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Astron. Astrophys. 326, 433-441 (1997)

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2. The probability of mass and position for a particle with a known mass function distribution and Boltzmann equilibrium.

HYPOTHESIS 1: We have a grand canonical ensemble of N point-particles, interchangeable with an external reservoir, with positions [FORMULA],..., [FORMULA] momenta [FORMULA],..., [FORMULA] and masses [FORMULA],..., [FORMULA] respectively. This means that the distribution of positions and momenta in the particles follows a Maxwell-Boltzmann distribution, i.e. they are relaxed.

The probability of a configuration in positions and masses is [FORMULA]. First, we define a general distribution of masses, and then, once we have the mass of each particle established a priori, the probability of a particle occupying a position depends on its mass:

[EQUATION]

[EQUATION]

where [FORMULA] is the chemical potential, [FORMULA] is a constant, [FORMULA] is the mass distribution function, and [FORMULA] the position distribution function which also depends on the masses as parameters (the probability of a configuration of positions is conditioned by the distribution of masses; the slash stands for "conditioned"). We have decomposed the probability [FORMULA] into the product of three probabilities: first, the probability of having N particles in a grand canonical ensemble (see for example Saslaw 1985, ch. 34); secondly, the probability of a mass configuration, [FORMULA] ; and, finally, once we know the mass of each particle, the probability of the positions configuration. In the following subsections I obtain [FORMULA] and [FORMULA].

2.1. Probability of positions

We have assumed that the number of particles is discrete and that the probability of that they be configured in a certain way is given by a function [FORMULA].

To analyse the form of [FORMULA] an assumption is needed on how position space (or the phase space that includes it) is populated, and this can be carried out by assuming a state of thermodynamic equilibrium, which was our first hypothesis. The Boltzmann distribution is characterized by a configuration probability proportional to the negative exponential of the Hamiltonian of the system, and the Hamiltonian is the sum of two terms, one depending on the momenta of the particles and other on the positions of the particles.

The Boltzmann probability distribution function for a fixed time (see for example Stanley 1971) is

[EQUATION]

[EQUATION]

[EQUATION]

where [FORMULA] is the Hamiltonian of the gravitational system, U the potential energy and [FORMULA] the kinetic energy. We integrate over the momentum space [FORMULA], thus obtaining the probability for the positions space configuration:

[EQUATION]

[EQUATION]

The masses [FORMULA] and [FORMULA] are constant so the integrals in ( 3 ) are constant and

[EQUATION]

2.2. Probability of masses

Obviously, if we have to take into account the difference of masses between particles, we must first derive the mass distribution. The usual tool for doing this is the mass distribution function for a particle, [FORMULA], i.e. the probability of a particle with mass between m and [FORMULA] is [FORMULA]. Actually, [FORMULA] represents an average over space of positions of the mass distribution. We could consider the distribution of mass to vary at different positions, i.e. the centre of a galaxy cluster with respect to another position, an effect which should indeed be taken into account. But this is wrong. Remember that by means of ( 1 ), we first chose the masses of the particles according to their probabilities, and then we assigned a position to the particle conditional on having a previously established mass.

So the probability of N particles having masses [FORMULA],..., [FORMULA], respectively, is

[EQUATION]

2.3. Normalized probability

Expressions ( 1 ), ( 4 ) and ( 5 ), once normalized, lead to

[EQUATION]

[EQUATION]

and the partition function of this canonical mass-dependent distribution is given by

[EQUATION]

[EQUATION]

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© European Southern Observatory (ESO) 1997

Online publication: October 15, 1997
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