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dynamics. Also, for simplicity, further we will ignore the

problem of “primordial binaries”, supposing that their

number is determined with the random numbers in the

initial conditions.

As the stellar systems are open, the achievement of the

exact equilibrium distribution function of the stars’ ve-

locities (in Boltzmann’s sense) is impossible, because the

tendency to statistical equilibrium leads to dissipation of

stars with large energies, which determines slow evolu-

tion of the system due to the “evaporation” of the stars

and the final collapse of the core of the system. The rate

of the evolution is determined with the relaxation time of

the system [4]

0.14

ln 0.4

rx cr

N

t

N

t

(2)

where N is the total number of stars in the system,

ln(0.4N) is the so-called “Coulomb logarithm” for stellar

systems, tcr—the crossing time, determined usually as the

time needed for a star to cross the region of the system

containing half of the system’s mass. Note that the nume-

rical coefficients in (2) are known with the precision ~

30% - 50% only; here we give the most commonly used

values.

Let us turn back to the question of equivalence of the

evolution processes in the systems with different (in the

sense of different realization of random numbers choos-

ing) initial conditions. Of course, the concrete orbits of

particles in the systems with different initial conditions

will be completely different, but the dynamical behavior

and the secular evolution of the distribution functions of

the physical parameters of the orbits in both systems will

be equivalent in statistical and dynamical sense, if we are

interested of distribution functions, but not of the con-

crete orbits. Note that the quality of modern computers is

such that the orbits of the particles, calculated in two

simulations with the same initial conditions will be ab-

solutely identical. In contrast, in real stellar systems there

are always some random factors, which lead to a mixing

(random small deviations) of the orbits. As a result, the

distribution functions only, not the concrete stellar orbits

may be compared between a model and real systems.

Consequently, one may speak not about identity, but only

about dynamical and statistical equivalence of a model

and real stellar systems. The situation is similar, in a

sense, (not literally!) to the quantum-mechanics des-

cription: Only the probability description is valid, so the

description with a single numerical calculation is “too

much deterministic” and, in that sense, non-adequate

(fuller analyses of some analogies in quantum mechanics

were discussed earlier in [5]).

In fact, the characteristic times of achievement of the

“chaotic” (statistical) equilibrium in stellar systems are

much shorter than the relaxation time (2), due to the

quick dynamical mixture of the systems [6]. As a result,

the solutions with different initial conditions in the same

system (the “microscopically” different solutions) are

supposed to be dynamically and statistically equivalent.

As a rule, the problem of different initial condition (in

the sense of different random realization) is usually even

not discussed, and the solution for the system evolution

is realized with a single initial condition of stars’ (par-

ticles’) coordinates and velocities distribution.

With particle numbers more than several thousands,

the probability of a deviation from “almost equilibrium”

initial distribution is diminishingly small, but the pro-

blem becomes actual for the systems with comparatively

small number of particles, and for the systems feebly

stable relative to the small perturbations. Because of that,

the question of equivalence of the numerical solutions

became important for comparison of the systems with

different numbers of particles, N1 and N2 << N1.

3. The Dynamical Equivalence of the

Systems with Different Particle Numbers

Let us consider the question of the dynamical equiva-

lence of the evolution of two systems with different par-

ticle numbers, N1 and N2 << N1. To compare the relaxa-

tion processes in the systems it is convenient to use di-

mensionless unit system (the N-body units, NBU [4]),

usually employed in model tasks. In these units, the total

mass of the system is M = 1, its characteristic size R = 1,

and the gravitation constant G = 1. In this system, the

characteristic particle velocity is V = (GM/R)1/2 = 1, the

characteristic particle crossing time Tcr = R/V = 1, and

the characteristic time of evolution of the system (the

relaxation time) is

0.14

ln 0.4

rx

N

TN

(3)

We define the concept of the dynamical equivalence of

the solutions for the systems with different particle num-

bers. It is naturally to suppose that the behavior of two

systems with different (but large enough) particle num-

bers N1 and N2 N1 will be dynamically equivalent, if

the physical parameters of the systems are compared at

equivalent moments of evolution, defined by equal evo-

lution time of systems Tev = t/Trx. The evolution times are

measured in N-body relaxation time units of both sys-

tems (one may call these units the NBEU, the “N-body

evolution units” system),

2211rxrx ev

tTtT T (4)

In the NBU system, the time (and the relaxation times)

for both systems are defined through the crossing times,

which are equal to unity (Tcr1 = Tcr2 = 1), and in the

NBEU the evolution time is defined through the relaxa-

tion time. The relation of the “equivalent dynamical evo-

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