We derive Bose-Einstein statistics and Fermi-Dirac statistics by Principle of Maximum Entropy applied to two families of entropy functions different from the Boltzmann-Gibbs-Shannon entropy. These entropy functions are identified with special cases of modified Naudts' $ϕ$-entropy.
We derive Bose-Einstein statistics and Fermi-Dirac statistics by Principle of Maximum Entropy applied to two families of entropy functions different from the Boltzmann-Gibbs-Shannon entropy. These entropy functions are identified with special cases of modified Naudts' ϕ-entropy.
Abstract Chapter 11 presents a statistical mechanical treatment of the quantum ideal gases, i.e., the ideal Boltzmann, fermion, and boson gases. The discussion begins with the microscopic description and the solution of the eigenvalue problem for a quantum ideal gas. It is argued that calculating the partition function is most readily accomplished in the grand canonical ensemble using a second-quantized formulation. When the particles are distinguishable, the equation of state is identical to that of a classical ideal gas. For fermions and bosons, however, the problem of computing thermodynamic properties is significantly more complex and can only be solved exactly in certain limits. Away from these limits approximations are needed and are discussed in detail. The relevant distributions - the Fermi-Dirac and Bose-Einstein distributions are derived. The local density approximation of density functional theory is derived for the ideal electron gas. For the ideal boson gas, the phenomenon of Bose-Einstein condensation is discussed
In a previous note (1) we argued that for a wavefunction W(x,t)= Sum over n an exp(iEnt) Wn(x), the entropy for a particle in a box with infinite potential walls from (2) is: S/(-k)= Sum over n |anan| ln |anan| ((1)) and should be modified to: S/(-k)= Sum over n |anan| ln|anan| + |anan| Sp(n,L) + |anan| Sx(L) ((2)) where Sp and Sx are the pure state momentum and spatial entropies i.e Sp/-k = Integral dp an(p)an(p) ln|an(p)an(p)| and Sx/(-k) = Integral dx W*n(x)Wn(x) ln[Wn*Wn|. In (2) the Clausius relationship: dQ/T = dS is used where dQ = Sum over n En d|anan| and T is temperature to find an expression for |anan|. In general this would be: |anan| = exp(-En/T) , but with ((2)) we find: |anan| = exp(-En/kT) exp(-1-B(n)) ((3)) where B(n)= Sp(n)+Sx. L portions form Sp and Sx cancel so there is no L dependence, L being the size of the box. Given the different form ((3)) we suggest that the Fermi-Dirac (FD) and Bose-Einstein (BE) distributions may change if one uses the grand partition function approach. For example, exp(-ni ei/kT) is replaced by: exp(-ni ei/kT) exp(-ni(1+B(n)). This leads to changes in the FD and BD distributions: zexp(-Ei/kT)exp(-1-B(i) / [1 +/- exp(-u/kT) exp(-ei/kT) exp(-1-B(i))] where z=exp(u/kT), u being the chemical potential. In the high n limit B(n)--> constant (3) so the Maxwell-Boltzmann (MB) distribution remains unchanged. The MB distribution has been verified to high precision and so modifications related to 1+B(n) might have been noticed had B(n) remained a function of n.
In thermodynamics and statistical mechanics entropy is the fundamental physical quantity to describe the evolution of a statistical ensemble. Its microscopic definition was provided by Boltzmann through the celebrated expression S = kB ln Γ, where kB is the Boltzmann constant and Γ is the number of microstates exploiting the given macroscopic properties. In this context, it is well known that in classical mechanics the entropy: i) allows the violation of the uncertainty principle [1], [2]; ii) can be considered as a special case of the so-called Boltzmann-Gibbs-Shannon entropy that enables one to apply results of information theory to physics [1], [3]. In particular, by introducing the principle of maximum entropy (MEP) it was found possible to derive rigorous hydrodynamic (HD) models based on the moments of the distribution function to all orders of a power expansion and including appropriate closure conditions [4], [5]. Accordingly, making use of the Lagrange multipliers technique, it was found possible to construct the set of evolution equations for the macro-variables of interest. Apart from some partial attempts [3], [6], this is no longer the case in quantum mechanics. Here, the main difficulties concern with: i) the definition of a proper quantum entropy that includes particle indistinguishability; ii) the formulation of a global quantum MEP (QMEP) that allows one to obtain a quantum distribution function both for thermodynamic equilibrium and nonequilibrium configurations. From one hand, in the framework of a nonlocal quantum theory, the generalization of the corresponding Lagrange multipliers is also an open problem. Fron another hand, a rigorous formulation of quantum HD (QHD) closed models is a demanding issue for many kinds of problems in quantum systems like, interacting fermionic and bosonic gases, quantum turbulence, quantum fluids, quantized vortices, nuclear physics, confined carrier transport in semiconductor heterostrucures, phonon and electron transport in nanostructures, nanowires and thin layers. The aim of this talk is to address the above drawbacks by defining a quantum entropy in terms of the reduced density matrix, thus explicitly incorporating quantum statistics into problems involving a system of identical particles. As a further step, with respect to the uncertainty principle, starting from the Wigner representation we formulate a quantum maximum entropy principle which, in the framework of information theory, allows us to obtain a non-local theory for the system under study. As a final step, we determine a closed quantum hydrodynamic model for the macroscopic variables used as constraints in the QMEP approach.
Abstract Attempts at generalizing Shannon entropy and Kullback-Leibler divergence (relative entropy) led to a plenthora of deformation models in theoretical physics, including q -model, κ -model, etc. Naudts and Zhang ( Inf. Geom. , 1 (2018) 79) established that these models can be unified under two notions: deformed ϕ -exponential family ( Naudts, J. , J. Inequal. Pure Appl. Math. , 5 (2004) 102) and conjugate -embedding ( Zhang J. , Neural Comput. , 16 (2004) 159) of probability functions. Conjugate -embedding has a gauge freedom which, upon its fixing, subsumes the U -model of Eguchi ( Sugaku Expositions , 19 (2006) 197) proposed in a statistical machine learning context. The generalization by -entropy, -cross-entropy, -divergence, when applied to the ϕ -exponential family, yields either a Hessian structure or a conformal Hessian structure under different gauge selections —this “splitting” is the hallmark when deforming the exponential family with its dually flat (Hessian) geometry. This letter provides a unified information geometric perspective of deformation of the exponential model, with calculations for Tsallis q -model.