Bose-Einstein statistics

Bose-Einstein statistics is a type of statistical mechanics applicable to the determination of the statistical properties of large sets of indistinguishable particles capable of coexisting in the same quantum state (bosons) in thermal equilibrium.. At low temperatures, bosons tend to have similar quantum behavior that can become identical at temperatures close to absolute zero in a state of matter known as a Bose-Einstein condensate and produced for the first time in the laboratory in the year 1995. The Bose-Einstein condenser operates at temperatures close to absolute zero, -273.15°C (0 kelvin). Bose-Einstein statistics were introduced to study the statistical properties of photons in 1920 by Indian physicist Satyendra Nath Bose and generalized to atoms and other bosons by Albert Einstein in 1924. This type of statistics is closely related to Maxwell statistics. -Boltzmann (initially derived for gases) and Fermi-Dirac statistics (applicable to particles called fermions over which the Pauli exclusion principle governs, which prevents two fermions from sharing the same quantum state).

The Bose-Einstein statistic reduces to the Maxwell-Boltzmann statistic for sufficiently high energies.

Mathematical formulation

The number of particles in an energy state i is:

${displaystyle n_{i}left(varepsilon _{i}{text{, }Tright)={frac {g_{i}}{e^{{left(varepsilon _{i}-mu right)}{k_{B}T};}1}}$

where:

• ${displaystyle n_{i},}$ is the number of particles in a state i,
• ${displaystyle g_{i},}$ is the quantum degeneration of the state i or number of different wave functions that possess such energy,
• ${displaystyle varepsilon _{i},}$ is state energy i,
• ${displaystyle mu ,}$ is the chemical potential,
• ${displaystyle k_{B},}$ It's Boltzmann's constant,
• ${displaystyle T,}$ It's the temperature.

The Bose-Einstein statistic reduces to the Maxwell-Boltzmann statistic for energies:

${displaystyle (epsilon _{i}-mu)。$

Derivation

Since bosonic systems are indistinguishable particle systems, the states whose only difference is the permutation of two particle states are identical. Thus, a state of the system will be uniquely defined by the number of particles found in a given energy state. It will be denoted by ${displaystyle epsilon}$ the energy state r-simo, by ${displaystyle n_{r}}$ the number of particles in the r-simo and R state each of the possible combinations of occupancy numbers. The partition function results:

${displaystyle {mathcal {Z}}=sum _{l}e^{-beta (E_{l}-mun_{l})}=sum _{R}e^{-beta sum _{r}{r}(epsilon _{r}{r}{r}{r !$

The previous expression contains all possible combinations ${displaystyle n_{r}}$ between 0 and ${displaystyle infty }$ (since in a bosonic system the number of particles by quantum state is not limited) so that it can be rewritten as follows:

${displaystyle {mathcal {Z}}=prod _{r}sum _{n_{r}=0}^{infty }e^{-beta (epsilon _{r}n_{r}}-mu n_{r}}}}}=prod _{r}{frac {1}{1-eFF}{$

Applying that:

${displaystyle Phi =k_{B}Tln{mathcal {Z}}}quad yquad {frac {partial Phi }{partial mu }}}=-N}$

You have to:

${displaystyle Phi =k_{B}Tln{mathcal {Z}}}=-k_{B}Tsum _{r}ln(1-e^{-beta (epsilon _{r}-mu)}})}$

${displaystyle quad Rightarrow quad {frac {partial Phi }{partial mu }}=-N=-sum _{r}n_{r}=-sum _{r}{c}{frac {e^{-beta (epsilon _{r}}{1-e{beta$

So:

${displaystyle n_{r}={frac {1}{e^{beta (epsilon _{r}-mu)}}}}}$

Because there can be different quantum states with the same energy the number of particles with a given energy will be given by:

${displaystyle n_{epsilon }={frac {g_{epsilon }{e^{beta (epsilon -mu)}}}}}{epsilon$

being ${displaystyle g_{epsilon }$ the degeneration of such energy.

In the previous expression it is observed that the chemical potential must be lower than all energies, otherwise the average number of particles in a state could be negative. This fact could also have been observed when adding the geometric series, since the previous condition is the condition for its convergence.

Applications

• The energy distribution of the radiation of the black body is deducted from the application of the Bose-Einstein statistics to the photons that make up the electromagnetic radiation.
• The heat capacity of the solids at both high and low temperatures can be deduced from the Bose-Einstein statistics applied to the fonones, quasi-particles that realize the excitations of the crystalline network. In particular the law of Dulong-Petit may be deduced from the Bose-Einstein statistics.
• Bose-Einstein's statistics predict the phenomenon of Bose-Einstein's condensation, also known as the fifth state of matter.