct definition of the rest frame, since in PRM the description of the interaction between particles can be made without the introduction of the field as something independent of the degrees of freedom of the particles.Now, as can be seen in Ref. 11, where the case without an external field is treated, q"p, and H are complicated functions of x~, v~e ven at first order in the coupling constants.This is true for both short-and long-range scalar and vector interactions and also for the interesting special case of the electromagnetic interactions.Then the calculation of Z becomes involved.Nevertheless, in the case of a dilute completely ionized plasma we guess that a remarkable simplification is pos- sible, as we will see in Sec.III.
In Sec.II we develop a general approach to classical relativistic statistical mechanics to treat both the equilibrium and nonequilibrium cases, which allows for practical calculations.We begin defining the generalized distribution function of Ã particles and then we obtain N continuity equations for this distribution function.
In Sec.III the case of a dilute relativistic plasma is considered.We work in the space of the three- positions x, and three-velocities u, =y,v, and ob- tain N Liouville equations from the N continuity equations although tbe (x,, u) are not canonical coordinates.Then we give the standard distribution function relative to the coordinates (x,, uJ for a dilute plasma in equilibrium in a frame where the system is at rest, and finally we give the relativistic extension of the Bogoliubov-Born- Green-Kirkwood-Yvon (BBGKY}hierarchy.
In Sec.IV the relativistic BBGKY hierarchy is cut off in a standard way and some general results are established for the resulting solutions.Finally, in Sec.V we calculate the two-body distri- bution function of a slightly relativistic plasma in order to get a simple test for the theory.

II. THE GENERALIZED DISTRIBUTION FUNCTION AND THE CONTINUITY EQUATIONS
We now consider a relativistic macroscopic system of N classical particles which are inter- acting among themselves, with or without an external field, in the framework of the PRM.Let us define its "generalized distribution function" P(t"x"u"t"x"u". . . ) = F(t,, x,, u) (u, is t-he three-vector consisting of the space components of u, '} as the probability density of finding particle 1 at xl with velocity u, at time t» particle 2 at x, with velocity u, at time t» and so on.That is, E 6P(t,, x,, u) = E(t,,x"u), d'x, d'u, (9) &=1 is the elementary probability of finding every particle z in the corresponding elementary volume d'x, d'u, at time t, .
Because of the Newtonian character of the equa- tions of motion (1) and because of the equations (3) which allow for the existence of dynamical trajectories such as (2), we have a deterministic dynamical problem with a finite number of initial- value data: (t,, x,, u), a=1, . . ., ¹ Then 5P(t,, x,, u) does not change by changing the argu- ments (t"x"u) to new arguments representing the same dynamical trajectories, or correlatively, 5P depends only on which ensemble of dynamical trajectories is considered.That is, These N conservation equations for 5I' are actually compatible since Eqs.(3), satisfied by the four- accelerations, guarantee that the integrability con- ditions of (10) are satisfied.
From the N conservation equations (10) we ob- tain the N continuity equations Bu, }, dt, that is, the latter part of the alphabet, R, S, . . ., take the values s+1, . . ., N. The integral is extended to all positions x~a nd to all velocities u~.Let us set g =R in Eq. ( 12) and integrate over d'x&d'u&, the integral being extended to all values of x~, u~.After applying Gauss's theorem we ob- tain in an obvious notation that is, for any reasonable asymptotic conditions for the macroscopic system f(t, x,, u) =F (t,=t, x,, u) .
(13) By adding the N continuity equations (12) and setting t, =t, Vb, we obtain the standard continuity equation for the standard distribution function If the macroscopic system is in equilibrium then F must be invariant by time translations and f does not depend on time.If besides being in equi- librium the system is homogeneous, then F must be also invariant by space translations.That is, in this case, F will depend on the x, only through the relative positions x, -x, .Of course if there is a container we will have some sort of inhomo- geneity near the walls.
From the generalized distribution function, F(t,, x, u), we obtain the reduced generalized distribution function of order s &N, N d'"(t, g") = I E xd'x"d'u".

