Henry E. Kandrup

Dynamical friction in a mean field approximation

An exact formula is derived for the average frictional force acting upon a ‘test’ star which moves along a prescribed trajectory amongst a collection of ‘field’ stars which are characterized by a Maxwellian distribution of velocities. In the limit that the actual stellar trajectories may be approximated by their average forms, as determined by the mean gravitational field, one obtains a relatively simple expression which establishes an important connection with the fluctuation-dissipation theorem. For the case of an infinite, homogeneous system, one recovers Chandrasekhars classical result. Alternatively, by allowing for the possibility of nearly periodic motion, one is led to new and intriguing phenomena.

Nonequilibrium statistical mechanics in the general theory of relativity. I: A general formalism

Abstract This is the first in a series of papers, the overall objective of which is the formulation of a new covariant approach to nonequilibrium statistical mechanics in classical general relativity. The object here is the development of a tractable theory for self-gravitating systems. It is argued that the “state” of an N-particle system may be characterized by an N-particle distribution function, defined in an 8N-dimensional phase space, which satisfies a collection of N conservation equations. by mapping the true physics onto a fictitious “background” spacetime, which may be chosen to satisfy some “average” field equations, one then obtains a useful covariant notion of “evolution” in response to a fluctuating “gravitational force.” For many cases of practical interest, one may suppose (i) that these fluctuating forces satisfy linear field equations and (ii) that they may be modeled by a direct interaction. In this case, one can use a relativistic projection operator formalism to derive exact closed equations for the evolution of such objects as an appropriately defined reduced one-particle distribution function. By capturing, in a natural way, the notion of a dilute gas, or impulse, approximation, one is then led to a comparatively simple equation for the one-particle distribution. If, furthermore, one treats the effects of the fluctuating forces as “localized” in space and time, one obtains a tractable kinetic equation which reduces, in the newtonian limit, to the standard Landau equation.

Non-equilibrium statistical mechanics in the general theory of relativity II. Linear fields in a kinetic approximation

The first paper in this series introduced a new, manifestly covariant approach to non-equilibrium statistical mechanics in classical general relativity. The object of this second paper is to apply that formalism to the evolution of a collection of particles that interact via linear fields in a fixed curved background spacetime. Given the viewpoint adopted here, the fundamental objects of the theory are a many-particle distribution function, which lives in a many-particle phase space, and a many-particle conservation equation which this distribution satisfies. By viewing a composite N-particle system as interacting one- and (N − 1)-particle subsystems, one can derive exact coupled equations for appropriately defined reduced one- and (N − 1)-particle distribution functions. Alternatively, by treating all the particles on an identical footing, one can extract an exact closed equation involving only the one-particle distribution. The implementation of plausible assumptions, which constitute straightforward generalizations of standard non-relativistic “kinetic approximations”, then permits the formulation of an approximate kinetic equation for the one-particle distribution function. In the obvious non-relativistic limit, one recovers the well-known Vlasov-Landau equation. The explicit form for the relativistic expression is obtained for three concrete examples, namely, interactions via an electromagnetic field, a massive scalar field, and a symmetric second rank tensor field. For a large class of interactions, of which these three examples are representative, the kinetic equation will admit a relativistic Maxwellian distribution as an exact stationary solution; and, for these interactions, an H-theorem may be proved.

Statistical mechanics of the gravitational field in a conformally static setting

This paper formulates a statistical description of a collection of N identical classical particles that interact relativistically via linear fields in a fixed background space‐time that admits a conformal timelike Killing field. Attention focuses upon the special cases of a simple scalar interaction and a linearized gravitation interaction which should suffice to model many systems of astrophysical interest. The fundamental object of the theory is a complicated distribution function that depends upon appropriate variables for both the particles and the fields. By assuming that, in a first approximation, this distribution factorizes into an infinite product of reduced distribution functions, one recovers the type of mean‐field theory developed by such authors as Ipser and Thorne. Alternatively, one may derive various exact and approximate relations which contain information about the interparticle correlations.

