Comments on the Extensivity of the Boltzmann Entropy

Abstract

In thermodynamics entropy S td is an extensive state function. Its derivation by statistical mechanics following Boltzmann and Gibbs with the famous formula S=k B lnW for a micro-canonical ensemble with N particles, k B the Boltzmann constant, and W the number of accessible micro-states is however in general not extensive unless the Stirling approximation given by lnN!-NlnN + N is used. Furthermore, at the thermodynamic limit with the number of particles N→∞ at constant density the Stirling approximation can not be used to show extensivity because lim N→∞ (lnN!-NlnN + N)=∞. Hence, the Boltzmann entropy S as shown here for the ideal gas is neither for a small system with N particles nor at the thermodynamic limit extensive. Thus, if strict extensivity for the entropy is requested the claim of statistical mechanics that the Boltzmann entropy is a microscopic description of its thermodynamic analog is challenged.

On the Entropy of a One-Dimensional Gas with and without Mixing Using Sinai Billiard

“…A physical meaning of this entity is, on the one hand, that the number of micro-states is proportional to the volume that a single particle is able to access on average, which is inversely proportional to the number of particles N , to the power of N . This finding also yields zero for the Boltzmann entropy of a fully occupied ideal gas (i.e., every micro-state is occupied by a particle) as one would expect, but which is actually not the case for the Boltzmann entropy calculated by the standard textbook approach (for details see [ 8 ]). On the other hand, since the volume for particle p decreases with increasing number of particles in the system, all the other particles in concert define, by colliding with it, the configurational space of particle p as shown in Figure 1 .…”

“…When time is defined as the metric of causality, the causal history of the system yields an entropy that describes the causal chain between cause and events [ 19 ]. While this proposition on the nature of entropy contrasts entirely to the concept of Boltzmann the mathematical formulas are very similar and even the same if in the case of the Boltzmann entropy only the first order term of the Stirling formula is applied, which has its problems though [ 8 ]. In this context, it is interesting to note, that the derivation of the Sackur–Tetrode entropy of the ideal gas using quantized space/energy by Sackur used also a volume per particle approach [ 20 ].…”

“…This yields an entropy description that is of extensive character for any number of particles, n , located in the middle of a large box such that boundary effects can be neglected. This is in contrast to the standard entropy of an ideal gas that holds only in the thermodynamic limit with (and this only under the assumption that the first order term of the Stirling formula can be used, but see [ 8 , 18 ]). The entropy calculated here is proportional to the number of particles of interest, n , and proportional to the logarithm of the particle density.…”

“…After applying (and only after applying) the first term in the Stirling formula (shown in brackets above) the entropy is of extensive nature [ 8 ], provided that the number of sites, M , is chosen proportional to the system size, i.e., the number of particles, N .…”

“…For a micro-canonical ensemble the entropy is given by with the Boltzmann constant and the number of accessible micro-states describing the macro-state [ 2 , 3 , 4 , 5 , 6 , 7 ]. In order to describe the entropy of a micro-canonical isolated system in textbooks simple probability theories are used with imaginary boxes as well as the use of the first order Stirling approximation, with an error proportional to , which poses issues in the thermodynamic limit ( ) [ 8 , 9 , 10 , 11 ], unless the concept of an entropy density [ 12 ], or even redefining the microscopic origin on entropy [ 13 , 14 ] is introduced. Instead of using these somewhat ad hoc models to describe a micro-canonical ideal gas, we study here a one-dimensional gas comprising N point molecules that undergo elastic collisions within a finite space because it can be analytically calculated using the Sinai billiard approach.…”