R=s+1 (15)
Here and in the following capital Latin letters from the first part of the alphabet, A, B, . . ., take the values 1, ... , s and capital Latin letters from where $, is the three-vector consisting of the space components of $", .
[Note that at variance with Eq, (10}, where the four components of u', are considered as independent, in Eq. ( 11) the identity ( 5) is taken into account.That is, the three components of u, are considered as inde- pendent while ug is given by ug =(1 + u, 3)'~3.] Thus, instead of having a unique continuity equa- tion for the standard distribution function f(t, x"u) depending on 6N+1 variables, we have N contin- uity equations for the generalized distribution function j(t,, x,, u) depending on 7N variables.
When we put t, =t, yg in the generalized distri- bution function F(t"x,, u,}, we obtain the standard distribution function and then BF (s)  =0 Bt in agreement with the notation in the left-hand side of ( 15), where the labels ts were omitted.Here e"e, are the charges of particles g and 0, ~, is the mass of particle g& and 8" is the func- tion Rgg-[(Xggub) -Xgg'] ' ~(21) For x"we ha e x"=x, -x, , and (u, u,}, (x"u,), (x"u,}, and x"' mean four-scalar products. When no bound states are present, formal expansions in the charges, which underly the method to obtain the Eqs.( 19) and (20), must be interpreted as expansions in the dimensionless parameter e, e, /mcgh, where m is m, or m, and I3 stands for the typical impact parameter of the collisions.
Let us consider a system of N interacting charged particles without any external field.(Later we will relax somewhat this condition when we consider a system in equilibrium, in which case a container may be necessary, but we may suppose that the container produces a surface effect which will be negligible for particles not too near the walls.) Then, to first order in the products of the charges, the physical solution to Eq. (3) gives" for the four- acceleration $, of the charge g Then, in order to assure the fast convergence of the expansions, we must have relat, ively high impact parameters, that is, we must have dilute enough systems.
For a dilute completely ionized plasma without external field, it is easy to verify that (22) with, $, given by Eqs. ( 19) and ( 20). [Note that Eq. (22} is true even in the presence of an external field.] Hence, in this plasma, the N continuity equations ( 12) can be written as the N Liouville equations BF ~» 8F -+v-+r '$. for the standard distribution function f(t, x,, u) on the space of the positions x, and velocities u, .
As we have mentioned in the Introduction, we have a canonical formalism for the dynamical system described by the instantaneous three-accelerations p, ', ~, , so that we can write directly the Liousville equation ( 25) sf ~dq, sf ~dp, &f (26} where (q,, p) are canonical coordinates and the standard distribution function, g=f(t, q"p,) is de- fined on phase space.
The standard distribution functions f and f are connected by the relation according to Eq. ( 4).
A generalized distribution function such as the one defined here, E(t,, x,, u), has been considered pre- viously by Hakim' and van Kampen." Both auth- ors give N Liouville equations for this distribution function but they limit themselves to the case when the particles are free or only a prescribed external field is present.%hen the interaction among the particles is taken into account we need the framework of PRM in order to assure the integrability conditions of the N Liouville or the N con- tinuity equations.In Ref. 15 the important result establishing the Lorentz invariance of the generalized distribution is derived.Starting from Eq. ( 14) and again taking into account Eq. ( 22), we obtain the Liouville equation ( 27) where D(x,, m, u,)/D(q, , p,) is the Jacobian deter- minant for the coordinate transformation (x,m u) -(q, p).Note that the determinant is an integral of the motion since in a dilute plasma the continui- ty equation ( 14) becomes the Liouville equation (25) in the (x,, u) space.Indeed, our guess is that the value of the determinant is const & exp [(P -P')H], where P = I/kT and P' is a con- stant.
