## 2. The probability of mass and position for a particle with a known mass function distribution and Boltzmann equilibrium.HYPOTHESIS 1: The probability of a configuration in positions and masses is . 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: where
is the chemical potential,
is a constant,
is the mass distribution function, and
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
into the product of three probabilities: first,
the probability of having ## 2.1. Probability of positionsWe 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 . To analyse the form of 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 where
is the Hamiltonian of the gravitational system,
The masses and are constant so the integrals in ( 3 ) are constant and ## 2.2. Probability of massesObviously, 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,
, i.e. the probability of a particle with mass
between So the probability of ## 2.3. Normalized probabilityExpressions ( 1 ), ( 4 ) and ( 5 ), once normalized, lead to and the partition function of this canonical mass-dependent distribution is given by © European Southern Observatory (ESO) 1997 Online publication: October 15, 1997 |