A covariant kinetic equation for a self-gravitating system

Recently, Israel and Kandrup have developed a new, manifestly covariant approach to non-equilibrium statistical mechanics in classical general relativity. One by-product of that approach has been the formulation of an approximate kinetic equation for the evolution of a self-gravitating system, valid in an “impulse” or “weak coupling” approximation in the limit that radiative effects may be neglected. This paper exploits the theory of random functions to present a much simpler derivation of that approximate kinetic equation.

Non-equilibrium statistical mechanics in the general theory of relativity III. Collisional stellar dynamics☆

Abstract This paper assembles and extends earlier results to formulate a coherent theory of relativistic stellar dynamics appropriate for comparatively small systems of stars in which relativistic effects can be important. The structure of the Newtonian theory is outlined, culminating in the “collisional Boltzmann” or Fokker-Planck equation appropriate for an unconfined system of point masses. The theory of relativistic Fokker-Planck equations is then developed for general Lorentz-covariant interactions such as electromagnetism or scalar fields. The basic physical ingredients of Newtonian stellar dynamics are identified, and it is indicated how they can be reformulated relativistically. These considerations are then used to construct a relativistic Fokker-Planck equation appropriate for the evolution of a collection of point mass stars. The analysis is then generalized to allow, both Newtonianly and relativistically, for the effects of direct physical collisions between stars of finite size. By way of conclusion and illustration, the theory is applied to the study of a prototypical dense galactic nucleus which could evolve to contain a massive black hole. The paper ends by enumerating a number of tractable unsolved problems deserving of further consideration.

Should a self-gravitating system relax towards an isothermal distribution?

Physical effects ordinarily neglected suggest that, even ignoring three-and higher-body ‘collisions’, a self-gravitating system of stars, such as a globular cluster, does not necessarily ‘want’ to relax completely towards an isothermal distribution. Even if one neglects evaporation and the gravothermal instability, one might anticipate deviations from a Maxwellian distribution of velocities manifest on a time scaletS∼(logN)tR, wheretR is the ordinary binary relaxation time andN is the number of stars.

Direct physical collisions and the evolution of dense stellar systems

This paper formulates kinetic equations to describe the destruction of stars via direct physical collisions, assumed characterized by a constant geometric cross section, which are valid for spherically-symmetric systems both in Newtonian gravity and general relativity. An ‘orbit averaging’ prescription is then used to formulate simpler equations for the evolution of an ‘average’ distribution function involving only the specific energyE, angular momentumJ, and time. If one supposes that, in the absence of physical collisions, the distribution would be stationary, it is easy to identify a collision time-scaletC associated with the stationary distribution that depends only uponE andJ. These relations assume especially simple forms in the limit that all the stars have vanishing angular momentum and follow radial trajectories: the analysis of physical collisions then reduces effectively to a one-dimensional problem. As a concrete example, one may suppose in this case that the stationary distribution corresponds to a gaussian distribution of radial velocities. The time-scaletC(E) then reduces to a single quadrature that involves the radial distribution of stars, whose asymptotic forms corroborate ones naive expectations.

Collisional Stellar Dynamics in General Relativity: An Overview

Recently, Israel and Kandrup (1984; Kandrup 1984 a,b,c,d) have formulated a new, manifestly covariant approach to non-equilibrium statistical mechanics in classical general relativity. The object here is to indicate how that formalism may be used to construct a theory of ‘collisional’ stellar dynamics, valid for a collection of point mass stars in the limit that incoherent radiative effects may be neglected.

Galaxy clustering in two dimensions: Some analytic results

The key results of Fall, Saslaw, and Kandrup regarding the early stages of galaxy clustering are reformulated here for the model problem of two-dimensional gravity with anr−1-interaction force. This should lead to new insights into the various physical processes involved and, more importantly, may prove of some use in interpreting numerical results obtained from two-dimensionalN-body simulations. The calculations suggest that, for a two-, rather than three-dimensional system, the correlational energy density should grow more rapidly with time, but should exhibit only a weak logarithmic dependence upon the typical particle speed. Various other measures of the clustering are independent of the dimensionality.