[Provided that we have a Liouville equation for the distribution function f(t, x,, u,), the same considerations which are used in Newtonian statistical mechanics (see e.g. , Ref. 16 for the New- tonian case) lead here to f = const && exp(-P'H) for an equilibrium system.On the other hand for canonical coordinates we have f = const x exp(-PH) and therefore, from Eq. ( 27), it follows the ex- pression constx exp(P -P'}H for the Jacobian de- terminant.Obviously in the limit of the density going to zero P' goes to P. In this case the de- terminant takes the value 1. ] Then we have for the partition function of a dilute plasma in equilibri- um Z = const x Jt exp(-P'H) 8F (a+1)   + ~d'x~d'u~= 0, B gA 29 where Eq. ( 19) has been taken into account.This hierarchy of equations corresponds to the BBGKY hierarchy for the standard reduced distribution functions of the nonrelativistic case." Here, for each value of s we have not one but s equations corresponding to the different values of the label A: 1, . . ., s.Furthermore, when one writes the nonrelativistic BBQKY hierarchy for interactions which depend on the particle velocities, it can be seen that each equation of the hierarchy, corresponding to each value of s, involves three re- duced standard distribution functions, f"', f'~", and f'"", instead of only the first two functions, as is the case when the accelerations do not de- pend on the velocities.We see in (32} that the where it is assumed that the total momentum and the intrinsic angular momentum are zero.Com- pared with Eq. ( 8) which gives Z in the general case, this expression for Z could represent an important progress as long as the actual calcula- tion of Z, for the dilute plasma, is concerned.
Let us now consider the definition (15) for p'»(t ",x", u").By integration over xa, us, we ob- tain from ( 24 "relativistic hierarchy" involves only two reduced generalized distribution functions in spite of the fact that relativistic accelerations do depend on velocitie s. As in the nonrelativistic case the determination of the reduced generalized distribution function E"' can only be made when the hierarchy is cut off somewhere, that is, when for some value s we give E""' as a function of the other E'"' functions with x& s+1.Let us remark that we have here restrictions on function E'"" that we do not have in the nonrelativistic case, since now s(s -1)/2 conditions of integrability, coming from (29), must be satisfied.To first order in the products of the charges these integrability conditions are on the four-accelerations, the integrability condi- tions for Eq. ( 24) are satisfied.
IV. APPROXIMATE SOLUTIONS FOR THE RELATIVISTIC BBGKY HIERARCHY In a completely ionized plasma, which is dilute enough, because of the long-range character of the interaction, the term ps)"sBF")/Bu" in Eq. ( 29) is small compared with the large collective effect represented by the integral term.Of course, this is only true as long as s«N.Then, accord-  ing to what is done in the nonrelativistic case" we set, whatever particles 1,2, 3 are, 8 [, BF"")) Of course, the integrability conditions for the whole hierarchy (29) are satisfied as long as we do not introduce any approximation.
The integrability condition of Eq. (34} is which is a nonlinear integrodifferential equation involving only E"'.
Let us consider the case of a homogeneous plas- ma in equilibrium.
One can be convinced of.theexistence of solutions of Eqs. ( 38) and (39) by working out the low-veloci- ty approximation.
To first order in 1/c we obtain from Eq. ( 20 Hence, taking into account Eq. ( 37), we obtain that the Debye-Hiickel solution (see, for example, Ref. 16) for a two-component plasma c(t, -tb) =xb -x~=x~b has been used.
In fact it can be seen that G(1, 2) given by Eqs.(41} is the only physical solution for Eqs. ( 38) and (39), when tQe interaction is given by (40).So to order 1/c, although the acceleration Eq. ( 40) does depend on f", the function G(1, 2) given in Eq. ( 41) does not depend on t12 That is, the relativistic corrections in G(1, 2) begin with 1/c terms, which incorporate the dependence on t12.

V. THE CASE OF A DILUTE SLIGHTLY RELATIVISITIC PLASMA IN EQUILIBRIUM
As a test for the theory we calculate in this section the standard two-particle distribution func- tion, f"'(x",u"u, ) = E"'(x, , u"u, ), , ", in the par- ticular case of a dilute, slightly relativistic, homo- geneous plasma in equilibrium, where only two different kinds of charges, e and -e, are present.
The plasma is supposed to be completely ionized.It will be seen that in the appropriate limit previously known results are recovered.
Let us multiply Eq. ( 39) by y"' and then sum the two equations that we obtain setting alternatively d4 =1, 2. When the system is in equilibrium, G(1, 2) depends on t" t2 only through its difference t».