Bose–Einstein statistics

In quantum statistics, Bose–Einstein statistics (or B–E statistics) describe one of two possible ways in which a collection of non-interacting, indistinguishable particles may occupy a set of available discrete energy states at thermodynamic equilibrium. The aggregation of particles in the same state, which is a characteristic of particles obeying Bose–Einstein statistics, accounts for the cohesive streaming of laser light and the frictionless creeping of superfluid helium. The theory of this behaviour was developed (1924–25) by Satyendra Nath Bose, who recognized that a collection of identical and indistinguishable particles can be distributed in this way. The idea was later adopted and extended by Albert Einstein in collaboration with Bose.Suppose we have a number of energy levels, labeled by index i {displaystyle displaystyle i} , each level having energy ε i {displaystyle displaystyle varepsilon _{i}} and containing a total of n i {displaystyle displaystyle n_{i}} particles. Suppose each level contains g i {displaystyle displaystyle g_{i}} distinct sublevels, all of which have the same energy, and which are distinguishable. For example, two particles may have different momenta, in which case they are distinguishable from each other, yet they can still have the same energy. The value of g i {displaystyle displaystyle g_{i}} associated with level i {displaystyle displaystyle i} is called the 'degeneracy' of that energy level. Any number of bosons can occupy the same sublevel.A much simpler way to think of Bose–Einstein distribution function is to consider that n particles are denoted by identical balls and g shells are marked by g-1 line partitions. It is clear that the permutations of these n balls and g − 1 partitions will give different ways of arranging bosons in different energy levels. Say, for 3 (= n) particles and 3 (= g) shells, therefore (g − 1) = 2, the arrangement might be |●●|●, or ||●●●, or |●|●● , etc. Hence the number of distinct permutations of n + (g-1) objects which have n identical items and (g − 1) identical items will be: S ( n , g ) = { ( m 1 , m 2 , … , m n ) | m i ≥ m i − 1 , m i ∈ { 1 , … , g } , ∀ i = 1 , … , n } . {displaystyle S(n,g)={Big {}left(m_{1},m_{2},dots ,m_{n} ight){Big |}{Big .}m_{i}geq m_{i-1},m_{i}in left{1,dots ,g ight},forall i=1,dots ,n{Big }}.} w ( n , g ) = ⟨ g n ⟩ = ( g + n − 1 g − 1 ) = ( g + n − 1 n ) = ( g + n − 1 ) ! n ! ( g − 1 ) ! {displaystyle displaystyle w(n,g)=leftlangle {egin{matrix}g\nend{matrix}} ight angle ={g+n-1 choose g-1}={g+n-1 choose n}={frac {(g+n-1)!}{n!(g-1)!}}} ∑ k = 0 n ( k + a ) ! k ! a ! = ( n + a + 1 ) ! n ! ( a + 1 ) ! . {displaystyle sum _{k=0}^{n}{frac {(k+a)!}{k!a!}}={frac {(n+a+1)!}{n!(a+1)!}}.} w ( n , g ) = ∑ k = 0 n w ( n − k , g − 1 ) = w ( n , g − 1 ) + w ( n − 1 , g − 1 ) + ⋯ + w ( 1 , g − 1 ) + w ( 0 , g − 1 ) {displaystyle displaystyle w(n,g)=sum _{k=0}^{n}w(n-k,g-1)=w(n,g-1)+w(n-1,g-1)+cdots +w(1,g-1)+w(0,g-1)} S ( n , g ) = ⋃ k = 0 n S ( n − k , g − 1 ) {displaystyle displaystyle S(n,g)=igcup _{k=0}^{n}S(n-k,g-1)} ; w ( n , g ) = ∑ k = 0 n w ( n − k , g − 1 ) {displaystyle displaystyle w(n,g)=sum _{k=0}^{n}w(n-k,g-1)} , w ( n , g ) = ∑ k 1 = 0 n ∑ k 2 = 0 n − k 1 ⋯ ∑ k g = 0 n − ∑ j = 1 g − 1 k j 1 , {displaystyle displaystyle w(n,g)=sum _{k_{1}=0}^{n}sum _{k_{2}=0}^{n-k_{1}}cdots sum _{k_{g}=0}^{n-sum _{j=1}^{g-1}k_{j}}1,} ∑ k = 0 q p = q p {displaystyle displaystyle sum _{k=0}^{q}p=qp} . In quantum statistics, Bose–Einstein statistics (or B–E statistics) describe one of two possible ways in which a collection of non-interacting, indistinguishable particles may occupy a set of available discrete energy states at thermodynamic equilibrium. The aggregation of particles in the same state, which is a characteristic of particles obeying Bose–Einstein statistics, accounts for the cohesive streaming of laser light and the frictionless creeping of superfluid helium. The theory of this behaviour was developed (1924–25) by Satyendra Nath Bose, who recognized that a collection of identical and indistinguishable particles can be distributed in this way. The idea was later adopted and extended by Albert Einstein in collaboration with Bose. The Bose–Einstein statistics apply only to those particles not limited to single occupancy of the same state—that is, particles that do not obey the Pauli exclusion principle restrictions. Such particles have integer values of spin and are named bosons, after the statistics that correctly describe their behaviour. There must also be no significant interaction between the particles. At low temperatures, bosons behave differently from fermions (which obey the Fermi–Dirac statistics) in a way that an unlimited number of them can 'condense' into the same energy state. This apparently unusual property also gives rise to the special state of matter – the Bose–Einstein condensate. Fermi–Dirac and Bose–Einstein statistics apply when quantum effects are important and the particles are 'indistinguishable'. Quantum effects appear if the concentration of particles satisfies where N is the number of particles, V is the volume, and nq is the quantum concentration, for which the interparticle distance is equal to the thermal de Broglie wavelength, so that the wavefunctions of the particles are barely overlapping. Fermi–Dirac statistics apply to fermions (particles that obey the Pauli exclusion principle), and Bose–Einstein statistics apply to bosons. As the quantum concentration depends on temperature, most systems at high temperatures obey the classical (Maxwell–Boltzmann) limit, unless they also have a very high density, as for a white dwarf. Both Fermi–Dirac and Bose–Einstein become Maxwell–Boltzmann statistics at high temperature or at low concentration. B–E statistics was introduced for photons in 1924 by Bose and generalized to atoms by Einstein in 1924–25. The expected number of particles in an energy state i for B–E statistics is with εi > μ and where ni is the number of particles in state i, gi is the degeneracy of energy level i, εi is the energy of the i-th state, μ is the chemical potential, kB is the Boltzmann constant, and T is absolute temperature. For comparison, the average number of fermions with energy ϵ i {displaystyle epsilon _{i}} given by Fermi–Dirac particle-energy distribution has a similar form: