In this paper, we consider the volume enclosed by the microcanonical ensemble in phase space as a statistical ensemble. This can be interpreted as an intermediate image between the microcanonical and the canonical pictures. By maintaining the ergodic hypothesis over this ensemble, that is, the equiprobability of all its accessible states, the equivalence of this ensemble in the thermodynamic limit with the microcanonical and the canonical ensembles is suggested by means of geometrical arguments. The Maxwellian and the Boltzmann-Gibbs distributions are obtained from this formalism. In the appendix, the derivation of the Boltzmann factor from a new microcanonical image of the canonical ensemble is also given.
Deep Dive into On the equivalence of the microcanonical and the canonical ensembles: a geometrical approach.
In this paper, we consider the volume enclosed by the microcanonical ensemble in phase space as a statistical ensemble. This can be interpreted as an intermediate image between the microcanonical and the canonical pictures. By maintaining the ergodic hypothesis over this ensemble, that is, the equiprobability of all its accessible states, the equivalence of this ensemble in the thermodynamic limit with the microcanonical and the canonical ensembles is suggested by means of geometrical arguments. The Maxwellian and the Boltzmann-Gibbs distributions are obtained from this formalism. In the appendix, the derivation of the Boltzmann factor from a new microcanonical image of the canonical ensemble is also given.
The microcanonical and the canonical ensembles represent two clearly different physical situations in a statistical system 1 . The microcanonical ensemble is presented in the literature as modeling an isolated system that conserves its energy in time. The canonical ensemble models a system in contact with a heat reservoir containing an infinite energy, which allows to fluctuate the energy of the system but maintains its mean value constant in time. In the thermodynamic limit, and under certain assumptions on the entropy function 2 , both formalisms converge and they give the same macroscopic statistical results 1,2 .
Here, we interpret the volume enclosed by the microcanonical ensemble in phase space as a statistical ensemble. It implies the existence of some kind of heat reservoir with an upper limit energy in order that the system can visit all the accessible states enclosed in that volume. The geometrical reason why this picture is equivalent to the microcanonical ensemble in the thermodynamic limit is discussed. Thus, if the microcanonical ensemble is supposed to be represented by the equiprobability over the hypersurface on which the system evolves as consequence of conserving an energy E (as recently explained in Refs. 3,5), then the volume-based ensemble can be interpreted as the equiprobability over the whole volume which is enclosed by that hypersurface. This means that, in the latter ensemble, the system can visit states with different energies, with an upper limit E given by the energy defined in its equivalent microcanonical picture. As we have said, we can think that in this image the system is exchanging energy with a heat (or energy) reservoir containing a maximum energy E. This constraint is removed in the thermodynamic limit when the number of degrees of freedom and the energy E are supposed to become infinite. Let us observe that this infinite limit establishes also the equivalence between this ensemble and the canonical ensemble (when certain conditions of smoothness in the entropy function are implicit 2 ), just because in this case the reservoir contains an infinite energy and then both pictures become identical. Hence, when the number of dimensions of the system increases infinitely, almost all the volume enclosed by that hypersurface is located in the vanishingly thin layer close to the hypersurface, and, in consequence, surface and volume, tend to coincide. This is the reason why the microcanonical ensemble and the volume-based ensemble, and by extension the canonical ensemble, give the same results for systems well-behaved 2 in the thermodynamic limit.
We proceed to obtain different classical results from this volume-based statistical ensemble. We start by deriving (recalling) the Maxwellian (Gaussian) distribution from geometri-cal arguments over the volume of an N-sphere. Following the same insight, we also explain the origin of the Boltzmann-Gibbs (exponential) distribution by means of the geometrical properties of the volume of an N-dimensional pyramid. We finish claiming a possible general statistical result that follows from the properties of the volume enclosed by a one-parameter dependent family of hypersurfaces, in which N-spheres and N-dimensional pyramids are included. In the appendix, an alternative microcanonical image of the canonical ensemble is also given.
Let us suppose a one-dimensional ideal gas of N non-identical classical particles with masses m i , with i = 1, . . . , N, and total maximum energy E. If particle i has a momentum m i v i , we define a kinetic energy:
where p i is the square root of the kinetic energy. If the total maximum energy is defined as
We see that the system has accessible states with different energy, which is supplied by the heat reservoir. These states are all those enclosed into the volume of the N-sphere given by Eq. (2). The formula for the volume V N (R) of an N-sphere of radius R is
where Γ(•) is the gamma function. If we suppose that each point into the N-sphere is equiprobable, then the probability f (p i )dp i of finding the particle i with coordinate p i (energy
) is proportional to the volume formed by all the points on the N-sphere having the ithcoordinate equal to p i . Our objective is to show that f (p i ) is the Maxwellian distribution, with the normalization condition
If the ith particle has coordinate p i , the (N -1) remaining particles share an energy less than the maximum energy R 2 -p 2 i on the (N -1)-sphere
whose volume is
It can be easily proved that
Hence, the volume of the N-sphere for which the ith coordinate is between p i and
We normalize it to satisfy Eq. ( 4), and obtain
whose final form, after some calculation is
with
For N ≫ 1, Stirling’s approximation can be applied to Eq. ( 9), leading to lim
If we call ǫ the mean energy per particle, E = R 2 = Nǫ, then in the limit of large N we have lim
The factor e -p 2 i /2ǫ is found when N ≫ 1 but, even for small N, it can be a good approximation f
…(Full text truncated)…
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