Thermodynamic probability Ω is defined as the number of alternative microscopic arrangements which correspond to the same macroscopic state.Thermodynamic probability Ω enables us to decide how much more probable certain situations are than others.The method of determining the number of microstates for a given macrostate decides the statistics, Bose-Einstein or Fermi-Dirac statistics; quantum or classical!The distribution of the particles in different microstates is demonstrated through animation with the help of computer graphics.
Thermodynamic probability Ω is defined as the number of alternative microscopic arrangements which correspond to the same macroscopic state.Thermodynamic probability Ω enables us to decide how much more probable certain situations are than others.The method of determining the number of microstates for a given macrostate decides the statistics, Bose-Einstein or Fermi-Dirac statistics; quantum or classical!The distribution of the particles in different microstates is demonstrated through animation with the help of computer graphics.
An interesting result arises in information theory, namely that if one considers a microstate with large N trials, each representing a state i with probability p(i), then even though this is only one of many possible microstates (each being a permutation of each other) it carries information about all of the permutations based on the average occupancy number. In particular, the probability of a microstate may be written as: Product over i p(ei) [power Np(ei) ] for large N. This holds for any microstate and so all have the same probability. Given that all of the microstates are permutations in the large N case, this probability is equivalent to 1/ {number of permutations} where Number of permutations = N! / { Product over i n(ei) }. We point out the above example, because it seems there are at least two main strategies in statistical mechanics. One is to begin with a set of average occupancy numbers n(ei/T) whose specific form is unknown. The idea then seems to be to seek degeneracy in the system, possibly among the arrangement of the n(ei)’s or degeneracy within single levels ei and to maximize the number of degenerate arrangements in order to account for as little information as possible. This approach may be used to calculate the Maxwell-Boltzmann, Fermi-Dirac and Bose-Einstein distributions. In the case of the latter two, degeneracy in the single energy levels ei is key. One then maximizes the ln of the degenerate arrangements subject to constraints such as Sum over i ei n(ei) and Sum over i n(ei) = N. The degeneracies themselves are associated with momentum degeneracies yielding the same energy. A completely different approach, it seems, is to consider exp(-(E-uN)/T) as representing the probability of a state with energy E and N particles and to use either the grand canonical ensemble i.e. sum the various exp(-(E-uN)/T)’s. These may then be broken into single particle sums using E(N) = sum over i ei n(ei) for n(ei) variable or sum over i ei for N fixed. Even though momentum degeneracy still exists, it is not part of the n sum of exp(-(E-uN)/T) factors which gives a single energy level probability distribution with no maximization necessary. Thus we point out that exp(-(E-uN)/T) for microstates seems to incorporate information about degeneracies without them needing to be explicitly added. On the other hand, if one starts with average n(ei) (without knowing their form), one must explicitly consider degeneracies in order to maximize so that there is no unnecessary information in the system. Both approaches should yield the same result. We consider the Bose-Einstein example of a photon gas as an example.
Abstract Part II of this two-part paper refutes the beliefs about the statistical interpretation of thermodynamics, and the association of entropy with disorder that are summarized in Part I. The refutation of the statistical approach is based on either a nonstatistical unified quantum theory of mechanics and thermodynamics, or an almost equivalent, novel, nonquantal exposition of thermodynamics. Entropy is shown to be: (i) valid for any system (both macroscopic and microscopic, including one-particle systems), and any state (both thermodynamic or stable equilibrium, and not stable equilibrium); (ii) a measure of the quantum-theoretic pliable shape of the molecules of a system; and (iii) a monotonic indicator of order. In contrast to statistics which associates a thermodynamic equilibrium macrostate with the largest number of compatible microstates, the second law avers that, for each set of values of energy, volume, and amounts of constituents of either a macroscopic or a microscopic system, there exists one and only one thermodynamic or stable equilibrium state. So, even if Boltzmann’s definition were used, a thermodynamic equilibrium state is one of perfect order.
In the previous chapter, we set up a basic framework for exploring thermodynamics in a quantum regime. We have seen that constrained particles “in a box” have quantized energies (energy levels) specified by a quantum number that helps us to index microstates. Under appropriate and limited circumstances, we can model work done by a change in the energy values of the different levels while heating the system shuffles the pack in terms of the probability for occupying the levels. In this chapter, we proceed to fill in some vital details. The idea of an ensemble is formalized, and two of the key examples explored. We also derive the Boltzmann distribution in two complementary ways. Finally, we turn the quantum version of an ideal gas, and in the process derive a formula for the entropy. Our first task, however, is to discuss the issues pertinent to our ability to count microstates .
This chapter discusses the main aspects of the Boltzmann distribution which played an important role for the discovery of Planck’s law of the spectral radiance of the thermal black-body experiment and Einstein’s equipartition as a derivation of Planck’s law. This chapter introduces the concepts of microstate, microstate, statistical weight of the distribution in order to obtain the Boltzmann distribution of distinguished particles. In the next section, it is discussed the statistics of undistinguishable particles of fermions (electrons, protons, neutrons) known as Fermi-Dirac statistics which began with Fermi’s approach of monoatomic molecules of an ideal gas where he applied the concept of quantum numbers associated with each molecule and Pauli’s exclusion principle. Afterwards, it is shown how Dirac amplified Fermi statistical distribution to many-particle systems. Dirac also used Pauli’s exclusion principle in order to arrive at the concept of antisymmetrical eigenfunctions and their determinantal form which later became known as Slater’s determinant. (151 words).
Abstract : The Fermi-Dirac, Bose-Einstein and, for completeness the Maxwell-Boltzmann, distributions are obtained respectively from considerations of binomial, negative binomial, and Poisson assemblies. The method used has the simplicity of the traditional derivations that are based on combinatorial considerations but involves neither the identification of most probable values with means values nor the invocation of large numbers of particles involved in the use of Stirling's approximation for factorials. The method thereby also relaxes the requirement for large numbers of particles needed in other available derivations that also use mean values, namely, the Darwin-Fowler method of steepest descents and the Khinchin method that employs limit theorems of the theory of probability.