If no property of a system of many particles discriminates among the particles, they are said to be indistinguishable. This indistinguishability is equivalent to the requirement that the many-particle distribution function and all of the dynamic functions for the system be symmetric. The indistinguishability defined in terms of the discrete symmetry of many-particle functions cannot change in the continuous classical statistical limit in which the number density n and the reciprocal temperature β become small. Thus, microscopic particles like electrons must (...) remain indistinguishable in the classical statistical limit although their behavior can be calculated as if they move following the classical laws of motion. In the classical mechanical limit in which quantum cells of volume (2πħ)3 are reduced to points in the phase space, the partition functionTr{exp(−βĤ) for N identical bosons (fermions) approaches (2πħ)−3N(N!) ∫ ... ∫ d3r1 d3p1 ... d3rN d3pN exp(−βH). The two factors, (2πħ)−3N and (N!)−1, which are often added in anad hoc manner in many books on statistical mechanics, are thus derived from the first principles. The criterion of the classical statistical approximation is that the thermal de Broglie wavelength be much shorter than the interparticle distance irrespective of any translation-invariant interparticle interaction. A new derivation of the Maxwell velocity distribution from Boltzmann's principle is given with the assumption of indistinguishable classical particles. (shrink)

New equations of motion for a Bloch electron [momentum p=h k,energy ε n(p),zone number n, charge -e]: $$m_j \frac{{dv_j }}{{dt}} = - e(E + v \times B)_j $$ are proposed, where v≡∂εn(p)/∂p is the velocity, and {mj}are the principal masses m j − 1=∂2εn/∂p j 2 along the normal and the two principal axes of curvatures at each point of the constant-energy surface represented by ε=εn(p).Their advantages over the prevalent equations of motion where the left-hand-side is replaced by hk j (...) are demonstrated by examining de Haas-van Alphen oscillations and orientation-dependent cyclotron resonance peaks. (shrink)

The Bose-Einstein condensation of free relativistic particles [ε=(M 2 c 4 +c 2 p 2 ) 1/2 −Mc 2 ] is studied rigorously. For massless bosons (ε=cp), the condensation transition of third (second) order occurs in2 (3) dimensions (D). The molar heat capacity follows the T 2 (T 3 ) law below the condensation temperature Tc [k B Tc=(2πħ 2 c 2 n/1.645) 1/2 [(π 2 ħ 3 c 3 n/1.202) 1/3 ], reaches4.38 (10.8) R at T=Tc, and approaches the (...) high-temperature-limit value2 (3) R with no jump (a jump equal to6.75R) in2 (3)D. For finite-mass (M) bosons, the phase transition occurs only in3D with the condensation temperature Tc always smaller than that of the corresponding nonrelativistic bosons [ε=(2M) −1 p 2 ]. If the mass M is reduced to zero, the condensation temperature Tc grows monotonically and reaches eventually that of massless relativistic bosons. This mass-dependence of Tc is therefore distinct from the case of nonrelativistic bosons, where Tc grows to infinity as M →0. A brief discussion is given for a possible connection with the normal-to-super transition of the independently moving Cooper pairs (bosons). (shrink)